Béatrice Rivière

Cédric Galusinski

Keywords: Richards' equation, seepage, weighted DG method, AMR, a posteriori error estimation, BDF, swash groundwater Richards' equation, seepage, weighted DG method, AMR, a posteriori error estimation, BDF, swash groundwater

Un Résumé étendu en français se trouve à la fin de la thèse dont il reprend l'introduction et la conclusion.

Foreword -Avant-propos

Abstract -Résumé

Flows in unsaturated porous media are modelled by the Richards' equation which is a degenerate parabolic nonlinear equation. Its limitations and the challenges raised by its numerical solution are laid out. Getting robust, accurate and cost-effective results is difficult in particular because of moving sharp wetting fronts due to the nonlinear hydraulic properties. Richards' equation is discretized by a discontinuous Galerkin method in space and backward differentiation formulas in time. The resulting numerical scheme is conservative, high-order and very flexible. Thereby, complex boundary conditions are included easily such as seepage condition or dynamic forcing. Moreover, an adaptive strategy is proposed. Adaptive time stepping makes nonlinear convergence robust and a block-based adaptive mesh refinement is used to reach required accuracy cost-effectively. A suitable a posteriori error indicator helps the mesh to capture sharp wetting fronts which are also better approximated by a discontinuity introduced in the solution thanks to a weighted discontinuous Galerkin method. The approach is checked through various test-cases and a 2D benchmark. Numerical simulations are compared with laboratory experiments of water table recharge/drainage and a largescale experiment of wetting, following reservoir impoundment of the multi-materials La Verne dam. This demanding case shows the potentiality of the strategy developed in this thesis. Finally, applications are handled to simulate groundwater flows under the swash zone of sandy beaches in comparison with experimental observations. Les écoulements en milieux poreux non-saturés sont modélisés par l'équation de Richards qui est une équation non-linéaire parabolique dégénérée. Ses limites et les défis que soulèvent sa résolution numérique sont présentés. L'obtention de résultats robustes, précis et efficaces est difficile en particulier à cause des fronts de saturation raides et dynamiques induits par les propriétés hydrauliques nonlinéaires. L'équation de Richards est discrétisée par une méthode de Galerkine discontinue en espace et des formules de différentiation rétrograde en temps. Le schéma numérique résultant est conservatif, d'ordre élevé et très flexible. Ainsi, des conditions aux limites complexes sont facilement intégrées comme la condition de suintement ou un forçage dynamique. De plus, une stratégie adaptative est proposée. Un pas de temps adaptatif rend la convergence non-linéaire robuste et un raffinement de maillage adaptatif basée sur des blocs est utilisée pour atteindre la précision requise efficacement. Un indicateur d'erreur a posteriori approprié aide le maillage à capturer les fronts de saturation raides qui sont également mieux approximés par une discontinuité introduite dans la solution grâce à une méthode de Galerkine discontinue pondérée. L'approche est validée par divers cas-tests et un benchmark 2D. Les simulations numériques sont comparées à des expériences de laboratoire de recharge/drainage de nappe et une expérience à grande échelle d'humidification, suite à la mise en eau du barrage multimatériaux de La Verne. Ce cas exigeant montre les potentialités de la stratégie développée dans cette thèse. Enfin, des applications sont menées pour simuler les écoulements souterrains sous la zone de jet de rive de plages sableuses en comparaison avec des observations expérimentales.

ACKNOWLEDGEMENTS -REMERCIEMENTS

fameux Bureau des Doctorants ! Aux anciens à qui je souhaite une très bonne continuation : Minh, Hadjer, Houssam, Moustoifa (quoi que tu fasses !), Yssouf et Mohamed Ali. À ceux qui soutiennent bientôt : Asma, Soufiane, courage, vous tenez le bon bout ! Aux nouveaux, je suis très honoré d'avoir fait votre connaissance et j'espère que vous passerez le flambeau pour les générations à venir : Youssef, Juliette, Bilal, Clémentin, Ali et Viktoriia. To those who were there briefly: Karol and Pepa. Special děkuji to Pepa who shared his hospitality by welcoming me in Prague for two months. I hope you're doing well and good luck for your Ph.D.! Dédicace aux stagiaires : bonne continuation ! Il y a forcément certaines personnes avec qui les atomes ont été plus crochus ! Merci à Hoàng Anh, doctorant hors-classe, qui est devenu au fil de cette thèse un ami formidable, plein d'énergie mais qui est surtout passé maître dans l'art de la négociation, l'humour et la cuisine ! Je pense pouvoir quand même deviner ton prochain repas du midi ! :p La thèse n'a été que plus agréable en ta compagnie. Merci Mai Anh, j'ai découvert une personne toujours rayonnante. Je te souhaite beaucoup de bonheur dans tes projets. Et merci à ton micro de qualité pour la soutenance ! ˆˆ" Obrigado à Matheus pour sa gentillesse dépassant l'entendement et son sens de l'amitié. Les discussions autour d'une bière et les sorties plage ont une saveur particulière grâce à lui. C'est tout naturellement que je dis dankeschön à Dorothea dont la joie de vivre, la culture et son regard sur le monde ont également rendu ces sorties spéciales. Je vous souhaite tous deux de vivre pleinement vos projets peu importe où le vent vous mène ! :) Merci à Manon pour les discussions toujours intéressantes au bureau et les moments de détente. Ça m'a fait beaucoup de bien en particulier les sorties d'escalade avec Romain ! :D Je remercie Ornella, la respo bureau, qui a toujours été à mi-cheval de me faire rire comme lorsqu'elle s'est retrouvée coincée dans le piège-à-Ornella. ;) Un grand merci pour son soutien et en particulier son aide à MT180 : ses talents ont fait merveilles pour numériser et coloriser ma planche ! Merci à Nicolas et Anna (et Sacha !) pour leur accueil et leur sens du jeu pendant le temps des confinements. Gate, Mr-2-Tot, tropfzahn, RaouldeCambrai, Ornella: thank you very much for the enjoyable games, 14! ˆÛn grand merci aux amis de l'école d'ingénieurs SeaTech, notamment la promo MOCA 2017 ; à jamais les meilleurs ! Leurs joutes verbales sont sources de franches rigolades et je souhaite bon courage à ceux qui finalisent leur thèse. Merci aussi aux autres doctorants ayant participé à MT180 d'avoir rendu ce moment magique.

Comment ne pas penser à la fine équipe : Thibaut, Célestin et Charline, François, Rémi !!! ˆˆQue de bons moments partagés ensemble qui ont permis de jalonner cette thèse ! Thibaut, je souhaite du fond du coe]ur que tu mènes ta thèse telle que tu la voies. Quant à tes élèves, Célestin, j'espère qu'ils feront beaucoup de thèses ; tu as une mission ! Julien, nous sommes passés à travers l'épreuve du feu de la prépa, et depuis, ça m'a fait plaisir de te revoir au gré de ton parcours international. ¡Cindy, no puedo esperar a que me muestres los secretos de México! Felicitaciones por su matrimonio. Enfin, Lucien et Vincent, ce fut à chaque fois très agréable de vous retrouver pour discuter avec vous ! Je souhaite remercier Marcel et Cathy pendant ces trois années de spéléo et canyon qui ont été essentielles à ma santé mentale ! :) Votre gentillesse et votre passion sont très inspirantes. Je ne pouvais pas rêver mieux que vous pour me faire découvrir cet univers aussi fabuleux que technique. Merci à toutes les personnes que j'ai croisées pendant ces sorties.

Je mesure la chance que j'ai d'avoir rencontré Maxime à qui je dédie des remerciements particulièrement appuyés. Son soutien sans faille et son humour imparable m'ont été d'une grande aide tout au long de cette thèse, sans compter les souvenirs inoubliables : merci, vraiment ! Onze ans, tu te rends compte ? Il est bien loin le temps des TPE ! ;p Je remercie de tout mon coeur ma famille -oncles, tantes, cousins, cousines, grands-parents -qui ont toujours été très encourageants ! Je tiens à exprimer toute la gratitude à mon papa, ma maman et ma III FOREWORD -AVANT-PROPOS soeur sans qui cette thèse n'aurait pu se faire. Les prémisses remontent sûrement à loin car c'est vous qui m'avez donné goût à cette curiosité et cette pensée critique qui font ce que je suis aujourd'hui ! Un grand merci à ma soeur qui est un exemple de courage et une source inépuisable de drôleries ! :-) JB, le [START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF]

Introduction

P redicting variably-saturated flows in porous media is a major issue for many fields in science and engineering. For example, such flows arise in soil physics, hydrogeology, environment, agriculture or oil industry for problems like subsurface contaminant transport, petroleum reservoir, water resources, surface ponding, etc. Experimental studies are a fundamental tool to describe such processes and have a good understanding. However, experimental studies are not always possible, nor feasible. Mathematical modelling and numerical simulation provide another possibility to explore and gain new insights into such physical systems.

Context and motivations

Understanding the dynamics of groundwater flow in coastal areas is of major interest in the fields of coastal engineering and sustainable development. In the present work, a particular attention will be paid on nearshore groundwater applications. A better understanding of the groundwater circulation is important for analysing and predicting a range of physical and bio-geochemical processes in coastal zones, such as sediment transport (bed stability), the diffusion of dissolved materials such as pollutants or nutrients, or mixing between continental (fresh) and marine (salt) water. Each of these issues requires special attention in the context of climate change and erosion [START_REF] Luijendijk | The State of the World's Beaches[END_REF][START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF] (sea level rise, submergence events, salinisation of arable land) and increasing anthropogenic pressure (coastal urbanisation). Photos of Fig. 1 show examples of such issues.

INTRODUCTION

of the physical processes governing groundwater flow dynamics in sandy beaches exposed to wave action. Recently obtained data during the ROUSTY2014 campaign will allow further analysis of these underground dynamics on a larger scale [START_REF] Sous | Groundwater fluxes within sandy beaches: swash-driven flow vs cross-barrier gradients[END_REF]. Despite their primary importance, experimental measurements remains very costly and are limited in time and space resolution and extent. A major scientific challenge is to develop robust and efficient numerical models to better anticipate, via the realisation of predictive scenarios, the evolution of the coastal zone. Few models have been presented so far and their scope of application remains limited, see e.g. [START_REF] Li | Wave-induced beach groundwater flow[END_REF][START_REF] Li | BeachWin: modelling groundwater effects on swash sediment transport and beach profile changes[END_REF]. Further numerical developments are needed to address the key issues of beach groundwater induced by the waves.

Many porous media flow models have been proposed and developed to simulate lots of different flows and transport processes at various scales. Multiphase flow problems present a more complex behaviour than single phase flow problem since phase dynamics influences each other. In this context, multiphase flow model may lead to solving difficulties or have a level of details too complex for the range of aimed applications. Simplified model considering reasonable hypotheses are very attractive alternative.

In the present work, flows in variably-saturated porous media are described by Richards' equation. What makes Richards' equation very appealing is that it models the porous medium as a whole part including both saturated and unsaturated zones. Despite neglecting air-phase, Richards' equations accounts for actions both from gravity and capillarity and so can model various and complex processes at different scales. Richards' equation is a nonlinear parabolic equation which can degenerate into an elliptic equation under complete saturation condition. The history of Richards' equation begins with Darcy's law formulated experimentally by Darcy in 1856 [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] for saturated porous media. Then, this result was extended to multiphase flows by Buckingham in 1907 [START_REF] Buckingham | Studies on the movement of soil moisture[END_REF] to give the Darcy-Buckingham law, which serves as cornerstone for the derivation of Richards' equation. The latter was first established by Richardson in 1922 [START_REF] Richardson | Weather Prediction by Numerical Process[END_REF] but was attributed solely to Richards who published independently the equation in 1931 [START_REF] Richards | Capillary conduction of liquids through porous mediums[END_REF]. First attempts to solve Richards' equation numerically go back to the late 1960s with Rubin [START_REF] Rubin | Theoretical Analysis of Two-Dimensional, Transient Flow of Water in Unsaturated and Partly Unsaturated Soils[END_REF] or Cooley [START_REF] Cooley | A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well[END_REF] for example. Then, from 1980s, Richards' equation was studied extensively, both from theoretical and numerical point of views. Results of existence, regularity and uniqueness were found [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF] and solution was assessed through simulations involving different numerical methods. One of the points of interest was nonlinear solver [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF]. Since 2000s, works focus mainly on Richards' equation stability and development of coupled and general simulations [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF]. In recent years, research consists in Richards' equation with hysteresis and advanced spatial/temporal discretizations [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Li | Adaptive local discontinuous Galerkin approximation to Richards' equation[END_REF] with adaptive approximations in size and/or in order.

Richards' equation has been extensively used for numerical simulations by the hydrogeology community, see e.g. [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF]. Despite this research effort, many numerical challenges remain for solving Richards' equation with some simulations being still unreliable and/or expensive [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF][START_REF] Zha | Review of numerical solution of Richardson-Richards equation for variably saturated flow in soils[END_REF]. This numerical complexity prevented a more general use of Richards-based model for a number of applications. Indeed, the solution of Richards' equation involves sharp wetting fronts which evolve both in space and time and are difficult to resolve. Besides, the simulation must treat simultaneously unsaturated/saturated regions of parabolic/elliptic natures, strong heterogeneity and anisotropy of the porous medium and possibly fast-changing boundary conditions. These aspects can also lead to steep gradients acting like discontinuities. Considering the set of nonlinear functions for hydraulic properties, it is seldom possible to foresee the behaviour of Richards' equation and getting numerical solutions can be hard to achieve. Especially, numerical schemes often fail to converge, requiring a careful numerical treatment, or need fine discretization, making computation costly. Many numerical methods are dealing with Richards' equation to treat wetting fronts and heterogeneous soils in various contexts. Each of them shows advantages and drawbacks and that is the reason why research is still undergoing to push back the limits for the use of Richards' equation. For instance, to handle layered soils, some recent publications include a Kirchhoff integral transform-based method [START_REF] Suk | Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media[END_REF], a transver-

OBJECTIVES AND SCOPE OF THE THESIS

sal method of lines [START_REF] Berardi | A mixed MoL-TMoL for the numerical solution of the 2D Richards' equation in layered soils[END_REF], a domain decomposition technique with L-scheme linearization [START_REF] Seus | A linear domain decomposition method for partially saturated flow in porous media[END_REF] and even upscaled models of Richards' equation for fractured porous media [START_REF] Kumar | Formal upscaling and numerical validation of unsaturated flow models in fractured porous media[END_REF].

Objectives and scope of the thesis

The objective of this thesis is to develop tools leading to a simulation of Richards' equation which is:

• accurate, in the sense that the numerical solution should be physically acceptable and converge to an exact solution when time steps and/or mesh resolution is refined;

• robust, in the sense that the method should provide a solution whatever the physical and numerical parameters for the problem are which means convergence reliability;

• efficient, in the sense that computation should be cost-effective for time and memory resources;

• suitable for beach groundwater induced by the waves.

The main targeted application of the present numerical research work is the simulation of groundwater flows within sandy beaches. Luijendijk et al. [START_REF] Luijendijk | The State of the World's Beaches[END_REF] found that more than one-third of the world's ice-free coastline are sandy. Satellite data analysis over a period of 33 years (1984-2016) indicates that a quarter of sandy beaches are eroding at rates exceeding 0.5 m/yr. Sandy shorelines have high socioeconomic benefits related to tourism and ecosystem services. They are a serious concern because of climate change since sea level rise put sandy beaches under threat of increasing erosion as Vousdoukas et al. asserted [START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF]. But Cooper et al. [START_REF] Cooper | Sandy beaches can survive sea-level rise[END_REF] qualified the disappearance of sandy beaches which can retreat inland. Nevertheless, sandy beaches remain subject to huge change and may disappear if urbanization blocks their migration. Besides, from a modelling point of view, sandy beaches are usually dissipative beach with weak slope where groundwater flow is known to be slow with a weak retroaction on the inducing waves. Then, a weak coupling of surface flow on groundwater flow can be considered to target applicative simulations. These reasons make the thesis prioritize the study of groundwater for sandy beaches.

For this purpose, a discontinuous Galerkin (DG) method is chosen to solve Richards' equation. DG methods are based on a variational formulation in an element-wise fashion, sharing advantages both with finite elements and finite volumes methods. In particular, they are locally conservative which is crucial in fluid dynamics [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF]. Moreover, the nature of DG formulation enables to work on nonconforming mesh and to change locally the degree of polynomial approximation. This is an important benefit since Adaptive Mesh Refinement (AMR), the so-called h-adaptation, and high-order accuracy, the so-called p-adaptation, become possible [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF], promoting a growing use of DG methods for transport phenomena in porous media such as two-phase flow problems [32,[START_REF] Bastian | A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure[END_REF].

Besides, Richards' equation is known to be a stiff differential equation with difficult convergence [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF] so the time discretization and the nonlinear solver have to be addressed carefully.

Keeping in mind that the numerical method should provide accurate and robust results, but efficient in terms of computational time, this thesis is an opportunity to contribute to update high-order adaptive DG methods for simulation of Richards' equation. Then, the following achievements are pursued:

• The Richards' equation is discretized by primal DG methods which lead to mass conservative, high-order and AMR to get accurate solution. Penalization weakly enforces boundary conditions in the DG formulation for robustness. Backward differentiation formulas are used for time discretization because they are known to be high-order and stable for stiff equation;

INTRODUCTION

• A fixed-point method together with adaptive time stepping make nonlinear iterations more reliable;

• An a posteriori error indicator is formulated for Richards' equation approximated by DG methods. This estimate is broken down into different contributions which guide mesh adaptation;

• The algorithm employs block-based AMR to save computational time.

Contributions and outline

This thesis proposes a novel strategy to investigate adaptive simulations in time and space for Richards' equation. AMR will be used to capture moving wetting fronts thanks to a posteriori estimation. h-Adaptation is employed for two-phase flow in porous media [32] but also for Richards' equation [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Li | Local discontinuous Galerkin approximations to Richards' equation[END_REF]. Making the most of DG methods flexibility, AMR is combined with a weighted discontinuous Galerkin (WDG) framework which allows discontinuity in the solution according to the nonlinear diffusivity. Following earlier works, this approach was formulated by Ern, Di Pietro and other collaborators [START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF][START_REF] Di Pietro | Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection[END_REF] and by Proft and Rivière [START_REF] Proft | Analytical and numerical study of diffusive fluxes for transport equations with near-degenerate coefficients[END_REF][START_REF] Proft | Discontinuous Galerkin methods for convection-diffusion equations with varying and vanishing diffusivity[END_REF]. Application of such strategy for Richards' equation, combining AMR, a posteriori estimate and WDG framework, is the main novelty of the study.

A dedicated code called Rivage has been designed, developed and benchmarked during this thesis. Rivage will serve in the laboratory for further study of Richards' equation but will also extended to other problems. Additional cases based on laboratory and largescale experiments are under construction and analysis.

The thesis is organized as follows.

Chapter I introduces Richards' equation in a comprehensive manner to get the model problem. Porous media are described to introduce basis notions and phenomenology of porous media flows. Upscaling methods are discussed as modelling tools to obtain fundamental relations for flow through porous media. In particular, they can serve to better understand the use of Richards' equation and are interesting working perspectives. The derivation of Richards' equation is given to draw the limitations of the model. Sections are devoted to hydraulic properties and the seepage boundary condition. They approximate specific processes. Besides, they hold nonlinearities responsible to numerical difficulties for Richards' equation which are discussed.

In Chapter II, Richards' equation is discretized. After introducing DG methods abilities, the space discretization is detailed. The use of weak penalization to enforce boundary conditions is highlighted. Time discretization is performed though implicit methods called backward differentiation formula.

The mass balance property of DG methods is exhibited. The nonlinear discrete problem is then solved by an iterative process based on Newton-Raphson method or fixed-point method. An heuristic adaptive time stepping is used to make nonlinear convergence more robust. At the end of this chapter, some implementations choices are provided given that DG methods offer several possibilities.

Chapter III is dedicated to the adaptive strategy proposed to improve solving of Richards' equation.

The block-based AMR is presented and justified for Richards' equation with a use of general meshes. Then, a gradient-based error indicator and estimation-based error indicator are introduced and motivated to drive mesh adaptation. They are designed to capture sharp fronts. Some variations for the numerical scheme are also proposed. In particular, jumps and flux are handled by the WDG method to better approximate sharp fronts with a discontinuity.

CONTRIBUTIONS AND OUTLINE

Several numerical tests are performed in Chapter IV. Numerical experiments allow to get some insights about DG methods. The model is validated against test-cases for 1D vertical infiltration involving different hydraulic properties and boundary conditions. Some specific behaviours of Richards' equation are noted and performance of mesh adaptation and weighted framework to capture wetting fronts are demonstrated. Tracy's benchmark provides an exact solution which is used to assess convergence and error indicators effectivity. The capabilities of the code are tested on laboratory experiments involving seepage, recharge and drainage for water table. Finally, the modelling strategy is tested on a large-scale challenging case with experimental dataset: the wetting of a multi-materials dam.

In Chapter V, some applications are made for groundwater in sandy beaches. After presenting notions about beach system involving forcing waves and groundwater circulation, comparisons are carried for three cases. An idealized beach is considered to see if the model is able to render main characteristics of beach groundwater observed in experiments from the domain.

At last, a section is devoted to some concluding remarks and perspectives for future works. N otions of hydrogeology and an historical perspective of study of flow in porous media are pro- vided in this chapter. In addition, some insights are given to better understand how it is possible to derive macroscopic flow relations for porous media from fundamental equations at pore scale. This overview enables to set useful concepts in order to describe the governing equation used in the next parts of the thesis to model unsaturated porous media flow: the Richards' equation. Constitutive laws and specific boundary conditions are also introduced to complete the model. Richards' equation is a well-known model, broadly employed for simulations of infiltration in many fields with a long background. However, this situation may be misleading for new arrivals in the domain because underlying hypotheses for its use are often ignored. Besides, solving Richards' equation may be very difficult largely due to nonlinearities of hydraulic properties. This is particularly the case when one wants to simulate non-standard phenomenon or reach accuracy, robustness and cost-effectiveness. So,

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA

solution of Richards' equation is discussed to highlight flow mechanisms that give rise to numerical challenges as well as computational developments commonly used to deal with them. A qualitative analysis is presented to attempt to sort Richards' equation behaviour.

I.1 Description of porous media

Most of the definitions and notions related here can be found in textbooks about flow in porous media, for instance [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF][START_REF] Bear | Dynamics of fluids in porous media[END_REF][START_REF] Helmig | Multiphase Flow and Transport Processes in the Subsurface[END_REF].

A porous medium is a material constituted of voids filled by fluids, either liquids or gas. The solid portion of material is called the skeleton or matrix while the void portions are called pores. Lots of natural and artificial structures can be considered as porous media. Besides, porous media are the place of many processes leading to many concepts for different fields of studies such as mechanics, geosciences, biology, material science or engineering. In particular, fluid flow through porous media is a subject of interest since it explains lots of observations made for natural structures or industrial systems. For example, knowing the way in which fluid flows through a porous medium is useful to describe water infiltration into a soil column or fluid moving in an absorbing sponge. A porous medium is characterised by many variables. Porosity is an essential property. It is a number between 0 and 1 which is the measure of pore volume over the total volume. Figure I.1 illustrates porous media with different porosity and pore structures. Indeed, other properties are implied to describe fully porous flow phenomena like mechanical dispersion: tortuosity (path length), viscosity (pore friction) and velocity (pore section). Then, porous media can have the same porosity but they exhibit different pore connectivity and types of porosity. Porous media present multi-scale structure. Indeed, pores may have various sizes and shapes resulting in different flow regimes. Fluid flow are mainly driven by two kinds of forces. Firstly, if pores are small enough, then liquid is displaced thanks to intermolecular forces. It is made possible by the combination of cohesive forces within the liquid (surface tension) and adhesive forces between the liquid and the surrounding solid matrix. This is called capillarity. Secondly, flow in large pores are subject to gravitational forces. Capillarity effects can be sufficiently strong to elevate fluid such as in plants to move water up or paper towels to absorb liquid.

A phase is a chemically homogeneous portion of the porous medium that is separated from other such portions by a definite physical boundary. Solid phase represents the skeleton while pores are filled by fluid phases. Fluid phases are immiscible if they maintain a boundary between them. Fluid flow in porous media can be either one-phase flow or multiphase flow. In this latter case, two-phase flow are particularly studied. To simplify description, a two-phase flow with water and air is considered from now on. Single-and multi-phase flow are outside the scope of the thesis. Most of the time, they can I.1. DESCRIPTION OF POROUS MEDIA be figured out from the following description but some additional phenomena may rise like osmotic forces or chemical reactions between different phases. In a two-phase system, there is a wetting (water) and a non-wetting (air) phase. Wettability definition is linked to the contact angle between fluid phases and the solid skeleton and plays a role in capillary forces.

Saturation represents the quantity of water contained in a porous medium. It is the volume fraction of water compared to the pore volume and ranges from 0 to 1. In practice, the range is smaller because of trapped air bubbles or liquid film around grains. In this case, residual saturation and maximum saturation defines a new normalized quantity called effective saturation. The porous medium is saturated if water fills fully the pore space. Otherwise, the porous medium is said to be unsaturated. Then, two different zones are distinguished: the saturated zone (or phreatic zone) and the unsaturated zone (or vadose zone). The water table is the surface delimiting these two zones. The capillary fringe is generally considered as a part of the unsaturated zone. It is the layer in which water seeps up from water table due to capillary forces. Capillary fringe exhibits a nearly-saturated state which may extend from few centimetres to several meters. Capillary fringe size depends strongly on local porosity. Figure I.2 illustrate the different zones of saturation as discussed above. Capillary fringe is showcased with an artesian well in which water cannot raise above water table with capillary forces. One should notice that type names refers to grain size but it does not mean that the particle is composed of this mineral even though it is often the case. Grain size can be used to guess how water infiltrates soil because it is linked to pore size which determines capillarity. However, other parameters contribute to water dynamics like pore shape, pore continuity or pore distribution. When air is present, retained water is attached to the grains and its thickness depends on molecular forces. It is made of hygroscopic water, which is in direct contact with the grain, immobile and can only be removed by desiccation or vacuum, and pellicular water, which adheres to hygroscopic water and moves around grains but can be extracted only with centrifugation. Retained water cannot be affected by gravity and does not transmit hydrostatic pressure. Capillary water is subject to tension forces at the interface between air and water. It can be partially extracted by gravity and transmit pressure. Gravitational water is able to be drained out by gravity and its flow moves under gravity and pressure fields. Capillary water and gravitational water are referred as free water and they define flow dynamics in unsaturated porous media. These regimes are sketched in Fig. I.7. Adsorbed regime corresponds to the retained water. At this stage, water can evaporate to make vapour in air phase. In pendular regime, the flow has no continuous paths so is considered as macroscopically immobile [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF]. The flow of liquid water is possible from the funicular stage. If the amount of water in the system is sufficient, pores can be filled with water which gives the capillary and saturated regime. Capillary regime contains occluded air bubbles and air phase losses its continuity. Saturated regime happens when remaining air is dissolved into water. From now on, evaporation and dissolution are neglected so we can focus on the funicular, capillary and saturated regimes. Adapted from Szymkiewicz [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF].

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA

The state of water phase is commonly described by the concept of energy potential. The total energy is divided into gravitational energy, pressure energy and velocity energy. This is related to the Bernoulli's equation (i.e. integrated momentum of the Euler equations), valid at any arbitrary point along a streamline S:

H = z + p ρg + cv 2 2g + 1 g S ∂ t v ds + h r , (I.1)
where

I.2 A brief derivation of subsurface flow relation

In this thesis, Richards' equation will be used, see Section I.3. However, the derivation of Richards' equation lies ultimately on Darcy's law which is the basis relation of flow in porous media. Then, it appears that assumptions to derive Darcy's law are important to know if one wants to employ Richards' equation.

I.2.1 Upscaling approaches

Porous media show heterogeneous structures at microscopic scale inducing spatial variability of their hydraulic properties and characteristics such as porosity, permeability or retention. These microstructures may have an influence at larger scale. However, simulating directly such features would require very fine grids resulting in excessive computational cost. Moreover, from an experimental viewpoint, the microscale properties are difficult to reach or not known with much detail. This makes not purposeful to represent the microscale field if one wants to simulate fields much larger than pore size. Instead, a macroscale representation might be assumed to reproduce the behaviour of the system. Upscaling approaches try to develop such model by the mean of averaged values from a volume element representative of microscale pattern.

Upscaling approches lie on the notion of hierarchy of scales ad spatial variability. Scales are usually separated into a microscopic (local) scale and a macroscopic (large) scale. More scales can be involved, for example, a mesoscopic scale for the volume element or for fractured porous media and even a

I.2. A BRIEF DERIVATION OF SUBSURFACE FLOW RELATION

megascopic scale for aquifers composed of several layers. Spatial variability is fundamental because it often defines the type of upscaling approaches. They can be sorted into three main ideas:

• homogenization methods with, for example, Hornung [START_REF]Homogenization and Porous Media[END_REF], Allaire [START_REF] Allaire | Homogenization and Two-Scale Convergence[END_REF][START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF] or Auriault [START_REF] Auriault | Heterogeneous medium. is an equivalent macroscopic description possible?[END_REF][START_REF] Auriault | Taylor dispersion in porous media: Analysis by multiple scale expansions[END_REF];

• volume average methods with many works from Whitaker [START_REF] Whitaker | Flow in porous media I: A theoretical derivation of Darcy's law[END_REF][START_REF] Whitaker | Flow in porous media II: The governing equations for immiscible, two-phase flow[END_REF][START_REF] Whitaker | Flow in porous media III: Deformable media[END_REF][START_REF] Whitaker | The Method of Volume Averaging[END_REF], Quintard [START_REF] Quintard | Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media II: Generalized volume averaging[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media v: Geometrical results for two-dimensional systems[END_REF], Gray [START_REF] Gray | Paradoxes and Realities in Unsaturated Flow Theory[END_REF], Neuman [START_REF] Neuman | Theoretical derivation of Darcy's law[END_REF] or Marle [START_REF] Marle | Écoulements monophasiques en milieu poreux[END_REF][START_REF] Marle | Les écoulements polyphasiques en milieu poreux: de l'échelle des pores à l'échelle macroscopique[END_REF];

• stochastic methods with authors like Dagan [START_REF] Dagan | The generalization of Darcy's Law for nonuniform flows[END_REF], Zhang [START_REF] Zhang | Stochastic Methods for Flow in Porous Media[END_REF] or Matheron [START_REF] Matheron | Genèse et signification énergétique de la loi de Darcy[END_REF][START_REF] Matheron | L'émergence de la loi de Darcy[END_REF].

Upscaling literature is extensive so a complete review is beyond the scope of the thesis. Consulted studies and general works are listed above. For further insight about authors and contributions of upscaling approaches, Davit et al. [START_REF] Davit | Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?[END_REF] made a state-of-the-art where a brief history is available and comparison of concepts is carried out like depicted in Fig. I.9. This tremendous amount of contributions leads to many theoretical framework and confusion. Despite evidence of links between these approaches, there is no answer for lots of questions and little effort to clarify/unify these theories as stated by Davit et al. [START_REF] Davit | Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?[END_REF]. Indeed, upscaling methods are not straightforward since they handle some concepts which may be complex and/or restrictive.

Figure I.9 -Diagram of coordinate systems and scale hierarchy for non-periodic media typical of volume averaging (left) and periodic media typical of homogenization (right). µ is pore-scale characteristic length, ℓ is the size of averaging volume, L is a macroscale characteristic lengths, x is spatial variable coordinate system, x * and y * are nondimensionalized coordinate systems with L and ℓ respectively. Taken from Davit et al. [START_REF] Davit | Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?[END_REF].

Homogenization is a very mathematical approach to derive Darcy's law. It is based on asymptotic de-

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA

velopments where two scales come into play. Some mechanisms make velocity converge to Darcy's velocity. The derivation is rigorous, well mastered with strong foundation. However, making a physical interpretation is quite difficult and underlying hypotheses are based on periodic or quasi-periodic structure for porous medium.

Volume average method is a more intuitive approach for Darcy's law derivation. It uses control volume (CV) in which physical quantities are averaged. These averaged quantities are reorganized and some of them are disregarded according to assumptions. The average velocity is obtained and corresponds to the Darcy's velocity. The physical meaning of the process is straightforward. This spontaneous interpretation makes it easy to apply for a wire range of problems. Nevertheless, some assumptions used to neglect terms lack strict justifications and the definition of control volume is not properly established. Indeed, the control volume needs to have the correct size to make averaging meaningful.

In this case, control volume is called a representative elementary volume.

Stochastic methods are designed with a statistical and probabilistic point of view. They are less used than the two aforementioned methods but are more fundamental. They work at micro-scale and analyse porous medium properties in a way similar to the ergodic hypothesis.

Due to the variability of spacelengths and timelengths scales of phenomena in porous media, the lack of accurate experimental measures and the wide range of problems, macroscopic description of the system by the mean of upscaling is determining. The boundary effects or the locality of processes define the macroscopic problem which is then used for numerical simulations.

I.2.2 Volume averaging method

Darcy's law is at the basis of the work of this thesis because the same remarks can be formulated when the relation is extended to a two-phase system, i.e. using the Darcy-Buckingham law. Indeed, the latter was accepted on experimental and deductive basis. So, as starting point, other derivations for Darcy's law may be proposed for specific problems or highlight "hidden" neglected terms. This makes the upscaling procedure both important and useful to assess the validity of a domain or find new equations. For example, it can be interesting to see effects related to anisotropy, tortuosity, anisotropy, inertia, non-Newtonian flow, etc. Upscaling for Darcy-Buckingham law would be a next step since upscaling is more complicated when multiphase flow system is in play [START_REF] Liu | Generalization of the Darcy-Buckingham Law: Optimality and Water Flow in Unsaturated Media[END_REF].

For this thesis, the volume averaging method was chosen to investigate Darcy's law derivation and draw clear hypotheses about its range of validity. Indeed, lots of works are available and its intuitive interpretation may serve as a guide. Many works were read to attempt this goal but those from Whitaker and Quintard were particularly used [START_REF] Whitaker | Flow in porous media I: A theoretical derivation of Darcy's law[END_REF][START_REF] Whitaker | Flow in porous media II: The governing equations for immiscible, two-phase flow[END_REF][START_REF] Whitaker | Flow in porous media III: Deformable media[END_REF][START_REF] Whitaker | The Method of Volume Averaging[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media II: Generalized volume averaging[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems[END_REF][START_REF] Quintard | Transport in ordered and disordered porous media v: Geometrical results for two-dimensional systems[END_REF]. The process of volume averaging consists in finding a macroscale description of the porous medium from an equation applied at microscale, see Fig. I.12.

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA

This transformation is made by the mean of an operator whose application range is called representative elementary volume.

Figure I.12 -Volume averaging procedure to get an equivalent homogeneous macroscale medium from the heterogeneous micro-structure. The red square stands for the representative elementary volume. Adapted from Losada et al. [START_REF] Losada | Modeling the Interaction of Water Waves with Porous Coastal Structures[END_REF].

Volume averaging method consists in averaging equations of porous medium over the Representative Elementary Volume (REV). In other words, it is a filtering of small space scales. The general approach aims to average equations to obtain terms which are estimated against each other. Then, given hypotheses at disposal, some of the terms are disregarded. The closure of the problem allows to model non-local terms and so, the rising average velocity is assimilated to the Darcy velocity.

In the following sections, fluid phase is denoted by F and the solid phase by S . Volume averaging method assumes that there are three space scales satisfying the ordered relation:

ℓ ≤ r 0 ≪ L, (I.3)
where L corresponds to the macroscopic characteristic length of the porous medium, r 0 is the mesoscopic characteristic length and ℓ is the microscopic characteristic length of the porous structures inside the medium. Let x ∈ Ω ⊂ R 3 be the global space variable sweeping the whole part of the porous medium and y the local space variable for the REV of center x. The space variable z is such that z = x + y. The notations, coordinates and scales in play are illustrated in Fig. I.13.

I.2. A BRIEF DERIVATION OF SUBSURFACE FLOW RELATION

.13 -Illustration of the coordinate system, notations and hierarchy of scale used for the volume averaging method used in this thesis.

REV size is crucial because it is determining in the averaging process of the quantities but also because it is defining the nature of the porous medium. Yet, it is difficult to give a precise value in practice. Indeed, the REV radius is usually set as the limit of the control volume radius for which the considered variable does not hold fluctuations induced by heterogeneities from small scales, see Fig. I.14. This definition is intuitive but there are several issues like the choice of the defining variable, the control volume shape or the intrinsic nature of the porous medium. Indeed, the porous medium may give rise to different scales where these conditions are met or even no appropriate scale at all. Despite this, the convenience of such a definition makes it useful. The VER can be assimilated into a ball of centre x and radius r 0 so its definition is:

∀x ∈ Ω, V = B(x, r 0 ) = V F (x) + V S (x) = Cst, (I.4)
where we can deduce that |V | ∝ r 3 0 . Similarly, inside a VER, we assume that the solid and fluid phases can be transformed thanks to a suitable function to correspond to balls whose definition is the

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA following: ∀x ∈ Ω, ∃Θ V |    V F (x) --→ Θ V B((x, r F (x)), |V F (x)| = |B((x, r F (x))|, (I.5) ∀x ∈ Ω, ∃Θ S |    V F (x) -→ Θ S B(x, s F (x)), |∂V F (x)| = |B(x, s F (x))|, (I.6)
where Θ V and Θ S are the transformation functions which preserve respectively the volume and the surface.

For this work, the considered variable to define micro-structures will be the porosity which is a classic choice:

Φ(x) = V poral (x) V total , (I.7)
where V poral is the poral space and V total is the total space of the porous medium.

Specific configurations can be distinguished thanks to porosity and characteristic lengths (I is the identity matrix):

If ℓ ∼ r 0 ≪ L, then the porous medium is said to be ordered. (I.8) If ℓ ≪ r 0 ≪ L, then the porous medium is said to be disordered.

(I.9) If ∇Φ ≪ I, then the porous medium is said to be homogeneous.

(I.10) If ∇Φ ≫ I, then the porous medium is said to be heterogeneous.

(I.11)

When the porous medium is ordered, it means that it follows a periodic or quasi-periodic structure.

Homogeneization is often preferred in this case. The more general hypothesis (I.9) is essential for the volume averaging method: it is called the scales separation hypothesis.

Let 1 F be the function defining the fluid phase:

1 F (x + y) = 1 if |y| ≤ r F (x), 0 otherwise. (I.12)
Since the work focus only the fluid phase, the subscript F will be omitted when it is not necessary.

We define two filtering operators. The first one is the phase average and for a quantity ξ, it writes:

⧼ξ⧽(x) = 1 r 0 3 B(0,r 0 ) 1 F (y)ξ(x + y) dy (I.13)
The second is the intrinsic average and writes:

ξ(x) = 1 r 3 (x) B(0,r 0 ) 1 F (y)ξ(x + y) dy (I.14)
We have the following relationship:

⧼ξ⧽(x) = Φ(x)ξ(x), (I.15)
where Φ(x) = r 3 (x) r 0 3 . Besides, it follows:

⧼1 F ⧽(x) = 1 r 0 3 B(0,r 0 ) 1 F (y)1 F (x + y) dy = 1 r 0 3 B(0,r 0 ) 1 F (y) dy = Φ(x) (I.16)

I.2. A BRIEF DERIVATION OF SUBSURFACE FLOW RELATION

We know that:

0 ≤ r(x) ≤ r 0 and 0 < ℓ ≪ r 0 ≪ L, ⇐⇒ 0 ≤ r(x) r 0 ≤ 1 and ℓ L = ϵ ℓ ≪ r 0 L ≪ 1, =⇒0 ≤ ϵ(x) ≤ ϵ 0 ≪ 1, (I.17)
where ϵ(x) = r(x) L and ϵ 0 = r 0 L .

Spatial averaging make use of an important result called the spatial averaging theorem [START_REF] Howes | The spatial averaging theorem revisited[END_REF] which means for a quantity ξ:

⧼∇ξ⧽ = ∇ξ + 1 r 0 3 Σ(0,s(x)) 1 F (y)ξ(x + y) dy (I.18)
where Σ(0, s(x)) is the surface of the ball of center 0 and radius s(x) corresponding to B(0, r(x)).

A short derivation is shown in Appendix A -Spatial averaging theorem with more general notations.

To begin the averaging process, we have to choose a local problem to describe the flow at microscale. Seen that flow is very often sufficiently slow in a porous medium, Navier-Stokes equation is simplified into the Stokes equation with a no slip condition:

-µ∆u + ∇p -ρg = 0, (I. [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF])

u • n = 0. (I.20)
The phase average is applied to the Stokes equation in Eq. (I. [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF]) and the spatial averaging theorem is used when it is possible. To facilitate visualization of calculation steps, viscous, pressure and gravity terms are treated separately. So it reads:

Pressure terms:

⧼∇p⧽(x) = ∇⧼p⧽ + 1 r 0 3 Σ(0,s(x)) 1 F (y)p(x + y)n dσ = Φ(x)∇p + p∇Φ(x) + 1 r 0 3 Σ(0,s(x))
1 F (y)p(x + y)n dσ, (I.21)

Viscous terms:

⧼ -µ∆u⧽(x) = -µ ⧼ ∇ • ∇u ⧽ (x) -µ∇ • ⧼ ∇u ⧽ - µ r 0 3 Σ(0,s(x)) 1 F (y)(∇u • n)(x + y) dσ = -µ∆ ⧼ u ⧽ + ∇ • - µ r 0 3 Σ(0,s(x)) u(x + y)n dσ =0 with Eq. (I.20) - µ r 0 3 Σ(0,s(x)) 1 F (y)(∇u • n)(x + y) dσ = -µ(Φ(x)∆u + u∆Φ(x) + 2∇Φ(x)∇u) - µ r 0 3 Σ(0,s(x)) 1 F (y)(∇u • n)(x + y) dσ, (I.22)
Gravity terms:

⧼ -ρg⧽(x) = -ρg ⧼ 1 ⧽ (x) = -ρgΦ(x). (I.23)
In the next step, variables will be decomposed with perturbation analysis to study the influence of small scales compared to a reference state. The most used decomposition is the so-called Gray's decomposition. For a variable ξ, it writes: ξ = ξ + ξ. Viscous terms:

⧼ -µ∆u⧽(x) = -µ(Φ(x)∆u + u∆Φ(x) + 2∇Φ(x)∇u) - µ r 0 3 Σ(0,s(x)) 1 F (y)(∇u • n)(x + y) dσ - µ r 0 3 Σ(0,s(x)) 1 F (y)(∇ u • n)(x + y) dσ (I.26)
The following step is a major stage of averaging process. It includes several techniques whose goal is to reject some terms and get a simplified problem by making assumptions for timescales/lengthscales and by approximating deviations. Finally, the last step will obtain a macroscopic equation from closure of the simplified problem. In order to investigate the method of volume averaging and make clear hypotheses about upscaling, a part of the thesis was dedicated to derive a different way to simplify the problem by giving clear assumptions. However, the averaging process could not be concluded during this thesis.

The main choice in this work was to process a nondimensionalization by the mean of several characteristic physical quantities, in particular two different characteristic space lengths. However, the star in superscript as well as dependence on x will be omitted in the following equations to preserve readability. Moreover, several quantities which may correspond respectively to deviation rate of velocity, deviation rate of pressure and compactness, are defined:

Υ = U U , Ψ = P P , η c (x) = s 2 (x) r 0 2 . (I.32)
So, we proceed to nondimensionalization and multiply by L ρU 2 which gives:

Pressure terms:

⧼∇p⧽(x) =Φ(x) P L ∇p + P L p∇Φ(x) + P s 2 (x) r 0 3 Σ(0,1) 1 F (s(x)y)p(x + s(x)y)n dσ + ps 2 (x) r 0 3 Σ(0,1) 1 F (s(x)y) p(x + s(x)y)n dσ ⧼∇p⧽(x) =Φ(x) P ρU 2 ∇p + P ρU 2 p∇Φ(x) + P Ls 2 (x)
ρr 0 3 U 2 Σ(0,1) 1 F (s(x)y)p(x + s(x)y)n dσ + pLs 2 (x)
ρr 0 3 U 2 Σ(0,1) 1 F (s(x)y) p(x + s(x)y)n dσ ⧼∇p⧽(x) = 1 Ma 2 Φ(x)∇p + 1 Ma 2 p∇Φ(x) + η c (x) ϵ 0 Ma 2 Σ(0,1) 1 F (s(x)y)p(x + s(x)y)n dσ non-local term + Ψη c (x) ϵ 0 Ma 2 Σ(0,1) 1 F (s(x)y) p(x + s(x)y)n dσ (I.33)
Viscous terms:

⧼ -µ∆u⧽(x) = - µU L 2 (Φ(x)∆u + u∆Φ(x) + 2∇Φ(x)∇u) - µs 2 (x)U Lr 0 3 Σ(0,1) 1 F (s(x)y)(∇u • n)(x + s(x)y) dσ - µs 2 (x) U r 0 4 Σ(0,1) 1 F (s(x)y)(∇ u • n)(x + s(x)y) dσ ⧼ -µ∆u⧽(x) = - µ ρLU (Φ(x)∆u + u∆Φ(x) + 2∇Φ(x)∇u) - µs 2 (x) ρr 0 3 U Σ(0,1) 1 F (s(x)y)(∇u • n)(x + s(x)y) dσ - µs 2 (x)L U ρr 0 4 U 2 Σ(0,1) 1 F (s(x)y)(∇ u • n)(x + s(x)y) dσ ⧼ -µ∆u⧽(x) = - 1 Re (Φ(x)∆u + u∆Φ(x) + 2∇Φ(x)∇u) Brinkman's correction - η c (x) ϵ 0 Re Σ(0,1) 1 F (s(x)y)(∇u • n)(x + s(x)y) dσ non-local term - Υη c (x) ϵ 0 2 Re Σ(0,1) 1 F (s(x)y)(∇ u • n)(x + s(x)y) dσ (I.34)
Gravity terms:

⧼ -ρg⧽(x) = - Lg U 2 Φ(x) ⧼ -ρg⧽(x) = - 1 Fr 2 Φ(x) (I.35)
The averaged equation corresponds to the boxed terms in yellow . One should notice that there are terms referred to the Brinkman's correction taking into account some boundary effects with inertia. Besides, non-local terms are present: dimensional analysis should go on with these terms by performing asymptotic developments related to the quantity x + s(x)y. We would obtain terms related to moments of different orders which can be estimated. Then, the terms highlighted in yellow should be kept in order to get an averaged equation defining Darcy's velocity. However, as said earlier, the different estimated weights are difficult to compare making these terms difficult to neglect. Finally, the deviation terms would be replaced by a closure problem to derive the Darcy's law.

This nondimensionalization is not standard compared to the one found in literature. Indeed, the core idea is to estimate the quantities in regard of several characteristic lengths following the concept of scale separation. However, the process was not able to complete successfully during the thesis. This prolongation constitutes a serious perspective of research whose goal would be to derive subsurface flow relations adapted according to dimensionless quantities defining different hydrodynamic regimes.

I.3 Governing equation

Richards' equation is a classic nonlinear parabolic equation used to describe flow occurring in unsaturated/saturated zones of an aquifer. This section is devoted to a synthetic recall of the derivation of Richards' equation, which is rather sparse in the literature [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF][START_REF] Baron | Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equation[END_REF]. In a porous medium, mass conservation principle stands for two-phase flow: a wetting phase and a non-wetting phase, usually water and air respectively. The following system is considered:

α ∈ {air; water},      ∂ t (ρ α ΦS α ) + ∇ • (ρ α q α ) = 0, q α = - k(S α ) µ α ∇(p α + ρ α gz), (I.36)
where ρ α denotes the density [M • L -3 ], Φ the porosity [-], S α the saturation [-], q α the Darcy velocity

[M • T -1 ], k the permeability tensor [L 2 ], µ α the dynamic viscosity [M • L -1 • T -1 ], p α the pressure [M • L -1 • T -2 ], t the time [T], z the elevation [L] and g the gravitational acceleration [L • T -2 ].
Each of these variables evolve in time inside a three-dimensional space for the general case. The Darcy velocity q α is modelled by the Darcy-Buckingham law as an extension of Darcy's law, initially devoted to single-phase flow, for a two-phase flow system . This modelling implies that both Darcy's law itself and its extension are valid, requiring in particular an assumption of local equilibrium [START_REF] Liu | Generalization of the Darcy-Buckingham Law: Optimality and Water Flow in Unsaturated Media[END_REF].

In the Equation system (I.36), if water contents and relative permeabilities are given, there are four unknowns for two equations: saturations and Darcy velocities for both air and water equations. The system is closed by two equations for saturation and capillary pressure:

S air + S water = 1, by definition, p air -p water = P c (S water ), (I.37)

where

P c is capillary pressure [M • L -1 • T -2 ],
an invertible function known from experiment.

A major hypothesis for Richards' equation is to eliminate the equation for air in the system (I.36). It requires to assume that the air phase is connected continuously with the atmosphere at every points inside the porous medium. Besides, air viscosity is about 50 times smaller than water viscosity [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF]. This means pressure gradients balance much faster in air than in water phase to equilibrate at atmospheric pressure. In other words, air pressure is hydrostatic and thus:

∇(p air + ρ air gz) = 0 (I.38)

⇐⇒ p air = p 0 -ρ air gz, (I.39)

where p 0 [M • L -1 • T -2 ]
is atmospheric pressure which is assumed to be constant at porous medium scale and during the observation time. It is worth mentioning that trapped air pockets and such effects, which do not satisfy the continuous connection assumption for air phase, cannot be treated by the Richards' approach.

Consequently, one can know air pressure everywhere and, if water saturation is known, air saturation is completely defined thanks to the relation (I.37). That is why the continuity equation for water phase CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA from the system (I.36) is now solely considered. Henceforth, the subscript □ water for water phase is omitted for the sake of clarity. So it writes:

∂ t (ρΦS) -∇ • ρ µ k(S)∇(p + ρgz) = 0. (I.40)
Some reasonable additional hypotheses are usually taken into consideration:

• (H1) water density is sufficiently homogeneous so that ∇ρ = 0;

• (H2) water is sufficiently incompressible so that ∂ t ρ = 0;

• (H3) the solid skeleton is sufficiently not deformable so that ∂ t Φ = 0.

Hypotheses (H1) and (H2) lead to rewrite the equation (I.40) such that:

∂ t (ΦS) -∇ • ρg µ k(S)∇ p ρg + z = 0. (I.41)
Here, a digression is undertaken to give an understanding about the underlying dynamics of Richards' equation. Knowing that air density is about 820 times smaller than water density at sea level at 15°C, the following quantity can be approximated:

∇ p ρg = ∇ p air -P c ρg = ∇ p 0 -ρ air gz -P c ρg ≈ -∇ P c ρg . (I.42)
This allows some manipulations from equation (I.41) to give:

∂ t (ΦS) - ρg µ ∇ • (k(S)∇z) + 1 µ ∇ • (k(S)∇P c ) = 0. (I.43)
This extended -but somehow approximated -formulation separates the convective term associated with gravitational forces and the diffusive term associated with capillary forces. This shows the complex convective-diffusive dynamics modelled by Richards' equation in spite of neglected air phase effects. Water content θ with hypothesis (H3) and hydraulic conductivity K depend only on the saturation S and not on the primary variable ψ, the pressure head, like it is usually done in Richards' equation. However, the writing is straightforward since the saturation is directly linked to pressure head though the invertible function of capillary pressure P c :

S = P c (p) = P c (ρgψ). (I.47)
This relation, often named water retention curve or soil moisture characteristic, is fundamental in study of porous media flow to define hydraulic properties. It will be discussed further in section Section I.4.

I.4. HYDRAULIC PROPERTIES

Three main formulations of Richards' equation can be established: the pressure-based formulation, the saturation-based formulation or the mixed formulation. The former two formulations contain a single unknown but they are either non-conservative or undefined for complete saturation and heterogeneous soils. The pressure-based formulation and the saturation-based formulation are written respectively as:

C(ψ)∂ t ψ -∇ • (K(ψ)∇(ψ + z)) = 0, (I.48) ∂ t θ -∇ • (D(θ)∇θ + K(θ)∇z) = 0, (I.49)
with the capillary capacity C [L -1 ] and the hydraulic diffusivity D [L 2 • T -1 ] which are respectively defined as:

C(ψ) = dθ(ψ) dψ , (I.50) D(θ) = K(θ) dψ(θ) dθ = K(θ) C(θ) . (I.51)
In the pressure-based formulation, fluid compressibility effects may be included and then accounting for hypothesis (H2). The usual capillary capacity definition is replaced by the following expression:

C(ψ) = dθ(ψ) dψ + S s S a , (I.52)
where S s denotes the specific storage [L -1 ] and S a the aqueous phase saturation [-].

The mixed formulation is selected in the present study owing to its versatility and better numerical behaviour in spite of holding two linked unknowns [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF]. The mixed formulation of Richards' equation can finally be written in pressure head form:

∂ t θ(ψ) -∇ • (K(ψ)∇(ψ + z)) = 0. (I.53)
The mixed formulation can be rewritten in the hydraulic head form which is more common in hydrology:

∂ t θ(h -z) -∇ • (K(h -z)∇h) =

I.4 Hydraulic properties

Solving Richards' equation (I.53) requires two constitutive laws: one for water content and one for hydraulic conductivity. Several models exist, depending on the hydraulic properties of the porous medium.

Usually, the tensor of permeability k, and so the hydraulic conductivity K, are supposed to react to saturation identically for each space direction. This leads to write:

k(ψ) =k s k r (ψ), (I.55) K(ψ) =K s K r (ψ), (I.56)
where k s denotes the intrinsic or saturated permeability tensor [L 2 ], k r the relative permeability [-], K s the intrinsic or saturated hydraulic conductivity tensor [L • T -1 ] and K r the relative hydraulic conductivity [-].

For practical purposes, the water content is often described in terms of effective saturation S e [-]:

S e (ψ) = θ(ψ) -θ r θ s -θ r , (I.57)
where θ s denotes the saturated water content [-] and θ r the residual water content [-], corresponding to the maximal and minimal saturations, respectively.

S e (ψ) =    1 if ψ ≥ ψ e ,
S ⋆ e (ψ) otherwise,

and K r (ψ) =    1 if ψ ≥ ψ e , K ⋆ r (ψ) otherwise. (I.58)
S ⋆ e and K ⋆ r are monotonic increasing functions of pressure head ψ in the unsaturated zone. Throughout the thesis, several constitutive laws will be used to model hydraulic properties in the unsaturated zone. They are compiled in Tab. I.1.

Name Expression Parameters

Gardner-Irmay relations (1958) [START_REF] Irmay | On the hydraulic conductivity of unsaturated soils[END_REF][START_REF] Gardner | Some Steady-State Solutions of the Unsaturated Moisture Flow Equation with Application to Evaporation from a Water Table[END_REF]]

S ⋆ e (ψ) = e αψ m K ⋆ r (ψ) = e αψ α: pore-size distribution [-] m: tortuosity [-]
Vachaud's relations (1971) [START_REF] Vachaud | Hysteresis During Infiltration and Redistribution in a Soil Column at Different Initial Water Contents[END_REF] S ⋆ e (ψ) =

C C + |ψ| D K ⋆ r (ψ) = A A + |ψ| B
S ⋆ e (ψ) = (1 + (α|ψ|) n ) -m K ⋆ r (ψ) = S ⋆ e l (ψ) 1 -1 -S ⋆
S ⋆ e (ψ) = (1 + (α|ψ|) n ) -m K ⋆ r (ψ) = S ⋆ e (ψ) S * l   1 -1 -S ⋆ e 1 m (ψ) m 1 -1 -S * 1 m m   2 S * = (1 + (-αψ s ) n ) -m [-] with ψ s ≤ ψ e = 0[-] l =    0.5 for Mualem [75] 1
for Burdine [START_REF] Burdine | Relative Permeability Calculations From Pore Size Distribution Data[END_REF] : pore connectivity [-] α: parameter linked to air entry pressure inverse

[L -1 ] n > 1: pore-size distribution [-] m = 1 - 1 n : pore-size distribution [-]
Table I.1 -Hydraulic relations used in this thesis. A well-known experimental fact is not taken into account by these models: hydraulic properties are hysteretic and dynamic, see Fig. I.16. Indeed, whether water is filling or draining the pores, different wetting (imbibition) and drying (drainage) curves may be distinguished. Besides, dynamics can appear due to memory effects or capillary actions. Equilibrium is located between the two main curves following history of wetting/drying scanning curves. The difference of water content between primary drying and main wetting curves is due to occluded air entrapped by fast infiltration. A similar effect can be observed for primary wetting curve with residual water. Each main curve can be approximated by asymptotics which define cut-off points: the air-entry value (drying) and the air-expulsion value (wetting). They delimit three zones along the curves which accounts for air/water phase continuity: the boundary effect zone (capillary regime), the transition zone (funicular regime) and the residual zone (pendular regime).

Even though the problem of hysteresis is well recognized, these phenomena are often ignored to keep the mathematical description of hydraulic properties simple. In other words, the hydraulic properties are approximated by constitutive laws showing a shape of transfer functions. Actually, hysteresis takes part in few situations, usually at small level, like for viscous fingering. Undergoing works try to integrate effectively these hysteretic and dynamic behaviours into hydraulic properties for Richards' equation [START_REF] Schweizer | The Richards equation with hysteresis and degenerate capillary pressure[END_REF][START_REF] Krämer | A discontinuous Galerkin scheme for the Richards equation: on dynamic capillarity effects in porous media[END_REF] but they are still limited and lack assessment.

It is worth noting that:

• under complete saturation, hydraulic properties become constant and Richards' equation degenerates into an elliptic equation characterised by fast diffusion;

• under almost complete unsaturation, hydraulic properties get very near-zero values which stop diffusion and may be inconvenient numerically;

• for particular set of parameters, when ψ → 0 -, constitutive laws may exhibit very steep gradients and even a numerical singularity. Nevertheless, in practice, these gradient-values can be large but are bounded.

These constitutive laws are mainly responsible for the numerical challenges of Richards' equation, such as nonlinearities, degenaracies and instabilities, which are often observed in the presence of material heterogeneities or dynamic boundary conditions like seepage boundary condition.

CHAPTER I. MODELLING OF FLOWS IN UNSATURATED POROUS MEDIA

Special treatment may be done to regularize the constitutive relations [START_REF] Jäger | Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes[END_REF][START_REF] Pop | Error Estimates for a Time Discretization Method for the Richards' Equation[END_REF][START_REF] Radu | Newton-Type Methods for the Mixed Finite Element Discretization of Some Degenerate Parabolic Equations[END_REF]. For instance, Dolejší et al. [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] modify slightly the functions to avoid some types of degeneracy mentioned above and improve the convergence properties. It can cause difficulties for numerical analysis but, in practice, it was found that the convergence of the solving algorithm described in Chapter II can be ensured choosing carefully the parameters of the computations. As a result, no modification on the constitutive relations is done in this study.

I.5 Seepage and dynamic forcing boundary conditions

The seepage boundary condition is specific to subsurface model [START_REF] Scudeler | Examination of the seepage face boundary condition in subsurface and coupled surface/subsurface hydrological models[END_REF]. This condition is used to model the interface between a porous medium and the atmosphere. If the porous medium is saturated and outflow occurs, then water pours out at atmospheric pressure: ψ = 0 ⇐⇒ h = z. Otherwise, the interface acts as an impervious boundary and there is no flux. This condition mimics an outflow condition. Figure I.17 depicts the situation. The treatment of the seepage boundary condition is difficult because the length of seepage face is unknown a priori: it depends on the sought solution. Therefore, seepage faces are associated to a dynamic nonlinear boundary condition. Paniconi and Putti [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF] reported that seepage condition can cause oscillations leading to convergence problems if rapid changes of saturation state occur near the seepage face. This holds no matter convergence requirements are relaxed or not for the nonlinear iterative solver.

There are many ways to express and realize the seepage boundary condition. A simple and common approach is to refer to its base definition by considering a switch between Dirichlet h D and Neumann q N boundary conditions: The realization of such condition should be done inside the nonlinear iterative process according to the previous solution guess at a local level. However, it is not always the case because of the involved numerical methods and discretization schemes. Then, the exit point is searched iteratively until a convergence criterion is reached according to different techniques. The seepage boundary condition is often written as well under the form of a Signorini's type condition [START_REF] Sochala | Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows[END_REF]:

h D = z if h ≥ z and -K(h -z)∇h • n > 0, q N = -K(h -z)∇h • n =
ψ ≤ 0, -K(h -z)∇h • n ≥ 0, (-K(h -z)∇h • n)ψ = 0.
(I.60)

I.6. NUMERICAL CHALLENGES OF RICHARDS' EQUATION

Another approach consists in treating variational inequalities in the weak formulation with specific schemes like in [START_REF] Helmig | Variational inequalities for modeling flow in heterogeneous porous media with entry pressure[END_REF][START_REF] Zheng | A variational inequality formulation for unconfined seepage problems in porous media[END_REF]. Alternatively, the seepage boundary condition may be interpreted as a nonlinear Robin boundary condition:

1 S (h)(h -z) -(1 -1 S (h))K(h -z)∇h • n = 0, (I.61)
where the seepage indicator function lying on the boundary Γ S is:

1 S : Γ S → {0, 1}, h →    1 if h ≥ z and -K(h -z)∇h • n > 0,

I.6 Numerical challenges of Richards' equation I.6.1 Behaviour analysis

The solution of Richards' equation holds typical features of groundwater flows. The key process concerns water moving through the unsaturated porous medium under the combined actions of gravity and capillarity.

When infiltration occurs, the interface between the initial state of soil and the newly wetted zone is called wetting front. The wetting front is moving dynamically and can be extremely sharp according to hydraulic properties and variable saturation conditions. It is linked to the nonlinearity varying diffusivity of hydraulic properties which control its steepness. Similar behaviour may be encountered for a drying front.

Another flow phenomenon is associated with porous medium heterogeneity and anisotropy. The interface between two materials of different hydraulic properties can lead to rapid change of saturation states called internal layer. This internal layer is static since it is attached to the material interface. It can be extremely sharp and is often linked to a discontinuity, in particular for water content. The jump depends to the saturation conditions and hydraulic properties on both sides. When a wetting front hits an internal layer, capillary forces can prevent water to infiltrate further. Water accumulates until it is able to break through. This is called the capillary barrier effect.

One may also want to connect the porous medium with another domain. For example, this is quite natural when a coupling with free surface water is considered though Beavers-Joseph-Saffman boundary condition. Then, the interface between the two domains should be checked carefully since it is related to some discontinuities and boundary layer phenomenon.

Spurious oscillations in the solution of Richards' equation are a non-physical effect and should not be present. They are found for the aforementioned flow patterns. These oscillations consist typically in undershoots/overshoots ahead of steep wetting fronts which are moving in an initially very dry zone.

Here, a brief qualitative analysis and some suggestions are outlined to understand the behaviour of Richards' equation. Richards' equation is classified as a quasilinear degenerate parabolic equation. It is quasilinear because nonlinearity depends only on the solution. It is degenerate parabolic because under complete saturation, hydraulic properties are constant so that Richards' equation becomes elliptic. Nevertheless, Richards' equation shares properties with convection-diffusion equation and can behave like them. For example, sharp wetting fronts and internal layers can occur which leads to instabilities in the form of spurious oscillations. This not so surprising if Richards' equation is considered as an approximation from a diphasic system of convection-diffusion equation [START_REF] Forsyth | Monotonicity Considerations for Saturated-Unsaturated Subsurface Flow[END_REF][START_REF] Kees | Locally conservative, stabilized finite element methods for variably saturated flow[END_REF]. Richards' equation is mainly diffusive as an elliptic-parabolic equation but convection and diffusion are locally in competition around these sharp fronts and layers. For this reason, making analogy between Richards' equation and heat equation is useful [START_REF] Bonelli | Résolution auto-adaptative par éléments finis de problèmes de diffusion fortement non linéaires[END_REF] but it is limited.

Richards' equation is predominantly diffusive. In saturated zones, it is elliptic and so purely diffusive: the time derivative is zero and there is no advection. Source terms and boundary conditions impose the solution state. In unsaturated zones, two states may be distinguished considering Eq. (I.43) which is a form of Fokker-Planck equation. Richards' equation is parabolic and mainly diffusive in unsaturated zone but Richards' equation may exhibit an hyperbolic behaviour where advection is dominant. This latter state occurs typically for sharp wetting fronts. In practice, the behaviours of Richards' equation are often mixed because hydraulic properties governing flow patterns are nonlinear. Then, the study and prediction of the dynamic response of the equation is difficult. Some attempts are found in literature to analyse Richards' equation fronts, for instance in [START_REF] Gilding | Qualitative mathematical analysis of the Richards equation[END_REF][START_REF] Caputo | Front Solutions of Richards' Equation[END_REF].

It is hard to represent Richards' equation locally by a linear approximation that is why Richards' equation is sometimes referred as highly nonlinear. Moreover, Richards' equation is known to be a stiff differential equation because some numerical schemes are numerically unstable unless the time stepsize is taken to be extremely small. Stiffness is difficult to define but means that the equation holds some terms leading to rapid variation in the solution.

I.6.2 Computational developments

The oscillations arise whatever the numerical method is used: finite difference, finite volume, finite element or discontinuous Galerkin methods. However, they are seldom observed in finite differences because a suitable scheme prevents them like upwinding. In finite volume and discontinuous Galerkin methods, tools like slope limiters make oscillation-free solution possible [32,[START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. These techniques may delay the wetting front and are not straightforward for multidimensional or general mesh because they depend on cells geometry. For finite elements, it is more difficult to eliminate the non-physical oscillations. Mass lumping [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF][START_REF] Pan | Finite element methods for modeling water flow in variably saturated porous media: Numerical oscillation and mass-distributed schemes[END_REF][START_REF] Ju | Mass types, element orders and solution schemes for the richards equation[END_REF][START_REF] Younes | Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media[END_REF] or mass distributed [START_REF] Pan | Finite element methods for modeling water flow in variably saturated porous media: Numerical oscillation and mass-distributed schemes[END_REF] schemes are usually employed because it was seen that they reduce oscillations but they may smear the wetting front in return. Additional techniques are often necessary to stabilize the solution: basis function respecting the maximum principle, multiscale formulation [START_REF] Kees | Locally conservative, stabilized finite element methods for variably saturated flow[END_REF], velocity field postprocessing [START_REF] Kees | Locally conservative, stabilized finite element methods for variably saturated flow[END_REF], etc.

The origin of these spurious oscillations is not well identified in literature since many explanations and analysis are given: saturation distribution, velocity field regularity, discrete hydraulic conductivity scheme [START_REF] Belfort | On equivalent hydraulic conductivity for oscillation-free solutions of Richard's equation[END_REF], M-matrix property [START_REF] Younes | Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media[END_REF]. Notably, refinement always eliminates the spurious oscillations around the wetting front and makes the numerical solution converge to the physical solution. This

I.6. NUMERICAL CHALLENGES OF RICHARDS' EQUATION

observation makes think that spatial resolution is not sufficient to capture the sharpness of the front.

Even though the solution is mathematically regular enough, the numerical solution is unstable and can even be discontinuous because of phenomena occurring at scale smaller than elements size [START_REF] Gresho | Don't suppress the wiggles-They're telling you something![END_REF]. Global refinement is not the answer because the needed element size is so small that the computing cost is irrelevant [START_REF] Kees | Locally conservative, stabilized finite element methods for variably saturated flow[END_REF]. That is why local and dynamic refinement is investigated for Richards' equation.

The development of such techniques is demanding because it must capture moving wetting fronts and add enough elements.

A useful property for quality assessment is the monotonicity preservation guaranteed by the maximum principle. Indeed, hydraulic conductivity and water content are bounded and (positive) monotonic increasing functions of pressure head. So, in the absence of source/sink terms, the solution should be locally bounded according to the boundary conditions [START_REF] Forsyth | Monotonicity Considerations for Saturated-Unsaturated Subsurface Flow[END_REF][START_REF] Younes | Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media[END_REF][START_REF] Forsyth | Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media[END_REF]. This is known as maximum principle. A monotonic solution respects the maximum principle which can serve for instance to detect non-physical solutions holding overshoots/undershoots.

Summary

Richards' equation is largely used to model flows in unsaturated porous media. It is very attractive since it allows to model the porous medium as a whole part, that is to say both the saturated and unsaturated zones, without actually using a two-phase system. However, some limitations can be drawn. For instance, the porous medium is considered incompressible and water is incompressible and homogeneous but theses assumptions can be removed with extended formulations of Richards' equation. Hydraulics properties of Richards' equation are modelled in an equilibrium state as well since they lack dynamic hysteresis effects. Recent developments of Richards' equation attempt to incorporate such dynamics to take into account capillary behaviour. Stronger assumptions imply that the porous medium is inert and water is isotherm. Moreover, Richards' equation lies on modelling hypotheses. Firstly, upscaling from microscale Navier-Stokes equation to macroscale Darcy-Buckingham law, which is the basis relation for Richards' equation, shows flow has to be sufficiently slow with small enough pores to neglect inertia terms. Secondly, air-phase effects, like trapped air pockets or bubbles, are completely neglected since air phase has to be at atmospheric pressure and continuously connected to be disregarded. Richards' equation is a parabolic equation, predominantly diffusive. It has been employed for a long time so community has gathered a lot of experience about its numerical solution. However, it can degenerate into an elliptic equation and holds nonlinear hydraulic properties. Then, Richards' equation exhibits different dynamics depending on the saturation state which makes its solution troublesome. The different formulations of Richards' equation involve different set of variables and are not equivalent from a numerical point of view. The mixed formulation has better properties because it is mass conservative and handle automatically saturatedunsaturated zones. Mass conservation is an important property to reduce mass balance errors and model properly water fluxes. One of the main numerical issue concerns moving sharp wetting fronts. They produce unphysical oscillations in the solution. A large number of techniques have been used to deal with them, sometimes in combination, but these approximations lack inter-comparison and assessments. So, it is difficult to know whether or not such techniques is sufficient or necessary. To contribute to the analysis of Richards' equation, its solution has been interpreted within the context of convection-diffusion equation. For this purpose, behaviour is sorted between pure, vanishing, heterogeneous, anisotropic and nonlinear varying diffusivity. T he numerical method used to solve Richards' equation is described in this chapter which can be formally divided into two parts.

II

The first part deals with the discretization of Richards' equation both in space and time. Provided numerical issues discussed in the previous chapter, the discretization of Richards' equation is still a matter of interest in order to get accurate and efficient results for a large range of problems which is pointed out by a short review. The space discretization method should hold a certain number of features to be interesting compared to numerous existing discretization for Richards' equation. In this work, discontinuous Galerkin (DG) methods are chosen because they provide flexibility in the numerical scheme and show promising developments. For instance, high-order approximation, adaptive mesh refinement or local mas balance are natural abilities of DG methods. For time discretization, implicit scheme are favoured for Richards' equation because they are stable for any time step. Backward difference formulas are chosen for the time dizcretization because they are known to be good for stiff partial differential equations and can reach high-order with less cost than other methods like Runge-Kutta methods.

The second part is about the treatment of nonlinearities from Richards' equation. In particular, the nonlinear solver based on the linearization of Richards' equation by the fixed-point iteration and Newton-Raphson methods is presented. The iterative procedure comes with adaptive time stepping in order to give reliability to the nonlinear convergence.

II.1 A brief review

II.1.1 General overview of discretization methods

Lots of discretization techniques were employed for Richards' equation with the use of finite difference (FD) methods until the 1980s. Here, some examples are listed. FD methods were used by Celia and Bouloutas [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] or Woodward and Dawson [START_REF] Woodward | Analysis of Expanded Mixed Finite Element Methods for a Nonlinear Parabolic Equation Modeling Flow into Variably Saturated Porous Media[END_REF] or Dogan and Motz [START_REF] Dogan | Saturated-Unsaturated 3D Groundwater Model. II: Verification and Application[END_REF]. Finite volume (FV) methods were used by Manzini and Ferraris [START_REF] Manzini | Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation[END_REF], Orgogozo et al. [START_REF] Orgogozo | An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale[END_REF] or Caviedes-Voullième et al. [START_REF] Caviedes-Voullième | Numerical simulation of groundwatersurface interactions by external coupling of the 3D Richards equation and the full 2D shallowwater equations[END_REF]. Finite element (FE) methods are found in Celia and Bouloutas [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF], Lehmann and Ackerer [START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF], Šimůnek [START_REF] Šimůnek | Development and Applications of the HYDRUS and STANMOD Software Packages and Related Codes[END_REF], Forsyth et al. [START_REF] Forsyth | Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media[END_REF], Katvetsi et al. [START_REF] Kavetski | Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow[END_REF]. Diersch and Perrochet [START_REF] Diersch | On the primary variable switching technique for simulating unsaturated-saturated flows[END_REF], Knabner and Schneid [START_REF] Knabner | Adaptive Hybrid Mixed Finite Element Discretization of Instationary Variably Saturated Flow in Porous Media[END_REF], Chavent and Roberts [START_REF] Chavent | A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems[END_REF] or Fahs et al. [START_REF] Fahs | An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards' Equation[END_REF] used mixed FE methods. Others techniques are more sparsely found like a multiscale FE methods [START_REF] He | A multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media[END_REF] and a lattice Boltzmann method [START_REF] Ginzburg | Lattice Boltzmann approach to Richards' equation[END_REF]. More information is available in the reviews from Farthing et al. [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF] and Zha et al. [START_REF] Zha | Review of numerical solution of Richardson-Richards equation for variably saturated flow in soils[END_REF].

Concerning time discretization, the common approach when working with Richards' equation is to use implicit schemes to avoid the Courant-Friedrichs-Lewy (CFL) condition which is very restrictive for some hydraulic properties. Implicit Euler or Crank-Nicolson schemes are widespread for solving Richards' equation. An ongoing trend is to use high-order accuracy in time with, for example, backward differentiation formula (BDF) [START_REF] Sochala | Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows[END_REF][START_REF] Kees | Higher order time integration methods for two-phase flow[END_REF]. In this study, the implicit Euler scheme and BDF methods will be used because they are suitable for stiff equations like Richards' equation. The first one is simple and shows a wide region of stability while the others give time high-order to balance space and time discretization errors. More information is available in the reviews from Farthing et al. [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF] and Zha et al. [START_REF] Zha | Review of numerical solution of Richardson-Richards equation for variably saturated flow in soils[END_REF].

Two nonlinear iterative solvers are presented in this study to solve the algebraic system of equations arisen from discretization: a damped Newton-Raphson method and a fixed-point method (or Picard's method). Previous studies assessed nonlinear behaviour of Richards' equation through various cases

II.1. A BRIEF REVIEW

and multiple numerical schemes [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF] in which other nonlinear solving techniques are also presented. Despite quadratic convergence for Newton-Raphson method, there is no clear advantage compared to fixed-point method for a certain number of cases, mainly because Newton-Raphson method often fails to converge due to poor initial estimate, derivative issues or solution overshoots. Fixed-point method appears as more robust. Moreover, time adaptation can help managing Richards' equation solution and relax the nonlinear iterative solver.

II.1.2 Principles of discontinuous Galerkin methods

DG methods are relatively recent in history of discretization methods since they date from 1973 when Reed and Hill introduced them in the context of neutron transport problem [START_REF] Reed | Triangular mesh methods for the neutron transport equation, Proceedings of the American Nuclear Society[END_REF]. They might be seen as a generalization of the FE or the FV frameworks. Indeed, they share properties from both of these methods which makes them attractive. Figure II.1 shows the typical differences of solution approximation between FE methods, FV methods and DG methods. FE methods are continuous approximation while FV methods are discontinuous approximation. In general, high-order approximation is doable with FE methods but much more difficult with FV methods. DG methods have high-order discontinuous approximation making mesh adaptation very easy. More precisely, DG methods have the following advantages:

• They give rise to an algebraic system of decoupled equations, that is to say matrices in the discrete system are sparse and structured by blocks. Each block has a size equals to the local number of degrees of freedom (belonging to one element). Simpler solving techniques can be employed and parallelization can be achieved;

• Stencil size is independent of the order of approximation. Then, solving on one element depends only on the solution of this element and its direct neighbours. The method is said to be compact. High-order approximation is easier to reach.

• Numerical approximation of the solution is only locally smooth that is to say over one element. First, discontinuity can be approximated like FV methods. Besides, solution sensibility to mesh regularity can be alleviated through the discontinuity between elements. In particular, this allows to work easily with meshes which are unstructured, non-conforming and hybrid (several elements geometry) giving path to mesh refinement;

• Boundary conditions are not imposed through the definition of approximation space but rather directly incorporated in the weak formulation. Study and implementation are then easier;

• DG methods are local making them very suitable for conservation laws. Numerical dissipation is low and they are locally mass conservative.

DG methods are very compliant methods handling many desirable features.

Yet, DG methods also have disadvantages:

• DG numerical solution can hold spurious oscillations which are not physical. This is due to numerical stability ensured by penalty or numerical dispersion at discontinuity. In this sense, DG methods lack robustness. Lots of techniques are developed to deal with these oscillations like slope limiters, artificial viscosity, penalty definition, numerical flux in the numerical scheme;

• The computational cost can be prohibitive because the number of degrees of freedom is multiplied compared to the standard FE method. For DG methods, the number of degrees of freedom is proportional to the number of elements depending on approximation order while, for FE methods, the number of degrees of freedom is linked to the number of vertices (and possibly elements);

• Implementation for DG methods is less spread than other discretization techniques due to its relatively recent use. Then, it can be harder to achieve for new-comers. Besides, comparatively to FE methods, degrees of freedom may have no physical meaning which may trouble new users;

• Even if DG methods analysis has made great progress, optimal error estimate are not fully derived and there are still gaps between theoretical rates of convergence and observed rates.

DG methods are more competitive nowadays since parallelization and computers are becoming more suited to have great computation and memory capacities. Besides, approximation can be better than FE methods but hold an equivalent number of degrees of freedom. For instance, DG methods can have less but higher order elements [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF] or use approximation space less rich than the ones needed in FE methods [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF]. One can also remark that numerical dispersion in DG methods stays locally restrained compared to FE methods [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF].

Further developments extended the use of DG methods to other study fields with elliptic and parabolic equations and new forms appeared such as primal and local formulations [117]. Despite this history of developments and their overall properties, DG methods were sparsely applied to Richards' equation up to now. Here, such studies are listed as exhaustively as possible. In 2007, Li et al. solved Richards' equation in 1D with a local DG method [START_REF] Li | Adaptive local discontinuous Galerkin approximation to Richards' equation[END_REF][START_REF] Li | Local discontinuous Galerkin approximations to Richards' equation[END_REF] which is a mixed formulation. Sochala discretized Richards' equation through a SIPG mixed DG formulation in 2008 [START_REF] Sochala | Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows[END_REF][START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. Krämer used a mixed DG formulation during his master thesis work in 2016 [START_REF] Krämer | A discontinuous Galerkin scheme for the Richards equation: on dynamic capillarity effects in porous media[END_REF]. Recently, in 2019, Dolejší et al. [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] proposed a space-time DG method for solving Richards' equation. Nevertheless, it should be noticed that DG methods are more and more used for complex flow and transport phenomena in porous media described by other models such as two-phase flow problems [32,[START_REF] Bastian | A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure[END_REF][START_REF] Sun | Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media[END_REF]. In this study, Richards' equation is approximated by primal interior penalty DG methods as matter of interest but also because this formulation is simpler.

Finally, making the most of DG methods, hp-adaptivity can improve solution quality of Richards' equation. So far, p-adaptation is not really adopted on its own for flows problems in porous media; whereas h-adaptation is employed largely for two-phase flow in porous media [32] and Richards' equation [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Li | Local discontinuous Galerkin approximations to Richards' equation[END_REF]; hp-adaptation has been exploited in many fluid dynamics applications but few works use it for porous media flow [START_REF] Kane | hp-Adaptive Discontinuous Galerkin Methods for Porous Media Flow[END_REF][START_REF] Kane | Adaptive Discontinuous Galerkin Methods for Flow in Porous Media[END_REF]. As far as author knows, for Richards' equation, three [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF]. These hp-adaptative algorithm are not equivalent since they aim for different goals, rely on distinct approaches to make decisions and use different techniques to refine/enrich the mesh/polynomial order.

In general, such questions remain open problems for hp-adaptation [START_REF] Mitchell | A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations[END_REF].

II.2 Space semidiscretization

Many DG methods [117] exist but they can be viewed according to two approaches: the primal formulation and the mixed formulation. The former corresponds to a model problem written as one single equation. The latter corresponds to a model problem broken down into a potential and flux parts so that it forms a system of two coupled equations which are approximated by two functional spaces linked by numerical flux. In this work, the Richards' equation is discretized by several DG methods. These methods are very similar to each other and can be described by the same primal weak formulation. In the following derivation of the variational formulation, all of them are considered. First of all, some notations and definitions are established for the spatial and temporal discretizations in order to get the numerical scheme. More careful developments can be found in Rivière [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF] or Dolejší and Feistauer [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF].

II.2.1 Fundamental concepts

Problem statement and notations

Let d ∈ {1, 2, 3} be the dimension. The porous medium is represented by the computational domain Ω ⊂ R d of boundary ∂Ω and the final time is T ∈ R * + . The boundary ∂Ω is subdivided into three mutually disjoint boundaries, ∂Ω = Γ D ∪ Γ N ∪ Γ S , corresponding to the Dirichlet, Neumann and seepage boundary conditions respectively. The following problem is considered:

find h(x, t) : Ω × (0, T ) -→ R such that                    ∂ t θ(h -z) -∇ • (K(h -z)∇h) = 0, in Ω × (0, T ), h = h 0 , in Ω × {0}, h = h D , on Γ D × (0, T ), -K(h -z)∇h • n = q N , on Γ N × (0, T ), 1 S (h)(h -z) -(1 -1 S (h))K(h -z)∇h • n = 0, on Γ S × (0, T ). (II.1)
The strong problem (II.1) has a solution h ∈ L 2 (Ω × (0, T )). The dependence of space and time is omitted for clarity. Let

h 0 ∈ L 2 (Ω), h D ∈ L 2 L 2 (Γ D ); 0, T , q N ∈ L 2 L 2 (Γ N ); 0, T . K is a monotonically increasing matrix-valued function of h: K : R -→ [R + ] d×d
which is symmetric, positive definite, bounded below and above uniformly. θ is a monotonically increasing function of h: θ : R -→ R + which is bounded below and above uniformly. K and θ are continuous functions of h for a given porous medium but can hold discontinuities at interface of heterogeneous materials. The existence, the uniqueness and the regularity for the solution of Richards' equation can be found in the work from Alt and Luckhaus [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF], Otto [START_REF] Otto | L¹-Contraction and Uniqueness for Quasilinear Elliptic-Parabolic Equations[END_REF][START_REF] Otto | L¹-Contraction And Uniqueness For Unstationary Saturated-Unsaturated Porous Media Flow[END_REF] or Schweizer [START_REF] Schweizer | The Richards equation with hysteresis and degenerate capillary pressure[END_REF][START_REF] Schweizer | Regularization of outflow problems in unsaturated porous media with dry regions[END_REF][START_REF] Pop | Regularization Schemes For Degenerate Richards Equations And Outflow Conditions[END_REF].

Geometry partition and definitions

The time duration (0, T ) is subdivided into N time intervals such that

0 = t 0 < t 1 < • • • < t N = T . Let n ∈ N, 0 ≤ n < N . If the time interval T n+1 = [t n , t n+1 ] is considered, the corresponding time CHAPTER II. NUMERICAL METHODS FOR RICHARDS' EQUATION step is τ n+1 = t n+1 -t n .
Ω is subdivided into N E mutually disjoint closed polygonal elements E forming a tessellation or mesh. The mesh at the time partition T n is denoted by

E n h = {E n } E⊂Ω . So, Ω = E∈E n h E. Moreover, one element E ∈ E n
h has a boundary denoted by ∂E, a roundness ρ E which is the diameter of the largest ball inscribed into E as well as a diameter h E defined by:

h E = diam(E) := sup x,y∈E |x -y|. (II.2)
In 2D, the diameter of a triangle is its largest edge and the diameter of a quadrangle is its largest diagonal or its largest edge. The aspect ratio For every mesh E n h , the space step is set as:

σ E = h E ρ E is a
h n := max E∈E n h (h E ). (II.3)
A face is defined as the intersection between exactly two elements or between one element and the boundary ∂Ω.

F n h,D := F ∈Γ D F, F n h,N := F ∈Γ N F, F n h,S := F ∈Γ S F, (II.4) F n h,B := F n h,D ∪ F n h,N ∪ F n h,S , F n h,I := F n h \ F n h,B . (II.5)
For each face F ∈ F n h , there is a unit normal vector n F . For F ∈ F n h,I , its orientation is arbitrary but kept fixed and, for F ∈ F n h,B , it is oriented outward.

Functional spaces and operators

Let s ∈ N + and 1 ≤ k ≤ ∞. The Sobolev space W s,p is introduced:

W s,k (Ω) = v ∈ L k (Ω) : ∀|α| ≤ s, D α v ∈ L k (Ω) , (II.6)
where α is a multi-index, D α v is a multidimesional partial derivative of order |α| in the sense of distribution and L k is a Lebesgue space. The Sobolev space is a Banach space. In the common case k = 2,

II.2. SPACE SEMIDISCRETIZATION

the Sobolev space is a Hilbert space: H s (Ω) = W s,2 (Ω). Sobolev spaces may have fractional (real) indices s obtained by interpolating the surrounding Sobolev spaces of integer indices. A theorem result states that a continuous representative is possible under the following conditions (see Theorem 2.4. in [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF]):

H s (Ω) ⊂ C 0 (Ω) if s > d 2 . (II.7)
DG methods use discontinuous approximations over the mesh E n h defining a so-called broken Sobolev space:

H s E n h = v ∈ L 2 (Ω) : v E ∈ H s (E), ∀E ∈ E n h . (II.8)
H s E n h denotes the space of functions whose restriction to each element E ∈ E n h belongs to the Sobolev space H s (E). There is

H s (Ω) ⊂ H s E n h .
The solution is sought in a finite-dimensional subspace of this broken Sobolev space, taken to be:

S p E n h := v ∈ L 2 (Ω) : v E ∈ P p (E), ∀E ∈ E n h , (II.9)
where P p (E) denotes the space of polynomial functions on E of total degree less than or equal to p ∈ N. P p (E) can be replaced by Q p (E) denoting degree less than or equal to p ∈ N. Obviously,

S p E n h ⊂ H s E n h Let two

II.2.2 Discontinuous Galerkin weak formulation

The problem (II.1) is multiplied by a test function v ∈ H s E n h and then integrated on each element E ∈ E n h . After using Green's theorem element-wise, it yields:

                                             E ∂ t θ(h -z)v dE + E K(h -z)∇h • ∇v dE - F ∈∂E F K(h -z)∇h • n F v dF = 0, E hv dE = E h 0 v dE, F ∈∂E∩F D h F hv dF = F ∈∂E∩F D h F h D v dF, - F ∈∂E∩F N h F K(h -z)∇h • nv dF = F ∈∂E∩F N h F q N v dF, F ∈∂E∩F S h F 1 S (h)(h -z)v dF - F ∈∂E∩F S h F (1 -1 S (h))K(h -z)∇h • nv dF = 0. (II.16)
Summing over all elements in E n h , it gives:

                                                             E∈E h E ∂ t θ(h -z)v dE + E∈E h E K(h -z)∇h • ∇v dE - F ∈F I h F K(h -z)∇h • n F v dF - F ∈F D h F K(h -z)∇h • n F v dF + F ∈F N h F q N v dF - F ∈F S h F 1 S (h)K(h -z)∇h • n F v dF = 0, E∈E h E hv dE = E∈E h E h 0 v dE, F ∈F D h F hv dF = F ∈F D h F h D v dF, F ∈F S h F 1 S (h)(h -z)v dF = 0.
(II.17)

II.2. SPACE SEMIDISCRETIZATION

The Neumann boundary condition has appeared naturally within the formulation and has been substituted. So it stands for the homogeneous Neumann part of the seepage boundary condition.

Assuming that

K(h -z)∇h • n F = 0, (II.18)
because this quantity is smooth enough, and using the identity:

∀u, v ∈ H s E n h , uv = ⦃u⦄ v + ⦃v⦄ u , (II.19)
it yields to:

                                                             E∈E h E ∂ t θ(h -z)v dE + E∈E h E K(h -z)∇h • ∇v dE - F ∈F I h F ⦃K(h -z)∇h • n F ⦄ v dF - F ∈F D h F K(h -z)∇h • n F v dF + F ∈F N h F q N v dF - F ∈F S h F 1 S (h)K(h -z)∇h • n F v dF = 0, E∈E h E hv dE = E∈E h E h 0 v dE, F ∈F D h F hv dF = F ∈F D h F h D v dF, F ∈F S h F 1 S (h)(h -z)v dF = 0.
(II.20)

II.2.3 Penalization

Now, two penalty terms are required to mimic the solution continuity by constraining the interior solution jump to be zero, and to enforce the Dirichlet boundary condition, including the remaining part of the seepage boundary condition. If the solution is a continuous function satisfying the Dirichlet boundary conditions, the two penalty terms are vanishing so they can be added to the formulation:

J I (h, v) = F ∈F I h F ϱ I F h v dF, (II.21) J D (h, v) = F ∈F D h F ϱ D F (h -h D )v dF. (II.22)
The interior and Dirichlet penalty weights are respectively set as:

ϱ I F := σ I F γ F µ F , (II.23) ϱ D F := σ D F γ F µ F , (II.24)
where the face measure is defined as µ F := h F β p F 2 and where σ I F , σ D F , β and γ F are suitably chosen positive constants. σ I F and σ D F are user-defined parameters which can be seen as a penalty speed. They must be above a threshold value to provide coercivity and guarantee stability for some of the DG methods [START_REF] Epshteyn | Estimation of penalty parameters for symmetric interior penalty Galerkin methods[END_REF]. At the same time, it should not be too large since the condition number increases with the penalty parameter leading to instabilities. This was studied by Castillo [START_REF] Castillo | Performance of Discontinuous Galerkin Methods for Elliptic PDEs[END_REF] and for layered reservoirs with high permeability contrast by Slingerland and Vuik [START_REF] Van Slingerland | Fast linear solver for diffusion problems with applications to pressure computation in layered domains[END_REF]. β should be one for standard methods but can take greater values to enforce continuity as in FE method but in response can also degrade numerical properties such as condition number. It can be seen as a penalty order and is referred as superpenalization in the literature. Finally, γ is a diffusion penalty coefficient set to one generally whereas some studies choose other definitions, see e.g. [START_REF] Schötzau | A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations[END_REF][START_REF] Dobrev | Preconditioning of Symmetric Interior Penalty Discontinuous Galerkin FEM for Elliptic Problems[END_REF].

Then, a first weak formulation rises:

                                                             E∈E h E ∂ t θ(h -z)v dE + E∈E h E K(h -z)∇h • ∇v dE - F ∈F I h F ⦃K(h -z)∇h • n F ⦄ v dF - F ∈F D h F K(h -z)∇h • n F v dF + F ∈F N h F q N v dF - F ∈F S h F 1 S (h)K(h -z)∇h • n F v dF + F ∈F I h F ϱ I F h v dF + F ∈F I h F ϱ D F hv dF + F ∈F S h F ϱ D F 1 S (h)hv dF = F ∈F D h F ϱ D F h D v dF + F ∈F S h F ϱ D F 1 S (h)zv dF, E∈E h E hv dE = E∈E h E h 0 v dE.
(II.25)

II.2.4 Symmetrization

The formulation is not symmetric between the unknown h and the test function v. Making the formulation symmetrical can be addressed in order to preserve the natural symmetry in the discrete diffusion operator, to use appropriate solvers or to enhance numerical properties of the formulation. This can be done by adding the corresponding symmetric term, -

F ∈F I h F ⦃K(h -z)∇v • n F ⦄ h dF ,
which vanishes because h = 0 if h is a sufficiently smooth function. The terms:

-

F ∈F D h F K(h -z)∇v • n F h dF - F ∈F S h F 1 S (h)K(h -z)∇v • n F h dF, (II.26)
are also added to get symmetry with their counterpart:

-

F ∈F D h F K(h -z)∇v • n F h D dF - F ∈F S h F 1 S (h)K(h -z)∇v • n F z dF. (II.27)
Finally, the space semidiscretization reads:

find h ∈ H s E n h such that ∀v ∈ H s E n h , m h,n (∂ t θ(h -z), v) + a h,n (h, v) = l h,n (v), (II.28)
where the DG bilinear forms, m h,n and a h,n , and linear form l h,n are: Several DG methods can be derived according to the penalty weights and the coefficient of symmetry as listed in Tab. II.1. Their numerical properties are slightly different. Some basic properties are listed in Tab. II.2. Further information is available in [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Dolejší | Discontinuous Galerkin Method[END_REF]. The IIPG method will be mainly chosen throughout the thesis because there is no additional terms from symmetrization which keeps the scheme simple which is interesting when the diffusion operator is nonlinear [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF]. The seepage boundary condition is treated numerically thanks to the indicator function 1 S which alternates dynamically between the Dirichlet part when 1 S = 1 and the Neumann part for 1 S = 0, directly in the usual weak framework of the DG methods. This treatment can model many physical situations, like multiple seepage faces simultaneously, because it does not require any assumption about the mesh or seepage face [START_REF] Scudeler | Examination of the seepage face boundary condition in subsurface and coupled surface/subsurface hydrological models[END_REF].

m h,n (∂ t θ(h -z), v) = E∈E h E ∂ t θ(h -z)v dE, (II.29) a h,n (h, v) = E∈E h E K(h -z)∇h • ∇v dE - F ∈F I h F ⦃K(h -z)∇h • n F ⦄ v dF + F ∈F I h F ϱ I F h v dF -Θ F ∈F I h F ⦃K(h -z)∇v • n F ⦄ h dF - F ∈F D h F K(h -z)∇h • n F v dF + F ∈F D h F ϱ D F hv dF -Θ F ∈F D h F K(h -z)∇v • n F h dF - F ∈F S h F 1 S (h)K(h -z)∇h • n F v dF + F ∈F S h F ϱ D F 1 S (h)hv dF -Θ F ∈F S h F 1 S (h)K(h -z)∇v • n F h dF , (II.30) l h,n (v) = F ∈F D h F ϱ D F h D v dF -Θ F ∈F D h F K(h -z)∇v • n F h D dF - F ∈F N h F q N v dF + F ∈F S h F ϱ D F 1 S (h)zv dF -Θ F ∈F S h F 1 S (h)K(h -z)∇v • n F z dF . (II.31) Θ = {-1, 0, 1}
Symmetrization Θ = -1 Θ = 0 Θ = 1 Penalization ∀F ∈ F h , σ I F = σ D F = 0 OBB method - global element method ∀F ∈ F h , σ I F ̸ = 0, σ D F ̸ = 0 NIPG IIPG
CHAPTER
This treatment was also done by Dolejší et al. in [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] where they pointed out this is a great benefit compared to conforming FE methods which need to modify the FE space or use numerical tricks, resulting in difficult implementation. However, the indicator function 1 S is not differentiable which causes difficulties for the numerical analysis. A regularization of this function can be done [START_REF] Schweizer | Regularization of outflow problems in unsaturated porous media with dry regions[END_REF][START_REF] Pop | Regularization Schemes For Degenerate Richards Equations And Outflow Conditions[END_REF]. For example, Dolejší et al. [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] smoothed the indicator function 1 F to avoid a brutal switch between the values 0 and 1. In this work, no regularization is performed because numerical experiments do not show major convergence solver problems. Besides, DG methods handle already discontinuities and weak penalty naturally. This makes the numerical scheme even more simple.

II.2.5 Algebraic representation

At a discrete level, the broken Sobolev space is approximated by a finite-dimensional subspace taken to be S p E n h which is defined in Eq. (II.9). The number of global degrees of freedom in the mesh E h is:

N dof := dim S p E n h = E∈E n h N E dof , (II.32)
where N E dof denotes the number of local degrees of freedom, that is to say those belonging to the element E. N E dof can vary from one element to another. For instance, it may depend on the chosen polynomial basis, the local polynomial degree p E and even on the element shape (triangle, quadrangle, etc). The sought solution could have be written as h h to highlight its discrete nature. It is expanded in the global polynomial basis functions Φ of the space S p E n h constructed from local basis functions ϕ of S p (E):

∀t ∈ (0, T ), ∀x ∈ Ω, h(x, t) = N E k=1 N E dof l=1 χ l k (t)Φ l k (x) where Φ l k (x) =    ϕ l k (x) if x ∈ E 0 otherwise .
(II.33) The degrees of freedom are denoted by χ l k , and are functions of time. The algebraic representation is interesting to make the link with programming described in Section II.7. It helps writing the matrix system originated from Eq. (II.28) and understanding the choice of polynomial basis functions.

II.3 Time discretization

To get the fully discrete DG formulation, the time derivative has now to be discretized. Such a procedure which takes care of space discretization before time discretization is called the method of lines. It is used commonly by others studies for Richards' equation, see e.g. [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Li | Adaptive local discontinuous Galerkin approximation to Richards' equation[END_REF][START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF]. Richards' equation is a stiff equation because of constitutive laws which usually dismisses explicit methods for time integration even though this choice can be done, for example in [START_REF] Maet | A fully-explicit discontinuous Galerkin hydrodynamic model for variably-saturated porous media[END_REF] under justifications of better parallel computing performances. For stability reasons, implicit methods are used for Richards' equation : backward Euler method, Crank-Nicolson method, backward differentiation formula methods, diagonally implicit Runge-Kutta (DIRK), etc. The use of low-order schemes is well-established but a trend can be oserved for higher order time schemes [START_REF] Kees | Higher order time integration methods for two-phase flow[END_REF].

The following notation will be used for any function u ∈ L 2 (0, T ; Ω):

∀n ∈ N, 0 ≤ n < N, u n (x) := u(x, t n ).
(II.34)

II.3.1 ϑ-Method

The ϑ-method (theta-method) is used to integrate partial differential equations in time and depends on the parameter ϑ ∈ R such that 0 ≤ ϑ ≤ 1. It is a generalization of the Euler methods since it represents the convex linear combination of the explicit Euler method (forward Euler method) and the implicit Euler method (backward Euler method). The later are simple and popular method of order 1. Besides, ϑ-method also encompasses the Crank-Nicolson method which can be viewed as the arithmetic mean of the Euler methods. Crank-Nicolson method is based on the trapezoidal rule providing second order convergence in time.

The DG formulation in Eq. (II.28) is integrated over T n+1 and the time integral is approximated by the ϑ-method:

find a sequence of (h n ) 0≤n<N ∈ S p E n h such that              h 0 = h 0 , ∀v ∈ S p E n h , m h,n θ(h n+1 ) -θ(h n ) τ n+1 , v +(1 -ϑ)(a h,n (h n , v) -l h,n (v, t n )) +ϑ a h,n h n+1 , v -l h,n v, t n+1 = 0. (II.35)
It can be noticed that ϑ = 1 leads to the implicit Euler method, ϑ = 0 to the explicit Euler method and ϑ = 1 2 to the Crank-Nicolson method.

Forward Euler scheme must be excluded because it is conditionally stable (explicit) under time steps restriction known as the CFL condition unless using tiny time stepsizes which is not viable. Crank-Nicolson scheme should be used carefully: despite being of second-order, it provides less accurate results than backward Euler scheme of order one. This is due to its semi-implicit nature which may cause oscillations. The latter may persist in the solution because they are damping weakly. Authors notice oscillations occur if time stepsize is too large or if there are jump discontinuities in the initial solution or between initial condition and boundary conditions [START_REF] Østerby | Five Ways of Reducing the Crank-Nicolson Oscillations[END_REF]. This situation is inherent to either Richards' equation or DG method, which may explain why Crank-Nicolson scheme is not a good choice.

Even though the backward Euler scheme has error of order one in time, it will be particularly used for Richards' equation in this study because the implicit nature of the scheme provides good stability properties. Indeed, the backward Euler scheme shows a huge region of absolute stability, making it A-stable, and even L-stable, which is suitable to integrate stiff equations. That is why, together with its simple implementation, this scheme is widespread for Richards' equation where good convergence behaviour is observed in [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF].

II.3.2 Backward differentiation formula methods

Backward differentiation formula methods (BDF) are implicit numerical integration methods especially used for stiff differential equations thanks to their stability properties. These linear multistep methods use q already computed solutions to produce a method of order q ≤ 6. Analysis can be found in [START_REF] Süli | An Introduction to Numerical Analysis[END_REF][START_REF] Dolejší | Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems[END_REF]. The DG formulation (II.28) is integrated over T n+1 and the time integral is approximated by the BDF methods of 1 ≤ q ≤ 6 steps with variable time steps [START_REF] Hay | hp-Adaptive time integration based on the BDF for viscous flows[END_REF]. The BDF method of order q is obtained from the Newton interpolation polynomial of degree q interpolating h j at time t j for j = n + 1, . . . , n + 1 -q. Divided differences are useful to express compactly variable stepsize BDF methods but also to compute efficiently and automatically coefficients using recursion. The backward divided difference of a given function y is defined by a recursive division process defined as:

      
y n+1 := y n+1 , y n+1 , . . . , y n+1-j := [y n+1 , . . . , y n+2-j ] -[y n , . . . , y n+1-j ] t n+1 -t n+1-j .

(II.36)

The implicit BDF method corresponds to:

∂ t θ(h -z) = q j=1   j-1 k=1 t n+1 -t n+1-k   θ h n+1 -z , . . . , θ h n+1-j -z (II.37) = q k=0 α s,k τ n+1 θ h n+1-k -z , (II.38)
where α s,k are the coefficients of the linear combination obtained from the coefficients of the divided differences of h, where q k=0 α s,k = 0. This provides an approximation of ∂ t θ h n+1 -z based on the θ h j -z and guarantees ∂ t θ h n+1 -z = 0 if the problem is steady. Besides, the last coefficient is computed as the opposite sum of the other coefficients:

α s,q = - q-1 k=0 α s,k . (II.39)
This ensures that a constant function is exactly integrated at machine precision adjusting the coefficients with the largest round-off error since it involves the highest number of levels of divided differences. When the time stepsize is fixed, the coefficients α s,k takes the classic values of the standard BDF method. They are given in Tab. II.3 because they may serve as a test for computation. Then, the complete discretized system with BDF methods reads:

II.3. TIME DISCRETIZATION

q 1 1 -1 2 3 /2 -2 1 /2 3 11 /6 -3 3 /2 -1 /3 4 25 /12 -4 3 -4 /
find a sequence of (h n ) 0≤n<N ∈ S p E n h such that        h 0 = h 0 , ∀v ∈ S p E n h , m h,n q k=0 α q,k τ n+1 θ h n+1-k -z , v + a h,n h n+1 , v = l h,n v, t n+1 . (II.40)
However, stability of BDF methods has some restrictions. Only BDF methods of order 1 and 2 are A-stable (and even L-stable) according to the second Dahlquist barrier for implicit linear multistep methods as defined and proved in [START_REF] Dahlquist | A special stability problem for linear multistep methods[END_REF]. BDF methods of order 3 to 6 are conditionally stable or A(α)-stable where α decreases with the order, see e.g. [START_REF] Hairer | Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems[END_REF][START_REF] Ascher | Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations[END_REF]. Then, time stepsizes have to be small enough to make the method stable. BDF methods of order q > 6 are unconditionally unstable so they cannot be used. To strengthen stability, one can use variable time steps. Roughly speaking, reducing time step length gives stability while increasing time step length degrades stability properties.

In practice, time step size variation are bounded above by a value called the swing factor to guarantee stability and give robustness. The restriction becomes more severe as the order q gets higher. Strategies are described in [START_REF] Hay | hp-Adaptive time integration based on the BDF for viscous flows[END_REF][START_REF] Lortie | Simulation d''écoulements á surface libre entre un liquide et un gaz compressible[END_REF] to adapt the time stepsize but also to select the appropriate order of BDF method according to stability indicator. Some theoretical bounds for swing factor exist [START_REF] Grigorieff | Stability of multistep-methods on variable grids[END_REF] but they are very severe since all sequences of time steps lengths are considered, included pathological cases. In practice, time stepping is usually constructed to vary smoothly so, in this study, following Hay et al. [START_REF] Hay | hp-Adaptive time integration based on the BDF for viscous flows[END_REF], practical maximum swing between two consecutive time steps are taken from Skelboe [START_REF] Skelboe | The control of order and steplength for backward differentiation methods[END_REF] and reported in Tab. II.4. They are still relatively pessimistic but favours stability with time stepping which can be adapted conveniently. The 1-BDF method is unconditionally stable because it is L-stable and has no swing limit. The 6-BDF method cannot increase rapidly time step lengths and thus is not very useful if time step has to adapt constantly during the simulation. Despite these stability restrictions, BDF methods behave better for stiff problems than other methods like Adams methods because they show a stiff decay property. This means BDF methods damp quickly numerical errors due to highfrequency oscillatory components in the solution if they are applied in their stability region. This is sometimes referred as neutral stability and is the reason why BDF methods are promoted for stiff problems.

Order q 1 2 3 4 5 6 Maximum swing τ n+2 /τ n+1 -2.6 1.9 1.5 1.2 1.05 Table II.4 -Maximum swing τ n+2 /τ n+1 for BDF methods with variable time stepsize. BDF methods are very interesting for Richards' equation known to be stiff in some cases. For instance, Crank-Nicolson scheme is the ϑ-scheme for ϑ = 1 2 and corresponds to Adams-Moulton method of order 2. Even though the method is semi-implicit, spurious oscillations may occur for large time stepsize under certain conditions because oscillations are not damped fast enough for all decaying components. Besides, coefficients of BDF methods are easily computed and the cost of integration stays almost constant when order gets higher since the only requirements is to have access to older already computed solutions, making them competitive compared to implicit Runge-Kutta methods.

II.3.3 Initialization

Initialization takes care of prescribing initial condition to start the first time step. This corresponds of the first line of Eq. (II.35) or Eq. (II.40). A direct and simple way is to write the corresponding discontinuous weak formulation:

∀v ∈ S p E 0 h , E∈E h E h 0 v dE = E∈E h E h 0 v dE. (II.41)
This consists in solving a system enforced by the right-hand side, which can be formally written as:

find h 0 ∈ S p E 0 h such that ∀v ∈ S p E 0 h , m h,0 h 0 , v = f h,0 (v), (II.42)
where m is the bilinear form already defined and f is the following linear form:

f h,0 (v) = E∈E h E h 0 v dE. (II.43)
One limiting aspect of BDF methods is for initialization seeing that a s-step BDF method needs s previous solutions to be computed. Then, the first time step given by the user is divided in s substeps. Starting with 1-BDF method, the order is progressively increased at each sub-steps to reach the specified s-steps BDF method for the user-defined initial time steps. If one works with BDF methods not based on variable stepsize, initialization and a change of time stepsize has to be treated with a more complicated technique. For a scheme with q-step BDF method, the first of the next time steps is subdivided into as many times as necessary to compute the sub-time steps with an increasing sequence of lower-order BDF methods. The remaining next macro-time steps are still subdivided while the number of computed solution, stored gradually, is not sufficient to go on with the q-step BDF method.

II.4 Local mass balance

Mass conservation is an important -necessary -property for flow problems in porous media since it can cause poor quality results or instabilities [START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF]. A numerical solution of Richards' equation is a good approximation if it is mass conservative [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF][START_REF] Rathfelder | Mass conservative numerical solutions of the head-based Richards equation[END_REF]. However, the inverse is not true [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] and additional criterion should assess whether a solution is correct or not such as monotony and maximum principle [START_REF] Forsyth | Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media[END_REF]. The mixed formulation of Richards' equation is usually preferred because it is physically mass conservative contrary to the other formulations [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] due to the chain differentiation rule for the time derivative. Then, discretization scheme should numerically preserve mass balance, at least globally, and if possible, locally. Unlike FE methods which are globally mass conservative over the computational domain, DG methods are particularly suitable for porous media flow problems [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF] because they satisfy naturally local mass conservation, that is to say for each element.

II.4. LOCAL MASS BALANCE

In order to introduce this property, let fix an element E and a test function v ∈ D p (E h ) that is equal to one on E and vanishes outside of E. This is possible with DG methods due to the lack of continuity constraints between the elements. The equation (II.28) reduces to:

E ∂ t θ(h -z) dE - F ∈∂E∩F I h F ⦃K(h -z)∇h • n F ⦄ dF - F ∈∂E∩F D h F K(h -z)∇h • n F dF - F ∈∂E∩F S h F 1 S (h)K(h -z)∇h • n F dF + F ∂E∩∈F I h F ϱ I F h dF + F ∈∂E∩F D h F ϱ D F h dF + F ∈∂E∩F S h F ϱ D F 1 S (h)h dF = F ∈∂E∩F D h F ϱ D F h D dF + F ∈∂E∩F S h F ϱ D F 1 S (h)z dF - F ∈∂E∩F N h F q N dF.
Then, let integrate the previous equation over time

T n+1 with n ∈ N, 0 ≤ n < N .
Here it is done with BDF methods to provide high-order accuracy but the procedure is similar with ϑ-method.

Reorganizing the terms, it yields:

m add t n+1 = m flux t n+1 + m art t n+1 , (II.44)
where the quantities m add , m flux and m art are defined by:

m add t n+1 = E q k=0 α q,k θ h n+1-k -z dE, (II.45)
m flux t n+1 = n l=0 τ l+1    F ∈∂E∩F I h F ⦃ K h l -z ∇h l • n F ⦄ dF + F ∈∂E∩F D h F K h l -z ∇h l • n F dF - F ∈∂E∩F N h F q N dF + F ∈∂E∩F S h F 1 S h l K h l -z ∇h l • n F dF    , (II.46)
m art t n+1 = n l=0 τ l+1   - F ∂E∩∈F I h F ϱ I F h l dF - F ∈∂E∩F D h F ϱ D F h l dF + F ∈∂E∩F D h F ϱ D F h D dF - F ∈∂E∩F S h F ϱ D F 1 S h l h l dF + F ∈∂E∩F S h F ϱ D F 1 S h l z dF    .
(II.47)

The total additional mass in the domain m add is a measure with respect to the initial mass in the system. If one wants to incorporate a sink/source term, it should be added to this quantity to take into account mass destruction/creation. The total net flux into the domain m flux is a measure of the mass flow rate passing through the boundaries ∂E. The additional artificial mass in the domain m art is a measure of the pure numerical mass created by penalty parameters. This quantity is vanishing if the penalty is zero. It can be removed if needed since this artificial mass can be exactly computed. For Richards' equation with primal DG methods, mass balance β m is defined with the following ratio:

β m t n+1 := m add (t n+1 ) m flux (t n+1 ) + m art (t n+1 ) . (II.48)
Error on mass balance ϵ m can be calculated and it is sometimes expressed as a rate τ m :

ϵ m t n+1 := 1 -β m t n+1 , (II.49) τ m t n+1 := 100ϵ m t n+1 . (II.50)
Mass balance must be equal to one. It means that the change of the fluid mass in the domain between two time steps should be equal to the integral of the mass flux over the domain boundary. Disregarding artificial mass, if the numerical approximation of the velocity -the total flux -is not locally conservative compared to water content change, the numerical solution of Richards' equation becomes unstable after a few time steps. This equilibrium could be violated because of inaccurate numerical integration for example, see [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF]. Mass balance is assessed in Section IV.2 and Section IV.3.

II.5 Nonlinear iterative process

Richards' equation is a nonlinear equation usually solved by an iterative procedure such as fixed-point iteration (also called Picard's) or Newton-Raphson method whose choice is determining for computation time performances and convergence. Lots of derived methods exist: modified Newton-Raphson, mixed Picard-Newton, quasi-Newton, L-scheme, etc. Studies have been carried out to compare these methods, and others, for solving Richards' equation for various problems [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF]. They emphasize that fixed-point iteration and even Newton-Raphson scheme are very sensitive and do not converge systematically according to Richards' equation's formulations, initial and boundary conditions and because of the nonlinearities of constitutive laws. In particular, the nonlinear iterative solver can oscillate between two solutions. However, these methods are problem-dependent and a conclusion cannot be generalized. There are still recent and significant works on this topic like Anderson acceleration [START_REF] Lott | An accelerated Picard method for nonlinear systems related to variably saturated flow[END_REF], nested Newton's type algorithm [START_REF] Casulli | A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form[END_REF] or modified L-schemes [START_REF] Mitra | A modified L-scheme to solve nonlinear diffusion problems[END_REF].

II.5.1 Newton-Raphson method

For this study, no source/sink term is considered but if one exists and depends on the solution, it should be incorporated in the linearization process and, then the linear form l may depend on h. A damped Newton-Raphson method and a fixed-point method were implemented. Let m be the number of the nonlinear iteration. The residual of Eq. (II.40) is expressed as:

r h,n h n+1 , v := m h,n q k=0 α s,k τ n+1 θ h n+1-k -z , v + a h,n h n+1 , v -l h,n v; t n+1 . (II.51)

Linearization

The difference between two successive iterations is written δ n+1,m h := h n+1,m+1 -h n+1,m so the iterative procedure reads:

     dr h,n (h n+1,m , v) dh n+1,m δ n+1,m h = -r h,n h n+1,m , v , h n+1,m+1 = h n+1,m + δ n+1,m h . (II.52)
In Eq. (II.52), dr h,n (h n+1,m , v) dh n+1,m corresponds to the Jacobian matrix, also known to be the tangent stiffness matrix. This term can be calculated with asymptotic developments. Indeed, provided that

II.5. NONLINEAR ITERATIVE PROCESS r h,n h n+1,m+1 , v = 0, one can write: r h,n h n+1,m+1 , v = r h,n h n+1,m + δ n+1,m h , v = r h,n h n+1,m , v + dr h,n (h n+1,m , v) dh n+1,m δ n+1,m h + o δ n+1,m h (II.53) ⇐⇒ dr h,n (h n+1,m , v) dh n+1,m δ n+1,m h = -r h,n h n+1,m , v + o δ n+1,m h . (II.54)
Then, the expression of Jacobian matrix of Eq. (II.52) can be deduced by identifying the term-wise asymptotic developments of r h,n h n+1,m+1 , v with Eq. (II.53). To do this, the following asymptotic developments should be used:

θ h n+1,m + δ n+1,m h -z = θ h n+1,m -z + dθ(h n+1,m -z) dh n+1,m δ n+1,m h + o δ n+1,m h , (II.55) K h n+1,m + δ n+1,m h = K h n+1,m + dK(h n+1,m ) dh n+1,m δ n+1,m h + o δ n+1,m h . (II.56)
Identification gives the following expression for the Jacobian matrix:

dr h,n (h n+1,m , v) dh n+1,m δ n+1,m h = E∈E h E dθ(h n+1,m -z) dh δ h v dE + E∈E h E K(h -z)∇δ h • ∇v dE + E∈E h E δ h dK(h n+1,m -z) dh ∇h • ∇v dE - F ∈F I h F ⦃K(h -z)∇δ h • n F ⦄ v dF - F ∈F I h F ⦃ δ h dK(h n+1,m -z) dh ∇h • n F ⦄ v dF - F ∈F D h F K(h -z)∇δ h • n F v dF - F ∈F D h F δ h dK(h n+1,m -z) dh ∇h • n F v dF - F ∈F S h F 1 S (h)K(h -z)∇δ h • n F v dF - F ∈F S h F 1 S (h)δ h dK(h n+1,m -z) dh ∇h • n F v dF + F ∈F I h F ϱ I F δ h v dF + F ∈F D h F ϱ D F δ h v dF + F ∈F S h F ϱ D F 1 S (h)δ h v dF -Θ F ∈F I h F ⦃K(h -z)∇v • n F ⦄ δ h dF -Θ F ∈F I h F ⦃ δ h dK(h n+1,m -z) dh ∇v • n F ⦄ h dF -Θ F ∈F D h F K(h -z)∇v • n F δ h dF -Θ F ∈F D h F δ h dK(h n+1,m -z) dh ∇v • n F h dF -Θ F ∈F S h F 1 S (h)K(h -z)∇v • n F δ h dF -Θ F ∈F S h F 1 S (h)δ h dK(h n+1,m -z) dh ∇v • n F h dF + Θ F ∈F D h F δ h dK(h n+1,m -z) dh ∇v • n F h D dF + Θ F ∈F S h F 1 S (h)δ h dK(h n+1,m -z) dh ∇v • n F z dF . (II.57)
In Eq. (II.57), the terms are coloured differently. Blue colour terms are zero-order since they does not involve any derivative. They match exactly the bilinear form a h,n . Whereas red colour terms are oneorder because they hold derivatives. Such a distinction is useful to know the definition of the tangent stiffness matrix in case of the fixed-point method described in Section II.5.2.

Derivatives of constitutive laws (hydraulic properties) present in the Jacobian matrix can be computed analytically. Nevertheless, the resulting expressions are heavy. That is why derivatives are approximated using the central difference formula for a scalar function u : R d -→ R evaluated at the point h:

du(h) dh = u(h + ε) -u(h -ε) 2ε , (II.58)
where ε = u(h) √ ϵ pr with ϵ pr the machine epsilon if u(h) ̸ = 0. Otherwise, ε = 3 √ ϵ pr . This rule of thumb aims for a small value of ε but not too small due to roundoff errors in the representation of real numbers.

Stopping criteria

Both Newton-Raphson and fixed-point methods require a first guess for the iterative process which is taken from the solution of the last time step. One important choice, determining accuracy for nonlinear iterative process, is the stopping criterion. For Richards' equation, the stopping criterion can be specified in terms of absolute error for pressure head or water content between two successive iterations [START_REF] Huang | A new convergence criterion for the modified Picard iteration method to solve the variably saturated flow equation[END_REF]. For this study, the stopping criteria read:

∥r h,n (h n+1,m , v)∥ L 2 (Ω) ∥a h,n (h n+1,m , v)∥ L 2 (Ω) < ε 1 and δ n+1,m h L 2 (Ω) ∥h n+1,m ∥ L 2 (Ω) < ε 2 , (II.59)
where ε 1 , ε 2 are a user-defined tolerances. These two criteria are relative in order to be independent from the characteristic quantities of the problem. The first criterion is residual and the second one is incremental. The numerical tests of Chapter IV were carried out with the fixed-point method because it was more robust concerning initialization.

II.5.2 Approximation techniques

Relaxation and damping

The convergence of the nonlinear process is known to be difficult for Richards' equation. In particular, the nonlinear iterative solver can oscillate between two solutions. This is due to steep nonlinearities of hydraulic properties in the capillary region under some circumstances. Many techniques try to avoid this situation to reach robustness. Most of them involve adaptive time stepping or underrelaxation. Adaptive time techniques are discussed in the next section. Under-relaxation techniques can affect:

• the nonlinear iterative solution with damping or Anderson acceleration;

• the hydraulic conductivity with chord slope approximation, polynomial spline or regularization;

• the nonlinear method with tangent matrix approximation or clever initial guessing.

General studies [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF] show that none of these techniques can solely make the solver robust. Some tests tend to indicate that a Newton-Raphson method in combination with a fixed-point method for the first iterations is sufficiently robust and efficient [START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF].

II.6. ADAPTIVE TIME STEPPING

The damped Newton-Raphson method relaxes the increment δ n+1,m h in Eq. (II.52) while the new residual is greater than the previous one. The relaxation of the increment is done thanks to line search. Such a procedure is described in [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] and has been implemented.

Fixed-point method

Newton-Raphson methods converge quadratically but they often fail for parabolic degenerate problem where the Jacobian matrix might become singular and because of poor initial guess (locally convergent) [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF]. Besides, Newton-Raphson methods require more computational resources. Therefore, fixed-point iteration methods, also known as Picard's method, are robust and popular techniques for Richards' equation even if they converge linearly. This is demonstrated in Chapter IV, particularly on the case of Section IV.7.

Underlying ideas are already described in [START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF] and have just been adapted here to the DG system. Roughly speaking, a fixed-point method is approximated from the Newton-Raphson method by avoiding the first derivative terms inside the Jacobian matrix, that is to say only the terms in blue are kept in Eq. (II.57).

II.6 Adaptive time stepping

Time adaptation is motivated by the convergence of the nonlinear solver. Indeed, nonlinearities may be so strong that the iterative process may oscillate between two solution states. This is even more the case because of stiff initialization, poor initial guess or severe boundary conditions. On one hand, transient simulations have difficulties to converge if the time step is too large but, on the other hand, shorter time steps mean more time steps and so, a longer computational time. That is the reason why time adaptation is very attractive and common for Richards' equation. Different strategies can be used to adjust the time step [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF]. They are either heuristic, mainly based on convergence performance of the nonlinear solver [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Helmig | Multiphase Flow and Transport Processes in the Subsurface[END_REF][START_REF] Bergamaschi | Mixed finite elements and Newton-type linearizations for the solution of Richards' equation[END_REF][START_REF] Šimůnek | The HYDRUS-1D Software Package for Simulating the Movement of Water, Heat, and MultipleSolutes in Variably Saturated Media[END_REF], or rational and based on error control [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Kavetski | Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow[END_REF][START_REF] Kavetski | Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation[END_REF][START_REF] Kavetski | Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards equation[END_REF][START_REF] Maina | Ross scheme, Newton-Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation[END_REF]. The latter ones are generally more efficient but heuristic methods may outperform them in regards of their simple implementation and so they remains a relevant approach for Richards' equation. For this study, the time step is adjusted heuristically according to the previous number of iterations N it from the nonlinear solver such as [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Bergamaschi | Mixed finite elements and Newton-type linearizations for the solution of Richards' equation[END_REF]. Indeed, the time stepsize has a direct effect on the convergence of the solver. Let n ∈ N, 0 ≤ n < N . The simulations begin with a time step τ 1 . Then, the future time steps are calculated accordingly to the following rule: each time step is kept if convergence at the previous time iteration is achieved between m it and W it nonlinear iterations, it is increased by an amplification factor λ amp > 1 if the convergence requires fewer than m it nonlinear iterations and it is decreased by a reduction factor λ red < 1 if the convergence requires more than M it nonlinear iterations. In the case where the convergence is not fulfilled because of a solver failure (poor initial guess, bad condition number) or because the nonlinear iterations go over a prescribed maximum bound W it , the time step is started again (back-stepping) using a stepsize reduced by λ red . The time stepping scheme to choose the next time step τ n+2 is calculated from the previous one τ n+1 can be summarised with:

             τ n+2 =        λ amp τ n+1 if N it ≤ m it , τ n+1 if m it < N it ≤ M it , λ red τ n+1 if M it < N it ≤ W it , τ n+1 = λ red τ n+1 if W it < N it
or if the solver has failed (time step is started again).

(II.60)

The factors 0 < λ red < 1 and 1 < λ amp as well as the threshold values 0 < m it ≤ M it < W it are prescribed by the user and found empirically. Default values are λ amp = 2, λ red = 0.5, m it = 3, M it = 7 and W it = 10. A minimum time step can be defined to avoid excessive small time steps.

With this approach, the nonlinear solver is more robust because the time step is adjusted until success of convergence independently on τ 1 . Thanks to the amplification/reduction coefficients, the time step is adjusted smoothly. Nevertheless, the method depends on fixed empirical parameters restricting the possible size of the new time step. This does not provide an optimal time step. The resulting loss/gain in computational time is difficult to assess in regards to the balance between the number of nonlinear iterations and time steps length.

II.7 Implementation choices II.7.1 Overall programming

Rivage code

In order to solve Richards' equation and simulate flows in unsaturated porous media, the development of a program was undertaken from scratch to implement the numerical methods described in this thesis. The code is written in Fortran 2008 and compiled with GFortran version 10.2. It is called Rivage which stands for « RIchards pour des VAgues par méthodes de Galerkine discontinuEs » (Richards for waves by discontinuous Galerkin methods). The goal of Rivage code is to provide a generic tool based on the DG method. The structures of the program and the variables are intended to be reused for other model equations than Richards' equation. The code can already be employed for a large range of features: non-conforming, unstructured and hybrid mesh of triangles and parallelograms, three basis functions, quadrature rules exact up to degree 20, three solvers based on different matrix storage, a clever numbering, etc. Moreover, the code development tried to follow as much as possible the quality requirements for programming. Among others, the user has many possibilities to choose for the numerical methods, gathered in a parameter file to make it more usable. A number of errors will result in messages in the terminal for the user who can find additional comments in the code. Finally, the code is easily maintainable and modular which makes future extensions possible. The structure and the subroutines were especially designed to work in 3D, to add hyperbolic term with numerical flux and to be parallelized.

The execution work with different executable files which communicate between them with binary files called rivbin containing necessary data. Executable files consist in:

• data2rivage which converts parameter file .prm into binary data readable by rivage;

• rivage which is the main executable file. It performs calculations for the simulation;

• rivage2visu which converts binary data exported by rivage into data file under the format .dat readable by visualization softwares like Tecplot or VisIt;

• adaptriv which can modify the mesh from a binary file of rivage;

• modifriv which can modify the parameters of the simulation from a binary file of rivage.

In addition, one file of extension .prm is mandatory to execute the program. The format is specific to Rivage and it contains all the physical and numerical parameters of the problem to perform the simulation. Within this file, it is possible to create a mesh made up of block-based grids. If one wants to specify particular mesh or boundary conditions, then optional files may be considered. A file of extension .msh can be read to create an original mesh. This is at the format up to version 2. For each nonlinear iteration, the linear system has to be constructed and solved. The construction step is called the assembly algorithm and is detailed in the next paragraph. The solution step is carried by different solvers let to user choice. There are three possibilities:

• the in-house solver SOLP which is a direct solver using fully stored matrices. It is based on Gaussian elimination with lower-upper (LU) decomposition. It can solve non-symmetric systems of linear equations. The algorithm can be found in [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF];

• the in-house solver SOLLC which is a direct solver using skyline matrix storage (SKS) also known as band matrix storage. It is based on Gaussian elimination with lower-upper (LU) decomposition. It can solve non-symmetric systems of linear equations. The algorithm can be found in [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF];

• the free academic version 3.2.1 of AGMG which stands for Iterative solution with AGgregationbased algebraic MultiGrid. This software library solves large systems of linear equations with an aggregation-based algebraic multigrid method. It is known to perform well, in particular for scalar second order elliptic partial differential equations, but being purely algebraic makes it robust on a large range of problems like refined unstructured grids, convection-diffusion equations or problems with big jumps and large anisotropy. It uses matrices represented with the compressed sparse row (CSR) format. AGMG software and documentation were realised by Yvan Notay and can be found in http://agmg.eu. AGMG main algorithms are presented and justified in [START_REF] Notay | An aggregation-based algebraic multigrid method[END_REF][START_REF] Notay | Aggregation-Based Algebraic Multigrid for Convection-Diffusion Equations[END_REF][START_REF] Napov | An Algebraic Multigrid Method with Guaranteed Convergence Rate[END_REF].

In this work, all numerical solutions were obtained using the direct methods because discrete systems remains usually small and we do not want to take care about convergence of iterative methods. Besides, we aim for the parallelization of the solver in the future.

Assembly algorithm

From the notations and notions introduced in Section II.2.5, the system of Eq. (II.40) can be rewritten as a matrix system. It is assumed that local degrees of freedom are numbered contiguously within every element, so that basis functions and degrees of freedom can be reindexed:

Φ l k : 1 ≤ k ≤ N E , 1 ≤ l ≤ N E dof = {Φ i : 1 ≤ i ≤ N dof }, (II.61) χ l k : 1 ≤ k ≤ N E , 1 ≤ l ≤ N E dof = {χ i : 1 ≤ i ≤ N dof }. (II.62)
Plugging the DG expansion Eq. (II.33) into the scheme Eq. (II.40), a nonlinear system of differential equations raises written with matrices which are defined such that, ∀1 ≤ i, j ≤ N dof :

M ij = m(Φ j , Φ i ), (II.63) A ij = a(Φ j , Φ i ), (II.64) L i = l(Φ i , t). (II.65)
where M , A and L are respectively called the mass matrix, the stiffness matrix and the load vector. These global matrices result from the assembly of local matrices corresponding to an element contribution. The stiffness matrix can be subdivided into three matrices with respect to volume, interior face and boundary face contributions highlighted by different colours in Eq. (II.28). The load vector might also be subdivided into two vectors: one for a boundary face contribution and one for a volume contribution if there is a source/sink term.

The assembly algorithm creates the matrix system to solve and consists in three loops: one loop for volume contributions, one for interior face contributions and one for boundary face contributions.

II.7. IMPLEMENTATION CHOICES

For each loop, the local matrix is computed before being assembled into the corresponding global matrix. Then, the latter form the matrix system to solve. This process is illustrated in Fig. II.5. It is similar to assembly algorithm from FE methods except there are face contributions in addition. Details can be found in Rivière [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF] or in Di Pietro and Ern [START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF].

Due to the local nature of element contributions, the global matrices are block-organized and mainly sparse. Diagonal blocks account for the element contribution while off-diagonal blocks account for interaction of elements through the face that connects them. Thus, numbering modifies directly the matrix structure since two neighbouring elements having distant numbering will produce far offdiagonal blocks. Reducing the bandwidth is important to decrease computational cost when specific storage are used for solvers. The standard numbering takes the element number progressively from their construction. The Cuthill-McKee algorithm is particularly used to form a small bandwidth matrix [START_REF] Cuthill | Reducing the bandwidth of sparse symmetric matrices[END_REF]. However, Rivage employs Morton's code to renumber elements with Z-order space filling curves [START_REF] Alauzet | On the Use of Space Filling Curves for Parallel Anisotropic Mesh Adaptation[END_REF]. The method is very easy to implement, cost-effective and provide good numbering preserving locality even though the resulting connectivity matrix is not as optimal as for Cuthill

II.7.2 Computing tools

Basis functions

Because there is no continuity constraint between elements for the test functions, their restriction to one element can be chosen independently from their restriction to other elements. Thereby, basis functions are specified to only one element which yields to a minimal stencil by reducing communications between elements. Besides, any polynomial functions basis is suitable which provides flexibility to DG methods. The choice of local basis is important because it affects the computational performance of DG methods, especially for high-order approximations. It exists two main families of basis functions: the nodal basis and the modal basis. The former give physical meaning to the degrees of freedom since they belong to nodes of elements such as for Fe methods. However, it is often difficult to have access to high-order because such basis require computations to find them. For instance, Lagrange or Hermite basis functions are nodal basis. One criterion to select them is the ease of integration. The modal basis are usually easy to implement for any polynomial degree. However, degrees of freedom have no meaning other than being coefficients in the DG expansion. Generally, the latter are preferred for DG methods because they can reach high-order accuracy easily. For instance, there are monomial, Taylor, Legendre or Dubiner basis. Two criteria are used to select them: orthogonality and hierarchism. If the basis is orthogonal or nearly orthogonal to respect of an inner product, this ensures that the condition number of the system does not grow too much with the polynomial degree and that some matrices are diagonal or nearly diagonal. A Gram-Schmidt procedure may be used for this purpose. If the basis is hierarchical, then, for a given polynomial degree p, the basis includes all the bases of polynomial degree k < p. This allows to effortlessly increase or decrease the polynomial degree locally.

In this work, a monomial basis is chosen because it is hierarchical and straightforward to implement to reach any order. The monomial basis principle is depicted in The Lagrange nodal basis -the classic FE basis -is also implemented in Rivage code, up to degree two for quadrangle and up to degree three for triangles. It can provide better approximations than II.7. IMPLEMENTATION CHOICES monomial basis because it holds additional degrees of freedom for the same order, especially for quadrangles. Indeed, polynomial belongs to Q p instead of P p . However, it is not hierarchical and reaching high-order is difficult. This is why the monomial basis is tensored to produce a new basis which is hierarchical and whose discrete space is enriched by tensor product, see Fig. II.7. The drawback is that computational cost increases due to higher dimension.

Mapping and shape functions

Like for the FE methods, it is worth to use a mapping from a reference element to the -actualphysical element. Indeed, computing integrals over each element in the mesh would be costly because numerical quadratures should be adapted at each time. A preferred approach is to use change of variables to compute integrals over a fixed element. The procedure will be explained for 2D but can be extended to 3D. For this purpose, every element has vertices which are ordered in counter-clockwise. Then, any element can be transformed from its corresponding reference element thanks to an affine transformation. In the case of Rivage, there are three reference elements: one for triangles, one for parallelograms and one for edges. Quadrangles are restricted to parallelograms because the mapping use an affine transformation which must preserve parallelism. For a given physical element E and its corresponding reference element Ê, they have N nod vertices (or nodes) whose 2D coordinates are respectively x i , y i and xi , ŷi , i = 1, . . . , N nod . There is a relation linking any point belonging to the physical element with the reference element:

x y = x 1 . . . x N nod y 1 . . . y N nod     φ1 (x, ŷ) . . . φN nod (x, ŷ)     .
(II.67)

The functions φi , i = 1, . . . , N nod are called the shape functions. They are specific to reference element shape (triangle, parallelogram, etc) and defines the mapping. Generally, they are preferred to be polynomial and come from the FE methods. They match the Lagrangian basis functions, see [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF].

CHAPTER II. NUMERICAL METHODS FOR RICHARDS' EQUATION

Formally, the mapping F E : Ê -→ E can be rewritten as:

x y = F E x ŷ (II.68) = J E x ŷ + x O y O , (II.69)
where x O , y O are the coordinates of the physical element point related to the reference element origin point. The reference element origin point is set directly from the definition of the reference element. This is a predefined vertex for triangles and the barycentre for parallelograms. J E is the Jacobian matrix of the mapping F E defined as:

J E = x 1 . . . x N nod y 1 . . . y N nod         φ1 (x, ŷ) ∂ x φ1 (x, ŷ) ∂ ŷ . . . . . . φN nod (x, ŷ) ∂ x φN nod (x, ŷ) ∂ ŷ         . (II.70)
If the determinant of J E does not vanish, which is always satisfied when E is convex, there exists an inverse transformation F E -1 : E -→ Ê such that:

x ŷ = F E -1 x y (II.71) = J E -1 x y -J E -1 x O y O . (II.72)
For a given function g, substitution theorem leads to a transformation of integral and the chain-rule gives transformation of derivatives:

E g(x, y) dE = Ê g • F E (x, ŷ)|det J E | d Ê, (II.73) E ∇g(x, y) dE = Ê J E -⊺ ∇g • F E (x, ŷ)|det J E | d Ê. (II.74)
When face integrals are considered, the reference face is taken to be the reference edge element (in 2D) as illustrated in Fig. II.8. For a given function g, the face integral is computed such that:

F g dF = F g • F E (x, ŷ) det F J E d F , (II.75)
where F E has to be understood as the mapping from the reference edge element (in 2D) to the physical face. The Jacobian matrix keeps the same definition. The last column of the Jacobian matrix will always hold zeros since face are (d -1)-objects. For this reason, the determinant is replaced with the following expressions from parametrization [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF]:

In 2D, det where j kl = J E (k, l).

F J E := j 11 2 +

II.7. IMPLEMENTATION CHOICES

Numerical integration

Quadrature rules will be used to calculate the numerical value of a definite integral. One particularity of DG methods is that integrals need to be evaluated exactly both over edges and elements, and for polynomial functions of various degrees, possibly high. There are many possible quadrature rules with their own advantages and drawbacks. The Gaussian quadrature rules is considered in this work because it is widespread and presents a good compromise between rule difficulty and accuracy. For edges in 1D, the rule is given by:

1 -1 f (x) dx = N G i=1 w i f (x i ), (II.78)
where w i and x i for i = 1, . . . , N G are respectively the Gauss quadrature weights and nodes. Their values can be found in many textbooks, for example in [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF]. General formulas are available but it is preferred to store values directly in the code to facilitate access during computation since this type of calculation is performed many times during simulation. Equation (II.78) is exact for polynomials of degree 2N G -1 or less and is sometimes referred as the Gauss-Legendre quadrature. Since DG methods reach easily high-order approximation, it is important to get high-order quadrature rules.

For parallelograms in 2D, the one dimensional Gauss quadrature rule is applied in each direction:

1 -1 1 -1 f (x) dx = N G i=1 N G j=1 w i w j f (x i , x j ). (II.79)
For triangles in 2D, the Gauss quadrature rule takes specific set of values. O ne of the main characteristics and quality of the DG method is to facilitate mesh adaptation by element refinement and coarsening (h-adaptation), as well as by enrichment of approximation order (p-adaptation). Combination of this two processes is called hp-adaptation.

III

In numerical simulation, mesh adaptation is an essential tool to get accuracy and cost-effectiveness. Indeed, local mesh modification can serve as a capturing technique to resolve physical phenomena or as a control technique to monitor numerical errors. Adaptation strategy is based on two key ingredients defining main goals of the mesh adaptation. Firstly, the overall process has to be chosen to target goals of adaptation. It is sometimes referred as adaptivity. Many different techniques exist to optimize

CHAPTER III. ADAPTIVE STRATEGY BASED ON DISCONTINUOUS GALERKIN METHOD

the mesh, select elements to adapt and refine/coarsen these elements [START_REF] Mitchell | A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations[END_REF]. Secondly, an error criterion has to be decided in order to localize zones to adapt. One possibility is to use an error indicator which defines a quantity of interest taken to be typical of the physics, very other related to solution gradient [START_REF] Johnson | Adaptive finite element methods for diffusion and convection problems[END_REF]. Another possibility is to use error estimator which comes from theoretical derivations of the numerical scheme [START_REF] Babuška | Error Estimates for Adaptive Finite Element Computations[END_REF][START_REF] Ainsworth | A Posteriori Error Estimation in Finite Element Analysis[END_REF]. Error estimation is said to be a priori, if estimate is established before the solution is known to exist, or a posteriori, if estimate uses the already computed solution. Besides, a posteriori estimation can be either explicit or implicit [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF]. Estimator is explicit when it can be computed directly from the numerical solution and the problem parameters, or implicit when its computation needs an additional problem to solve. Lots of a posteriori error estimations are found in literature [START_REF] Johnson | Adaptive finite element methods for diffusion and convection problems[END_REF][START_REF] Ainsworth | A Posteriori Error Estimation in Finite Element Analysis[END_REF][START_REF] Verfürth | A Posteriori Error Estimation Techniques for Finite Element Methods[END_REF] and they are sometimes classified according to their underlying principle. For instance, they are said to be hierarchical [START_REF] Demkowicz | Toward a universal adaptive finite element strategy, part 1. constrained approximation and data structure[END_REF][START_REF] Oden | Toward a universal adaptive finite element strategy, part 2. a posteriori error estimation[END_REF][START_REF] Rachowicz | Toward a universal adaptive finite element strategy part 3. design of meshes[END_REF] when it measures the difference between the numerical solution and a reference numerical solution obtained with an enhanced scheme like a more refined region [START_REF] Šolín | Higher-Order Finite Element Methods[END_REF][START_REF] Šolín | Goal-oriented hp-adaptivity for elliptic problems[END_REF]. Others are based on residuals [START_REF] Ainsworth | A unified approach to a posteriori error estimation using element residual methods[END_REF], flux reconstruction [START_REF] Ern | An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF], dual spaces [START_REF] Becker | An optimal control approach to a posteriori error estimation in finite element methods[END_REF], etc.

The strategy presented in this chapter aims to optimize a compromise between feasibility, relevance, accuracy and computation time. This is a block-based mesh refinement which is not applied systematically to save computational time. Then, two criteria are introduced: one is an error indicator and the other is an estimation-based error indicator. The latter is called in this way because no theoretical derivation supports it but rather an heuristic comparison from other error estimators. Finally, some variations on the numerical scheme are devised. They are taking the most of the DG flexibility to propose improvements. One is the use of the weighted DG method which relaxes average and penalty definitions, and the other is an extrapenalty term for the jumps of total flux (derivatives).

III.1 Adaptive mesh refinement

III.1.1 Adaptivity

In this part, the h-adaptation tool that has been developed is presented. Mesh refinement consists in concentrating the discretization points (element, volume, ...) at the places where it is necessary, in a sense that must be defined. Adaptive Mesh Refinement techniques (AMR) are now widely used and have since proven their efficiency. As said earlier, mesh refinement may target different goals and so use different strategies defining adaptivity. To get an idea of the process for a posteriori estimation, several steps can be sketched, see Fig. First of all, the problem is solved to get the numerical solution (solution). Then, error estimates are computed for each element (estimation). Next, elements are eventually marked for refinement/coarsening (selection). Last, the mesh is adapted accordingly (correction). Each of this step can

III.1. ADAPTIVE MESH REFINEMENT

be performed by many different techniques: evaluation is proper to the numerical method, estimation is based on the chosen error indicator/estimator, selection may be proportional or absolute, userdefined or automatic and correction depends on the element geometry and mesh structure. In addition, Fig. III.1 illustrates the two main adaptive approaches. Sequential adaptivity is done throughout the simulation when it is considered needed whereas iterative adaptivity is done systematically for every computed solution. Usually, sequential adaptation wants to stay efficient in term of computational time and iterative adaptation wants to reach optimality by finding the mesh minimizing numerical errors.

AMR techniques are able to work with many different meshes: on 2D Cartesian grids for quadtree [START_REF] Berger | Local adaptive mesh refinement for shock hydrodynamics[END_REF][START_REF] Yiu | Quadtree grid generation: Information handling, boundary fitting and CFD applications[END_REF][START_REF] Min | A second order accurate level set method on non-graded adaptive cartesian grids[END_REF][START_REF] Zhang | A two dimensional hydrodynamic and sediment transport model for dam break based on finite volume method with quadtree grid[END_REF], on 3D Cartesian grids for octree [START_REF] Losasso | Simulating water and smoke with an octree data structure[END_REF][START_REF] Fuster | Numerical simulation of droplets, bubbles and waves: state of the art[END_REF], and on anisotropic mesh [START_REF] Coupez | Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing[END_REF][START_REF] Hachem | Immersed stress method for fluid-structure interaction using anisotropic mesh adaptation[END_REF][START_REF] Bahbah | Conservative and adaptive level-set method for the simulation of two-fluid flows[END_REF]. As shown in Fig. III.2 (right), anisotropic mesh refinement distorts the mesh to concentrate the elements where they are needed. They are sometimes called r-adaptation or relocation refinement and belong to moving mesh methods. This method is used for stabilized finite elements on any conforming mesh and it allows very local refinement, but at the price of frequent remeshing in order to follow the physics of the phenomenon. Examples for porous media flow include [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF][START_REF] Dong | Adaptive moving grid methods for two-phase flow in porous media[END_REF][START_REF] Huang | Adaptive moving mesh modeling for two dimensional groundwater flow and transport[END_REF]. The refinement by quadtree or octree is mainly used for finite volumes on structured but non-conforming Cartesian mesh. Very local refinement is also possible. The tree-like structure of the mesh data allows, through appropriate algorithms, a quick access to a cell or its neighbours.

III.1.2 Block-based structure

There is a number of AMR types that have been developed [START_REF] Zheng | Block-Based Adaptive Mesh Renement Finite-Volume Scheme forHybrid Multi-Block Meshes[END_REF][START_REF] Giuliani | Adaptive mesh refinement on graphics processing units for applications in gas dynamics[END_REF][START_REF] Dunning | Adaptive mesh refinement in the fast lane[END_REF], see Fig.

CHAPTER III. ADAPTIVE STRATEGY BASED ON DISCONTINUOUS GALERKIN METHOD

In this work, applications with complex geometries are aimed, or later, couplings of models on hybrid mesh. This is why a non-conforming unstructured mesh is used like in [START_REF] Altazin | Numerical investigation of BB-AMR scheme using entropy production as refinement criterion[END_REF]. Although computing time may be lost compared to fully structured Cartesian codes, block-based AMR is adopted for several reasons. First, block-based mesh can be viewed as a macroscale mesh surrounding large regions of the domain. Provided that physical phenomena like moving fronts stay in the block, the block refinement is kept avoiding frequent remeshing. This idea follows some remarks found in [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF]. Secondly, block-based structure provides a basis to parallelize the code since only inter-block connectivity is required. Finally, block-based structure leads to solving techniques like local time stepping or domain decomposition which can be useful for Richards' equation [START_REF] Kuraz | Solving the nonlinear Richards equation model with adaptive domain decomposition[END_REF].

III.1.3 Selection procedure and corrective mechanism

In accordance with the simulations presented in this work, only the bidimensional case is presented. The strategy adopted is then as follows and illustrated in the The unstructured mesh is composed of quadrilateral and/or triangular elements, where each element, as in [START_REF] Altazin | Numerical investigation of BB-AMR scheme using entropy production as refinement criterion[END_REF], defines a root element or block, see • For β c ≤ η < β r , the block remains unchanged and so for the related elements;

• For β r ≤ η, the block is refined so its elements are split into four isotropic subelements;

• For η < β c , the block is coarsened so four neighbouring subelements are merged into one element.

Limitations of such selection are described in [START_REF] Pons | Adaptive mesh refinement method. Part 1: Automatic thresholding based on a distribution function[END_REF] as well as a procedure to overcome them. The mesh refinement level is adapted according to the rule that the ratio of mesh refinement should not exceed 2 between two neighbouring blocks producing a graded mesh. As illustrated in [START_REF] Ersoy | Adaptive multiscale scheme based on numerical density of entropy production for conservation laws[END_REF], this constraint allows smooth transitions between refined and unrefined regions.

III.2. A POSTERIORI ERROR ESTIMATION

The mesh is then built using a quadtree graph (available for quadrangle and triangle) in order to define easily and precisely the neighbouring elements of each face. This procedure is pursued until the desired level of mesh refinement is reached as shown in Fig. III.4c. In the case of evolutionary problems, this implies frequent remeshing to follow the studied phenomenon. This is why we prefer to widen the stencil with a coarse root mesh, namely blocks, and therefore to use the whole quadtree graph, in order to remesh less often. It is illustrated for quadrangles and triangles in

III.1.4 Projection by local DG problem

During a simulation, if a mesh refinement or coarsening process takes place, a new evaluation of the unknown field must be carried out to keep solution information. The prolongation (refinement) and restriction (coarsening) process are facilitated by the fact that the quadtree graph connect explicitly the "mother" element to its four "daughter" elements. To do that, one may use local interpolation/extrapolation. This is not straightforward in practice because it requires to identify the number of points and their position on the element to construct a right-determined system giving a fairly close solution. In the present study, the projection is made by solving a local DG problem. This technique gives good degrees of freedom by performing calculations at quadrature points. The element-wise weak formulation corresponding to h Enew = h E old reads:

Enew h Enew v dE new = Enew h E old v dE old . (III.1)
Then, mass matrix can be used to project the solution on the new mesh. This method is also used to enforce the initial condition h = h 0 , see Section II.3.3.

III.2 A posteriori error estimation III.2.1 Gradient-based error indicator

An a posteriori error indicator is employed to show how a simple physical quantity is able to provide information about refinement. This error indicator is inspired from Miller et al. [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF] where they

CHAPTER III. ADAPTIVE STRATEGY BASED ON DISCONTINUOUS GALERKIN METHOD

use changes in effective saturation. Here, the error indicator is based directly on changes in the solution that is to say the gradient of hydraulic head h. This allows to work both for the saturated and unsaturated regions. So, it reads:

C E = 1 max E∈E h 1 |E| ∥∇h∥ L 2 (E) 1 |E| ∥∇h∥ L 2 (E) . (III.2)
The error indicator is normalized so that 0 ≤ C E ≤ 1.

III.2.2 Estimation-based error indicator

Adaptive refinement aims to control numerical approximation within a prescribed tolerance employing the least computational resources. That is why it is essential to have an a posteriori error estimation which gives information about the true error only from the computed numerical simulation and the given data of the problem. The true error for a scalar function u : R d -→ R is defined as:

e n h := u -u n h (III.3)
The computation of the a posteriori error estimation should be as economical as possible compared to the numerical solution. Besides, other features are desired for the error estimator:

• locality, namely, the error estimator gives information element-wise about the error distribution which is relevant for local adaptation;

• reliability, namely, the error estimator gives an upper bound on the true error up to a multiplicative constant. This ensures that the true error is below the tolerance prescribed for the error estimator;

• efficiency, namely, the error estimator gives a lower bound on the error up to another multiplicative constant. This ensures that the true error is not too far from the error estimator to achieve the prescribed tolerance cost-effectively.

The product of the two aforementioned constants measures the error estimator quality analogously to a condition number. This quantity gives details about error estimation calibration in the sense that it should stay bounded regardless of the parameters involved in the differential equation [START_REF] Verfürth | A Posteriori Error Estimation Techniques for Finite Element Methods[END_REF]. Another powerful property is the derivation of a guaranteed a posteriori error estimate where guaranteed means the upper bound of the error is explicitly computable that is to say the reliability constant is guaranteed to be one. Ern and Vohralík achieved this goal using divergence-conforming flux reconstruction for mixed h-adaptive DG methods, see for example [START_REF] Ern | Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems[END_REF] for convection-diffusion-reaction problems. Even though residual-based error estimation overestimates the true error by a significant factor, this approach is interesting because it is simple and well-developed for lots of problems, including time or hp-adaption.

Another a posteriori error indicator is introduced but it is based on error estimation. Indeed, Verfürth [START_REF] Verfürth | A Posteriori Error Estimation Techniques for Finite Element Methods[END_REF] developed a posteriori error estimators based on residual and derived thanks to suitable norms in the context of finite elements methods for many equations. These estimates are valid for many equations and he worked in particular on nonlinear parabolic problems such as Richards' equation. For this kind of problem, Verfürth [START_REF] Verfürth | A posteriori error estimates for nonlinear problems: L r (0, T ; W 1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations[END_REF] used general variational approach based on the concept of very weak solutions to prove reliability as well as global and local efficiency for non-standard time discretization such as implicit Runge-Kutta schemes. However, this analysis is sophisticated and subject to step

III.2. A POSTERIORI ERROR ESTIMATION

size restriction in space and time. Then, he removed this constraint in [START_REF] Verfürth | A posteriori error estimates for finite element discretizations of the heat equation[END_REF][START_REF] Verfürth | A posteriori error estimatesfor non-linear parabolic equations[END_REF] for standard time discretization, ranging from implicit Euler scheme to Crank-Nicolson scheme and midpoint rule, thanks to energy norm augmented by dual norm of time derivative. It results in truly reliability and efficiency, provided that solving an additional discrete problem is agreed. Later on, this type of residual-based derivation was adapted by Melenk and Wohlmuth [START_REF] Melenk | On residual-based a posteriori error estimation in hp-FEM[END_REF] to hp-adaptive finite elements methods for elliptic problems generalizing the Clément/Scott-Zhang interpolation operator. Afterwards, Houston et al. [START_REF] Houston | Energy Norm A Posteriori Error Estimation of hp-Adaptive Discontinuous Galerkin Methods for Elliptic Problems[END_REF] developed residual-based a posteriori error estimation for hp-adaptive interior penalty DG methods applied to elliptic problems exploiting the DG energy norm. Finally, Schötzau and Zhu [START_REF] Schötzau | A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations[END_REF][START_REF] Zhu | A robust a posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations[END_REF] extended this approach to convection-diffusion equations both for non-and hp-adaptive DG methods.

Greatly inspired by these works and supported by an heuristic analysis, the following a posteriori estimation-based error indicator is used:

(η n E ) 2 = η n E,R 2 + η n E,F 2 + η n E,J 2 , (III.4)
where η n E,R , η n E,F and η n E,J are respectively the element residual, the face residual and the solution jump sub-estimates. ϱ I F and ϱ D F were defined in Eq. (II.23) and (II.24). The quantities λ m (K) and λ M (K) stand respectively for the lowest and largest eigenvalue of K on E. For a face F of normal n, the minimum of normal component from neighbouring K is chosen:

κ m := min(κ l , κ r ) where κ = n ⊺ • K u n+1 h
• n. So sub-estimates write:

η n E,R 2 = h 2 E p 2 E λ m (K) θ u n+1 h -θ u n h τ n -∇ • K u n+1 h ∇u n+1 h 2 L 2 (E) , (III.5) η n E,F 2 = F ∈∂E∩F I h h F 2p F κ m K u n+1 h ∇u n+1 h • n 2 L 2 (F ) + F ∈∂E∩F N h h F p F κ l q N -K u n+1 h ∇u n+1 h • n 2 L 2 (F ) , (III.6) η n E,J 2 = F ∈∂E∩F I h 1 2 ϱ I F + h F p F κ m u n+1 h 2 L 2 (F ) + F ∈∂E∩F D h ϱ D F + h F p F κ l u D -u n+1 h 2 L 2 (F ) . (III.7)
Here, the time discretization is based on the implicit Euler scheme but the a posteriori estimationbased error indicator can be determined for other time schemes such as in [START_REF] Verfürth | A posteriori error estimates for nonlinear problems: L r (0, T ; W 1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations[END_REF]. Global estimates are obtained with:

(η n ) 2 = E∈E n h (η n E ) 2 . (III.8)
Unlike the aforementioned papers, there is no rigorous mathematical proof for this residual-based energy norm a posteriori error estimation. Extension to the case of nonlinear and time-dependant parabolic equation solved by hp-adaptive DG methods remains an open problem. However, some remarks can be drawn. Firstly, the numerical treatment of the time dependence and of the nonlinearity resorts to consider a sequence of linear and steady problems. Secondly, as noted by Verfürth [START_REF] Verfürth | A Posteriori Error Estimation Techniques for Finite Element Methods[END_REF], the element residual term is related to the residual of the numerical solution with respect to the strong form of the equation which may be viewed as the error from the nonlinear process. The face METHOD residual term is related to the boundary operator which is associated with the strong and weak forms of the differential equation. It reflects, on one hand, that the numerical solution gradient -the flux -is discontinuous and on the other hand, that Neumann boundary conditions may not be satisfied. The solution jump term is related to the penalization which are associated with the weak form of the equation. It reflects that the numerical solution is discontinuous at interior faces in the DG framework and Dirichlet boundary conditions may be slightly violated according to the penalty coefficients. Moreover, the second term and third parts of the estimation-based error indicators measure how valid the hypotheses on the seek solution are to derive the DG formulation.

III.3 Variation on the numerical scheme III.3.1 Weighted discontinuous Galerkin framework

In the context of convection-diffusion equations or coupling of parabolic/hyperbolic domains, the solution may hold spurious oscillations at internal layers for vanishing or varying diffusion [START_REF] Ern | Multi-algorithmic methods for coupled hyperbolic-parabolic problems[END_REF]. This situation is typical of porous media problems where there are material heterogeneity and degenerate diffusion. Even in the case where diffusion is nonzero but small, the numerical solution imitates this limiting case. One interpretation is the flux stays continuous but the solution mimics a discontinuity. Then, even though the mathematical description has some regularity, the numerical scheme is unable to recognize the sharp internal layer leading to numerical instabilities. This can trouble mass conservation.

Even though this phenomenon may be resolved by mesh refinement, the corresponding computational cost is substantial to maintain a physically acceptable solution. Slope limiters are another possibility to control these undershoots/overshoots but further computational developments are needed to cope with the geometry of elements in high-dimension [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. A satisfactory approach would be to design a DG method that can handle internal layers in an automated fashion. That is the reason why the weighted DG framework was introduced. Following earlier works [START_REF] Houston | Discontinuoushp-Finite Element Methods for Advection-Diffusion-Reaction Problems[END_REF][START_REF] Croisille | Analysis and simulation of a coupled hyperbolic/parabolic model problem[END_REF][START_REF] Burman | A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems[END_REF], Proft and Ern proposed one possible remedy in [START_REF] Ern | Multi-algorithmic methods for coupled hyperbolic-parabolic problems[END_REF]. This work was continued by Proft and Rivière on one hand [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Proft | Analytical and numerical study of diffusive fluxes for transport equations with near-degenerate coefficients[END_REF][START_REF] Proft | Discontinuous Galerkin methods for convection-diffusion equations with varying and vanishing diffusivity[END_REF], and by Ern, Di Pietro, Zunino, Stephansen, Guermond on the other hand [START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF][START_REF] Di Pietro | Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection[END_REF][START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF][START_REF] Ern | Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems[END_REF]. Proft and Rivière called their two proposed methods improved and adapted DG methods where they focus on exploring different average flux definitions. Nevertheless, their methods can be viewed as particular cases from Ern-Di Pietro work referred as the weighted discontinuous Galerkin methods (WDG).

WDG methods work thanks to two key ingredients which can be seen as a way to incorporate into the definition of the scheme some partial knowledge of the solution. The first one is the use of weighted averages instead of the standard arithmetic average inside the discontinuous formulation. The amount of diffusive flux for each side of a face is controlled and is used to formulate the consistency terms in the method analysis. The second ingredient is to relax the face penalization used for continuity constraint by a coefficient depending on the harmonic mean of the diffusivity of the neighbouring elements. Such a penalty strategy turns out to tune automatically the amount of local penalty to regulate the degree of smoothness of the approximate solution. Analysis and details can be found in [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF] and related. One should note that the weighted framework was developed for convection-diffusion equation and not parabolic equation like Richards' equation. However, Proft and Ern demonstrate how it is useful even for purely parabolic problems when the diffusion coefficient varies spatially [START_REF] Ern | Multi-algorithmic methods for coupled hyperbolic-parabolic problems[END_REF].

Internal layers depend on the spectral structure of the diffusivity, especially the degree of anisotropy is related to the condition number. The hydraulic conductivity is a matrix-valued function which

III.3. VARIATION ON THE NUMERICAL SCHEME

corresponds to an anisotropic diffusion. Its degree of anisotropy is

∆ K := λ M (K) λ m (K)
. Weights across a face are positive numbers such that: ω l + ω r := 1.

(III.9)

The weighted average and the conjugate weighted average are respectively defined as:

⦃u⦄ ω := ω l u l + ω r u r , (III.10)

⦃u⦄ ω := ω r u l + ω l u r . (III.11)
Then, the relation ∀u, v ∈ S p E n h , uv = ⦃u⦄ ω v + u ⦃v⦄ ω is replacing Eq. (II. [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF]) in the weak formulation derivation. One interpretation of this weighted average is that the mortar between elements is proportional to the stiffness of two springs in series. This choice of average is employed to find an equivalent hydraulic conductivity for Richards' equation, see [START_REF] Belfort | On equivalent hydraulic conductivity for oscillation-free solutions of Richard's equation[END_REF]. Moreover, the weighted average defines a numerical flux which is related to the ones from the Local Discontinuous Galerkin method which is known to better behave with discontinuous solutions.

The weighted diffusion penalty coefficient γ F is taken to be the harmonic mean of the neighbouring normal hydraulic conductivities across the face [START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF]:

γ F = 2κ l κ r κ l + κ r . (III.12)
Standard DG methods are designed to weakly enforce the continuity of the numerical solution between mesh elements. However, excessive penalty results in spurious oscillations for very small and vanishing diffusivity. Allowing stronger discontinuities leads to smaller overshoots/undershoots and reduces the spurious oscillations in the vicinity of the internal layer by preventing their propagation. The stability of the scheme is improved. It is interesting to note that several authors consider a specific definition for the penalty parameter to adjust the size of the jumps according to problem variables, space dimension, approximation order and element shape, see [START_REF] Bastian | A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure[END_REF][START_REF] Sun | Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media[END_REF][START_REF] Shahbazi | An explicit expression for the penalty parameter of the interior penalty method[END_REF][START_REF] Hartmann | An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations[END_REF] where penalty treats problems with high heterogeneity or discontinuity.

Standard DG methods assume ω l = ω r = 1 2 , which reduces the weighted average to the arithmetic average, and γ F = 1. The WDG method chooses other definitions. Following definitions in [START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF], let the weights be:

       ω l = κ r κ l + κ r , ω r = κ l κ l + κ r if κ l + κ r ̸ = 0, ω l = ω r = 1 2 otherwise. (III.13)
With this definition choice for weights, γ F results to be equivalent to the weighted average of diffusivities:

γ F = 2κ l κ r κ l + κ r = ⦃κ⦄ ω . (III.14)
WDG methods assume that diffusivity discontinuities fit the mesh [START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF]. This is generally the case for heterogeneous media but it is not true for nonlinear diffusivity such as in Richards' equation. Then, sharp internal layers move dynamically without a priori knowledge of the location and may occur inside an element. So the weighted framework is not expected to work in this situation. Nevertheless, a suitable AMR as described previously can be used as a capturing technique. This idea is one of the main novelties proposed in this thesis. The weighted framework and mesh adaptation are working dynamically in synergy: the former changes the smoothness nature of the numerical solution while the latter tracks the internal layer linked to nonlinearity thanks to refinement driven by the a posteriori estimation-based error indicator of Eq. (III.4).

III.3.2 Extrapenalization

Rivière introduced an additional penalty term in the DG formulation [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF]:

J X (h, v) = F ∈F I h F ϱ X F K(h -z)∇h • n K(v -z)∇v • n dF (III.15)
It constrains the jump of derivatives of the function to be zero. Therefore, it weakly imposes the continuity of the numerical total flux which is wanted from a physical point of view but also mathematically to be able to derive the weak formulation by subtracting it Eq. (II.18). That is why it is referred as an extra stabilization term. Nevertheless, the numerical analysis of the method is independent of this term and so Rivière discarded it without further use or explanation.

The total flux regularity comes into play to explain the presence of wetting front/internal layer due to nonlinearity and heterogeneity. The total flux becomes discontinuous if sharp wetting fronts produce spurious oscillations which is not desirable. As a result, an extrapenalty term is interesting for Richards' equation to preserve continuity of the Darcian flux. Moreover, this extrapenalty term controls the jump of the flux which is measured by one part of the a posteriori error estimator, see Eq. (III.6) and so have influence on mesh adaptation.

Summary

DG methods offer great abilities to make mesh adaptation by working with local approximation. This enables non-conforming mesh and element-wise different basis functions, resulting in the so-called h-and p-adaptation. This thesis focuses on h-adaptation by proposing an adaptive strategy whose goals are to minimize computational cost and reaching accuracy by capturing physical phenomena of Richards' equation.

A block-based AMR is chosen because it allows to save computational time. First, blocks partition the domain into large regions where the solution is expected to behave similarly. Internal layers due to heterogeneous diffusion are attached to blocks interface while moving fronts propagate through the domain but should stay within a block for a while. Then, if the front is well resolved by the block discretization, less frequent remeshing is needed so computational time is saved. Second, the block-based structure leads to other advanced solving making domain decomposition and parallelization easier. Two estimates are proposed to drive mesh adaptation which is used as a capturing technique for sharp fronts rather than a control technique to reduce numerical errors. With this in mind, an error indicator based on hydraulic head gradients is proposed as well as an error indicator based on a posteriori error estimation. Their performance will be evaluated in the next chapter for Haverkamp's test-case and Tracy's benchmark.

Taking the most of DG flexibility, two modifications are also proposed for the numerical scheme. One is to use the weighted DG method which has demonstrated efficiency in the context of convection-diffusion equation to resolve sharp internal layers due to heterogeneous and vanishing diffusivity. A diffusivity-based weighted framework is applied to flux average and penalty definition. This allows for jumps in the solution which are expected to help numerical approximation to better resolve sharp fronts. Besides, thanks to weak penalization, an extrapenalty term is introduced to control jump of derivatives and guarantee total flux continuity which is important both for the numerical scheme and Richards' equation behaviour. F ollowing the numerical methods presented in the previous chapters, the Rivage code and the adaptive strategy are validated against test-cases and experimental comparisons.

First of all, few numerical experiments are carried out with artificial cases built on the analytical solution of Poisson's equation. They allow to get some basic behaviours of stability and convergence for the DG methods.

Then, a series of test-cases are used to validate the implementation of the numerical methods. All of them are 1D infiltrations into a soil column but they hold different hydraulic properties and various boundary conditions. The solution of Richards' equation is more or less difficult to get and these cases are found largely in the literature, providing reference numerical solutions. That is why these infiltration test-cases are good points of comparison to validate the solution of Richards' equation by the newly developed code. In particular, Polmann's test-case illustrates the importance of mesh resolution to get oscillation-free solutions of Richards' equation. In this case, the WDG method can be useful to improve the approximation quality.

Finally, performance of the adaptive strategy proposed in this thesis is demonstrated. A 2D benchmark with a known exact solution for Richards' equation is employed to show convergence and the error estimation efficiency to drive mesh adaptation. Laboratory experiments involving 2D water table control, as well as their related simulations available in the literature, serve as a keystone to know if the method is able to simulate complex porous media flow dynamics with recharge, drainage and seepage. Finally, experimental data from the reservoir impoundment of La Verne dam are used to assess the accuracy and the robustness of the method. This is a demanding case because there are several materials, stiff hydraulic properties, dynamic boundary condition and seepage. The results of the simulation are discussed to highlight the potential of the solving strategy.

IV.1 Numerical experiments

These test-cases aim to highlight some properties of DG methods for simple problems whose analytical solution is known. They are built upon the Poisson's equation. In particular, convergence and stability tests are targeted.

IV.1.1 One-dimensional analytical tests

For this first test-case, theoretical convergence of the DG methods are checked and numerical stability is evaluated with respect to penalty values. The following one-dimensional problem is considered:

         -∆u(x) = 4x 3 -4x 2 -6x + 2 exp -x 2 , ∀x ∈ (0, 1) u(0) = 1, u(1) = 0. (IV.1)
The analytical solution is u ex (x) = (1 -x) exp -x 2 . Numerical errors and convergence order in L 2 -norm are computed for NIPG, SIPG and IIPG methods with linear to cubic polynomial approximations. Penalty values are such that = σ D = 1 for SIPG with p = 3, NIPG and IIPG but σ I = σ D = 2 for SIPG with p < 3. Uniform meshes of size h are considered. The numerical error is defined as:

e h = u ex -u h . (IV.2)
The L 2 -norm is:

∥e h ∥ L 2 (0,1) = 1 0 e h 2 (x) dx 1 2 . (IV.3)
Orders of convergence are obtained numerically by computing log(∥e

h 1 ∥/∥e h 2 ∥) log(h 1 /h 2 )
. Theoretical convergence rates of primal DG method for uniform meshes can be derived. In one dimension , they are given for example in [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF][START_REF] Dolejší | Discontinuous Galerkin Method[END_REF], and reported in Tab. IV. The numerical experiments are in good agreement with theoretical order and with those made in [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF] on the same problem. This demonstrates the capability to reach high order approximation using DG methods. The code is written to work either from monomial basis or Lagrangian basis functions (classic finite element basis functions). Results do not differ in both case which is in accordance with DG methods because basis functions of S p E n h have their support contained in one single element. This raises the question of the existence of a set of polynomial basis functions more suitable for a given problem or a specific use. A discussion about this topic can be found in [START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF]. In particular, modal basis like monomial basis are preferred to reach high order even though degrees of freedom loose their physical meaning. Orthogonality property is also often desired to diagonalize the mass matrix for instance.

Penalty values were varied during tests on the problem Eq. (IV.1). One observation is that there is a well-known threshold value below which instability happens, see [START_REF] Epshteyn | Estimation of penalty parameters for symmetric interior penalty Galerkin methods[END_REF] for details. Moreover, another behaviour was observed: some spurious effects occur for great penalty values beyond a threshold value. Numerical results for SIPG method with linear approximation on a grid with 32 elements are presented in Fig. IV.1 for two cases with different penalty values. As shown in Fig. IV.1b, the numerical solution becomes unstable. One possible explanation is that condition number of the system worsen the numerical scheme as stated in [START_REF] Caputo | Front Solutions of Richards' Equation[END_REF]. This aspect should be tackled in future investigations.

IV.1.2 Two-dimensional analytical tests

Now, this second experiment focuses on the property of DG methods to treat non-conforming mesh.

Only SIPG method will be treated seeing that there is no difference between the different methods in this case. High-order elements could be used but the experiment will be limited to linear approximation for brevity. This second experiment is considering the following two-dimensional problem, where (x, y) ∈ Ω = [0, 1] × [0, 1]:

-∆u(x) = 8π 2 sin(2πx) sin(2πy), in Ω, u = 0, on ∂Ω. Numerical comparison with the example given in [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF] is in good agreement even though a quadrilateral mesh is used instead of triangles. An interesting point of view about the influence of mesh nature

IV.2. HAVERKAMP'S TEST-CASE

can be found in Wirasaet et al. [START_REF] Wirasaet | A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals[END_REF]. These results illustrate that DG methods can handle easily non-conforming mesh, giving path to mesh adaptation.

IV.2 Haverkamp's test-case

From now on, simulations are carried for Richards' equation given in the problem Eq. (II.1). A series of 1D infiltration cases is considered to validate the code and assess the method.

This numerical test-case was firstly introduced by Celia et al. [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF]. It is based on an experiment from Haverkamp et al. [START_REF] Haverkamp | A Comparison of Numerical Simulation Models For One-Dimensional Infiltration[END_REF] who mention the existence of quasi-analytical solution given by Philip [START_REF] Philip | The theory of infiltration: 1. the infiltration equation and its solution[END_REF].

It was taken afterwards by others like [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF][START_REF] Manzini | Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation[END_REF][START_REF] Rathfelder | Mass conservative numerical solutions of the head-based Richards equation[END_REF][START_REF] Li | A simplified Newton Iteration Method with linear finite elements for transient unsaturated flow[END_REF][START_REF] El-Kadi | The Courant and Peclet Number criteria for the numerical solution of the Richards Equation[END_REF] and corresponds to a class of classical test cases, for example, see [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF]. Although the case is simple, it allows for understanding fundamental physics of a wetting front caused by infiltration. This is a first step to validate the code. In particular, the ability of the error indicator Eq. (III.2) to follow the front will be evaluated.

IV.3 Polmann's test-case

This test-case considers a soil from New Mexico whose hydraulic properties are provided by Polmann et al. [START_REF] Polmann | Application of Stochastic Methods to the Simulation of Large-scale Unsaturated Flow and Transport[END_REF]. A 1D vertical sample of this soil subject to downward infiltration was simulated by Celia et al. [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF], Manzini and Ferraris [START_REF] Manzini | Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation[END_REF] as well as Sochala [START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. This test-case can be challenging because, on one hand, the prescribed pressure head on both sides of the 100 cm soil column has a difference of 925 cm resulting in steep solicitation, and, on the other hand, hydraulic conductivity shows strong variations under the set of values taken by pressure head. The computational domain Ω is a rectangle (0, 20) × (0, 100) cm. The test-case is solved for pressure head ψ during T = 172800 s (48 h) with a constant time step τ = 120 s and the initial condition is ψ 0 = -1000 cm. Implicit Euler scheme is used with p = 1, σ I = σ D = 100. Computation is done for two meshes: M100 is a coarse mesh of 100 elements and M1000 is a refined mesh of 1000 elements. The test-case configuration is sketched in For the mesh M100, nonlinear iterations have difficulties to reach the requirements of stopping criteria and the solution holds spurious oscillations, in particular an undershoot ahead of the sharp wetting front as shown in Fig. IV.9. This behaviour was already noticed by Celia et al. [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] and Sochala [START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. The latter decides to eliminate this undershoot by adding a slope limiter which causes a small delay compared to the non-limited solution. In addition, the WDG method from Section III.3.1 is tried without any adaptation. The mesh M100 is used. Results in Fig. IV.10 show that the undershoot is still present but reduced to one single element. It is relevant to point out that the solution holds jumps where the wetting front is steep. They prevent oscillations to propagate and make solution catch the front steepness. The remaining undershoot depends on the wetting front localization in relation with the mesh geometry. This is due to the nonlinear nature of diffusivity and that is why it is important to associate mesh adaptation.

IV.3. POLMANN'S TEST-CASE

The extrapenalty term from Eq. (III.15) has no repercussion. Oscillations are not absorbed or reduced, no matter if it is employed solely or in conjunction with WDG method or mesh adaptation. However, convergence is affected: reaching stopping criteria is more difficult.

IV.4. VOGEL-VAN GENUCHTEN-CISLEROVA TEST-CASES

lations of the sharp wetting front. Indeed, they follow the wetting front when it hits dry regions and are reduced with refinement. This indicates insufficient discretization for the scheme to respect mass balance in this particular case where high water content variations reach water-less regions. Roughly speaking, information should propagate from upwind to downwind flow but the downwind answer is not zero in the numerical scheme because of the sharpness of the front compared to the local mesh.

IV.4 Vogel-Van Genuchten-Cislerova test-cases

Vogel et al. [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF] made three test-cases whose principle is the same: a 1D infiltration into a column of soil. After them, Sochala [START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF] used these tests to validate his code, Ginzburg et al. [START_REF] Ginzburg | Lattice Boltzmann approach to Richards' equation[END_REF] assess a lattice Boltzmann method for Richards' equation on one of them and De Maet et al. [START_REF] Maet | A fully-explicit discontinuous Galerkin hydrodynamic model for variably-saturated porous media[END_REF] demonstrate their numerical method with a similar case from the code HYDRUS [START_REF] Šimůnek | The HYDRUS-1D Software Package for Simulating the Movement of Water, Heat, and MultipleSolutes in Variably Saturated Media[END_REF]. Even if these test-cases are one-dimensional infiltration, they are interesting for two main reasons apart from being quite classical. One is because they employ other types of boundary conditions like a non-zero Neumann boundary condition or Dirichlet boundary conditions resulting in upward infiltration. The second reason is because they employ hydraulic properties based on non-standard relations, called modified Van Genuchten-Mualem relations (see Tab. I.1), including a parameter capable of affecting their shape. Vogel et al. [START_REF] Vogel | Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions[END_REF] introduce this so-called minimum capillary height ψ s to provide more flexible relations for hydraulic properties and model better the capillary zone behaviour, when the porous medium is almost fully saturated but with negative pressure head. This parameter acts as the air entry pressure ψ e but it can take smaller values, about centimetres, which corresponds to maximum pore size. Use and validity of these relations are not discussed here Ginzburg et al. [START_REF] Ginzburg | Lattice Boltzmann approach to Richards' equation[END_REF] and Sochala [START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF]. These three test-cases are interesting to evaluate the performance of nonlinear solving for the DG scheme applied to Richards' equation. Indeed, for ψ s = 0 cm, hydraulic conductivity holds strong variations when the porous medium is near saturation, illustrating what a capillary-barrier effect is. For ψ s = -2 cm, the wetting front propagation is fast due to a higher hydraulic conductivity. It has been noted that the stopping criteria may have different order of magnitude between the cases which may trouble convergence if the user gives arbitrary values. The

IV.4. VOGEL-VAN GENUCHTEN-CISLEROVA TEST-CASES

cases with ψ s = 0 cm are more demanding since hydraulic conductivity shape is steep. So, nonlinear iterations struggle to converge and adaptive time stepping is then suitable to reach convergence. It is particularly the case for the flux-controlled downward infiltration with ψ s = 0 cm. Besides, the model manifests some front instabilities for downward infiltrations with ψ s = 0 cm which confirms that something is needed when front is steep with insufficient discretization. No oscillation occurs for upward infiltration because the flux is against gravity taking values outside of the most nonlinear portion of hydraulic conductivity.

CHAPTER IV. VALIDATION AND NUMERICAL RESULTS

Some additional results are provided in Appendix B -Complementary results to look at the results of this section against the number of nonlinear iterations and the time stepsize.

IV.5 Tracy's benchmark

This test-case has an analytical solution given by Tracy [START_REF] Tracy | Clean two-and three-dimensional analytical solutions of Richards' equation for testing numerical solvers[END_REF] for 2D and 3D problems which can be used as a benchmark for Richards' equation [START_REF] Tracy | Three-dimensional analytical solutions of Richards' equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top[END_REF][START_REF] Tracy | Testing Computational Algorithms for Unsaturated Flow[END_REF]. Tracy's benchmark is particularly relevant because it is transient, has a simple parameter to vary nonlinearities for Richards' equation and holds differentiated steep regions. It can serve to check the nonlinear solver robustness, convergence properties and AMR. Then, it is very convenient to assess the solution quality and compare codes: Šolín and Kuraz [START_REF] Šolín | Solving the nonstationary Richards equation with adaptive hp-FEM[END_REF] and Dolejší et al. [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF] used it to evaluate the performance of their respective method. The treatment of initial condition is not straightforward. First, because the initial condition does not respect the top boundary condition: they are inconsistent. This issue can be troublesome for discretizations like finite elements methods [START_REF] Šolín | Solving the nonstationary Richards equation with adaptive hp-FEM[END_REF] but it is natural for DG methods where boundary conditions are usually enforced by weak penalization [START_REF] Dolejší | Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows[END_REF]. Second, because the top boundary condition is particularly stiff compared to the initial state. Therefore, the very first moments are the most interesting to simulate in order to address the time-boundary inconsistency and the early stiffness. Nonlinearities are controlled by the parameter α in Eq. (IV.5). The larger α, the faster the relative hydraulic conductivity drops to nearly zero, and so, nonlinearities are increased for Richards' equation [START_REF] Tracy | Three-dimensional analytical solutions of Richards' equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top[END_REF]. Large values of α mean greater errors in the solution unless to reduce the time step [START_REF] Tracy | Testing Computational Algorithms for Unsaturated Flow[END_REF]. It challenges the nonlinear solver so much that the Newton-Raphson method struggles to converge [START_REF] Tracy | Testing Computational Algorithms for Unsaturated Flow[END_REF]. Moreover, errors are linked to the top corners since the top boundary condition induces a steep slope at these corners. This is particularly the case for smaller values of α and transient state but errors can also lie at the bottom of the grid for largest values of α and steady state [START_REF] Tracy | Three-dimensional analytical solutions of Richards' equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top[END_REF][START_REF] Tracy | Testing Computational Algorithms for Unsaturated Flow[END_REF]. As Tracy stated, these features make this problem a good option to investigate AMR.

ψ ts = 2 1 -e αψr bγ T sin πx a e α 2 (b-y) ∞ k=1 (-1) k λ k µ k sin(λ k y)e -µ k t , (IV.8) β T = α 2 2 + π a 2 , (IV.9) γ T = α(θ s -θ r ) b , (IV.10) λ k = kπ b , (IV.11) µ k = β 2 T + λ k 2 γ , (IV.
The classic L 2 -norm and, following Rivière [START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF], Schötzau and Zhu [START_REF] Schötzau | A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations[END_REF][START_REF] Zhu | A robust a posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations[END_REF] or Dolejší et al. [START_REF] Dolejší | hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems[END_REF], the energy norm, also called the DG intrinsic norm, are introduced for a scalar function u : R d -→ R:

∥u∥ 2 L 2 (E) := E u 2 dE |||u||| 2 E(E) := ∥u∥ 2 R(E) + F ∈∂E ∥u∥ 2 J(F ) , (IV.13)
where the residual seminorm and the jump seminorm are respectively:

∥u∥ 2 R(E) := E K(u)∇u • ∇u dF = (K(u)) 1 2 ∇u 2 L 2 (E) , (IV.14) ∥u∥ 2 J(F ) := F ϱ F u 2 dF = ϱ F ∥ u ∥ 2 L 2 (F ) . (IV.15)
Errors between numerical and exact solution are computed for a time step such that:

∥e∥ X = ∥ψ ex -ψ∥ X with X = L 2 , E, R, J . (IV.16)
To measure how effective the estimation-based error indicator is, the effectivity indices for the estimationbased error indicator (III.4), the element and face residuals (III.5, III.6), and the solution jump (III.7) will be used:

I eff E := η n E |||e||| E(E) , I eff R := η n E,R 2 + η n E,F 2 1 2 ∥e∥ R(E) , I eff J := η n E,J ∥e∥ J(F ) . (IV.17)
Due to the lack of rigorous derivation, the effectivity index is not expected to represent properly the true error. The estimator efficiency is rather appreciated to set up a mesh refinement strategy.

First of all, several simulations are performed to show convergence properties. Because of symmetry, only one half vertical plan of the domain is considered to speed up the simulation. Duration is set to T = 10 -4 day to focus on small times. The computations are carried out with the 6-step BDF and time steps are set to τ = 10 -6 day. This is needed to reduce time discretization errors expecting that they will be negligible compared to space discretization errors in order to observe space convergence. Indeed, according to theoretical error estimates [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF], error in L 2 -norm and H 1 -seminorm behave such that:

∥e∥ L 2 ≈ c h h p + c τ τ q , (IV.18) ∥e∥ E ≈ c ′ h h p+1 + c ′ τ τ q , (IV.19)
where c h , c ′ h , c τ and c ′ τ are constants independent of h and τ . Numerical experiments involve four meshes of 50, 200, 400 and 3200 quadrilateral elements. Order approximation varies from one to four. Figure IV.17 shows the convergence of the error e h for the L 2 -and energy norms with respect to the number of degrees of freedom. It is interesting to note that space error are so small for the most dense mesh with p = 4 that time error saturates the convergence. 18 by varying the order of approximation and the mesh size. Since they do not represent properly the true error, observations are difficult to relate with the true error. However, some remarks can be drawn. Firstly, these diagrams show the convergence up to a certain extent. Secondly, one can see that the volume contribution -the residual indicator from Eq. (III.5) -behaves quite differently compared to the surface contributions -the flux and jump indicators from Eq. (III.5) and (III.6). The convergence is globally maximised with p-refinement for the surface contributions while h-refinement seems to improve convergence better for the volume contribution. This observation should be assessed in more details but could drive the hp-decision making.

Error estimation is evaluated in terms of ability to drive the mesh adaptation. The distribution of ∥e∥ L 2 (E) with no mesh adaptation is given in

IV.6 Comparison with Vauclin's studies

Vauclin, Vachaud and Khanji made a series of laboratory experiments in the 1970's whose description can be found in [START_REF] Vachaud | Étude expérimentale des transferts bidimensionnels dans la zone non saturée application à l'étude du drainage d'une nappe à surface libre[END_REF][START_REF] Vauclin | Two dimensional numerical analysis of transient water transfer in saturated-unsaturated soils[END_REF][START_REF] Vachaud | Comments on 'A numerical model based on coupled one-dimensional Richards and Boussinesq equations[END_REF][START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF]. They imply water table recharge and drainage in a slab of sandy soil.

IV.6.1 Water table recharge

Vauclin et al. [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF] experimented and simulated water flow recharge through a slab of soil. It consists in a box of 6 m by 2 m whose only one half is simulated because of symmetry. Sides are impervious on the left, at the top for x > 50 cm and at the bottom: ∇h • n = 0. A constant flux is prescribed on the top for x ≤ 50 cm: -K(ψ)∇h • n = 14.8 cm/h. The water level is kept constant in the ditch on the right for z ≤ 65 cm: h = 65 cm. The remaining boundary on the right for z > 65 cm accounts for a seepage boundary condition. Initial state is at hydrostatic equilibrium with water table at z = 65 m.

Steady state

The following steady-state Richards' equation is considered: Water table, initially at z = 65 cm, raises because of the recharge which moistens the unsaturated zone. Compared to experiment [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF], distribution of flux as well as positions of water table and capillary fringe are in good agreement even though a small shift is to notice. Distribution of hydraulic head is consistent with experiment within the saturated zone and under the recharge zone. However, on the top left-hand area, hydraulic head is higher than one measured and obtained by Vauclin et al. [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF].

-∇ • (K(h -z)∇(h + z)) = 0. (IV.

Transient state

The complete simulation of water table recharge by Vauclin et al. [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF] was used by many other studies to assess their method, see [START_REF] Dogan | Saturated-Unsaturated 3D Groundwater Model. II: Verification and Application[END_REF][START_REF] Clement | A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow[END_REF][START_REF] Shen | A process-based, distributed hydrologic model based on a largescale method for surface-subsurface coupling[END_REF][START_REF] Twarakavi | Evaluating Interactions between Groundwater and Vadose Zone Using the HYDRUS-Based Flow Package for MODFLOW[END_REF][START_REF] Xu | Integration of SWAP and MODFLOW-2000 for modeling groundwater dynamics in shallow water table areas[END_REF][START_REF] Zha | A generalized Ross method for two-and three-dimensional variably saturated flow[END_REF]. The validation of the code MODFLOW is partly done with this experiment dataset [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF]. One can notice that the majority of these simulations does not take into account the seepage boundary condition and prescribe an impervious wall instead which is in contradiction with the experimental set-up and numerical studies of Vauclin et al. In [START_REF] Clement | A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow[END_REF], the authors argue that a no-flow boundary can be used because the problem shows experimentally and numerically a steady-state solution without the development of a significant seepage face.

The simulation is carried for Richards' equation given in the problem Eq. (II.1) during T = 8 h. Slightly different parameters are changed compared to the steady-state simulation. The mesh is a regular grid ∆x = ∆z = 5 cm. Monomial basis is used with p = 2. The penalty parameters are σ I = σ D = 100. The implicit Euler scheme is used with adaptive time stepping, starting with τ = 0.003 h and not exceeding 0.1 h. The transient position of the water table is reported every 2 hours, as shown in Fig. IV.26, and compared with experimental results taken from [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF]. The results obtained from the numerical model closely agree with the experimental profile. The growth of water table mound is well simulated. Small discrepancies are observed in the middle of water table which is underestimated by numerical results. This underestimation is observed by other study using different models [START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF][START_REF] Twarakavi | Evaluating Interactions between Groundwater and Vadose Zone Using the HYDRUS-Based Flow Package for MODFLOW[END_REF] and may be explained by the sandy soil which is non-perfectly isotropic and homogeneous as stated by hydraulic properties. Indeed, the recharge flux is redistributed horizontally by diffusion in the unsaturated zone before reaching the water table. Then, the unsaturated zone stores a portion of water inflow which moistens dry regions and does not contribute to the elevation of water table in this region. Similar observations are made in [START_REF] Shen | A process-based, distributed hydrologic model based on a largescale method for surface-subsurface coupling[END_REF][START_REF] Xu | Integration of SWAP and MODFLOW-2000 for modeling groundwater dynamics in shallow water table areas[END_REF]. In addition, it may be difficult to identify experimentally the exact position of the water table because of the capillary fringe. One can see that seepage face length is very small and so, it may be disregarded for this experiment as stated in [START_REF] Clement | A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow[END_REF]. . This shows that it is difficult to evaluate the attainment of a steady state due to the slow dynamics of porous media flow. In addition, one should note that the use of different basis functions and approximation order for quadrangles are determinant for the solution quality. Indeed, quadratic monomial basis functions are necessary to get equivalent results to linear Lagrangian basis functions. This is explained by the tensored term.

IV.6.2 Water table drainage

Lowering

Vauclin et al. made a drainage experiment of a slab of soil in which experimental details and positions of water table can be found in [START_REF] Vauclin | Two dimensional numerical analysis of transient water transfer in saturated-unsaturated soils[END_REF] and [START_REF] Vachaud | Comments on 'A numerical model based on coupled one-dimensional Richards and Boussinesq equations[END_REF]. A water table lowering occurs after an instantaneous properly. Another reason is the determination of hydraulic properties which was done by correlation from measurements with some fairly important dispersion, due to natural heterogeneity/anisotropy and by neglecting hysteresis [START_REF] Vauclin | Two dimensional numerical analysis of transient water transfer in saturated-unsaturated soils[END_REF]. This would explain some discrepancies. In every instance, it is not linked to discretization because results does not change whatever the mesh and order approximation are chosen. Other numerical studies based on this experiment are reported [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Vauclin | Two dimensional numerical analysis of transient water transfer in saturated-unsaturated soils[END_REF][START_REF] Clement | A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow[END_REF]. They also struggle to match water table profiles near the seepage.

Drawdown

Vachaud et al. made a drainage experiment of a slab of soil in which experimental details and positions of water table can be found in [START_REF] Vachaud | Étude expérimentale des transferts bidimensionnels dans la zone non saturée application à l'étude du drainage d'une nappe à surface libre[END_REF] and [START_REF] Vachaud | Comments on 'A numerical model based on coupled one-dimensional Richards and Boussinesq equations[END_REF]. A water table drawdown occurs after an instantaneous drop of the imposed hydraulic head. The box is 200 cm × 300 cm. Hydraulic properties are taken from Vachaud's relations (see Tab. together with the experimental data from [START_REF] Vachaud | Étude expérimentale des transferts bidimensionnels dans la zone non saturée application à l'étude du drainage d'une nappe à surface libre[END_REF]. Results show that the model is able to reproduce satisfactorily the transient water table dynamics. However, discrepancies are observed between numerical and experimental profiles, in particular near the seepage boundary (left) with a small delay for numerical results. The same explanation than the previous can be made.

IV.7 Assessment of La Verne dam wetting IV.7.1 Experimental configuration

The model is here used to simulate the full-scale wetting of a multi-materials dam: the La Verne dam. La Verne dam was constructed in 1991 to supply Gulf of Saint-Tropez (south of France) with drinkable water. The dam is 42 m high and peaks at 90 m above sea level. Its width ranges from -102.3 m to 110.5 m. The reservoir is on the left in Fig. IV. [START_REF] Li | Local discontinuous Galerkin approximations to Richards' equation[END_REF]. The genuine La Verne dam inner section shows a complicated heterogeneous material patterns, partially described in [START_REF] Bonelli | Résolution auto-adaptative par éléments finis de problèmes de diffusion fortement non linéaires[END_REF][START_REF] Fleureau | Ministère de la Recherche et de la Technologie/CNRS. Rapport Scientifique du GRECO "Géomatériaux[END_REF]. The dam is an earthfilled embankment dam. A clay core allows for impermeability while outer zones are semi-permeable alluvial shells. The dam is protected by rip-rap upstream. Filters and drains of very permeable gravellike materials secure the dam inner saturation. The dam foundation lies on an impervious rock stratum thanks to concrete injections. A fine loam layer with vegetation covers the downstream dam part. The filling of La Verne dam was controlled with instruments [START_REF] Fleureau | Ministère de la Recherche et de la Technologie/CNRS. Rapport Scientifique du GRECO "Géomatériaux[END_REF] during 40 days, which provides a unique and challenging dataset to test the model. In particular, the reservoir height evolution is available as well as three groundwater hydraulic head measurements from the core, see ). Since comparison focusses on the core of the dam, it is assumed that outer high permeable and/or fine layers materials do not have a significant impact on inner groundwater flows and saturation. The downstream thin loam layer, the filter-isolated toe drain and the lower submerged rip-rap are therefore not represented by the simulation and assimilated to the adjacent materials (see Tab. IV.3 for description of numerical dam structure). A Dirichlet boundary condition for hydraulic head is prescribed at the downstream boundary to model remaining water in the filter which is diverted to exit horizontally. The upper rip-rap and the cofferdam are simulated as such to assess the method robustness owing to their direct contact with the upstream dynamic boundary condition (forcing) and their hydraulic properties are steep. Rock/concrete foundations are supposed to be perfectly impervious which means a zero-valued Neumann boundary condition can be prescribed.

Table IV.3 -Materials of La Verne dam

The simulation of La Verne dam case is a very challenging benchmark for numerical models, combining heterogeneous materials, steep constitutive laws and dynamic boundary conditions. However, despite of its interest, the La Verne dam benchmark remains only partly documented by in-situ instruments, precluding a comprehensive quantitative confrontation between observations and model. In particular, the hydraulic properties, which have a drastic effect on flow dynamics, remain approximative due to the absence of direct in-situ characterization. In addition, note that no initial in-situ values for water table elevation and saturation are available inside the dam, which can also strongly affect the subsequent evolution. For the present simulation, the initial water table is imposed at 67 m to fit the initial values of experimental hydraulic head. Finally, Bonelli et al. pointed out that hydromechanical coupling should be considered to be fully consistent with the real case, see [START_REF] Fleureau | Ministère de la Recherche et de la Technologie/CNRS. Rapport Scientifique du GRECO "Géomatériaux[END_REF]. It is recalled that initial reservoir free surface and water table elevation are at 57 and 67 m, respectively. Note that the water table is not displayed in Two major stiff periods for the simulation can be noticed in Fig.

IV.7.2 Numerical results

IV.37: one is approximately between 17 and 21 days and the other between 29 and 35 days. They match a decrease of the time stepsize showing rapid variations. This means nonlinear iterative process struggles to converge. After growing a lot at the very beginning, the number of elements stabilizes around 4100 elements during the simulation despite the global error estimate is increasing for the two stiff periods. This is because mesh refinement is unable to control errors where it is needed because the maximum refinement level has already been reached in these regions. Usually, it comes with the development of overshoots/undershoots which are discussed later. andc). This reveals that, even if both m2-and m3-materials show the same hydraulic conductivity at saturation, the difference in constitutive laws induces a radically different dynamics of the capillary fringe. At S4, the head remains nearly constant during the adjustment phase while the water content is very low. The m3-material is here nearly desaturated, well above a thin capillary fringe. By contrast, S5 is within the large capillary fringe associated to the m2-material, providing a much higher water content. The hydraulic head shows first a small decay, probably due to the drain proximity, before starting to increase in response to the reservoir filling. Around day 14, both S4 and S5 sensors show a simultaneous regular rising, slightly lagged from the reservoir level. A strong jump is observed at S4 (Fig. When compared to hydraulic head experimental data (see Fig. IV.38a), the overall time evolution and order of magnitude are well represented. Some discrepancies are observed, in particular the late increase of head which is experimentally observed on each sensor around 30 days, which tends to be underestimated by the model. However, recalling the lack of experimental control evoked before, more detailed comparison should be made with caution. For the sake of comparison, the previous numerical results presented by Bonelli et al. [START_REF] Bonelli | Résolution auto-adaptative par éléments finis de problèmes de diffusion fortement non linéaires[END_REF] are recalled in Fig. IV.38 in green lines. While the overall time evolution is rather similar between models, a better agreement with experimental data is obtained with the present simulation.

IV.7.3 Discussion and investigation

The proposed modelling strategy for Richard's equation showed very satisfactory performance in reproducing analytical and laboratory test cases and in simulating complex full-scale experiment. In this latter case, the model is able to capture automatically wetting fronts which are moving dynamically thanks to AMR. In particular, the fronts associated with heterogeneity are well resolved by the WDG technique. This is illustrated by the discontinuous water content distribution in Fig. IV.40a. Moreover, the simulation is robust since it is able to compute completely without user intervention and handle many features: degeneracy for saturated/unsaturated zones, dynamic forcing boundary condition and seepage. However, the simulation struggles to capture wetting fronts associated with nonlinear varying diffusivity resulting in severe overshoots/undershoots for the transition between saturated and unsaturated zones in very permeable materials. To have a better idea of the situation, Fig.

IV.39 shows an elevated-view of hydraulic head at t = 30 days. The view illustrates the jumps in the solution introduced by the WDG method, the adapted mesh and the aforementioned overshoots.

Further insight on this behaviour can be discussed using the Péclet number framework. Richards' equation is an elliptic-parabolic equation predominantly diffusive, it shares properties with convectiondiffusion equation and can behave like them. For example, sharp wetting fronts and internal layers can be reinterpreted within this context. The Péclet number represents the ratio of the rate of advection to the rate of diffusion. This dimensionless number is useful to analyse locally the nature of the flow regime and determine whether advection or diffusion is dominant. Following ideas for subsurface flow from Huyakorn and Pinder [START_REF] Huyakorn | The Computational Methods in Subsurface Flow[END_REF] or remarks from [START_REF] Gresho | Don't suppress the wiggles-They're telling you something![END_REF], the oscillations in the solution for Galerkin finite element method are linked to the Péclet (and Courant) number. Sharp fronts carry short wavelength harmonics of the Fourier series not recognised by the numerical scheme for the current discretization [START_REF] Pinder | Finite Element Simulation in Surface and Subsurface Hydrology[END_REF]. El-Kadi and Ling [START_REF] El-Kadi | The Courant and Peclet Number criteria for the numerical solution of the Richards Equation[END_REF] derive a Péclet number and a Courant number for Richards' equation in 1D to estimate the size of spatial and temporal local steps for acceptable accuracy of the numerical simulation. However, the Péclet number definition is slightly different from Eq. (IV. [START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF]. A recent study tries to select suitable approximation for Richards' equation based on the Péclet number [START_REF] Yao | Time Compression Approximation Relationship for Infiltration in the Presence of a Shallow Water Table: Evaluating the Role of Péclet Number[END_REF].

For this study, the Péclet number is extracted from the saturation-based formulation of Richards' equation given in Eq. (I.49). The latter is similar to a convection-diffusion equation with the water content θ as single unknown. This is a Fokker-Planck equation in essence [START_REF] Philip | Theory of Infiltration[END_REF] whose form is useful to study stability and investigate travelling front solutions of wetting and drying fronts [START_REF] Caputo | Front Solutions of Richards' Equation[END_REF]. This formulation is undefined in saturated zones because hydraulic diffusivity D(ψ) = K(ψ) C(ψ) tending to infinity but this zone is not of interest for the Péclet number since Richards' equation is purely diffusive. Moreover, it is assumed that θ ̸ = 0. This assumption is not restrictive because if water content drops to zero, nothing happens and, in practice, there is often a residual water content. The total flux can be separated into a diffusive flux defined by hydraulic diffusivity and driven by capillarity, However, as soon as the specified maximum level of refinement is reached within a block, the error is increasing freely. It would be interesting to attempt different process for the selection of elements to be refined/coarsened.

The simulation of La Verne dam filling shows many difficult points which lead to two unwanted effects. The first one is the development of non-physical oscillations around steep wetting fronts associated with nonlinear varying diffusivity which affect solution accuracy. The other one is nonlinear convergence possible only with very small time steps so that the calculation duration is high. These challenges illustrate numerical issues of Richards' equation for heterogeneous media with high-varying diffusivity. The improvements offered by the augmented simulation support the idea that a series of heuristic tools (threshold values, maximum refinement level, refinement frequency, adaptive time steps) should be investigated to improve accuracy while maintaining cost-effectiveness.

Summary

The work done in this chapter has shown validations for the Rivage code as well as numerical results supporting the proposed solving strategy. A series of 1D infiltration in a soil column illustrates behaviour of Richards' equation solution according to different relations for the hydraulic properties. It has been seen that wetting fronts induced by the hydraulic properties may be sharp and move fast causing spurious oscillations which are not physical. Global mass balance is respected but element-wise mass balance should be assessed more when oscillations occur.

To deal with these instabilities, the proposed strategy is promising: AMR is capturing the moving fronts while the WDG framework accounts for jumps in the solution to better approximate the fronts. Thanks to Tracy's benchmark, the error indicator based on error estimation is chosen for its ability to drive mesh adaptation in synergy with the weighted approximation. It produces cost-effective mesh compared to the gradient-based error indicator. Besides, the subpart od the estimation-based error indicator suggest a way to make hp-adaptation in future studies. Finally, Rivage code is examined for 2D unsaturated porous media flow with laboratory and large-scale experiments. First of all, a set of water table recharge and drainage validate the model for saturated-unsaturated conditions and complex flow dynamics involving the seepage boundary condition. In particular, the adaptive time stepping is mandatory to get convergence robustness. Last, experimental data from La Verne dam wetting after reservoir impoundment provide a challenging case to simulate. Indeed, the earth-filled dam has a multi-material structure and the reservoir filling induces complex dynamics with multiple wetting fronts, dynamic forcing and seepage. The proposed strategy is very satisfactory to reproduce experimental data and computational time stays reasonable. Moreover, sharp internal layers due to material heterogeneity are well resolved. However, the simulation struggles to capture moving wetting fronts associated with nonlinear varying diffusivity resulting in severe overshoots/undershoots. The Péclet number gives interesting information: it happens at transition between saturated and unsaturated zones in very permeable materials. A more refined simulation of La Verne dam wetting suggests that the strategy can be improved by investigating heuristic tools on which adaptation relies. G roundwater dynamics in sandy beaches is a matter of interest since beach represents interface between land and open sea and accounts for a wide range of processes: biogeochemical cycles, sediment transport, diffusion of dissolved materials, fresh/salt water exchanges between ocean and coastal aquifer, etc. It is therefore directly involved in the anthropogenic issues in nearshore areas in the global context of climate change and growing urbanization: beach erosion, submersion, aquifer salinization, etc. Many laboratory and field studies have been devoted to understand groundwater flows inside sedimentary beaches [START_REF] Steenhauer | Subsurface processes generated by bore-driven swash on coarse-grained beaches[END_REF][START_REF] Heiss | Coupled surface-subsurface hydrologic measurements reveal infiltration, recharge, and discharge dynamics across the swash zone of a sandy beach[END_REF]. Observational approaches are however still limited to resolve the full spatio-temporal variability of governing variables, in particular the fields of head, saturation and flux. Numerical modelling stands out as a promising prospect, allowing to evaluate more accurately water table fluctuations, flow velocities or infiltrations/exfiltrations during swash events which are for now very difficult to measure directly. Nevertheless, up to now, few models have been devoted to the simulation of wave-driven groundwater flows and their scope remains limited [START_REF] Li | BeachWin: modelling groundwater effects on swash sediment transport and beach profile changes[END_REF][START_REF] Malott | Dynamic groundwater flows and geochemistry in a sandy nearshore aquifer over a wave event[END_REF].

V

CHAPTER V. APPLICATIONS TO SWASH GROUNDWATER DYNAMICS IN SANDY BEACHES

The main purpose of this chapter is to present the application of Rivage code and models described in the previous chapters for beach groundwater in the swash zone. The action of waves is modelled by a weak coupling with the dynamic forcing boundary condition described in Section I.5. First of all, a synthetic description of the swash zone is given highlighting the roles of waves on groundwater. An application aims to see if the code is able to simulate the main groundwater features on an idealized beach case. Additional cases are under consideration to look further into swash groundwater dynamics. They are under construction and analysis.

V.1 Wave-forced beach groundwater dynamics

V.1.1 Waves hydrodynamic forcing

Several large families of surface waves can be considered in the coastal zone. Roughly speaking, it can be distinguished:

• the gravity waves of period included in 3 and 20-22 s. There are two sub-categories: the swell for periods higher than 6 s, the sea of the wind below;

• the tide and its harmonics, with typically diurnal or semi-diurnal periods;

• in between, it is the domain of long waves, which include in particular seiche and infragravity waves (between 20 and 500 s);

• surge waves linked to variations in atmospheric pressure, with day to week periods.

Swash is the up and down propagation of bores on the beach face after an incoming wave has broken, see Fig. V.1. A swash event shows generally two basic phases: uprush (onshore flow) and backwash (offshore flow). An example is given by Fig. V.2. Generally, uprush has higher velocity and shorter duration than backwash. Uprush starts with great velocities diminishing until its end, whereas backwash velocity increase towards its end. The swash zone extends from the lower level reached by backwash (run-down) to the upper level reached by uprush (run-up). The swash zone hydrodynamics involves various processes characterized by turbulence, sediment transport and the interaction of the sheet of water with the porous beach. The dynamics of the interaction between surface waves and the beach depends on the forcing conditions (type of waves: period, amplitude) the slope of the beach and the grain size of the sediment. In nature, the beach morphology will adapt to the hydrodynamic forcing conditions and the grain size, whose spatio-temporal distribution can also adapt to the forcing. For the present applications, it will first be assumed that grain size is homogeneous (constant and isotropic porosity). There is therefore a link between granulometry, forcing and slope. This allows to define two characteristic cases: dissipative beach and reflective beach, see Fig. Dissipative beaches are generally characterised by a fine sediment (sand or loamy sand) and gentle slopes. On these beaches, it is for excepted very small waves, offshore-generated waves break well before the shore (spilling waves). Throughout the surf zone, these waves dissipate a part of their energy

CHAPTER V. APPLICATIONS TO SWASH GROUNDWATER DYNAMICS IN SANDY BEACHES

(friction, turbulence) and transfer it to longer waves, the so-called infragravity waves. The more dissipative the conditions are (gentle slope, large waves), the greater the dissipation and transfers are. Close to the shore, the sea state combines a variable amount of remaining post-breaking bore-shaped gravity waves and much longer infragravity waves, depending on the beach profile. An example is illustrated in Fig. V.4a. Within the beach, the velocities are about six orders of magnitude lower than the free surface velocities (1/10-1/100 mm/s), with mainly pressure waves that propagate. Saturation is highly variable and linked to sand properties (porosity) and local pressure field (capillary processes, formation of meniscuses, etc.). Then, large seepage face and capillary fringe may develop. The porous medium acts as a low-pass filter: the infragravity waves will have much more influence (deeper, stronger) than other waves.

Reflective beaches are generally characterized by coarser sediments, such as coarse sand or gravel, and steep slopes. There is little dissipation and energy transfer to the low frequencies (weak infragravity energy waves). The wave breaking occurs close to the shore, often forming plunging or surging breakers. It results in a thin tongue of turbulent water sheet whose tip is propagating rapidly with a lot of sediment transport. The phenomenon is analogous to a dam-break flow over a dry bed. An example is illustrated in Fig. V.4b. Due to much higher porosity than sandy beaches, there is a significant infiltration of water. Groundwater flows in coarse-grained beaches are fast, and the saturation is binary: either 100% or nothing.

V.1.2 Issues on groundwater beach dynamics

Beach groundwater dynamics has received a growing attention during the last two decades due to its importance in a number of physical, biological and chemical processes. The determination of groundwater transport through beaches or coastal barriers remains a key challenge in the prediction of salinity, contaminants and nutrients exchanges between the open sea, the lagoon and the aquifer which in turn control crucial biogeochemical cycles, see e.g. [START_REF] Robinson | Driving mechanisms for groundwater flow and salt transport in a subterranean estuary[END_REF][START_REF] Burnett | Quantifying submarine groundwater discharge in the coastal zone via multiple methods[END_REF][START_REF] Anschutz | Tidal sands as biogeochemical reactors[END_REF][START_REF] Sawyer | Stratigraphic controls on fluid and solute fluxes across the sediment-water interface of an estuary[END_REF][START_REF] Charbonnier | Aerobic respiration in a highenergy sandy beach[END_REF][START_REF] Mcallister | Dynamic hydrologic and biogeochemical processes drive microbially enhanced iron and sulfur cycling within the intertidal mixing zone of a beach aquifer[END_REF][START_REF] Heiss | Physical Controls on Biogeochemical Processes in Intertidal Zones of Beach Aquifers[END_REF]. The beach groundwater circulation is mainly controlled by the combination of the cross-barrier gradients due to tides or large-scale fluctuations of the mean water level and the wave forcing at the shoreline, including swell and long

V.1. WAVE-FORCED BEACH GROUNDWATER DYNAMICS

infragravity waves [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF][START_REF] Sous | Groundwater fluxes within sandy beaches: swash-driven flow vs cross-barrier gradients[END_REF][START_REF] Nielsen | Tidal dynamics of the water table in beaches[END_REF][START_REF] Bertin | Infragravity waves: From driving mechanisms to impacts[END_REF]. Furthermore, at smaller scale, the percolation flows in the swash zone have early been suspected to affect swash zone sediment transport and beach morphodynamics. The first effect is the loss of volume of the swash tongue, only active on coarse-grained beach. The two other processes involved in sediment transport modification by in/exfiltration flows through the beachface are, on one hand, the modification of the boundary layer shear stress (the "ventilated bed" effect [START_REF] Chen | Velocity Distribution of Turbulent Open-Channel Flow with Bed Suction[END_REF][START_REF] Conley | Ventilated oscillatory boundary layers[END_REF][START_REF] Corvaro | Flow dynamics on a porous medium[END_REF][START_REF] Corvaro | Fluid-particle interaction and generation of coherent structures over permeable beds: an experimental analysis[END_REF]) and, on the other hand, the change of the relative weight of sediment. Significant research efforts have been engaged to estimate and model the net effect of these processes on sediment dynamics [START_REF] Li | Wave-induced beach groundwater flow[END_REF][START_REF] Li | BeachWin: modelling groundwater effects on swash sediment transport and beach profile changes[END_REF][START_REF] Turner | Swash infiltration-exfiltration and sediment transport[END_REF][START_REF] Butt | The influence of swash infiltration-exfiltration on beach face sediment transport: onshore or offshore?[END_REF][START_REF] Masselink | The role of swash infiltration in determining the beachface gradient: a numerical study[END_REF][START_REF] Karambas | Modelling of Infiltration-Exfiltration Effects of Cross-Shore Sediment Transport in the Swash Zone[END_REF][START_REF] Austin | Swash-groundwater interaction on a steep gravel beach[END_REF][START_REF] Bakhtyar | Wave-induced water table fluctuations, sediment transport and beach profile change: Modeling and comparison with large-scale laboratory experiments[END_REF][START_REF] Masselink | Large-scale laboratory investigation into the effect of varying backbarrier lagoon water levels on gravel beach morphology and swash zone sediment transport[END_REF][START_REF] Chardón-Maldonado | Advances in swash-zone research: Small-scale hydrodynamic and sediment transport processes[END_REF][START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF]. Overall, while the impact is major for coarse-grained beaches, it is now generally acknowledged that the ex/infiltration effects on sediment transport for fine sand beach are overpowered by the other physical processes acting in the swash zone morphodynamics [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Sous | Swash-groundwater dynamics in a sandy beach laboratory experiment[END_REF]. This is in line with the mixed success met by beach drainage systems in trying to alter beach morphodynamics by controlling groundwater head [START_REF] Horn | Measurements and modelling of beach groundwater flow in the swash-zone: a review[END_REF][START_REF] Turner | Beach Dewatering as a 'Soft' Engineering Solution to Coastal Erosion-A History and Critical Review[END_REF].

Owing to the difficulty to measure experimentally the groundwater parameters, and in particular to monitor their spatio-temporal dynamics over a relevant area, a series of numerical approaches have been proposed during the last few decades. First models based on Darcy's theory and purely horizontal groundwater dynamics (Dupuit-Forchheimer assumption) [START_REF] Nielsen | Tidal dynamics of the water table in beaches[END_REF][START_REF] Kang | Watertable Dynamics in Coastal Areas[END_REF] allowed to describe the tidal dynamics. Similar assumptions of purely horizontal flows were made in the coupled approach using Boussinesq model for wave propagation and Darcy-Forchheimer within the porous medium [START_REF] Karambas | Modelling of Infiltration-Exfiltration Effects of Cross-Shore Sediment Transport in the Swash Zone[END_REF][START_REF] Karambas | Prediction of sediment transport in the swash-zone by using a nonlinear wave model[END_REF]. Finer vertically-resolving model have been proposed based on the Darcy-related 2D Laplace law to model groundwater dynamics forced by tides and waves [START_REF] Li | Wave-induced beach groundwater flow[END_REF][START_REF] Li | Numerical modelling of tide-induced beach water table fluctuations[END_REF] and further refined to describe in/exfiltration bed exchanges [START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF]. The groundwater circulation cell highlighted by coupled VOF-RANS model [START_REF] Bakhtyar | Wave-induced water table fluctuations, sediment transport and beach profile change: Modeling and comparison with large-scale laboratory experiments[END_REF] has been later confirmed by laboratory [START_REF] Sous | Swash-groundwater dynamics in a sandy beach laboratory experiment[END_REF] and field [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF] experiments. In coarsegrained sand or gravel beaches, the Darcy theory is still used but extended beyond its laminar theoretical framework with the Darcy-Forchheimer approximation [START_REF] Van Gent | The modelling of wave action on and in coastal structures[END_REF][START_REF] Steenhauer | Numerical model of swash motion and air entrapment within coarse-grained beaches[END_REF][START_REF] Desombre | SIMULTANEOUS SURFACE AND SUBSURFACE AIR AND WATER FLOWS MODELLING IN THE SWASH ZONE[END_REF]. Quasi-3D models have been proposed, relying on an a priori parabolic profile of the non-hydrostatic groundwater pressure field [START_REF] Mccall | MODELLING OVERWASH AND INFILTRATION ON GRAVEL BARRIERS[END_REF][START_REF] Mccall | Modelling storm hydrodynamics on gravel beaches with XBeach-G[END_REF].

Although significant effort has been deployed during the last two decades in the understanding and the modelling of beach groundwater dynamics, most of the studies have been (i) based on Darcy-Forchheimer approximation, i.e. totally ignoring the partial saturation and capillarity effect which are essential in fine-grained sand beaches [START_REF] Heiss | Coupled surface-subsurface hydrologic measurements reveal infiltration, recharge, and discharge dynamics across the swash zone of a sandy beach[END_REF] and (ii) mainly focused on horizontal dynamics, although detailed field measurements demonstrated the presence of vertical head gradients and related circulation cells in sandy beaches [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF]. Aiming to tackle these two major challenges in the fine sand beach context, the present study aims to model wave-resolved variably-saturated groundwater flow based on Richards' equation. Generalized Richards' equation are generally used to model surface/subsurface flow in hydrological large-scale context (see e.g. [START_REF] Weill | A generalized Richards equation for surface/subsurface flow modelling[END_REF][START_REF] Pachepsky | Generalized Richards' equation to simulate water transport in unsaturated soils[END_REF]), but very few attempts have been made to implement Richards' equation in wave-forced environments [START_REF] Caviedes-Voullième | Numerical simulation of groundwatersurface interactions by external coupling of the 3D Richards equation and the full 2D shallowwater equations[END_REF].

V.1.3 Beach response to a single swash event

Sedimentary beaches are considered as a porous medium in which there is a water table: the beach water table. Waves are covering and uncovering alternately the beach whose boundary is mobile, porous and characterized by complex flows of various natures which affect groundwater. That is why grasping groundwater dynamics as a whole in the swash zone is difficult. Laboratory experiments and field studies have revealed a typical circulation pattern during the swash cycle illustrated in Fig. V.5. The flow is mainly directed offshore, with inflow in the upper part of the swash zone and exfiltration in the lower part [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF][START_REF] Sous | Swash-groundwater dynamics in a sandy beach laboratory experiment[END_REF]. However, such features are expected to strongly depend on the saturation state of the beach as well as the duration and location of uprush/backwash phase, with significant V.2. IDEALIZED BEACH equations including non-hydrostatic pressure. The wave-resolving model is used here in a two-layer configuration, all parameters being kept by default [START_REF] Zijlema | SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters[END_REF]. The space and time resolved bed pressure are extracted in order to force Richards' equation by a space and time dependant Dirichlet boundary condition for hydraulic head. Data are interpolated linearly from the space-time grid of SWASH to the space-time grid of Rivage. The swash tip is treated specifically by extrapolation to correspond to the closest discretization of Rivage. To complete this approach, a so-called seepage boundary condition [START_REF] Scudeler | Examination of the seepage face boundary condition in subsurface and coupled surface/subsurface hydrological models[END_REF] is prescribed wherever there are dry conditions, beyond the swash tip. In this way, the model can capture seepage faces, between surface water and a higher phreatic surface. The seaward seepage flow along beachface, as water is discharging, does not influence surface water. This forced coupling corresponds to a dynamic forcing boundary condition described in Section I.5.

V.2 Idealized beach

In the present study, an idealized long wave swash event is tested to evaluate the ability of the model to simulate groundwater response to swash cycles. To this end, a weak coupling is considered: waveresolving bed pressure data provided by the open-source SWASH code [START_REF] Zijlema | SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters[END_REF] is used as a beachface boundary condition in the Rivage code. Qualitative comparisons are carried out between numerical results from the model and observations from laboratory and experimental measurements found in the literature [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF][START_REF] Li | Wave-induced beach groundwater flow[END_REF][START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF]. Distinctive features of the unconfined coastal aquifer during swash cycles are selected and commented. In particular, the analysis focuses on the evolution of seepage location across the beachface as well as global and local pattern of pressure and velocity field. The goal of this study is to present a first case handled by a promising numerical tool. The findings will assist future studies to quantify effects of swash on groundwater flow in sandy beaches for more realistic cases.

The beach considered is a straight slope of ratio 1:35 which spans over 105 meters. These types of waves have been observed to be a major driver of beach groundwater dynamics of the Rousty microtidal dissipative beach [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF]. This results in two swash events on the beachface. Simulated time lasts 120 s.

An overview of groundwater features during a swash cycle is presented. Then, results focus on uprush and backwash events. Some selected snapshots from the simulation show interesting observations for the idealized beach case. Surface water and water table are depicted by blue and white solid lines respectively. Hydraulic head contours are displayed together with velocities vectors. Throughout the simulation, a nearly hydrostatic state (no head gradients) is observed in the upper beach area. Moreover, flow divides into seaward and landward components near the uprush limit. Velocities magnitude is rather small everywhere, about 1 cm/h, except for uprush infiltration at the top where it reaches up to 160 cm/h. Around exit point of water table, a local overpressure is observed. It is probably caused by numerical artefacts due to the lack of mesh resolution for the seepage boundary condition. An overall pattern can be depicted for swash cycles. Two recirculation zones are attached respectively to the back and the front of the swash tongue. These two groundwater cells are interacting together and are evolving during uprush and backwash, growing or diminishing. Sometimes one vanishes completely, sometimes another one gets trapped by recharge induced by water table elevation.

During uprush, there is downward infiltration while the swash tongue is covering the beach. It is mainly concentrated in the upper zone, through the boundary in contact with the fully saturated part of the beach. Infiltration induces a raise of the water table near the exit point, with, as seen in Fig.

V.8.
During backwash, a low hydraulic head zone emerges at the base of the swash zone where exfiltration occurs. Velocities are mainly seaward under the swash tongue moving back. Meanwhile, a small region of high hydraulic head persists in the corner of the saturated zone where water In the simulation, the offshore water level is similar to the inland water table, therefore no steady pressure gradient is observed across the beachface by contrast to field or laboratory studies where crossbeach gradients can be observed depending on boundary conditions [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF]. Within the beach, overall velocity magnitude is about 1 cm/h which is fairly comparable to measurements made in Sous et al.

(2016). The only exception is for infiltration/exfiltration where maximum velocity order is about 100 cm/h. This seems to be less compared with the experiments of Turner et al. [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF]. Further tests are now engaged in the exact configurations of the laboratory [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF] and field [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF] experiments to allow direct comparisons. As described by Perera et al. [START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF], the moving bore is carrying a sharp hydraulic front. Exfiltration occurs at its front and infiltration at its back when it meets the groundwater conditions of the precedent swash. A local superelevation of the water table has also been revealed in these observational studies due to the water table charge by short or IG wave forcing under the swash zone. Perera et al. showed a domed water table in their study [START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF]. The present benchmark case will be extended to a longer duration to test if the model is able to represent such feature. Water table elevation induces charge in this zone as long as the low-pressure zone does not affect it with the ongoing uprush. As a consequence, infiltration with flow division occurs near the uprush limit. Laboratory experiments and numerical simulations [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Li | Wave-induced beach groundwater flow[END_REF] both predict this feature but Sous et al. do not observe it [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF]. During the simulation, seepage face has small length because water is able to pour out from sand quite fast following surface water. As a consequence, a small hump-like structure appears during backwash because of seepage along the boundary. This last feature is not described in any study found, but further tests are ongoing with more realistic configurations and longer wave forcings to better understand involved processes. The two circulation zones with infiltration/exfiltration in the upper/lower part of the beach are also identified in the simulation from Li and Barry [START_REF] Li | Wave-induced beach groundwater flow[END_REF] and experiments from Turner et al. [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF]. Infiltration and exfiltration involve greater pore velocities than in other parts of the beach. Like it is depicted by Perera et al. [START_REF] Perera | Surface-groundwater flow numerical model for barrier beach with exfiltration incorporated bottom boundary layer model[END_REF], gradients brought by the wave are much larger than the average values in the beach due to larger differences in water depth across the wave.

V.3 Swash groundwater dynamics prospects

The idealized beach case gives confidence, at least qualitatively and for order of magnitude, in the ability of Rivage model to treat swash groundwater dynamics in sandy beaches. Two additional cases V.3. SWASH GROUNDWATER DYNAMICS PROSPECTS are considered. They are under construction and analysis.

The first case is provided by the laboratory swash experiments of [START_REF] Steenhauer | Subsurface processes generated by bore-driven swash on coarse-grained beaches[END_REF]. A series of dambreak-generated bore-driven swash have been performed to study subsurface processes in the swash zone. The experimental set-up has been described in details in a series of studies, see e.g. [START_REF] Steenhauer | Subsurface processes generated by bore-driven swash on coarse-grained beaches[END_REF][START_REF] Kikkert | Experimental study of bore-driven swash hydrodynamics on permeable rough slopes[END_REF][START_REF] Kikkert | Experimental study of bore-driven swash hydrodynamics on impermeable rough slopes[END_REF]. The initial interest is mainly placed on coarse-grained beaches [START_REF] Steenhauer | Numerical model of swash motion and air entrapment within coarse-grained beaches[END_REF] but finer sediment is also tested [START_REF] Steenhauer | Subsurface processes generated by bore-driven swash on coarse-grained beaches[END_REF]. This experiment remains a test-case of primary interest because it is one of the rare wave-resolved experiments on swash groundwater dynamics with detailed visualisation. Waves induce a mound in water table due to recharge. The development of corner wetting fronts above beach surface with sharp gradients in saturation and pressure head is a challenging issue for model validation.

The second case concerns the BARDEX II experiments. They were performed in July 2012 at the Delta Flume in the Netherlands [START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF][START_REF] Masselink | Large-scale Barrier Dynamics Experiment II (BARDEX II): Experimental design, instrumentation, test program, and data set[END_REF]. One of the main goals of this large scale experimental project was to characterize, under controlled laboratory conditions, the effect of groundwater flow on the hydro-and morphodynamics in a sand barrier. In particular, we focus here on a series of experiments dedicated to the effect of waves and cross-barrier mean water level gradients on the groundwater dynamics.

Finally, Rivage model will be used for in-situ experimental data collected during the ROUSTY campaign [START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF][START_REF] Sous | Groundwater fluxes within sandy beaches: swash-driven flow vs cross-barrier gradients[END_REF], see Fig.

CHAPTER V. APPLICATIONS TO SWASH GROUNDWATER DYNAMICS IN SANDY BEACHES

Summary

This chapter deals with a numerical model supported by the Rivage code and built around Richards' equation so that it can simulate wave-driven groundwater flows in sandy beaches. By generating waves, the SWASH code computes pressure on the bed to enforce Richards' equation with a Dirichlet boundary condition. For this forced coupling, groundwater does not hold sway over surface water but this influence can be neglected if pore velocity is slow compared to wave velocities as it happens for beaches with fine grains. What is more, exfiltration by seepage can be modelled through the seepage boundary condition prescribed on the beachface wherever there is no enforced pressure. Based on the simulation results from an idealized beach case, the following outlines can be drawn for a swash cycle: One, associated with low pressure and exfiltration, is attached to the swash base during backwash and to the forefront of the next wave during uprush. The other one, associated with high pressure and infiltration, is attached to water table recharge during backwash and to the swash wave during uprush. This general scheme is consistent with observations made by experimental studies in the literature. Further investigations are needed to assess, on the one hand, numerical effects on the solution, such as discretization level or boundary conditions and, on the other, physical parameters such as constitutive laws, beach topography or incoming waves.

Chapter II introduced the numerical methods used for Richards' equation. Space discretization was done by DG methods. DG weak formulation give flexibility to handle hybrid non-confirming unstructured mesh, local and arbitrary approximation order, special boundary conditions. Backward differentiation formula methods are used for time discretization to reach high-order and provide stability. Nonlinear solving lies on fixed-point iterations whose convergence is controlled by an heuristic adaptive time stepping. Some implementations choices in Rivage code were presented for solvers, basis functions or numerical integration.

Chapter III proposed the adaptive strategy based on AMR. A block-based structure saved computational time and prepares for future parallelization. Mesh adaptation is driven by error indicators supported by either problem physics or a posteriori estimation. They are combined with WDG framework to monitor jumps and flux in the solution. Wetting fronts may be so sharp that a discontinuity better approximate Richards' equation solution.

Chapter IV used several test-cases. Various 1D infiltration into soil column allowed to catch Richards' equation dynamics, to validate the code and to test capturing techniques by mesh adaptation and the weighted framework. In particular, jumps were introduced by the weighted penalty which prevents oscillation propagation when the front is sharp. The estimation-based error indicator revealed to perform very well to drive mesh adaptation cost-effectively. In addition, high-order time scheme were necessary to reach high-order accuracy when spatial errors are small. Adaptive time stepping was needed to make nonlinear convergence reliable. The code was able to simulate various dynamics (recharge/drainage) of laboratory experiments. The multi-materials La Verne dam wetting is a demanding case with experimental data. Comparison showed very good agreement with the system dynamics. Besides, it demonstrated the strategy ability to deal with sharp fronts from heterogeneity or nonlinear varying diffusivity. Mesh adaptation captured fronts with the estimation-based error indicator and the WDG method adjusted jumps size. Without providing a perfect solution, this strategy is very cost-effective compared to the mesh refinement required to improve the results.

Chapter V applied the method to groundwater in sandy beaches. The Richards' equation model was able to render main features observed experimentally even though only a forcing coupling is used. This is demonstrated by an idealized beach case where global and local space-time scales are modelled properly.

Perspectives

The strategy leads to mass conservative, cost-effective and reliable solution of Richards' equation. However, the method holds heuristic parameters so that questions remain open regarding the numerical analysis, the hp-decision making, the nonlinear convergence or the error balance. All these aspects have great potential of improvements and will be investigated in future studies to optimize the solving of Richards' equation.

The following suggestions are provided for future research:

• A work of validation and confrontation is under way with cases involving laboratory and largescale experiments for groundwater in sandy beaches. An application to a field experiment on Rousty beach is aimed to provide new insights about swash groundwater;

• Volume averaging method would be interesting to continue in order to derive clear assumptions about flow regime and porous medium. First, it will contribute to the field of volume averaging by proposing dimensional analysis based on different characteristics lengths. Secondly, it will precise the range of validity for the fundamental relation modelling porous medium flux which VALORIZATIONS is then used in Richards' equation. Finally, volume averaging is also a powerful tool to elaborate coupling models between subsurface and surface flows. This will contribute building a coupling more elaborate than a forcing;

• Penalty parameters provide stability for DG methods and enforce boundary conditions. However, they can also degrade numerical properties with arbitrary values, particularly condition number, leading to instabilities. Besides, they offer a way to incorporate into the scheme some partial knowledge about the solution. Numerical analysis and further tests would be carried on to derive a definition convenient for the problem or, at least, suitable bounds;

• Adaptive time stepping controls nonlinear iterative convergence. However, the method does not prevent the time step to decrease so much that computation is not feasible. Besides, the method is heuristic and it is difficult to know if computational time is saved since swing factor is a user-defined parameter. One possibility would be to use a time error estimator. Another possibility would be to study convergence of the nonlinear procedure which depends on the time step and to deduce a method. Local time stepping might be an idea to save computational time but it is difficult to make with implicit schemes;

• Time order selection is fixed for now in the code. This part should be fully automatic providing the best BDF method to guarantee stability. This is usually done by the mean of error estimators or stability analysis;

• hp-Adaptation would be a profitable for Richards' equation solution by performing mesh refinement or order enrichment following the one is better. Lots of techniques are already available in literature to do hp-decision making. Some remarks in this thesis make think that subparts of the estimation-based error indicator may drive the decision between h-and p-adaptation.

Even though this hp-adaptation will not be optimal, the technique would be very simple and cost-effective. Further tests are needed and a true a posteriori error estimation analysis could help;

• Adaptivity is an issue for the refinement frequency and the selection of elements to refine. Automatic thresholding based on a distribution function would be a possibility to mark elements for refinement/coarsening. Deriving velocity of wetting fronts would be a way to define refinement frequency given the block size;

• Parallelization and 3D are natural extensions for Rivage code to handle more complex cases and save computational time. Even though the code is prepared for this (block-based structure leading to domain decomposition and general implementation), it will be not straightforward since 3D involves lots of element geometries and time scheme is implicit. Besides, adding a module to treat hyperbolic equation would be a possibility to investigate in order to incorporate surface flow models and make a proper coupling.

Valorizations

Several parts of the thesis were presented during conferences and seminars whose some of them provided proceedings. One article has been proposed to a journal for publication where the adaptive DG strategy is described to get Richards' equation solution. Another one is planned to introduce and discuss groundwater applications to sandy beaches. These works are listed in List of Publications, page 145.

1 |V | if |y| ≤ r 0 , 0 otherwise. (V.2)
Since m is an even function, it writes:

⧼ξ⧽(x) = R 3 m(y)1 F (x + y)ξ(x + y) dy. (V.3)
The average of a constant quantity ξ 0 should be ξ 0 . Yet, it stands:

⧼ξ 0 ⧽(x) = m * (1 F ξ 0 ) x = ξ 0 m * (1 F ) x = ξ 0 ⧼ 1 ⧽ (x) ̸ = ξ 0 . (V.4)
That is the reason why a second filtering operator is needed. It is called intrinsic average:

ξ(x) = ⧼ξ⧽(x) ⧼ 1 ⧽ (x) = ⧼ξ⧽(x) Φ(x) . (V.5)
It is interesting that the intrinsic average is idempotent, that is to say the intrinsic average of the intrinsic average is equal to the intrinsic average. However:

ξ = ξ + ξ = ξ + ξ. (V.6)
In order to have ξ = ξ, it is necessary that ξ = 0. In other words, the intrinsic average of the spatial deviation is zero. However, to be in accordance with the idempotency property, there are:

Φ(x)ξ(x) = ⧼ξ⧽(x) = R m(y)1 F (x + y)ξ(x + y) dy = R m(y)1 F (x + y)ξ(x) dy + R m(y)1 F (x + y) ⊺ ∇ξ(x)y dy + R m(y)1 F (x + y) 1 2 ⊺ yH ξ (x)y dy + O ( ⊺ yy) = ⧼ 1 ⧽ (x) Order 0 moment ξ(x) + ⧼y⧽(x) Order 1 moment ∇ξ(x) + 1 2 ⧼ ⊺ yy⧽(x)
Order 2 moment

H ξ (x) + O ( ⊺ yy) = Φ(x)ξ(x) + ⧼y⧽∇ξ(x) + 1 2 ⧼ ⊺ yy⧽H ξ (x) + O ( ⊺ yy) = -Φ(x) ξ = 0 . (V.7)
A dimensional analysis gives:

⧼y⧽(x)∇ξ(x) = O r 0 Φ(x)ξ L , (V.8) ⧼ ⊺ yy⧽(x)H ξ (x) = O r 0 Φ(x)ξ L 2 . (V.9)
So the intrinsic average is idempotent provided that r 0 L ≪ 1.

Demonstration of spatial averaging theorem:

∇ x ⧼ξ⧽ x = ∇ x R 3 m(y)1 F (x + y)ξ(x + y) dy (V.10) = ∇ x R 3 m(z -x)1 F (z)ξ(z) dz (V.11) = R 3 ∇ x [m(z -x)]ind F (z)ξ(z) dz (V.12) = - R 3 ∇ z [m(z -x)]1 F (z)ξ(z) dz (V.13) = - R 3 (∇ z [m(z -x)1 F (z)ξ(z)] -m(z -x)∇ z [1 F (z)ξ(z)]) dz (V.14) = R 3 m(z -x)∇ z [1 F (z)ξ(z)] dz -m(z -x)1 F (z)ξ(z)n dσ (V.15) = R 3 m(y)∇ z [1 F (x + y)ξ(x + y)] dy -m(z -x)1 F (z)ξ(z)n dσ (V.16) = ⧼∇ x ξ⧽ x -m(z -x)1 F (z)ξ(z)n dσ (V.17)
And thus:

⧼∇ x ξ⧽ x = ∇ x ⧼ξ⧽ x + m(z -x)1 F (z)ξ(z)n dσ (V.18)

Résumé étendu en français

Présentation générale L a prédiction des écoulements dans les milieux poreux variablement saturés est un problème ma- jeur pour de nombreux domaines en science et en ingénierie. Ces écoulements apparaissent par exemple dans la physique des sols et de l'environnement, l'hydrogéologie, l'agriculture ou l'industrie pétrolière pour des problèmes comme le transport de contaminants, les réservoirs de pétrole, les ressources en eau, les affleurements de nappe, etc. Les études expérimentales sont un outil fondamental pour décrire de tels processus et obtenir une bonne compréhension. Cependant, les études expérimentales ne sont pas toujours possibles, ni réalisables. La modélisation mathématique et la simulation numérique offrent une autre possibilité pour explorer et acquérir de nouvelles connaissances sur ces systèmes physiques.

La compréhension de la dynamique des écoulements souterrains dans les zones côtières est d'un intérêt majeur dans les domaines du génie côtier et du développement durable. Dans le présent travail, une attention particulière sera apportée aux applications des écoulements souterrains littoraux. Une meilleure compréhension de la circulation de ces écoulements est d'une importance primordiale pour analyser et prévoir un ensemble de processus physiques et biogéochimiques des zones littorales, tels que le transport des sédiments (stabilité du lit), la diffusion de matières dissoutes telles que les polluants ou les nutriments ou encore le mélange entre les eaux continentales (douces) et marines (salées). Chacune de ces questions requiert une attention particulière dans le contexte du changement climatique et de l'érosion [START_REF] Luijendijk | The State of the World's Beaches[END_REF][START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF] (élévation du niveau de la mer, évènements de submersion, salinisation des terres arables) et de la pression anthropique croissante (urbanisation côtière).

Des études récentes en laboratoire [START_REF] Steenhauer | Subsurface processes generated by bore-driven swash on coarse-grained beaches[END_REF][START_REF] Turner | Groundwater fluxes and flow paths within coastal barriers: Observations from a large-scale laboratory experiment (BARDEX II)[END_REF] et sur le terrain [START_REF] Heiss | Coupled surface-subsurface hydrologic measurements reveal infiltration, recharge, and discharge dynamics across the swash zone of a sandy beach[END_REF][START_REF] Sous | Field evidence of swash groundwater circulation in the microtidal rousty beach, France[END_REF] ont permis d'acquérir une bonne connaissance expérimentale des processus physiques régissant la dynamique des écoulements souterrains dans les plages sbleuses exposées à l'action des vagues. Les données récemment obtenues au cours de la campagne ROUSTY2014 permettront d'approfondir l'analyse de cette dynamique souterraine à plus grande échelle [START_REF] Sous | Groundwater fluxes within sandy beaches: swash-driven flow vs cross-barrier gradients[END_REF]. Malgré leur importance primordiale, les mesures expérimentales restent très coûteuses et sont limitées en termes de résolution et d'étendue temporelle et spatiale. Un défi scientifique majeur consiste à développer des modèles numériques robustes et efficaces pour mieux anticiper, via la réalisation de scénarios prédictifs, l'évolution de la zone littorale. Peu de modèles ont été présentés jusqu'à présent et leur champ d'application reste limité, voir e.g. [START_REF] Li | Wave-induced beach groundwater flow[END_REF][START_REF] Li | BeachWin: modelling groundwater effects on swash sediment transport and beach profile changes[END_REF]. De nouveaux développements numériques sont nécessaires pour aborder les questions clés des écoulements souterrains de la plage induits par les vagues.

De nombreux modèles d'écoulements en milieux poreux ont été proposés et développés pour simuler les flux et processus de transport à différentes échelles. Les problèmes d'écoulement multiphasique présentent un comportement plus complexe que les problèmes d'écoulement monophasique car la dynamique des phases s'influence mutuellement. Dans ce contexte, le modèle d'écoulement multi-phasique peut conduire à des difficultés de résolution ou avoir un niveau de détails trop complexe pour la gamme d'applications visées. Un modèle simplifié tenant compte d'hypothèses raisonnables constitue une alternative très intéressante.

Dans le présent travail, les écoulements dans des milieux poreux à saturation variable sont décrits par l'équation de Richards. Ce qui rend l'équation de Richards très attrayante est qu'elle modélise le milieu poreux dans son ensemble, y compris les zones saturées et non-saturées. Bien qu'elle néglige la phase d'air, l'équation de Richards tient compte des actions de la gravité et de la capillarité et peut donc modéliser des processus variés et complexes à différentes échelles. L'équation de Richards est une équation parabolique non-linéaire qui peut dégénérer en une équation elliptique dans des conditions de saturation complète. Son histoire commence avec la loi de Darcy formulée expérimentalement par Darcy en 1856 [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF] pour les milieux poreux saturés. Puis, ce résultat a été étendu aux écoulements multiphasiques par Buckingham en 1907 [START_REF] Buckingham | Studies on the movement of soil moisture[END_REF] pour donner la loi de Darcy-Buckingham, qui sert de pierre angulaire pour la dérivation de l'équation de Richards. Cette dernière a été établie pour la première fois par Richardson en 1922 [START_REF] Richardson | Weather Prediction by Numerical Process[END_REF] mais a été attribuée uniquement à Richards qui a publié indépendamment l'équation en 1931 [START_REF] Richards | Capillary conduction of liquids through porous mediums[END_REF]. Les premières tentatives de résolution numérique remontent à la fin des années 1960 avec Rubin [START_REF] Rubin | Theoretical Analysis of Two-Dimensional, Transient Flow of Water in Unsaturated and Partly Unsaturated Soils[END_REF] ou Cooley [START_REF] Cooley | A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well[END_REF] par exemple. Puis, à partir des années 1980, l'équation de Richards a été étudiée de manière approfondie, tant du point de vue théorique que numérique. Des résultats d'existence, de régularité et d'unicité ont été trouvés [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF] et la solution a été évaluée par des simulations impliquant différentes méthodes numériques. Un des points d'intérêt concernait le solveur non-linéaire [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF]. Depuis les années 2000, les travaux portent principalement sur la stabilité de l'équation de Richards et le développement de simulations couplées et générales [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF]. Ces dernières années, les recherches portent sur l'équation de Richards avec hystérésis et discrétisations spatiales/temporelles avancées [START_REF] Miller | A spatially and temporally adaptive solution of Richards' equation[END_REF][START_REF] Li | Adaptive local discontinuous Galerkin approximation to Richards' equation[END_REF] avec des approximations adaptatives en taille et/ou en ordre. L'équation de Richards a été largement utilisée pour des simulations numériques par la communauté d'hydrogéologie, voir e.g. [START_REF] Thoms | User's guide to the Variably Saturated Flow (VSF) process to MODFLOW[END_REF][START_REF] Szymkiewicz | Modelling Water Flow in Unsaturated Porous Media[END_REF]. Malgré cet effort de recherche, de nombreux défis numériques restent à relever pour résoudre l'équation de Richards, certaines simulations étant encore peu fiables et/ou coûteuses [START_REF] Farthing | Numerical Solution of Richards' Equation: A Review of Advances and Challenges[END_REF][START_REF] Zha | Review of numerical solution of Richardson-Richards equation for variably saturated flow in soils[END_REF]. Cette complexité numérique a empêché une utilisation plus générale du modèle de Richards pour un certain nombre d'applications. En effet, la solution de l'équation de Richards implique des fronts de saturation raides difficiles à résoudre car ils évoluent à la fois dans l'espace et le temps. En outre, la simulation doit traiter simultanément des régions non-saturées/saturées de nature parabolique/elliptique, une forte hétérogénéité et anisotropie du milieu poreux et éventuellement des conditions aux limites dynamiques. Ces aspects peuvent également conduire à de forts gradients agissant comme des discontinuités. Compte tenu des fonctions non-linéaires pour les propriétés hydrauliques, il est rarement possible de prévoir le comportement de l'équation de Richards et il peut être difficile d'obtenir des solutions numériques. En particulier, les schémas numériques ne convergent pas systématiquement, ce qui nécessite un traitement numérique minutieux, ou une discrétisation fine, rendant le calcul coûteux. L'objectif de cette thèse est de développer des outils conduisant à une simulation de l'équation de Richards qui est :

• précise, dans le sens que la solution numérique doit être physiquement acceptable et converger vers une solution exacte lorsque les pas de temps et/ou la résolution du maillage sont raffinés;

• robuste, dans le sens que la méthode doit fournir une solution quels que soient les paramètres physiques et numériques du problème;

• efficace, dans le sens que le calcul doit être rentable en termes de temps de calcul et de ressources mémoire ;

• adapté aux écoulements souterrains de la plage induits par les vagues.

Le présent travail de recherche numérique vise la simulation des écoulements souterrains dans les plages sableuses comme principale application. [START_REF] Dolejší | Discontinuous Galerkin Method[END_REF]. Cel favorise une utilisation croissante des méthodes de GD pour les phénomènes de transport dans les milieux poreux tels que les problèmes d'écoulement diphasique [32,[START_REF] Bastian | A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure[END_REF].

En outre, l'équation de Richards est connue pour être une équation différentielle raide avec une convergence difficile [START_REF] Paniconi | A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[END_REF][START_REF] Lehmann | Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media[END_REF][START_REF] List | A study on iterative methods for solving Richards' equation[END_REF]. La discrétisation temporelle et le solveur non-linéaire doivent donc être traités avec soin.

En gardant à l'esprit que la méthode numérique doit fournir des résultats précis et robustes, mais efficaces en termes de temps de calcul, cette thèse constitue une opportunité de contribuer à l'avancement des méthodes GD adaptatives d'ordre élevé pour la simulation de l'équation de Richards. Ainsi, les réalisations suivantes sont attendues :

• L'équation de Richards est discrétisée par des méthodes GD primales qui conduisent à une formulation conservative, d'ordre élevé et un RAM pour obtenir une solution précise. La pénalisation impose faiblement les conditions aux limites dans la formulation GD pour la robustesse. Les formules de différenciation à rebours sont utilisées pour la discrétisation temporelle car elles sont connues pour être d'ordre élevé et stables pour les équations rigides;

• Une méthode à point fixe associée à un échelonnement adaptatif du temps rend les itérations non-linéaires plus fiables;

• Un indicateur d'erreur a posteriori est formulé pour l'équation de Richards approchée par des méthodes de GD. Cette estimation est décomposée en différentes contributions qui guident l'adaptation du maillage;

• L'algorithme utilise une RMA par blocs pour gagner du temps de calcul.

Cette thèse propose une nouvelle stratégie pour étudier les simulations adaptatives en temps et espace pour l'équation de Richards. Le RMA sera utilisé pour capturer les fronts de saturation en mouvement nages de nappe phréatique. Enfin, la stratégie de modélisation est testée sur un cas difficile à grande échelle avec un ensemble de données expérimentales : la saturation d'un barrage multi-matériaux.

Dans le Chapitre V, des applications sont menées pour les écoulements souterrains des plages sableuses. Après avoir présenté les notions de système de plage impliquant le forçage des vagues et la circulation des écoulements souterrains, des comparaisons sont effectuées pour trois cas. Une plage idéalisée est considérée pour voir si le modèle est capable de rendre les principales caractéristiques des écoulements souterrains de plage observées dans les expériences disponibles dans la littérature.

Synthèse conclusive

D ans cette thèse, le but principal consistait à développer des simulations précises, robustes et effi- caces de l'équation de Richards pour modéliser les écoulements souterrains des plages. À cette fin, une nouvelle stratégie a été formulée pour la solution de l'équation de Richards sur la base de la méthode GD avec des FDR et un RMA.

L'approximation de l'équation de Richards peut atteindre un ordre élevé à la fois en espace et en temps, traiter des conditions de suintement, des conditions aux limites dynamiques, des milieux poreux hétérogènes et des fronts de saturation/désaturation raides. Les questions exposées précédemment ont été abordées et discutées plus en détail tout au long de la thèse, de sorte que les remarques suivantes peuvent être tirées en guise de conclusion :

• L'équation de Richards a été dérivée en soulignant des hypothèses claires pour la modélisation;

• La condition aux limites de suintement a été directement incorporée dans la formulation GD faible, ce qui rend son traitement naturel et simple.

• Le pas de temps adaptatif a permis au solveur itératif non-linéaire de converger, ce qui lui confère une grande robustesse;

• L'adaptation du maillage a été utilisée pour contrôler les erreurs spatiales de l'équation de Richards en capturant les fronts de saturation dynamiques grâce à un indicateur d'erreur a posteriori basé sur l'estimation;

• Le RAM a été basé sur une structure par blocs pour délimiter de grandes régions où les fronts de saturation se déplacent, afin de gagner du temps de calcul en évitant un raffinement systématique. Elle prépare également une parallélisation future;

• Les oscillations des fronts de saturation raides de l'équation de Richards ont été interprétées dans le contexte d'une équation de convection-diffusion. Les fronts de saturation ont été considérés comme des couches internes en raison de la diffusivité variable non-linéaire, voire dégénérée, ainsi que de l'hétérogénéité et de l'anisotropie des matériaux. Un nombre de Péclet local a été présenté pour mettre en évidence les régions problématiques. Une méthode GDP a été utilisée pour permettre des sauts dans la solution qui approximent mieux les couches internes raides. L'adaptation du maillage et le cadre pondéré fonctionnent en synergie pour capturer et résoudre les fronts de saturation raides grâce à l'indicateur d'erreur a posteriori proposé.

Le Chapitre I a apporté des éclaircissements sur les possibilités de modélisation de l'équations de Richards. Cette équation parabolique non-linéaire simule un système air-eau, mais en ne considérant que la phase liquide. Ainsi, elle est très utile pour modéliser les écoulements en milieux poreux variablement saturés, mais il existe des limitations comme la phase gazeuse qui doit rester continuellement connectée. Les propriétés hydrauliques supposent que les phases sont à l'équilibre et ne tiennent pas compte de l'hystérésis. La modélisation de l'équation de Richards reste riche en prenent en compte les phénomènes de gravité et les effets capillaires. Ce double comportement associé à la non-linéarité rend l'équation de Richards difficile à résoudre numériquement pour un large éventail d'applications.

Le Chapitre II a présenté les méthodes numériques utilisées pour l'équation de Richards. La discrétisation de l'espace a été réalisée par des méthodes de GD. La formulation faible de la GD donne la flexibilité nécessaire pour traiter des maillages hybrides non-conformes non-structurés, des ordres d'approximation élevés et des conditions aux limites spéciales. Des FDR sont utilisées pour la discrétisation en temps afin d'atteindre un ordre élevé et d'assurer la stabilité. La résolution non-linéaire repose sur la méthode du point fixe dont la convergence est contrôlée par un pas de temps adaptatif heuristique. Certains choix d'implémentation dans le code Rivage ont été présentés pour les solveurs, les fonctions de base ou l'intégration numérique.

Le Chapitre III a proposé la stratégie adaptative basée sur le RAM. Une structure basée sur des blocs a permis d'économiser du temps de calcul et de préparer une future parallélisation. L'adaptation du maillage est déterminée par des indicateurs d'erreur appuyés soit par la physique du problème, soit par une estimation a posteriori. Ils sont combinés avec le cadre GDP pour contrôler les sauts et les flux dans la solution. Les fronts de saturation peuvent être si raides qu'une discontinuité approxime mieux la solution de l'équation de Richards.

Le Chapitre IV a utilisé plusieurs cas-tests. Diverses infiltrations 1D dans une colonne de sol ont permis de saisir la dynamique de l'équation de Richards, de valider le code et de tester les techniques de capture par l'adaptation du maillage et le cadre pondéré. En particulier, des sauts ont été introduits par la pénalité pondérée qui empêche la propagation des oscillations lorsque le front est raide. L'indicateur d'erreur basé sur l'estimation s'est révélé très performant pour conduire l'adaptation du maillage efficacement. En outre, un schéma temporel d'ordre élevé a été nécessaire pour atteindre une précision élevée lorsque les erreurs spatiales sont faibles. Un pas de temps adaptatif a été nécessaire pour rendre la convergence non-linéaire fiable. Le code a permis de simuler diverses dynamiques (recharge/drainage) d'expériences de laboratoire. La saturation du barrage multi-matériaux de La Verne est un cas exigeant présentant des données expérimentales. Une comparaison a montré un très bon accord du modèle avec la dynamique du système. Qui plus est, elle a démontré la capacité de la stratégie à traiter des fronts raides causés par l'hétérogénéité ou la diffusivité variable non-linéaire. L'adaptation du maillage a permis de capturer des fronts avec l'indicateur d'erreur basé sur l'estimation et la taille des sauts ajustée par la méthode GDP. Sans apporter de solution parfaite, cette stratégie est très rentable par rapport au raffinement du maillage requis pour améliorer les résultats.

Le Chapitre V a appliqué la méthode aux écoulements souterrains dans les plages sableuses. Le modèle basé sur l'équation de Richards a pu rendre les principales caractéristiques observées expérimentalement même si seul un couplage forcé est utilisé. Ceci est démontré par un cas idéalisé de plage.

Cette stratégie conduit à une solution conservative, efficace et fiable de l'équation de Richards. Cependant, la méthode contient des paramètres heuristiques de sorte que des questions restent ouvertes concernant l'analyse numérique, la prise de décision hp, la convergence non-linéaire ou le bilan d'erreurs. Tous ces aspects ont un grand potentiel d'amélioration et seront étudiés dans de futurs travaux afin d'optimiser la résolution de l'équation de Richards.

Les suggestions suivantes sont fournies pour les recherches futures :

• Un travail de validation et de confrontation est en cours avec des cas impliquant des expériences en laboratoire et à grande échelle pour les écoulements souterrains dans les plages sableuses. Une application à une expérience de terrain sur la plage de Rousty est visée pour apporter de nouvelles connaissances sur les écoulements souterrains de jet de rive;

• Il serait intéressant de poursuivre la méthode de la prise de moyenne volumique afin de dégager des hypothèses claires sur le régime d'écoulement et le milieu poreux. Tout d'abord, cela contribuera au domaine des méthode de changment d'échelle en proposant une analyse dimensionnelle basée sur différentes longueurs caractéristiques. Deuxièmement, cela précisera la plage de validité de la relation fondamentale modélisant l'écoulement du milieu poreux, qui est ensuite utilisé dans l'équation de Richards. Enfin, la prise de moyenne volumique est également un outil puissant pour élaborer des modèles de couplage entre les écoulements souterrains et de surface. Cela contribuera à construire un couplage plus élaboré qu'un forçage; Mots-clés : Équation de Richards, suintement, méthode GD pondérée, raffinement de maillage, estimation d'erreur a posteriori, BDF, hydrodynamique souterraine du jet de rive.

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Numerical simulation of flows in unsaturated porous media by an adaptive discontinuous Galerkin method: application to sandy beaches

Abstract: Flows in unsaturated porous media are modelled by the Richards' equation which is a degenerate parabolic nonlinear equation. Its limitations and the challenges raised by its numerical solution are laid out. Getting robust, accurate and cost-effective results is difficult in particular because of moving sharp wetting fronts due to the nonlinear hydraulic properties. Richards' equation is discretized by a discontinuous Galerkin method in space and backward differentiation formulas in time. The resulting numerical scheme is conservative, high-order and very flexible. Thereby, complex boundary conditions are included easily such as seepage condition or dynamic forcing. Moreover, an adaptive strategy is proposed. Adaptive time stepping makes nonlinear convergence robust and a block-based adaptive mesh refinement is used to reach required accuracy cost-effectively. A suitable a posteriori error indicator helps the mesh to capture sharp wetting fronts which are also better approximated by a discontinuity introduced in the solution thanks to a weighted discontinuous Galerkin method. The approach is checked through various test-cases and a 2D benchmark. Numerical simulations are compared with laboratory experiments of water table recharge/drainage and a largescale experiment of wetting, following reservoir impoundment of the multi-materials La Verne dam. This demanding case shows the potentiality of the strategy developed in this thesis. Finally, applications are handled to simulate groundwater flows under the swash zone of sandy beaches in comparison with experimental observations.

κFFϑcs 3 v

 3 janvier 2021, à La Garde. -I am partially supported by a grant from Provence-Alpes-Côte d'Azur region, France. -More information about what is MT180 and my participation is available in Appendix C -Three Minute Thesis -Ma thèse en 180 secondes. Normal component of hydaulic conductivity for a face κ m Minimum of neighbouring normal hydaulic conductivities of a face λ amp Amplification factor for time stepping λ k Parameter for Tracy's solution λ M Largest eigenvalue of the hydraulic conductivity tensor -Penalty weight for an interior face L•T -1 ϱ X Extrapenalty weight for an interior face L•T -1 Σ(x, y) Surface of the ball of center x and radius y -Parameter for the ϑ-method -Width and height of rectangular domain L a h,n Bilinear form for stiffness terms L d+1 •T -1 A ij Stiffness matrix -B(x, r 0 ) Ball of center x and radius r 0 -Coefficient expressing the fact that a part of the kinetic energy of the real fluid turns into heat -Additional artificial mass L d+1 m flux Total net mass flux L d+1 m h,n Bilinear form for mass terms L d+1 •T -1 Number of Gauss quadrature nodes -N it Number of nonlinear iterations -N nod Number of vertices of an element -P p Space of polynomial functions of total degree less than or equal to p -Space of polynomial functions of degree less than or equal to p -Fractional index of Sobolev space s Radius for the fluid ball preserving surface L Flow speed at a point on a streamline L•T -1

Figure 1 -

 1 Figure 1 -Examples of coastal urbanization and threat of sea level rise for beaches with high socio-economic services.

"I. 1 8 I. 2 A 12 I. 2 . 1 12 I. 2 . 2 15 I. 3 23 I. 4 25 I. 5 28 I. 6 29 I. 6 . 2

 182122112221532342552862962 La science est faite d'erreurs, mais d'erreurs qu'il est bon de commettre, car elles mènent peu à peu à la vérité." -Jules Verne, Voyage au centre de la Terre, 1864 Description of porous media . . . . . . . . . . . . . . . . . . . . . . . . . brief derivation of subsurface flow relation . . . . . . . . . . . . . . . . Upscaling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume averaging method . . . . . . . . . . . . . . . . . . . . . . . . Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seepage and dynamic forcing boundary conditions . . . . . . . . . . . . . Numerical challenges of Richards' equation . . . . . . . . . . . . . . . . . 29 I.6.1 Behaviour analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational developments . . . . . . . . . . . . . . . . . . . . . . 30

Figure I. 1 -

 1 Figure I.1 -Schematic representation of a part from two different porous media.

Figure I. 2 -

 2 Figure I.2 -The different zones of saturation in a porous medium.

Figure I. 3 -

 3 Figure I.3 -Scale of grain size.

Figure I. 4 -

 4 Figure I.4 -Photos with a schematic close-up of a sample of sand-, silt-, clay-sized particles.

Figure I. 5 -

 5 Figure I.5 -Soil texture classification diagram from USDA. Textures are determined by the proportions of sand-, silt-, and clay-sized grains.

Figure I. 6 -

 6 Figure I.6 -Basic concepts for different types of water in soils

  H [L] denotes the total head, z [L] the point elevation with respect to an arbitrary datum, p [M•L -1 •T -2 ] the pressure, ρ [M•L -3 ] the water density, v [L•T -1 ] the flow speed at a point on a streamline, g [L•T -2 ] the acceleration due to gravity, c the coefficient expressing the fact that a part of the kinetic energy of the real fluid turns into heat and h r [L] the head loss due to the irreversible transformation of mechanical energy into thermal energy through the mechanism of frictional resistance. According to this equation, groundwater flow can be divided in different head components shown in Fig. I.8.Usually, Bernoulli's principle is only applicable for isentropic flows that is to say for both adiabatic and reversible processes. Then, c = 1 and h r = 0. When porous media is considered, pore flow is very slow so velocity head and inertial head are often neglected.

Figure I. 8 -

 8 Figure I.8 -Components of the head for water.

  A typical example concerns Darcy's law which is a fundamental subsurface flow relation in hydrogeology. Formally, Darcy's law links, on one hand, the flow q [L•T -1 ] of a fluid with viscosity µ [M•L -1 •T -1 ] trough a porous medium of permeability k [L 2 ], with, on the other hand, the pressure gradient ∇p [M•L -2 •T -2 ]. This relation is explicit and linear: 's law is valid to describe the so-called Darcy flux only if the flow is laminar [40]. Indeed, Darcy's law is the restriction from a much broader relation linking flow velocity with pressure gradient as illustrates in Fig. I.10.

Figure I. 10 -

 10 Figure I.10 -Range of validity of Darcy's law. Darcy zone delimits the linear approximation of the flow-pressure gradient relation. Outside, nonlinear deviation are observed.

Figure I. 11 -

 11 Figure I.11 -Comparison of different interface models for porous media flow. Taken from Fujisawa and Murakami [67].

Figure I. 14 -

 14 Figure I.14 -Concept for a definition of the Representative Elementary Volume (REV).

(I. 24 )

 24 Gray's decomposition consists to separate a variable into an intrinsic average component, accounting for slow variations, according to L, and a spatial deviation component, driven by fast variations, according to ℓ. The principle of the process is illustrated in Fig. I.15.

Figure I. 15 - 1 F 1 F

 1511 Figure I.15 -Principle of Gray's decomposition between average and deviation fields.

  Going back to equation (I.41), water content (θ, [-]), pressure head (ψ, [L]) and hydraulic conductivity tensor (K, [L • T -1 ]) are introduced:

A

  , C: empirical shape parameters [L B;D ] B, D: empirical shape parameters [-]Van Genuchten-Mualem relations (1980)[START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils[END_REF]

  parameter linked to air entry pressure inverse [L -1 ] n > 1: pore-size distribution[-]

Figure I. 16 -

 16 Figure I.16 -Typical soil water characteristic curve with key elements.

Figure I. 17 -

 17 Figure I.17 -Principle of the seepage boundary condition for the pressure head ψ.

"II. 2 II. 4 II. 5 II. 6 II. 7

 24567 Tú, que me lees, ¿estás seguro de entender mi lenguaje?" -Jorge Luis Borges, The Library of Babel, 1941 II.1 A brief review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1.1 General overview of discretization methods . . . . . . . . . . . . . . . II.1.2 Principles of discontinuous Galerkin methods . . . . . . . . . . . . . . Space semidiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . II.2.2 Discontinuous Galerkin weak formulation . . . . . . . . . . . . . . . . II.2.3 Penalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2.4 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2.5 Algebraic representation . . . . . . . . . . . . . . . . . . . . . . . . . II.3 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.3.1 ϑ-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.3.2 Backward differentiation formula methods . . . . . . . . . . . . . . . . II.3.3 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear iterative process . . . . . . . . . . . . . . . . . . . . . . . . . . II.5.1 Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . II.5.2 Approximation techniques . . . . . . . . . . . . . . . . . . . . . . . . Adaptive time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.7.1 Overall programming . . . . . . . . . . . . . . . . . . . . . . . . . . . II.7.2 Computing tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Figure II.1 -Typical example of a 2D-solution representation arising from different methods of approximation.

  measure of element non-degeneracy and defines the mesh regularity. The geometry definitions and notations are shown in Fig. II.2.

  Figure II.2 shows a particular example. The set of all open faces of all elements E ∈ E n h is denoted by F n h . Furthermore, the definitions for the set of Dirichlet, Neumann, seepage, boundary and interior faces stand:

Figure II. 2 -

 2 Figure II.2 -Examples of mesh geometry for discontinuous Galerkin methods.

Figure II. 3 -

 3 Figure II.3 -Notations for two elements E l and E r sharing one face F ∈ F n h,I and one element E l with a boundary face F ∈ F n h,D in 2D.

Figure II. 4 -

 4 Figure II.4 -Definitions of the mean and jump operators for two elements E 1 and E 2 in 1D.

  is a constant leading to the different symmetric versions of the DG formulation. In Eq. (II.29), Eq. (II.29) and Eq. (II.29), the terms are coloured differently according to their nature. Blue colour corresponds to element volume terms and green colour to interior face terms. Other colours are for boundary face terms: magenta is for Neumann boundary condition, red for Dirichlet boundary condition and orange for seepage boundary condition.

  16.0 of Gmsh, an open source 3D FE mesh generator. In option, a file .lim can be provided to take into account dynamic Dirichlet boundary conditions monitored by data. They are written in a format only readable by Rivage and can serve as prescribing dynamic forcing boundary condition. The core algorithm of Rivage is described in Fig. II.5. There are two embedded loops. The main one corresponds to the time cycle where each iteration goes through several steps: the solution of Richards' equation, adaptation of time stepsize, remeshing and eventually data export for visualization. The second loop concerns the nonlinear iterative process to get the solution of Richards' equation.

Figure II. 5 -

 5 Figure II.5 -General architecture of the core algorithm for the numerical solving from Rivage code.

  -McKee algorithm. Effects of the reordering are observed in Fig. II.6 for mass matrix and tangent stiffness matrix taken from a simulation made with Rivage.

Figure II. 6 -

 6 Figure II.6 -Sparsity patterns of the tangent stiffness matrix (top) and mass matrix (bottom) for standard numbering (left) and Morton-based numbering (right). nz stands for the number of non-zero values in the matrix.

Figure II. 7 -

 7 Figure II.7 -Principle of the construction of the basis functions based on the monomial basis. The triangle highlights the order hierarchy of the basis. Monomial basis corresponds to each stage while the tensor product of monomial basis corresponds to each coloured V -shaped mark.

  The reference element used in Rivage are depicted in Fig. II.8.

Figure II. 8 -

 8 Figure II.8 -Three reference elements (red) with an example of related physical elements (blue): edge (left), triangle (middle) and parallelogram (right). V i are the counter-clockwise vertices of the element and the origin O is at (0, 0).

  III.1.

Figure III. 1 -

 1 Figure III.1 -Different types of adaptation strategy

Figure III. 2 -

 2 Figure III.2 -Example of mesh refinement applied dam break benchmark. Using Basilisk code with quadtree and finite volume (left) and using finite element and anisotropic mesh adaptation [184] (right).

III. 3 .

 3 In cell-based AMR, refinement occurs element-wise and is generally built upon a parent-child data structure. Cellbased AMR would be interesting since DG methods provide local features suitable for such AMR. Patch-based AMR groups a certain number of elements into a patch where the requested refinement is done, providing regular grids. Finally, block-based AMR predefines coarse elements in which a refinement is performed. Block-based is suitable for structured Cartesian mesh.

Figure III. 3 -

 3 Figure III.3 -Cell-, block-and patch-based AMR techniques for Cartesian grids

  Fig. III.4.

Figure III. 4 -

 4 Figure III.4 -AMR Strategy: (a) Block mesh; (b) Level of mesh refinement; (c) mesh generation; (d) Morton numbering.

  Fig. III.4a. Each of this block can be refined appropriately into an isotropic grid of elements. All elements generated by the blocks are gathered into a list. It is not an element-wise parent-child data structure but blocks know their elements likewise. Mesh refinement level is defined for each block, initially by the user and thereafter in accordance with the chosen mesh refinement criterion, see Fig. III.4b. Coarsening and refinement are decided according to threshold values 0 < β c ≤ β r . If η denotes the block-level error estimate, three situations are encountered:

  Fig. III.5. Finally, a space filing curve using Morton numbering (Z-order) is built in order to number easily the degrees of freedom line in Fig. III.4d.

Figure III. 5 -

 5 Figure III.5 -Mesh refinement for quadrangle and triangle using quadtree.

"IV. 1 74 IV. 1 . 1 74 IV. 1 . 2 76 IV. 2 77 IV. 3 79 IV. 4 83 IV. 5 86 IV. 6

 174117412762773794835866 We are reaching the stage where the problems that we must solve are going to become insoluble without computers. I do not fear computers. I fear the lack of them." -Isaac Asimov, The Computer Society: The Age of Miracle Chips (Time), 1978 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional analytical tests . . . . . . . . . . . . . . . . . . . . . Two-dimensional analytical tests . . . . . . . . . . . . . . . . . . . . . Haverkamp's test-case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polmann's test-case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vogel-Van Genuchten-Cislerova test-cases . . . . . . . . . . . . . . . . . . Tracy's benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison with Vauclin's studies . . . . . . . . . . . . . . . . . . . .

15 Figure IV. 1 -

 151 Figure IV.1 -Numerical error for model problem Eq. (IV.1) with SIPG method and p = 1.

(IV. 4 )

 4 The analytical solution is u ex (x, y) = sin(2πx) sin(2πy). Computations are done for SIPG method with σ I = σ D = 20. First of all, the numerical solution is computed on a conforming mesh of 1024 elements on which ∥e h ∥ E h = 1.6085 × 10 -3 (Fig. IV.2a). Then, a non-conforming mesh is used. The half part on the right is refined. Mesh is made of 696 elements and ∥e h ∥ E h = 3.2272 × 10 -3 (Fig. IV.2b). Numerical results are presented in Fig. IV.2.

  Figure IV.2 -Numerical error for model problem Eq. (IV.4) with SIPG method and p = 1.

  This problem deals with an 1D-infiltration into a soil column of 40 cm height and 8 cm width. On the top and bottom, Dirichlet boundary conditions for hydraulic head are prescribed: h top = 19.3 cm, h bottom = -61.5 cm which results in a cumulative downward infiltration. Sides are impervious. The initial condition is h 0 = -61.5 + z cm. Simulation parameters are p = 1, σ I = σ D = 100 and T = 600 s. Threshold values for AMR selection from Section III.1.3 are: β c = β r = 0.5. The Haverkamp's test-case is sketched in Fig. IV.3.

Figure IV. 3 -

 3 Figure IV.3 -Haverkamp's test-case configuration.

( a )

 a Figure IV.4 -Vachaud's relations for Haverkamp's test-case.

  Figure IV.5 -Results for Haverkamp's test-case.

  Fig. IV.7.

Figure IV. 7 -

 7 Figure IV.7 -Polmann's test-case configuration.

  Figure IV.8 -Van Genuchten-Mualem relations for Polmann's test-case.

Figure IV. 9 -

 9 Figure IV.9 -Pressure head along the vertical for Polmann's test-case with the mesh M100.

Figure IV. 10 -

 10 Figure IV.10 -Comparison of pressure head at t = 24 h for Polmann's test-case: IIPG on mesh with 100 elements (M100), IIPG on mesh with 1000 elements (M1000), IIPG with adaptive mesh refinement (AMR), WDG with 100 elements (WDG).

  Figure IV.11 -Mass balance for Polmann's test-case. Blue values are for the mesh M100 and red values for the mesh M1000.

  . The goal is to evaluate the model answer to such nonlinear relations. If ψ s = 0, the modified Van Genuchten-Mualem relations are equivalent to the standard Van Genuchten-Mualem relations. Geometry is the same for the three cases with Ω being a rectangle (0, 20) × (0, 100) cm. Simulations use τ = 120 s, p = 1, σ I = σ D = 100 and 100 elements distributed vertically. The initial condition is hydrostatic: h 0 = -1000 cm. Boundary conditions are different which results in different infiltrations defining the three test-cases (see Fig. IV.12): • First test-case: upward infiltration. The simulation lasts T = 86400 s = 24 h. Pressure head is plotted every 17280 s = 0.2 d; • Second test-case: ponded downward infiltration. The simulation lasts T = 8640 s = 0.1 d. Pressure head is plotted every 43200 s = 12 h; • Third test-case: flux-controlled downward infiltration. The simulation lasts T = 8640 s = 0.1 d. Pressure head is plotted every 43200 s = 12 h.

  Figure IV.12 -Vogel-Van Genuchten-Cislerova test-cases configuration.

Figure IV. 14 -

 14 Figure IV.14 -Results for Vogel-Van Genuchten-Cislerova test-cases for ψ s = -2 cm (blue) and ψ s = 0 cm 'red) with upward infiltration (top), ponded downward infiltration (middle), flux-controlled downward infiltration (bottom).

  Tracy's benchmark employs Gardner-Irmay relations with K s = 1.1 m•d -1 , θ s = 0.5, θ r = 0, α = 0.1 m -1 and m = 1. It is solved for pressure head ψ. The residual pressure head ψ r is a parameter. The computational domain Ω is a square (0, a) × (0, b) m. Here, a = b = 1 m and ψ r = -10 m. A specific Dirichlet boundary condition is prescribed on the top edge with the function: ψ top = 1 α log e αψr + 1 -e αψr sin πx a . (IV.5) For other edges, a constant Dirichlet boundary condition is imposed: ψ = ψ r . The initial condition is ψ 0 = ψ r . Tracy gives the 2D exact solution ψ ex [213]: ψ ex = 1 α log e αψr + ψ ts + ψ ss , (IV.6) ψ ss = 1 α log e αψr + 1 -e αψr sin πx a e α 2 (b-y) sinh β T y sinh β T b , (IV.7)

  12) where ψ ss denotes the steady-state solution and ψ ts the transient counterpart for the exact solution. The configuration is sketched in Fig. IV.15. Figure IV.16 shows the numerical solution obtained for a fixed grid with τ = 10 -4 day and T = 10 -2 day.

Figure IV. 15 -

 15 Figure IV.15 -Tracy's benchmark configuration.

Figure IV. 16 -

 16 Figure IV.16 -Hydraulic head for Tracy's benchmark with p = 1 and 6-step BDF.

  Figure IV.17 -Numerical convergence for Tracy's benchmark. P1, P2, P3 and P4 stands respectively for linear, quadratic, cubic and quartic approximations.

  Figure IV.18 -Effectivity indices for different indicators (left) and convergence diagrams for the three parts of the indicator η n E (right). P1, P2, P3 and P4 stands respectively for linear, quadratic, cubic and quartic approximations.

  Fig. IV.19 for reference to localize the zones of interest. It is above the error measured with energy norm. In Fig. IV.20, it is worth to notice that the estimationbased error indicator η n E is able to give the shape (but not the magnitude) of the true error measured with the energy norm, particularly for steady state, even though it was not expected because there is no theoretical proof. Besides, during transient state, the estimation-based error indicator spots the error in the top corner where Tracy stated there were problems for this test-case, see Fig. IV.19 at t = 10 -4 d.

Figure IV. 19 -

 19 Figure IV.19 -True error distribution in the L 2 -norm on Tracy's benchmark.

  Figure IV.20 -Comparison between distributions of true error in energy norm and the indicator η n E

  Figure IV.22 -Comparison with energy norm between C E -based adaptation (left) and η n E -based adaptation on Tracy's benchmark.

Figure IV. 23 -

 23 Figure IV.23 -Water table recharge test-case configuration.

20 )

 20 Numerical simulation is performed with Lagrangian basis functions, linear approximation p = 1, σ I = σ D = 20 on a mesh of 2367 elements. Zones around the expected position of water table and boundary recharge flux are refined, see Fig. IV.24. Water table location matches ψ = 0 and capillary fringe location is established at θ(ψ) = 0.98θ s = 0.294. Numerical results are shown in Fig. IV.25. Values and red contour lines of hydraulic head h together with distribution of flux are displayed.

Figure IV. 24 -

 24 Figure IV.24 -Mesh used for the steady-state simulation of watertable recharge test-case.

Figure IV. 25 -

 25 Figure IV.25 -Steady-state simulation of water table recharge with distributions of hydraulic head h and flux. Hydraulic head contours are in red. Calculated water table and capillary fringe are respectively represented by a blue solid line and a blue dotted line. Those of Vauclin et al. [220] are represented by an orange solid line and an orange dotted line.

Figure IV. 26 -

 26 Figure IV.26 -Numerical simulation of two-dimensional recharge with water table position compared to experimental data from Vauclin et al. [220].

Figure IV. 27 -

 27 Figure IV.27 -Numerical results from Vauclin et al.[START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF] for the distribution of hydraulic head and flux, the location of water table (solid line) and capillary fringe (dotted line) at t = 3 h (top) and t = 8 h (bottom). The numbers with crosses represent experimental data of hydraulic head (cm). After Vauclin et al.[START_REF] Vauclin | Experimental and numerical study of a transient, twodimensional unsaturated-saturated water table recharge problem[END_REF].

Figure IV. 28

 28 Figure IV.28 illustrates a typical field distribution of fluxes and hydraulic head as well as the location of water table and capillary fringe (set at θ = 0.29), at time t = 3 h and t = 8 h. Hydraulic head values have been turned upside down to correspond with a downward vertical axis and to facilitate comparison with the numerical and experiment results from Vauclin et al. [220] provided in Fig. IV.27. At t = 8 h, Vauclin et al. [220] stated that a steady state was reached experimentally. Numerically, the present results are not in a steady state because if the simulation is continued, one reaches results in Fig. IV.[START_REF] Suk | Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media[END_REF]. This shows that it is difficult to evaluate the attainment of a steady state due to the slow dynamics of porous media flow. In addition, one should note that the use of different basis functions and approximation order for quadrangles are determinant for the solution quality. Indeed, quadratic monomial basis functions are necessary to get equivalent results to linear Lagrangian basis functions. This is explained by the tensored term.

  drop of the imposed hydraulic head. The box is 200 cm × 300 cm. Hydraulic properties are taken from Vachaud's relations (see Tab. I.1) with A = 359720.49, B = 4.5, C = 40000, D = 2.9, K s = 40 cm/h, θ s = 0.3 and θ r = 0. This experiment is used as a test-case to show a transient simulation with seepage boundary condition. The domain is initially in hydrostatic equilibrium: h 0 = 145 cm. Boundary conditions at the bottom, at the top, on the right and on the left for z > 145 cm are ∇h • n = 0 (impervious). On the left, the water level is kept constant in the ditch for z ≤ 75 cm, h = 75 cm. A seepage boundary condition is prescribed for 75 cm< z ≤ 145 cm. Simulation lasts T = 20 h. The configuration is sketched in Fig. IV.29.

Figure IV. 29 -

 29 Figure IV.29 -Water table lowering test-case configuration.

Figure IV. 30 -

 30 Figure IV.30 -Numerical simulation of two-dimensional lowering with water table position compared to experimental data from Vauclin et al. [218].

Figure IV. 31

 31 Figure IV.31 illustrates a typical field distribution of fluxes and hydraulic head as well as the location of water table and capillary fringe (set at θ = 0.29), at time t = 0.1 h and t = 1 h.

Figure IV. 31 -

 31 Figure IV.31 -Distribution of hydraulic head h (turned upside down) and flux, location of water table (white line) and capillary fringe (magenta line) at t = 0.1 h (top) and t = 1 h (bottom) for water table lowering. Hydraulic head contours are in red.

  I.1) with A = 359720.49, B = 4.5, C = 40000, D = 2.9, K s = 40 cm/h, θ s = 0.3 and θ r = 0. This experiment is used as a test-case to show a transient simulation with seepage boundary condition. The domain is initially in hydrostatic equilibrium: h 0 = 143 cm. Boundary conditions at the bottom, at the top, on the right for z > 143 cm and on the left for z > 143 cm are ∇h • n = 0 (impervious). The water level is kept constant in the right ditch for z ≤ 143 cm, h = 143 cm, and, in the left ditch for z ≤ 80 cm, h = 80 cm. A seepage boundary condition is prescribed for 80 cm< z ≤ 143 cm. Simulation lasts T = 10 h. The configuration is sketched in Fig. IV.32.

Figure IV. 32 -

 32 Figure IV.32 -Water table drawdown test-case configuration.

Figure IV. 33 -

 33 Figure IV.33 -Numerical simulation of two-dimensional drawdown with water table position compared to experimental data from Vauclin et al. [217].

Figure IV. 34

 34 Figure IV.34 illustrates a typical field distribution of fluxes and hydraulic head as well as the location of water table and capillary fringe (set at θ = 0.29), at time t = 1 h.

  Fig. IV.35 and IV.38. The three sensors are at z = 55 m and their x-coordinates are -10.5 m, -3 m and 5 m. For the simulation, the reservoir height is used as a forced boundary condition in the form of a Dirichlet boundary condition monitored by a function based on experimental data (see Fig. IV.36

  Finally, seepage boundary condition are prescribed everywhere else because outflow can drain from the exposed upstream and downstream shell slopes. The actual simulated configuration of La Verne dam is sketched in Fig. IV.35. Five different materials are simulated (Tab. IV.3), with hydraulic properties provided by Bonelli et al. [91]. Their quotation marks designation in Tab. IV.3 does not reflect the actual soil texture but can serve as a guide. The dam wetting simulation involves the main tools presented in this thesis for the adaptive strategy: the WDG method in combination with the AMR technique driven by the estimation-based error indicator. Mesh adaptation is done every five time steps. Fixed-point iteration is used together with adaptive time stepping. Order of approximation is quadratic for space and one for time (implicit Euler) in order to have a robust and cost-effective computation. Penalty parameters are σ I = σ D = 100.

Figure IV. 35 - 1 64 A 64 A

 3516464 Figure IV.35 -Geometry, materials and boundary conditions (BC) of La Verne dam for the numerical case. S1, S2, S3 are numerical sensors corresponding to the experimental sensors while S4 and S5 are additional purely numerical sensors.

Figure IV. 36

 36 Figure IV.36 depicts hydraulic head (color contours), water table (white line) and adaptive mesh for both simulations every 10 days. Number of elements, magnitude of time steps and the value of global error estimate during the simulation are shown in Fig. IV.37. Time evolution of simulated hydraulic head and water content are displayed in Fig. IV.38 at the position of the three experimental sensors (S1 to S3) and for two additional relevant points called two additional numerical sensors (S4 and S5).

Figure IV. 36 -

 36 Figure IV.36 -Time evolution of hydraulic head h, mesh and water table (white line) during LaVerne dam wetting.

  Fig. IV.36 at some points due to insufficient resolution and oscillations w.r.t the front sharpness, which are detailed in Section IV.7.3 thereafter. The numerical model is observed to provide a good overall description of the dam wetting process. Figure IV.36 shows that the propagation of the wetting wave inside the dam is strongly dependent on the inner materials properties.

  Figure IV.37 -Evolution of some numerical variables during La Verne simulation.

  IV.38c), indicating the crossing of a sharp capillary fringe. While rising, the water table remains nearly horizontal in the upstream m2-and m3-materials zone but a sharp front is observed to develop across the left inner drain (m5-material), see Fig. IV.36. Small numerical oscillations in hydraulic head are observed in Fig. IV.38 but do not affect the global dynamics. Further downstream and lower within the dam, the hydraulic head simulated at S1, S2 and S3 shows the slow wetting wave propagation, with attenuation and damping (Fig. IV.38b). At S1, S2 and S3, the water content remains constant during the whole simulation, in accordance to the sensors position below water table.

Figure IV. 38 -

 38 Figure IV.38 -Time evolution of inner dam hydraulic parameters. a) Hydraulic head at sensors S1 (short-dashed lines), S2 (dash-dotted lines) and S3 (long-dashed lines) from the present simulation (red), the experimental data (light blue) and the previous results of Bonelli et al. [91]. The reservoir height is depicted in dark solid line. b) Hydraulic head at numerical sensors S4 (orange) and S5 (yellow). c) Water content at S1 to S5 sensors (note that S1, S2 and S3 show similar values).

IV. 7 .

 7 Figure IV.39 -Elevated-view of hydraulic head h with mesh for La Verne dam wetting at t = 30 days. Point of view is from above, downstream.

Figure IV. 42 -

 42 Figure IV.42 -Time evolution of hydraulic head h, mesh and water table (white line) during LaVerne dam wetting for the augmented simulation.

Figure IV. 43

 43 Figure IV.43 illustrates the issues of La Verne dam simulation with the augmented computation. Time step is decreasing at 10 days when water table meets the m3-material. Two drastic reductions happen next: around 15-20 days and 28 days. This corresponds to the water table moving through m4material and reaching the filters made of m5-material. The solver struggles in these conditions to reach convergence which makes time step decrease drastically. At the end of the simulation, time step is increasing largely which matches drainage with the drop of the reservoir level. One can also remark that the number of elements is increasing at the beginning of the simulation in the course of successive refinements. The errors are controlled which is shown by the global estimator remaining quite low. However, as soon as the specified maximum level of refinement is reached within a block, the error is increasing freely. It would be interesting to attempt different process for the selection of elements to be refined/coarsened.

  Figure IV.43 -Evolution of some numerical variables during La Verne simulation with augmented computation.

ChapterV. 1 112 V. 1 . 1 112 V. 1 . 2 114 V. 1 . 3 116 V. 2 V. 3

 111211112121141311623 Applications to swash groundwater dynamics in sandy beaches"La mer, la mer, toujours recommencée !" -Paul Valéry, Le cimetière marin, 1920 Wave-forced beach groundwater dynamics . . . . . . . . . . . . . . . . . . Waves hydrodynamic forcing . . . . . . . . . . . . . . . . . . . . . . . Issues on groundwater beach dynamics . . . . . . . . . . . . . . . . . . Beach response to a single swash event . . . . . . . . . . . . . . . . . . 115 V.1.4 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idealized beach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Swash groundwater dynamics prospects . . . . . . . . . . . . . . . . . . . 120

Figure V. 1 -

 1 Figure V.1 -Sketch of the swash event consisting in uprush and backwash.

Figure V. 2 -

 2 Figure V.2 -Swash on Coumeenoole beach, Ireland.

V. 3 .

 3 In reality, there is obviously a whole range of intermediate conditions.

Figure V. 3 -

 3 Figure V.3 -Beach classification taken from Wright and Short [232]. The classification depends on the angle of the beach slope β and on the wave condition.

  Figure V.4 -Examples of hydrodynamic processes in the swash zone.

  Figure V.6 shows the beach profile and the mesh used. Y-axis is multiplied by 20 to make it more visible.

Figure V. 6 -

 6 Figure V.6 -Topography of the idealized beach and mesh used for the simulation.

A

  first observation concerns wave propagation. When a wave is moving, groundwater flows in the saturated zone respond immediately. Hydraulic head spreads vertically quite sharply under the wave front. Infiltration/exfiltration occurs at the back/front of the bore respectively. It is particularly visible in Fig. V.7 even if this phase is completely ideal because it is about the first wave moving through hydrostatic conditions.

Figure V. 7 -

 7 Figure V.7 -Snapshot of hydraulic head contours at t = 12.5 s during the first wave propagation on calm waters.

Figure V. 8 -

 8 Figure V.8 -Snapshot of hydraulic head contours at t = 37.5 s for the end of the first uprush.Velocities range from about 0.3 to 0.6 cm/h in the beach. Maximum velocity is 35 cm/h for infiltration.

Figure V. 9 -

 9 Figure V.9 -Snapshot of hydraulic head contours at t = 57.5 s for the end of the first backwash.Most velocities are less than 1 cm/h.

1 Figure V. 11 -

 111 Figure V.11 -Examples of experimental data collected during the ROUSTY campaign. Taken from Sous et al. [6].

1

  Examples of coastal urbanization and threat of sea level rise for beaches with high socio-economic services. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1 Schematic representation of a part from two different porous media. . . . . . . . . I.2 The different zones of saturation in a porous medium. . . . . . . . . . . . . . . . I.3 Scale of grain size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.4 Photos with a schematic close-up of a sample of sand-, silt-, clay-sized particles. . . . I.5 Soil texture classification diagram from USDA. Textures are determined by the proportions of sand-, silt-, and clay-sized grains. . . . . . . . . . . . . . . . . . . . . I.6 Basic concepts for different types of water in soils . . . . . . . . . . . . . . . . . . I.7 Saturation regimes in porous medium showing spatial configurations of air and water. Adapted from Szymkiewicz [22]. . . . . . . . . . . . . . . . . . . . . . . . . I.8 Components of the head for water. . . . . . . . . . . . . . . . . . . . . . . . . . I.9 Diagram of coordinate systems and scale hierarchy for non-periodic media typical of volume averaging (left) and periodic media typical of homogenization (right). µ is pore-scale characteristic length, ℓ is the size of averaging volume, L is a macroscale characteristic lengths, x is spatial variable coordinate system, x * and y * are nondimensionalized coordinate systems with L and ℓ respectively. Taken from Davit et al. [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.10 Range of validity of Darcy's law. Darcy zone delimits the linear approximation of the flow-pressure gradient relation. Outside, nonlinear deviation are observed. . . . . . I.11 Comparison of different interface models for porous media flow. Taken from Fujisawa and Murakami [67]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.12 Volume averaging procedure to get an equivalent homogeneous macroscale medium from the heterogeneous micro-structure. The red square stands for the representative elementary volume. Adapted from Losada et al. [69]. . . . . . . . . . . . . . . I.13 Illustration of the coordinate system, notations and hierarchy of scale used for the volume averaging method used in this thesis. . . . . . . . . . . . . . . . . . . . . I.14 Concept for a definition of the Representative Elementary Volume (REV). . . . . . I.15 Principle of Gray's decomposition between average and deviation fields. . . . . . . I.16 Typical soil water characteristic curve with key elements. . . . . . . . . . . . . . . I.17 Principle of the seepage boundary condition for the pressure head ψ. . . . . . . . . II.1 Typical example of a 2D-solution representation arising from different methods of approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2 Examples of mesh geometry for discontinuous Galerkin methods. . . . . . . . . . IV.18 Effectivity indices for different indicators (left) and convergence diagrams for the three parts of the indicator η n E (right). P1, P2, P3 and P4 stands respectively for linear, quadratic, cubic and quartic approximations. . . . . . . . . . . . . . . . . . . . . IV.19 True error distribution in the L 2 -norm on Tracy's benchmark. . . . . . . . . . . . IV.20 Comparison between distributions of true error in energy norm and the indicator η n E on Tracy's benchmark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.21 Comparison with L 2 -norm between C E -based adaptation (left) and η n E -based adaptation on Tracy's benchmark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.22 Comparison with energy norm between C E -based adaptation (left) and η n E -based adaptation on Tracy's benchmark. . . . . . . . . . . . . . . . . . . . . . . . . . IV.23 Water

LISTS IV. 38

 38 Time evolution of inner dam hydraulic parameters. a) Hydraulic head at sensors S1 (short-dashed lines), S2 (dash-dotted lines) and S3 (long-dashed lines) from the present simulation (red), the experimental data (light blue) and the previous results of Bonelli et al.[START_REF] Bonelli | Résolution auto-adaptative par éléments finis de problèmes de diffusion fortement non linéaires[END_REF]. The reservoir height is depicted in dark solid line. b) Hydraulic head at numerical sensors S4 (orange) and S5 (yellow). c) Water content at S1 to S5 sensors (note that S1, S2 and S3 show similar values). . . . . . . . . . . . . . . . . IV.39 Elevated-view of hydraulic head h with mesh for La Verne dam wetting at t = 30 days. Point of view is from above, downstream. . . . . . . . . . . . . . . . . . . . IV.40 Heterogeneity and nonlinear varying diffusivity highlighted respectively by water content and Péclet number for La Verne dam simulation with the reference simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.41 Hydraulic properties functions and Péclet number (with L = 1) for m2-and m3materials (Van Genuchtem-Mualem and Vachaud's relations respectively) for La Verne dam in the horizontal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.42 Time evolution of hydraulic head h, mesh and water table (white line) during La Verne dam wetting for the augmented simulation. . . . . . . . . . . . . . . . . . IV.43 Evolution of some numerical variables during La Verne simulation with augmented computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.1 Sketch of the swash event consisting in uprush and backwash. . . . . . . . . . . . V.2 Swash on Coumeenoole beach, Ireland. . . . . . . . . . . . . . . . . . . . . . . . V.3 Beach classification taken from Wright and Short [232]. The classification depends on the angle of the beach slope β and on the wave condition. . . . . . . . . . . . . V.4 Examples of hydrodynamic processes in the swash zone. . . . . . . . . . . . . . . V.5 Conceptual view of the main flow patterns of beach groundwater induced by swash. Blue lines and arrows are for surface dynamics, black arrows for groundwater dynamics, shaded area depicts the saturated beach and red arrows for infiltration/seepage. . V.6 Topography of the idealized beach and mesh used for the simulation. . . . . . . . . V.7 Snapshot of hydraulic head contours at t = 12.5 s during the first wave propagation on calm waters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.8 Snapshot of hydraulic head contours at t = 37.5 s for the end of the first uprush. Velocities range from about 0.3 to 0.6 cm/h in the beach. Maximum velocity is 35 cm/h for infiltration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.9 Snapshot of hydraulic head contours at t = 57.5 s for the end of the first backwash. Most velocities are less than 1 cm/h. . . . . . . . . . . . . . . . . . . . . . . . . . V.10 Snapshot of hydraulic head contours at t = 67.5 s for the next incoming uprush. Most velocities are less than 0.8 cm/h. . . . . . . . . . . . . . . . . . . . . . . . . V.11 Examples of experimental data collected during the ROUSTY campaign. Taken from Sous et al. [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.12 Time stepsisize (orange) and number of nonlinear iterations (purple) variations through the simulation for Vogel-Van Genuchten-Cislerova test-cases. . . . . . . . . . . . . V.13 Illustration of the slide during the talk for MT180. "But what is really happening under beach sand?", "You shall not pass!!! (sand grain)", "Yeeees!!!". . . . . . . . .

  The air entry pressure is denoted by ψ e[L]. It defines saturated and unsaturated zones inside the porous medium. Most of the time, ψ e is set to 0 in the literature. The saturated zone of the porous medium corresponds to ψ > ψ e while the unsaturated or vadose zone corresponds to ψ < ψ e . The water table corresponds to ψ = ψ e by definition and is considered belonging to the saturated zone. The capillary fringe is the layer above water table where water is raised due to capillary actions. It belongs to the unsaturated zone and there is no standard definition for its upper limit. The hydraulic properties present two different behaviours depending on whether the porous media is saturated (ψ ≥ ψ e ) or not (ψ < ψ e ):

Table II .

 II II. NUMERICAL METHODS FOR RICHARDS' EQUATION 2 -Selected properties of some of the discontinuous Galerkin methods. σ * is a threshold value for the penalty parameters σ.

		Coercivity (stability)	Error estimation in L 2 -norm	Remarks
	IIPG	σ > σ *	Suboptimal	Simplest version
	SIPG	σ > σ *	Optimal	Symmetric consistent version
	NIPG	∀σ	Suboptimal	Optimal error estimation with superpenalization

  The 1-step BDF method is the backward Euler scheme. BDF methods were sparsely used for Richards' equation. From author's knowledge, there are mainly a study of Sochala et al.[START_REF] Sochala | Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows[END_REF][START_REF] Sochala | Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement[END_REF] with 1-, 2-and 3-order BDF methods and a series of papers from collaborating authors[START_REF] Li | Adaptive local discontinuous Galerkin approximation to Richards' equation[END_REF][START_REF] Li | Local discontinuous Galerkin approximations to Richards' equation[END_REF][START_REF] Kees | Higher order time integration methods for two-phase flow[END_REF][START_REF] Tocci | Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines[END_REF][START_REF] Farthing | Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow[END_REF] with 1-to 6-order BDF methods. It may be possible to find few other examples of 2-order BDF methods for Richards' equation. For this study, BDF methods are interesting because they provide high-order accuracy in time which is suitable if one wants to reach high-order in space without loss of gain. In other words, the BDF methods will allow to balance space and time errors.

  3 1 /4 5 137 /60 -5 5 -10 /3 5 /4 -1 /5 6 49 /20 -6 15 /2 -20 /3 15 /4 -6 /5 1 /6 Table II.3 -Coefficients α q,k for all BDF methods with fixed time stepsize.

	α s,k	0	1	2	3 k	4	5	6

68 III.3 Variation on the numerical scheme . . . . . . . . . . . . . . . . . . . . . . 70 III

  There is nothing like looking, if you want to find something. You certainly usually find something, if you look, but it is not always quite the something you were after." .3.1 Weighted discontinuous Galerkin framework . . . . . . . . . . . . . . 70 III.3.2 Extrapenalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

	Chapter
	Adaptive strategy based on
	discontinuous Galerkin method

"-J. R. R. Tolkien, The Hobbit, or There and Back Again, 1937 III.1 Adaptive mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 III.1.1 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 III.1.2 Block-based structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 III.1.3 Selection procedure and corrective mechanism . . . . . . . . . . . . . . 66 III.1.4 Projection by local DG problem . . . . . . . . . . . . . . . . . . . . . 67 III.2 A posteriori error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 67 III.2.1 Gradient-based error indicator . . . . . . . . . . . . . . . . . . . . . . 67 III.2.2 Estimation-based error indicator . . . . . . . . . . . . . . . . . . . . .

95 IV.7 Assessment of La Verne dam wetting . . . . . . . . . . . . . . . . . . . . . 99 IV

. . 91 IV.6.1 Water table recharge . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 IV.6.2 Water table drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.1 Experimental configuration . . . . . . . . . . . . . . . . . . . . . . . 99 IV.7.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 IV.7.3 Discussion and investigation . . . . . . . . . . . . . . . . . . . . . . . 104

Table IV

 IV

						IV.1. NUMERICAL EXPERIMENTS
			Method		NIPG	SIPG	IIPG
			Theoretical order	p + 1 if p odd p if p even	p + 1	p + 1 if p odd p if p even
		Table IV.1 -Theoretical convergence rates for 1D uniform meshes
			Linear approximation Quadratic approximation Cubic approximation
	Method	h	∥e h ∥ L 2 (0,1)	Order ∥e h ∥ L 2 (0,1)	Order	∥e h ∥ L 2 (0,1) Order
		1/2	9.7104E-02	-	1.2899E-02	-	6.2398E-04	-
		1/4	2.6019E-02 1.8999 4.1564E-03	1.6339	4.2126E-05 3.8887
	NIPG	1/8	6.5796E-03 1.9835 1.2318E-03	1.7545	2.7853E-06 3.9188
		1/16 1.6361E-03 2.0078 3.2297E-04	1.9314	1.7792E-07 3.9685
		1/32 4.0652E-04 2.0088 8.1698E-05	1.9830	1.1217E-08 3.9875
		1/2	1.6443E-01	-	2.0337E-02	-	6.0659E-04	-
		1/4	3.4034E-02 2.2724 2.4644E-03	3.0448	2.3031E-05 4.7191
	SIPG	1/8	7.0833E-03 2.2645 1.7036E-04	3.8545	1.1354E-06 4.3423
		1/16 1.3566E-03 2.3844 1.4595E-05	3.5450	5.3777E-08 4.4000
		1/32 2.4925E-04 2.4444 1.2907E-06	3.4992	2.4664E-09 4.4465
		1/2	1.37511E-01	-	4.4606E-02	-	4.4968E-03	-
		1/4 3.53440E-02 1.9600 7.3239E-03	2.6066	2.2775E-04 4.3034
	IIPG	1/8 8.67816E-03 2.0260 1.5210E-03	2.2676	1.3408E-05 4.0863
		1/16 2.11935E-03 2.0338 3.5286E-04	2.1079	8.1252E-07 4.0445
		1/32 5.20962E-04 2.0244 8.5108E-05	2.0517	4.9747E-08 4.0297

1. Numerical errors and convergence rates are shown in Tab. IV.2. .2 -Numerical errors and order of convergence for solution of problem Eq. (IV.1)

  table have been raisedV.[START_REF] Vousdoukas | Sandy coastlines under threat of erosion[END_REF]. IDEALIZED BEACH by the precedent uprush. It results in recharge and, as water is percolating inside the porous medium, flow divides between seaward and landward direction. Moreover, above the unsaturated zone which has been covered by uprush, water is infiltrating downward in a sharp wetting front. While the swash toe retreating to the sea, a seepage face is appearing which drops, locally but quickly, the water table what gives form to a small hump-like structure. Figure V.9 presents these observations. The next incoming wave arrives just after the backwash. It pushes the region of low hydraulic head up. As this zone swept away, it reaches the remaining small region of high hydraulic head induced by the water table elevation and makes it disappear temporally. The incoming wave is bringing a new high hydraulic head front. This causes infiltration which arises water table again, ending seepage and erasing the hump-like structure of water table. Then, a new cycle begins. This intermediate state is shown in Fig. V.10.

  • Groundwater pressure stays nearly hydrostatic under dry area, in the upper beach. Incoming waves propagate groundwater pressure front; • The swash tongue causes overall infiltration during the bed flooding what elevates water table next to beachface and creates sharp wetting front in the unsaturated zone lapped by swash. For charge induced by water table, flow divides into landward and seaward components; • During backwash, seaward groundwater can be observed under the swash tongue moving back to the sea. Water table exit point follows the swash toe with delay because of sand hydraulic properties, resulting in seepage; • Two localized groundwater circulation zones evolve and interact during the simulation.

  Luijendijk et al. ont constaté que plus d'un tiers des côtes sont sableuses. L'analyse des données satellitaires sur une périodede 33 ans (1984-2016) indique qu'un quart des plages de sable s'érodent à des taux supérieurs à 0,5 m/an. Les littoraux sableux présentent des avantages socio-économiques liés au tourisme et aux services écosystémiques. Ils font l'objet de vives préoccupations en raison du changement climatique, car l'élévation du niveau de la mer menace les plages sableuses d'une érosion, comme affirmé par Vousdoukas et al.. Cooper et al.[START_REF] Cooper | Sandy beaches can survive sea-level rise[END_REF] ont nuancé la disparition des plages sableuses qui peuvent se retirer à l'intérieur des terres. Les plages sableuses restent soumises à d'importants changements et peuvent disparaître si l'urbanisation bloque leur migration. En outre, du point de vue de la modélisation, les plages sableuses sont généralement des plages dissipatives à faible pente pour lesquelles les écoulements souterrains sont connus pour être lents avec une faible rétroaction sur les vagues incidentes. C'est pourquoi un couplage faible de l'écoulement de surface sur l'écoulement souterrain peut être envisagé pour cibler des simulations applicatives. Ces raisons font que la thèse priorise l'étude des écoulements souterrains pour les plages sableuses.À cette fin, une méthode de Galerkine discontinue (GD) est choisie pour résoudre l'équation de Richards. Les méthodes GD sont basées sur une formulation variationnelle élément par élément, partageant des avantages des méthodes par éléments finis et par volumes finis. En particulier, elles sont localement conservatives, ce qui est crucial en mécanique des fluides[START_REF] Rivière | Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations[END_REF]. De plus, la nature de la for-

mulation GD permet de travailler sur des maillages non-conformes et de modifier localement le degré d'approximation polynomiale. C'est un avantage important car le Raffinement du Maillage Adaptatif (RMA), appelé adaptation h, et la des méthodes d'ordre élevé, appelée adaptation p, deviennent possibles

  • Les paramètres de pénalisation assurent la stabilité des méthodes GD et imposent les conditions aux limites. Cependant, ils peuvent également dégrader les propriétés numériques avec des valeurs arbitrairement grandes, en particulier le conditionnement, ce qui entraîne des instabilités. De plus, ils offrent un moyen d'incorporer dans le schéma une connaissance partielle de la solution. Une analyse numérique et d'autres tests seraient effectués afin de dériver une définition adaptée au problème ou, du moins, des bornes appropriées; • Le pas de temps adaptatif contrôle la convergence itérative non-linéaire. Cependant, la méthode n'empêche pas le pas de temps de diminuer au point de rendre le calcul impossible. Qui plus est, la méthode est heuristique et il est difficile de savoir si du temps de calcul est économisé puisque le facteur d'amplication/réduction est un paramètre défini par l'utilisateur. Une possibilité serait d'utiliser un estimateur d'erreur temporelle. Une autre possibilité serait d'étudier la convergence de la procédure non-linéaire qui dépend du pas de temps et d'en déduire une méthode. Un pas de temps local pourrait être une idée pour économiser du temps de calcul mais la mise en oeuvre est difficile avec des schémas implicites; • La sélection de l'ordre temporel est pour l'instant fixée dans le code. Cette partie devrait être entièrement automatique et fournir la meilleure FDR pour garantir la stabilité. Cela se fait généralement au moyen d'estimateurs d'erreurs ou d'une analyse de stabilité; • L'adaptation hp serait une solution efficace pour la solution de l'équation de Richards en effectuant un raffinement du maillage ou un enrichissement de l'ordre suivant la meilleure possibilité. De nombreuses techniques sont déjà présentes dans la littérature pour prendre des décisions de type hp. Certaines remarques dans cette thèse laissent penser que les sous-parties de l'indicateur d'erreur basé sur l'estimation peuvent conduire à la décision entre l'adaptation h et l'adaptation p. Même si cette adaptation hp ne sera pas optimale, la technique serait très simple et rentable. D'autres tests sont nécessaires et une véritable analyse d'estimation d'erreur a posteriori pourrait être utile; • L'adaptabilité est un problème pour la fréquence de raffinement et la sélection des éléments à raffiner. Un seuillage automatique basé sur une fonction de distribution serait une possibilité pour décider des éléments à raffiner/déraffiner. La dérivation de la vitesse des fronts de saturation serait un moyen de définir la fréquence de raffinement en fonction de la taille du bloc; • La parallélisation et la 3D sont des extensions naturelles du code Rivage pour traiter des cas plus complexes et gagner du temps de calcul. Même si le code est préparé pour cela (structure basée sur des blocs conduisant à la décomposition de domaine et implémentation générale), ce ne sera pas simple car la 3D implique de nombreuses géométries d'éléments et que le schéma temporel est implicite. En outre, l'ajout d'un module pour traiter des équations hyperboliques serait une possibilité à étudier afin d'incorporer des modèles d'écoulement de surface et effectuer un couplage adéquat. Plusieurs parties de la thèse ont été présentées lors de conférences et de séminaires dont certains ont donné lieu à des actes. Un article a été proposé pour publication à un journal dans lequel la stratégie adaptative GD est décrite afin de simuler l'équation de Richards. Un autre article est prévu pour présenter et discuter des applications des écoulements souterrains aux plages sableuses. Ces travaux sont répertoriés dans la List of Publications, page 145.

  table recharge test-case configuration. . . . . . . . . . . . . . . . . . . . . IV.24 Mesh used for the steady-state simulation of water table recharge test-case. . . . . . IV.25 Steady-state simulation of water table recharge with distributions of hydraulic head h and flux. Hydraulic head contours are in red. Calculated water table and capillary fringe are respectively represented by a blue solid line and a blue dotted line. Those of Vauclin et al. [220] are represented by an orange solid line and an orange dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.26 Numerical simulation of two-dimensional recharge with water table position compared to experimental data from Vauclin et al. [220]. . . . . . . . . . . . . . . . . IV.27 Numerical results from Vauclin et al. [220] for the distribution of hydraulic head and flux, the location of water table (solid line) and capillary fringe (dotted line) at t = 3 h (top) and t = 8 h (bottom). The numbers with crosses represent experimental data of hydraulic head (cm). After Vauclin et al. [220]. . . . . . . . . . . . . . . . IV.28 Distribution of hydraulic head h (turned upside down) and flux, location of water table (white line) and capillary fringe (magenta line) at t = 3 h (top) and t = 8 h (bottom) for water table recharge. Hydraulic head contours are in red. . . . . . . . IV.29 Water table lowering test-case configuration. . . . . . . . . . . . . . . . . . . . . IV.30 Numerical simulation of two-dimensional lowering with water table position compared to experimental data from Vauclin et al. [218]. . . . . . . . . . . . . . . . . IV.31 Distribution of hydraulic head h (turned upside down) and flux, location of water table (white line) and capillary fringe (magenta line) at t = 0.1 h (top) and t = 1 h (bottom) for water table lowering. Hydraulic head contours are in red. . . . . . . . IV.32 Water table drawdown test-case configuration. . . . . . . . . . . . . . . . . . . . IV.33 Numerical simulation of two-dimensional drawdown with water table position compared to experimental data from Vauclin et al. [217]. . . . . . . . . . . . . . . . . IV.34 Distribution of hydraulic head h (turned upside down) and flux, location of water table (white line) and capillary fringe (magenta line) at t = 1 h for water table drawdown. Hydraulic head contours are in red. . . . . . . . . . . . . . . . . . . . . . IV.35 Geometry, materials and boundary conditions (BC) of La Verne dam for the numerical case. S1, S2, S3 are numerical sensors corresponding to the experimental sensors while S4 and S5 are additional purely numerical sensors. . . . . . . . . . . . . . .

IV.

[START_REF] Ern | A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity[END_REF]

Time evolution of hydraulic head h, mesh and water table (white line) during La Verne dam wetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.37 Evolution of some numerical variables during La Verne simulation. . . . . . . . . .

Baptiste CLÉMENT Simulation numérique des écoulements en milieu poreux non-saturés par une méthode de Galerkine discontinue adaptative : application aux plages sableuses Résumé :

  LISTS • WCCM-ECCOMAS Congress 2020, hp-Adaptation for Discontinuous Galerkin Methods Applied to Variably-Saturated Flows, Paris, France, 12 January 2021. Re-scheduled to January 2021 due to COVID-19 pandemic (virtual event). • TPFM 2020, Adaptive Discontinuous Galerkin Method for Richards Equation, Prague, Czech Republic, 20 February 2020. • Workshop GLADYS New Eve 2020, Simulation numérique des écoulements insaturés dans les plages sableuses, Université de Montpellier/MUSE, France, 7 January 2020. • Current Problems in Numerical Analysis, seminar from Institute of Mathematics of the Czech Academy of Sciences, Richards equation for flows in sandy beaches: from modelling to Discontinuous Galerkin simulation, Prague, Czech Republic, 25 october 2019. • TPFM 2019, Discontinuous Galerkin Method for Steady-State Richards Equation, Prague, Czech Republic, 20 February 2019. • IMATH laboratory seminar, Écoulements en milieux poreux non-saturés par méthode de Galerkine discontinue, La Garde, France, 7 February 2019. Les écoulements en milieux poreux non-saturés sont modélisés par l'équation de Richards qui est une équation non-linéaire parabolique dégénérée. Ses limites et les défis que soulèvent sa résolution numérique sont présentés. L'obtention de résultats robustes, précis et efficaces est difficile en particulier à cause des fronts de saturation raides et dynamiques induits par les propriétés hydrauliques non-linéaires. L'équation de Richards est discrétisée par une méthode de Galerkine discontinue en espace et des formules de différentiation rétrograde en temps. Le schéma numérique résultant est conservatif, d'ordre élevé et très flexible. Ainsi, des conditions aux limites complexes sont facilement intégrées comme la condition de suintement ou un forçage dynamique. De plus, une stratégie adaptative est proposée. Un pas de temps adaptatif rend la convergence non-linéaire robuste et un raffinement de maillage adaptatif basée sur des blocs est utilisée pour atteindre la précision requise efficacement. Un indicateur d'erreur a posteriori approprié aide le maillage à capturer les fronts de saturation raides qui sont également mieux approximés par une discontinuité introduite dans la solution grâce à une méthode de Galerkine discontinue pondérée. L'approche est validée par divers cas-tests et un benchmark 2D. Les simulations numériques sont comparées à des expériences de laboratoire de recharge/drainage de nappe et une expérience à grande échelle d'humidification, suite à la mise en eau du barrage multi-matériaux de La Verne. Ce cas exigeant montre les potentialités de la stratégie développée dans cette thèse. Enfin, des applications sont menées pour simuler les écoulements souterrains sous la zone de jet de rive de plages sableuses en comparaison avec des observations expérimentales.

	Jean-

I.1 Hydraulic relations used in this thesis. . . . . . . . . . . . . . . . . . . . . . . . .

Summary

Since the solution of Richards' equation is not available analytically for most cases, numerical methods are necessary. DG methods based on penalty are proposed in this study because they show useful mathematical properties. Indeed, they rely on an element-wise weak formulation which turns to be flexible, both mathematically and numerically. This allows to handle complexed modelling features and advanced computational methods. In particular, the treatment of seepage boundary condition or dynamic forcing condition is natural since they are directly incorporated into the weak formulation thanks to penalization. Moreover, penalty and symmetry terms provide stability and continuity at different level of control following the variants of the formulation. Finally, the lack of continuity constraint between elements gives the possibility to use different polynomial approximations and non-conforming mesh which are the basis properties to reach high-order approximation and use adaptive mesh refinement. The scheme is mass conservative at machine precision thanks to natural local mass balance. Implicit schemes are chosen for time discretization to guarantee stability. Backward difference formula methods are chosen because they are known to be good to deal with stiff equations and they provide high-order time scheme which is convenient if spatial errors are greatly reduced. Due to nonlinearities present in water content and hydraulic conductivity, solving Richards' equation requires a robust iterative process. Even though the linearization by the classic Newton-Raphson method is quadratic convergent, it suffers from reliability because of poor initial guess or singular Jacobian matrix. The fixed-point method is preferred because it is more robust despite being linearly convergent. Robustness is strengthened thanks to an heuristic adaptive time stepping based on nonlinear iterations. The numerical methods were implemented in Fortran from scratch to build a computational code named Rivage. This program was thought to be as general as possible to handle a large range of user choices: non-conforming/unstructured/hybrid mesh of triangles/quadrangles, solvers based on different matrix storage, high-order approximation, several basis functions, etc. Future extensions should be done naturally like parallelizing and 3D. Figure IV.41 illustrates several hydraulic properties for m2-and m3-materials in the horizontal direction. m3-Material holds stronger gradients than m2-material in the vicinity of zero (capillary fringe).

In particular, hydraulic diffusivity D shows a singularity for ψ -→ 0 -. Besides, the Péclet number is higher for m3-material with a maximum corresponding to the foot of hydraulic conductivity.

In order to overcome these limitations, an additional ad-hoc configuration, named augmented simulation, has been implemented. This one does not use WDG framework and other difference lies in the mesh and tolerances for nonlinear convergence criterion. Finer discretization is allowed in permeable material that is to say the m1-, m2-and m3-materials. The mesh is also refined around water table and both for the gradient-based and estimation-based error indicators. Instead of the reference simulation, 4-order BDF is employed and hydraulic properties are relaxed with a numerical minimal value to avoid complete degeneracy. The augmented computation allowed to eliminate spurious oscillations from the solution thanks to a more refined mesh, which explains why WDG method is not needed. As the mesh is extremely refined, the augmented simulation takes approximately 13.

V.1.4 Coupling

Surface and subsurface flows are a unified system by nature. For technical reasons (space/time scales, computational solvability), they are often divided into two sub-models which work together through interface conditions making a coupling. Numerically, different coupling schemes are possible, see the review of Furman [START_REF] Furman | Modeling Coupled Surface-Subsurface Flow Processes: A Review[END_REF] or the work of Caviedes-Voullième et al. [START_REF] Caviedes-Voullième | Numerical simulation of groundwatersurface interactions by external coupling of the 3D Richards equation and the full 2D shallowwater equations[END_REF]:

• forced coupling, also known as, degenerated uncoupling, external coupling or weak coupling.

It is a one-way coupling without feedback, from surface to subsurface flow. This type of coupling only requires (spatially and temporally variable) input data for interface conditions (boundary conditions);

• parallel coupling. Surface and subsurface models are solved separately with their own time step but with the interface conditions (source terms) evolving according to the updated results. This type of coupling presents a feedback but it is always shifted, in a kind of interlacing;

• iterative coupling, also known as alternating coupling or strong coupling. Solution of surface and subsurface models are harmonised at each time step via an interface condition with an iterative procedure. This coupling is the most complete but require additional algorithmic and computational efforts;

• full coupling. Solution of surface and subsurface models together with interface conditions are obtained simultaneously.

Moreover, theoretically, interface conditions can come from different ideas, for instance mass conservation principle or Beavers-Joseph-Saffman conditions.

The simplest one has been chosen for this study: the forced coupling. Based on the solution of surface water sub-model, a boundary condition is prescribed for the groundwater model. It means that the surface model is solved before the groundwater model, and so, the surface sub-model cannot be corrected by feedback. One benefit of forced coupling is that implementation is straightforward because no infiltration/exfiltration model is necessary. This technique can approximate sandy beaches where pore velocities are too slow to affect surface water (dissipative beach). Nevertheless, it goes without saying that coupling should gain in maturity by reckoning with an infiltration model or mass conservative principle, even without considering retroaction.

The open-source SWASH code simulates the propagation of free surface waves according to selected input parameters at the offshore boundary. The governing equations are the nonlinear shallow water

Conclusion

I n this thesis, the main goal consisted in developing accurate, robust and efficient simulations of Richards' equation to model beach groundwater. For this purpose, a new strategy has been formulated for the solution of Richards' equation on the basis of DG method with BDF and AMR.

Summary

The Richards' equation approximation can reach high-order both in space and time, handle seepage, dynamic boundary conditions, heterogeneous porous media and deal with sharp drying/wetting fronts. Issues outlined in the introduction have been tackled and further discussed throughout the thesis so the following concluding remarks can be drawn:

• Richards' equation has been derived underlying clear hypotheses for the modelling;

• The seepage boundary condition has been directly incorporated into the DG weak formulation which makes its treatment natural and simple;

• The adaptive time stepping has allowed the nonlinear iterative solver to converge giving robustness;

• Mesh adaptation has been employed to monitor spatial errors of the Richards' equation by capturing the moving wetting fronts thanks to an estimation-based a posteriori error indicator;

• The AMR has been based on a block structure to surround large regions where wetting fronts move aiming to save computational time by avoiding systematic refinement. It also prepares a future parallelization;

• Oscillations in sharp wetting fronts of Richards' equation have been interpreted within the context of convection-diffusion equation. Wetting fronts have been considered as internal layers due to the nonlinear varying, and possibly vanishing, diffusivity as well as material heterogeneity and anisotropy. A local Péclet number has been exhibited to highlight troublesome regions. A WDG method has been used to allow for jumps in the solution which better approximate the sharp internal layers. Mesh adaptation and the weighted framework work in synergy to capture and resolve sharp wetting fronts through the proposed a posteriori error indicator.

Chapter I brought some insights about Richards' equation modelling possibilities. This nonlinear parabolic equation simulates an air-water system but considering only water phase. Thus, it is very useful to model flow in variably-saturated porous media but there are limitations such as air phase must stay continuously connected. Hydraulic properties suppose porous medium phases at equilibrium, disregarding hysteresis. Richards' equation modelling is still rich taking into account gravity and capillary effects. This double behaviour associated with nonlinearity make Richards' equation difficult to solve numerically for a wide range of applications.

Appendix A -Spatial averaging theorem

The derivation of the spatial averaging theorem may use different routes. The one presented here is done in the framework of convolution. The demonstration involves change of variables. That is why derivatives operators are indexed by the variable at stake.

A filtering operator is defined as the convolution product between a mask function m and the considered intrinsic quantity ξ F = 1 F ξ. This is called phase average:

The mask function m defines a weighted window which acts as a kernel in volume averaging. An analogy is possible with the theory of image processing. So m is set as:

Appendix B -Complementary results

Additional

Appendix C -Three Minute Thesis -Ma thèse en 180 secondes

Ma thèse en 180 secondes (MT180) is an annual popular science competition accessible to French speaking Ph.D. students worldwide. The concept is inspired from Three minute thesis (3MT) created by the University of Queensland in Australia. Participants present their research subject in French to a general and diversified audience with no background in the research area. The challenge is each student has three minutes exactly to give an intelligible and convincing talk with only one slide illustration. Competition is held in each university. Then, selected candidates competes each other successively at regional, national and international level.

I participated in Well, yes, there is water in the sand of the beach. Everybody knows that. But wait... What is all this water really doing in the sand of the beach? That is a riddle wrapped in a mystery inside an enigma. I'm going to tell you. Each wave, through its rising flow -infiltration -and its descending flowexfiltration -creates a slow underground circulation, so slow that experiments date back to not so long ago. We're talking about centimetres per hour. And now? Now, we want to know more. Does it depend on the type of waves or sand? What is the influence on erosion? And that's where my thesis comes in. I am interested in computer simulation. Ladies and gentlemen, welcome to the wonderful world of numerical data! Be careful, not the digital on the Internet or the digital on your tablet. No, it's applied mathematics. Does that scare you? Don't be scared, because there's a little magic down there. I warned you, we stayed in childhood. To make a good numerical simulation, we need to combine three ingredients. First, a model, a magic formula. The star of my thesis, I chose Richards' equation, which specifically describes the movement of water in sand even though air is present. Beware of modelling errors! Then, I need a spell, a numerical method that transforms my equation to solve it on a computer. After consulting the books of spells, I set my sights on a rather original method that requires sacrifices -computing ones -in time and memory, but thanks to its greater flexibility allows to achieve remarkable precision. Finally, you have to prepare the potion in a very special cauldron, a computer program called a calculation code. The whole process is tricky because I have to make sure that computation is done in a reasonable time and that the metamorphosis does not degrade the properties of the equation, otherwise I will get false results.