There is a lot of people that I would like to thank for a support during this thesis. First of all this thesis would not be possible without prof. Sonja Zlatovi and prof. Jolanta Lewandowska, their friendship provided me an opportunity to enroll doctoral study in Montpellier. There I was welcomed by prof. Philippe Gouze who was my mentor for three years. To Philippe I have to thank really a lot, he was not just my advisor who guided me through studies he also helped me a lot in my private life. Without him I wouldn't be able to rent a flat, solve numerous administrative problems, visit Barcelona, Gottingen, Padova, learn to wakeboard and actually he made this three years in Montpellier much nicer. Since, I started to talk about administrative problem I would like to express my gratitude to Bernadette Marie and Dominique Arnau who helped me overcome all administrative obstacles (and there is a lot of them in French administration).

I would also like to thank all the team (Richard, Gerard and Olivier (the big one)) in MSE that was really kind to me, and our conversations during the coffee breaks helped me to progress in French, beside helping me with technical parts of my manipulations or computers. Charlotte Garing was the most dedicated teacher trying to teach me French, I don't know why but she especially focused that J'ai and Je are not pronounced in a same way :D. At that time I was mostly hanging out with Ousman. He is a really great guy and big tnx to him for introducing me to all his friends, taking me out and for sure we had a great year in Montpellier. This chapter would not be complete without second (the small one) Olivier. This guy showed me the best hikes, MTB trails and other spots around Montpellier and also really helped me with programming, tnx big guy.

During my first year in Mtp a new post doc (Anna Russian) came from Barcelona and during the thesis she become one of my advisors and biggest support in critical points of the thesis. So thank you Anna because without you this thesis would never been finished. Discussions with Anna, Marco Dentz

Table of figures

Figure 3.13 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, injected tracer concentration 50 ppm, injected tracer volume 10 L. Differences between BTC's are in duration of push and resting period. Red curve -push duration 900s, resting duration 300s; green curve -push duration 1320s, resting period 0s; blue curve-push duration 1320s, resting period 300s. Times noted in legend are in minutes, SW stands for duration of push and ST for resting period. Figure 3.17 Scaled BTC according to the estimated tracer volume that is injected (a); red line 10 cm 3 /h, blue line 2 cm 3 /h. Compared with results of transport simulations performed on XRMT images of the same rock (b) at (purple symbols) and (green symbols). Black rectangular limits resolution that can be observed in the experiments. Unconnected symbols denotes simulations in which mass transfer to microporosity is neglected, while connected symbols represents simulations that account for mass transfer into the microporosity -original figure is in section 5.3, Figure 11

Introduction

Porous media are all around us and our everyday life is imbued with the interaction with them. All solid or semisolid materials with exception of metals, some very dense rocks and plastics are in fact porous media and moreover often permeable. Therefore, understanding the hydrodynamic processes occurring in porous media is in the scientific focus since a long time.

One of the vital processes in the permeable media is the transport of dissolved substances (i.e. solute transport). It plays an important role in a whole range of natural processes, many of which are in everyday engineering or scientific scope, such as: transport of nutrients, oxygen, medicines and/or poisons through human/animal tissues in biological and medical sciences accessibility of nutrients and spreading of the pesticides in the agriculture enhanced oil recovery in petroleum engineering CO 2 underground storages and nuclear waste disposal areas in geology and environmental engineering pollutant transport and remediation in hydrogeology Transport in porous media is driven by diffusion and advection, causing spreading of solute plume. First mathematical description of diffusion has been given by German physician and psychologist Adolf Eugen [START_REF] Fick | Uber Diffusion[END_REF]. He observed movement of solute due to differences in the concentration. Development of advection transport theory began in the 1911 with simplest case of capillary flow [START_REF] Griffiths | On the Movement of a Coloured Index along a Capillary Tube, and its Application to the Measurement of the Circulation of Water in a Closed Circuit[END_REF], however it was [START_REF] Taylor | Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube[END_REF] who gave a mathematical description of the phenomenon.

Traditional theory of dispersion in porous media has been developed through the work of several authors [e.g., [START_REF] Scheidegger | General Theory of Dispersion in Porous Media[END_REF][START_REF] Saffman | A theory of dispersion in porous media[END_REF][START_REF] Saffman | Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries[END_REF][START_REF] Bear | A generalized theory on hydrodynamic dispersion in porous media[END_REF]. This theory describes dispersion of a conservative (ideal) tracer in the fully saturated porous medium and with flow condition within the limits where Darcy's law is applicable by the advection dispersion equation (ADE):

(1.1)
where denotes the concentration, is the time, is the Darcy velocity and is the dispersion tensor. The solution of the ADE is diffusion like (Fickian) for the scale of the Representative Elementary Volume (REV) at which full mixing of the solute is assumed.

In order to capture true behavior of the dispersion there have been many field experiments of the nonreactive transport in the porous media in the aquifers with different properties around the world. Ranging from relatively homogeneous aquifers (e.g., sand aquifer at Borden [START_REF] Mackay | A Natural Gradient Experiment on Solute Transpoty in a Sand Aquifer 1. Approach and Overview of Plume Movement[END_REF]; and sand gravel at Cape Code [START_REF] Leblanc | Large Scale Natural Gradient Tracer Test in Sand and Gravel, Cape Code, Massachusetts 1. Experimental Design and Observed Tracer Movement[END_REF]); over more heterogeneous ones such as Macrodispersion Experimental (MADE) Site at Columbus [START_REF] Adams | Field Study of Dispersion in a Heterogeneous Aqifer 2. Spatial Moments Analysis[END_REF], Ses Sitjoloes at Mallorca island in Spain [Gouze et al., 2008a], to fractured aquifers such as ones in New Mexico [START_REF] Meigs | Tracer test in a fractured dolomite 1. Experimental design and observed traver recoveries[END_REF] and at Grafton County [START_REF] Becker | Tracer transport in fractured crytalline rock: evidence of nondiffusive breakthrough tailing[END_REF]. Observations from all this experiments indicate that the transport in real systems differs from Fickian assumption made by ADE. Moreover laboratory tracer experiments in bead column/packs [START_REF] Kandhai | Influence of Stagnant Zones on Transient and Asymptotic Dispersion in Macroscopically Homogeneous Porous Media[END_REF][START_REF] Moroni | Analysis of dispersion in porous media via matched particle tracking velocimetry experiments[END_REF]Bijeljic et al., 2011a], natural rocks [START_REF] Scheven | Quantitative nuclear magnetic resonance measurements of preasymptotic dispersion in flow through porous media[END_REF][START_REF] Gouze | CoFIS and TELog: New downhole tools for characterizing dispersion processes in aquifers by single well injection withdrawal tracer tests[END_REF] and uniformly [Cortis et al., 2004] and randomly packed sands [START_REF] Silliman | Laboratory Evidence of the Scale Effect in Dispersion of Solutes in Porous Media[END_REF][START_REF] Levy | Measurement and analysis of non Fickian dispersion in heterogeneous porous media[END_REF] prove that anomalous (non Fickian) transport is rather omnipresent than exceptional. Breakthrough curves (BTC) from all this experiments are highly asymmetric, with signatures of anomalous transport such as early breakthrough, long tailing (concentration usually decreases as a power law of time) and anomalous scaling of the mean square displacement. In Figure 1.1 one can see a comparison between the BTC obtained by the laboratory experiment and the best fitted solution of the ADE. This BTC is obtained as the result of a flow through experiment on a Berea sandstone core within experimental part of our work; details of the experiments are presented in the Chapter 3. ADE fails to predict the transport behavior for the most homogeneous natural aquifers [START_REF] Freyberg | A Natural Gradient Experiment on Solute Transport in a Sand Aquifer 2. Spatial Moments and the Advection and Dispersion of Nonreacitve Tracers[END_REF][START_REF] Garabedian | Large Scale natural Gradient Tracer Test in Sand and Gravel, Cape Cod, Massachusetts 2. Analysus of Spatial Moments for a Nonreactive Tracer[END_REF]. The constant dispersion coefficients used in ADE fails to capture the real dispersivity occurring in the porous medium that is dependent on the time and/or the observed scale. The spatial difference in the local dispersivity coefficients is usually explained by the flow field heterogeneity that exists even in the macroscopically completely homogeneous media [START_REF] Datta | Spatial Fluctuations of Fluid Velocities in Flow through a Three Dimensional Porous Medium[END_REF]. Accordingly, non Fickian dispersion is a pre asymptotic regime; dispersion becomes Fickian at long times if the scale of the heterogeneity is finite, i.e. when the tracer has visited the heterogeneity at all scales. Yet, in natural media displaying heterogeneity at virtually all scales, the effective dispersion regime is in general non Fickian.

The non Fickian dispersion properties, their origin and their relation to the geological heterogeneity are still debated. During the last several decades many authors have more or less successfully developed different models to simulate and predict anomalous transport. Many of these models heavily rely on the ADE formulation and try to fit experimental results by spatial variations of dispersivity. Major drawbacks of these models are that parameters that are strictly connected with the properties of the porous medium (e.g. dispersivity) are often used as fitting parameters and that transport is not necessarily Fickian even over very small domains [Cortis et al., 2004]. Two major groups of these models are based on the volume averaging and the stochastic approach.

Volume averaging models are based on solving ADE at smaller scale, usually smallest possible REV and then use these solutions to obtain results on larger scale. In these approaches one of the biggest challenges is the choice of the volume average operator to move from one scale to another [START_REF] Berkowitz | Modeling non Fickian transport in geological formations as a continuous time random walk[END_REF]. For an example [START_REF] Cushman | On Unifying the Concepts of Scale, Instrumentation, and Stochastics in the Development of MuTransport Theory[END_REF] and [START_REF] Quintard | Transport in Ordered and Disordered Porous Media I: The Cellular Average and the Use of Weighting Functions[END_REF] propose expressing the volume average operator in terms of convolution products of the spatial distribution functions. Most prominent representatives of the volume averaging models are the homogenization models and the models base on the renormalization techniques.

A stochastic approach is in the basis of many similar models that rely on the ADE [START_REF] Dagan | Flow and Transport in Porous Formations[END_REF]. Stochasticity is introduced by spatial and temporal variations of the medium properties, such as hydraulic conductivity, porosity, etc. The resulting transport is the ensemble average of the transports in all the different realizations.

There are some alternative methods to model non Fickian transport, such as multi rate mass transfer (MRMT) and continuous time domain random walk (CTRW) models. It is worth noticing here that [START_REF] Dentz | Transport behavior of a passive solute in continuous time random walks and multirate mass transfer[END_REF] have proven that there is equivalence between these models for special cases, and also included mobile immobile mass transfer into the CTRW framework. These models aims at describing the anomalous behavior based on the assumption that non Fickian dispersion comes from a broad distribution of the transition times that solute particles encounter in the porous medium. The transition time distribution is obviously related to the heterogeneity of the medium that triggers a broad range of possible flow paths for the tracer while diffusion tends to smooth the apparent differential tracer velocities. It follows that, a priori, the asymptotic Fickian dispersion regime should be recovered fast in highly diffusive systems [START_REF] Liu | Applicability of the Dual Domain Model to Nonaggregated Porous Media[END_REF].

The MRMT models [START_REF] Haggerty | Multiple rate mass transfer for modeling diffusion and surface reactions in media with pore scale heterogeneity[END_REF][START_REF] Carrera | On matrix diffusion: formulations, solution methods and qualitative effects[END_REF][START_REF] Haggerty | On the late time behavior of tracer test breakthrough curves[END_REF]] assume existence of two domains with completely different properties, relying on the idea of dual porosity models [START_REF] Barenblatt | Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (Strata)[END_REF]. The mobile domain where the transport is governed by advection is modeled by classical ADE with the constant average velocity and the dispersion coefficient, while the diffusive transport is dominant in the immobile domain. Anomalous transport arises from the mobile immobile mass exchange since tracer "particles" that enter the immobile domain on average experience significantly larger transient times than ones passing only through the mobile domain. Heterogeneity in this models in introduced by size and spatial disposition of the mobile and immobile domains and their mutual interfaces. Modeling of the MRMT is usually done by adding a source sink term in the ADE to account matrix diffusion [e.g., [START_REF] Carrera | On matrix diffusion: formulations, solution methods and qualitative effects[END_REF] or by CTRW as stated earlier. A major weakness of the MRMT approach is that it does not account for the effect of the velocity field heterogeneity in the mobile domain that may participate together with the immobile domain to control the observed non Fickian dispersion. Moreover, the memory function is often used as a fitting parameter without any connections to measurable properties of the porous medium, although there were some attempts to connect the memory function with the tortuosity and properties of the microporous phase from X ray microtomography images [e.g., Gouze et al., 2008b]. CTRW is a flexible framework that has proven capabilities to provide a convincing modeling of non Fickian transport in the porous medium [START_REF] Berkowitz | Anomalous transport in laboratory scale, heterogenous porous media[END_REF]Cortis and Berkowitz, 2004; Le Borgne and Gouze, 2008;[START_REF] Cortis | Continuous time random walk analysis of solute transport in fractured porous media[END_REF]. In general CTRW is used to provide 1D upscaled model where the macroscopic behavior is reproduced only (i.e. all the details linked to the pore scale mechanisms that act as producing fluctuation around the macroscopic behavior are not included). It is based on the basic mass transfer equations given by Scher and Lax [1973], who generalized Montroll Wiesss continuous time random walk [START_REF] Montroll | Random walks on lattices, II[END_REF]] on a lattice. The main input parameter in the CTRW is a probabilistic density function (pdf) of transition times, containing all the important properties of the porous medium influencing the macroscopic behavior of tracer transport. Accordingly, the macroscopic transport behavior in CTRW framework is determined by the shape of the transition time pdf, therefore it is essential to find pertinent correlation between the pdf shape and measurable physical parameters, but to the best of our knowledge this point has poorly been investigated so far. One of the most interesting CTRW models which appeared in recent years is the correlated CTRW. Le Borgne et al. [2011] have shown that anomalous transport in the simple 2D structures can be simulated by CTRW from correlated Lagrangian velocities along the streamlines. This approach was later successfully applied on Berea sandstone [START_REF] Kang | Pore scale intermittent velocity structure underpinning anomalous transport through 3 D porous media[END_REF]. This approach sound interesting for relating the effective dispersion to the properties of the velocities along the flow paths that can be computed from solving pore scale flow. It is also promising for introducing the distinction between mixing and spreading; or in other words the distinction between reversible and irreversible mechanisms of dispersion, such as measurable comparing push pull and flow through tracer tests [Gouze et al., 2008b].

The synthetic presentation of the actual dispersion models above indicates that they are based on the assumptions that anomalous transport arises from different transit times that the solute tracer experiences when travelling through the pore space. The recent developments in computing resources gave us the opportunity of investigating mechanisms that control transport on the pore scale. For instance, reliable techniques for the three dimensional imaging of pore space structures and flow showed rapid developments. Nuclear magnetic resonance (NMR) [e.g., [START_REF] Scheven | Pore Scale Mixing and Transverse Dispersivity of Randomly Packed Monodisperse Spheres[END_REF]] and X ray microtomography (XRMT) [e.g., [START_REF] Fourar | Non Fickian Description of Tracer Transport Through Heterogeneous Porous Media[END_REF][START_REF] Klise | Comparison of laboratory scale solute transport visualization experiments with numerical simulation using cross bedded sandstone[END_REF] have been used to directly visualize transport effects in the natural and artificial porous media. Among other results, many of these investigations showed wide distributions of velocities [e.g., [START_REF] Seymour | Generalized Approach to NMR Analysis of Flow and Dispersion in Porous Media[END_REF]]. However, these methods have limitations in terms of the resolution, both of the solute and velocity distribution, because of the necessity of performing fast acquisition in order to tackle the dynamics of the transport processes. Although it is important to note that the (ubiquitous) diffusion mechanisms cannot be tackled accurately at small scale. For example, at a resolution of 5 microns, the characteristic time of diffusion through a pixel is smaller than 0.1 second which is far smaller than the actual imaging capabilities for X ray tomography. The contribution of these methods for characterizing multiscale dispersion mechanisms is more qualitative than quantitative, similar to other imaging techniques developed for investigation of dispersion in transparent porous media. Nevertheless these methods are unmatched for characterizing the average macroscopic behavior and test models. An example of dispersion investigation performed by the visible light transmission technique on a glass bead pack (transparent porous media) can be seen on Figure 1.2. [START_REF] Zinn | Experimental Visualization of Solute Transport and Mass Transfer Processes in Two Dimensional Conductivity Fields with Connected Regions of High Conductivity[END_REF] created porous media of uniformly sized glass beads, and then inserted circular regions filled with uniformly sized glass beads but with significantly smaller diameter to obtain less permeable regions. Figure 1.2 clearly shows that heterogeneity in region permeability leads to different rates of the transport in different regimes, as well as shows different transport mechanisms occurring due to differences in the permeability.

Conversely, a detailed investigation of the transport processes is now possible by solving it numerically directly on the pore structure. Standard approach starts with the acquisition of high resolution XRMT images of the pore structure. For reservoir rocks the optimal resolution (voxel size) is in the order of few micrometers. These raw images of the X ray absorption in the sample are then processed (segmentation) to distinguish different phases (usually rock and void). Processed images are then used to mesh the pore space for flow and transport calculations using computed fluid dynamic codes [START_REF] Ovaysi | Pore scale modeling of dispersion in disordered porous media[END_REF][START_REF] Blunt | Pore scale imaging and modelling[END_REF]. Many recent studies have been investigating spreading of solute in the relation of the velocity distribution in the flow field in different porous media ranging from simple structures [e.g., Le Borgne et al., 2011] and glass beads packs [e.g., [START_REF] Maier | Sensitivity of pore scale dispersion to the construction of random bead packs[END_REF], to natural rocks with different level of complexity such as, sandstones [e.g., [START_REF] Kang | Pore scale intermittent velocity structure underpinning anomalous transport through 3 D porous media[END_REF] and carbonates [e.g., [START_REF] Bijeljic | Insights into non Fickian solute transport in carbonates[END_REF]. This approach gives useful insight into the processes happening at the pore scale and its main limitations are related to the quality and resolution of XRMT images and to the computational limitations for solving flow on large domains. Also, results are very sensitive to the accurate segmentation of XRMT images [START_REF] Schlüter | Image processing of multiphase images obtained via x ray micropomography: A review[END_REF], accuracy of the computational simulations and on the representativeness of the observed medium [START_REF] Arns | Pore Scale Characterization of Carbonates Using X ray Microtomography[END_REF] The work presented in this manuscript focuses on the investigation of passive (i.e. non reactive) tracer transport through reservoir rocks. We mainly used Berea sandstone samples. Berea sandstone is a relatively homogeneous rock that is used as a proxy to poorly altered sandstone reservoirs by the petroleum industry. Therefore it has been thoroughly investigated and its properties were well documented by several authors [e.g., [START_REF] Curcher | Rock Properties of Berea Sanstone, Baker Dolomite, and Indiana Limestone[END_REF][START_REF] Øren | Reconstruction of berea sandstones and pore scale modelling of wettability effects[END_REF].

Theoretical background of the processes that control flow and transport in the porous media is presented in Chapter 2. Chapter 3 is dedicated to the experiments, starting from the experimental setup and with an explanation of all procedures and assumptions applied during the design of the experiment. At the end of the chapter results are presented and discussed on the base of the analysis of the breakthrough curves (BTC). The numerical modeling methodology is presented in Chapter 4 with detailed analysis of the flow field properties focusing on the mesh resolution importance. Developed approach is incorporated in the transport simulations Chapter 5 and as well as a base for the development of the Darcy Brinkman solver presented in Chapter 6. General conclusion with a prospective of future work is given at the end.

Transport in porous media

To understand solute transport on the large scale first one has to understand the transport mechanisms at the pore scale. Two basic modes of solute displacement in porous media are miscible and immiscible displacement. Miscible transport is characterized by only one fluid phase present in the porous medium, but its properties or composition can vary in space and time. On the other hand, in the immiscible displacement there are two or more different fluid phases and between these phasess there is distinct fluid fluid interface.

In this work we are only interested in the miscible transport. It may be a case of two miscible liquids (e.g., fresh water and salt water) or a substance completely dissolved in a liquid. Mechanisms of solute spreading through completely saturated porous medium are described in the following section.

Diffusion

The molecular diffusion is an ubiquitous transport mechanism occurring in the porous medium as soon as the solute concentration varies spatially. It occurs regardless of fluid motion and it is driven exclusively due to differences in the concentration. Substance moves from a region of high concentration to a region of low concentration. A concentration distribution change caused by diffusive transport without any influence of flow movement is illustrated in Figure 2.1. One can see that as time passes, concentration at the place of injection decreases and the plum is spreading in the space, with diminution of relative differences in the concentration in the space. The solute concentration spatial distribution is normal (Gaussian). Fick's second law predicts changes in the concentration with time caused by diffusion

(2.2)
Figure 2.2 illustrates the effect of the diffusion in liquids without any advection. In Figure 2.2 a) one can see spreading of particles in the space due to the concentration difference. However, spreading is mitigated in the porous media (Figure 2.2 b)), due to the complexity of the pore structure and smaller contact area between volumes with different concentrations. To account for slower diffusion in the porous media the molecular diffusion coefficient must be replaced by the effective diffusion coefficient ( ) (detailed discussed in section 5.2.3)

(2.3)
where is a coefficient related to the structure properties of the porous media (e.g. porosity, tortuosity..). The values of the diffusion coefficient in water for the main cations and tracer molecules range from 2 10 10 to 10 9 m 2 /s.

Advective transport

Advection (or convection) is a process in which dissolved substances are carried along with the flowing water. It is easiest to imagine advection as a displacement of solute along the flow direction as illustrated in Figure 2.3. The total amount of transported solute is the function of its concentration and the total amount of flowing water. The total amount of flowing water is proportional to the average linear velocity and the effective porosity. The effective porosity consists of the interconnected pores in which water can flow. It is always lower than the total porosity since unconnected pores are excluded from the effective porosity. Because of the heterogeneity and anisotropy of pore network in rocks, the effective porosity can be different depending on the sampled volume and the geometry of the boundary conditions. We will see later that algorithms can be used to determine the connected porosity of imaged rock samples. The average linear velocity is representing the velocity at which solute is transported through the porous medium; it can be calculated from the Darcy's law

(2.4)
where denotes permeability, fluid viscosity, pressure and is the distance between pressure observation points. The one dimensional mass flux of solute , due to advection can be calculated by equation (2.5):

Asymptotic Fickian hydrodynamic dispersion

The illustration of a theoretical purely advective transport given in Figure 2.3 is made under assumptions that the flow field in the porous medium is homogeneous and that the diffusion is null. However, these assumptions are obviously not applicable to porous media. Water is moving through the porous medium at different rates using different flow path because of the heterogeneity of the pore space (Figure 2.4). This causes hydrodynamic spreading of the tracer "particles", while diffusion triggers tracer "particle" to jump from a given flow line to an (adjacent) other producing mixing. The mechanical or kinematic dispersion is a common name for all mixing and spreading that happens due to advection (i.e parabolic velocity distribution in pores, different velocities in different flow paths and different lengths of flow paths covering same linear distance (Figure 2.5)). Elongation of the solute plume in the direction of flow is called longitudinal dispersion. Solute plume also have the tendency to spread in the directions normal to the flow direction. This happens due to diverging flow paths as illustrated on Figure 2.4. Spreading of the solute in the directions normal to flow paths is called transverse dispersion. Both longitudinal and transverse dispersion occur even in the most homogeneous porous medium, however spreading in the transverse direction is often 5 to 100 times less than in the flow direction [START_REF] De Marsily | Quantitative Hydrology: Groundwater Hydrology for engineers[END_REF]]. However, dispersion in the porous media with flowing water is influenced by both diffusion and mechanical dispersion and they cannot be separated. Therefore both of these influences are defined together as hydrodynamic dispersion. Hydrodynamic dispersion coefficient as a function of mechanical dispersion and diffusion can be expressed as:

(2.6) (2.7)
where is the hydrodynamic dispersion coefficient in the direction of the flow, is the hydrodynamic dispersion coefficient perpendicular to the flow direction, is the effective diffusion coefficient, is the dynamic dispersivity along the flow direction, is the dynamic dispersivity transverse to the flow direction and is the Darcy velocity. Classically hydrodynamic dispersion in the homogenous porous medium is described by the advection dispersion equation (ADE) (2.8). The assumptions for the derivation of the ADE are that the porous medium is homogeneous, isotropic and fully saturated with fluid, also flow conditions have to be in the limits where Darcy's law is valid [START_REF] Bear | Dynamics of fluids in porous media[END_REF]. Macroscopically dispersion is assumed to be diffusion like (Fickian) process (Figure 2.6). With the solute concentration [g/m 3 ], the ADE is as follows

(2.8)
where denotes time [s], is the macroscopic (Darcy) velocity [m/s] and is the dispersion tensor. The dispersion tensor expressed in its principal directions of anisotropy, is limited to three components (2.9) while the average macroscopic velocity can be calculated from the Darcy's law by equation (2.4).

Two classical examples of interlaced influences of both diffusion and mechanical dispersion in the porous medium are Taylor and holdup dispersion. Taylor dispersion is a very simple and illustrative example of an evolution of the hydrodynamic dispersion starting as a pure mechanical dispersion and evolving due to the diffusion. First evidence of the dispersion due to the heterogeneity of a flow field in a capillary was published by [START_REF] Griffiths | On the Movement of a Coloured Index along a Capillary Tube, and its Application to the Measurement of the Circulation of Water in a Closed Circuit[END_REF]. However it was [START_REF] Taylor | Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube[END_REF][START_REF] Taylor | The dispersion of matter in turbulent flow through a pipe[END_REF] who mathematical described phenomenon later named after him.

Origin of Taylor dispersion is in the parabolic distribution of velocity in a capillary tube, given by Hagen Poisseuille law is the pressure difference between tube ends, is the tube length, is the interior tube radius and is the viscosity of the fluid. Solute in the vicinity of the central axis travels significantly faster than one located near the boundaries. This causes spreading on solute plume in the longitudinal direction, but in the same time causing the difference in the concentration perpendicular to the flow direction. Due to this difference in the concentration solute starts to diffuse towards tube boundaries at the front edge of the solute plume and towards the central axis in back of the solute plume. [START_REF] Aris | On the dispersion of a solute in a fluid flowing through a tube[END_REF] has extended Taylor's theory and proved that dispersion in a capillary tube will ultimately achieve Gaussian distribution if all conditions remain stable. Magnified area on Figure 2.5 shows parabolic velocity distribution that occurs in the pore. Hold up dispersion is typical for the porous media. It is caused by solute diffusing into stagnant zones (e.g. dead end pores). Stagnant zones are areas where there is virtually no fluid flow or where its velocity is negligible compared to the fluid velocity in the flow paths. These zones exist in all porous media, even in the most homogeneous ones such as packed beads [e.g., [START_REF] Sederman | Magnetic resonance imaging of liquid flow and pore sturcture within packed beds[END_REF][START_REF] Reynolds | Flow and dispersion through a close packed bed of spheres[END_REF], and their influence on the dispersion cannot be disregarded. [START_REF] Zinn | Experimental Visualization of Solute Transport and Mass Transfer Processes in Two Dimensional Conductivity Fields with Connected Regions of High Conductivity[END_REF] provides particularly nice visualization of influence of the stagnant regions over different transport regimes. Figure 2.7 illustrates typical dead end pore. Solute is transported by convection through the flow path and some particles diffuse into the dead end pore. Particles which are in the stagnant zones can rejoin flow path only by diffusion. Since diffusive transport happens from higher concentration to lower concentration, solute can diffuse back into the flow path only when solute plume has passed. Also this release from stagnant zones is very slow due to small concentration difference.

Characterization of dispersion regime

Relative influence of the kinematic dispersion and diffusion on the hydrodynamic dispersion depends on the transport regime. Transport regime can be quantified by dimensionless Peclet number ( )

(2.11)
where denotes mean microscopic velocity, is the characteristic length and is the molecular diffusion coefficient. Relationship between dispersion coefficient and Peclet number has been established from experiments, exhaustive overview is given by [START_REF] Delgado | Longitudinal and transverse dispersion in porous media[END_REF]. Based on these experimental results several authors defined five distinct transport regimes [e.g., de Marsily, 1986], corresponding to different relative influence of kinematic dispersion and diffusion. Some of experimental results are presented in Figure 2.8. On the same figure limits between different regimes are indicated with dashed lines. In each of these regimes empirical relation between longitudinal ( ) and transversal ( ) dispersion coefficients and Peclet number is found. [START_REF] Sahimi | Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing[END_REF] even gave approximate limits between regimes; however these limits are just tentative.

Figure 2.8 Experimental results showing relation between the longitudinal dispersion coefficient ( ) and Peclet number ( ). Dashed lines indicate limits between five different transport regimes -reprinted from Sahimi [1993]

i) Pe<0.3 Pure molecular diffusion regime is characterized with very slow convection, and dispersion is almost completely controlled by diffusion. Since there is no convection the dispersion is isotropic and the dispersion coefficients are given by (2.12)

where and are defined as

(2.13)
is the formation factor, and is the total connected porosity of the medium. The formation factor is a property that is originally defined from electrical conductivity measurements of fluid saturated porous media. It is defined as the ratio of the resistivity of porous material, saturated with an electrolyte, to the bulk resistivity of the same electrolyte [START_REF] Archie | The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics[END_REF]. The ratio is defined by the properties of the porous medium and commonly ranges between 0.15 and 0.7.

ii) 0.3<Pe<5 Intermediate zone between pure diffusion and predominant kinematic dispersion, in which both convection and diffusion influences on the dispersion. In this regime the dispersion coefficient increases with increase of convection influences. To this day we still do not have valid formula that describes evolution of the dispersion coefficients in terms of the Peclet number in this transitory regime. iii) 5<Pe<300 This regime is characterized by predominant kinematic dispersion, however effect of diffusion cannot be neglected. Usually it is referred to as power law regime since relation between the dispersion coefficients and the Peclet number are described by the empirical relations

(2.14) (2.15)
where and [m] are known as intrinsic dispersion coefficients or dispersivity. [START_REF] Sahimi | Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing[END_REF] points out that the average values obtained from experiments are 1.2 for and 0.9 for . Saffman discovered existence of a diffusive layer near the solid surface; therefore, sometimes this regime is referred to as boundary layer dispersion regime [START_REF] Saffman | A theory of dispersion in porous media[END_REF]. iv) 300<Pe<10 5 For a porous medium that does not contain any stagnant zones (e.g., Figure 2.7) in this regime, transport is characterized as pure kinematic dispersion (i.e. completely dominated by convection). Due to negligible influence of diffusion, transport in this regime is usually referred to as mechanical dispersion, and the dispersion coefficient scales linearly with the Peclet number:

(2.16)

(2.17)

However stagnant zones exist in almost all porous medium, and actually contribute to the pore space with large percentage. Therefore application of power law regime with its governing equations (2.14) and (2.15) is extended to this Peclet number range.

v) 10 5 <Pe This regime is characterized by a turbulent flow, in contrast to the previous ones where a transport took place under laminar flow conditions. Kinematic dispersion is outside the domain where Darcy's law is valid. Since the flow turbulence also contributes to the dispersion, the dispersion coefficients depend on both Peclet and Reynolds numbers (see section 4.3.3 and equation (4.7)).

Yet, it is worth noticing that regimes iv and v are usually not encountered in natural systems, but may occurs very locally close to pumping well, for example.

Alternative Approaches

As mentioned earlier (Chapter 1) standard theory of transport in the porous medium fails to describe the real nature of dispersion for most of the natural conditions. Certainly the main issue is its inability to describe the effects of the flow field heterogeneity triggered by the more or less complex geometry of the porosity that occurs at virtually all scales. For a porous medium that one wishes to model at a given scale, there is always some scales below which heterogeneities are unresolved, and several studies stressed that these unresolved heterogeneities can have a key influence on the overall transport behavior [START_REF] Berkowitz | Modeling non Fickian transport in geological formations as a continuous time random walk[END_REF]. Consequently the use of average local properties (e.g., mean velocity and dispersion) measured at the modeling scale seems to be unjustified. This led to the development of alternative models. Most of these models are based on non local in time representations of hydrodispersive transport to take into account the fact that local mixing and spreading at a given time depend on previous conditions. The models closest to the "classical" ADE model are stochastic ADE models [e.g., Morales [START_REF] Morales Casique | Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: Computational analysis[END_REF]. They are based on the premises that the transport can be accurately described at a given scale and that non Fickianity arises from time and space depended fluctuations of the flow and transport properties. Averaging over an ensemble of velocity fields results in the representation of the non Fickian transport behavior. Similarly Lagrangian framework models [e.g., [START_REF] Cushman | Nonlocal dispersion in media with continuously evolving scales of heterogeneity[END_REF] average the solute particle motion over a random velocity field. Transport is then modeled by the ensemble average of nonlocal space time representations. Non local in space models have been developed for simulating long range correlations using for instance fractional ADE models [e.g., [START_REF] Benson | The fractional order governing equation of Lévy motion[END_REF]]. Yet, the most used framework are the multi rate mass transfer (MRMT) models (section 2.5.1) and the continuous time random walk (CTRW; section 2.5.2) that will be also at the center of the works presented in this document. Readers are directed towards detailed overviews given by [START_REF] Berkowitz | Modeling non Fickian transport in geological formations as a continuous time random walk[END_REF] and [START_REF] Neuman | Perspective on theories of non Fickian transport in heterogeneous media[END_REF] for additional details on the non Fickian transport model theories; in the following MRMT and CTRW approaches will be briefly presented.

Multi Rate Mass Transfer Model (MRMT)

Molecular diffusion from the water flowing portion of porous media to the stagnant zones (i.e. matrix diffusion) has been recognized and widely cited as a critical transport mechanism controlling the residence time. Prolonged residence time (e.g. tailing observed in the BTCs) is one of the most recognizable features of the anomalous transport. This premise led to development of simple transport models to account matrix diffusion, which can be divided into two main groups, first order mass transfer models [e.g., [START_REF] Coats | Dead end pore volume and dispersion in porous media[END_REF][START_REF] Van Genuchten | Mass Transfer Studies in Sorbing Porous Media, I. Analytical Solutions[END_REF]] and rate limited mass models [e.g., [START_REF] Rao | Solute Transport in Aggregated Porous Media: Theoretical and Experimental Evaluation[END_REF][START_REF] Toride | The CXTFIT Code for Estimating Transport Parameters from Laboratory or Field Tracer Experiments[END_REF]. In both cases mass transfer between mobile and immobile domain is simulated with a single mass transfer type and rate at the time. Mathematically mass transfer is coupled with standard advection dispersion equation by adding sink/source term. However in porous medium there is a large variety of different immobile domains (e.g. microporosity, stagnant zones, sorption...) with different properties and geometry (Figure 2.9a). Thus, they cannot be represented with a single mass transfer rate.

The MRMT models allow numerous types and rates of mass transfer to occur simultaneously [START_REF] Haggerty | Multiple rate mass transfer for modeling diffusion and surface reactions in media with pore scale heterogeneity[END_REF][START_REF] Carrera | On matrix diffusion: formulations, solution methods and qualitative effects[END_REF]] as illustrated on Figure 2.9b. The mathematical expression of MRMT model is similar to his simpler predecessors. Moreover it has been demonstrated that spherical, cylindrical and layered diffusion models are just a special case of the MRMT model. The main difference is in the formulation of the sink/source term that accounts for the effect of the immobile domain. Usually mass transfer between mobile and immobile domain is modeled by linear mass exchange process, which can be expressed as the convolution product of a time dependent variations in concentration in the mobile domain and the memory function. The memory function (expressed as the inverse of time and noted hereafter G(t)) is a time dependent function that denotes the intrinsic transport property of the immobile domain.

As stated earlier basic equation expressing the average solute concentration in the mobile domain ( ) is standard ADE equation with added sink/source term ( ), representing mass transfer between mobile and immobile domain [START_REF] Carrera | On matrix diffusion: formulations, solution methods and qualitative effects[END_REF][START_REF] Haggerty | Tracer tests in a fractured dolomite 2. Analysis of mass transfer in single well injection withdrawal tests[END_REF] (2.18) where denotes the volume fraction of the mobile domain which is equivalent to the macroscopic porosity (important to remind that this is only water flowing portion of the porosity), is the time, is the dispersion tensor and is the Darcy velocity [m/s]. The sink/source term for linear mass transfer processes can be expressed as a convolution with the memory function ( ):

(2.19)
where denotes volume fraction of immobile domain (microporosity, dead ends, stagnant and trapped water). The memory function depends on the properties of the immobile domain and specifically its heterogeneity. Heterogeneously distributed immobile domains with different mass transfer rates ( can be replaced with the homogeneous immobile domain with the probability distribution of the mass transfer rate (

). expresses the probability that the particle is in the immobile domain in a given time interval, generally it is controlled only by the volumetric distribution of the immobile region. Diffusive transfer rate between mobile and heterogeneous immobile domain is given by weighted average over the local memory function (

) [START_REF] Haggerty | On the late time behavior of tracer test breakthrough curves[END_REF].

(2.20)

Several authors have successfully used MRMT model to fit experimental data, trying to connect memory function with measurable rock properties [e.g., [START_REF] Haggerty | Tracer tests in a fractured dolomite 2. Analysis of mass transfer in single well injection withdrawal tests[END_REF][START_REF] Haggerty | What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results[END_REF]Gouze et al., 2008b;[START_REF] Berkowitz | Non Fickian transport and multiple rate mass transfer in porous media[END_REF]. However, the link between intrinsic properties of the rock and memory function is still debated. This is not surprising, knowing that mass transfer rate depends on many different factors, which contribution cannot be separated and evaluated individually. The MRMT class of model was developed for systems where it is possible to assume that two distinctly different transport modes coexist: a mobile domain (advection dominant) and an immobile domain (diffusion). In general these domains correspond to physical regions of the porous media (e.g., macroporosity and porous cement). Attempts have been made to use this approach to model the effects of the flow field heterogeneity, which means that the mobile and immobile volume fractions depend on the Peclet number [START_REF] Liu | Applicability of the Dual Domain Model to Nonaggregated Porous Media[END_REF].However, the justification of distinguishing a purely advective domain and a purely diffusive domain from a continuous distribution of velocities seems difficult and we should restrict the use of MRMT for cases where mobile and immobile domains can be clearly distinguished.

Continuous Time Random Walk (CTRW)

The CTRW framework has demonstrated an ability to successfully characterize non Fickian transport behavior at different scales, as well as in both porous and fractured medium [e.g., [START_REF] Berkowitz | Anomalous transport in laboratory scale, heterogenous porous media[END_REF]Cortis and Berkowitz, 2004; Le Borgne and Gouze, 2008;[START_REF] Cortis | Continuous time random walk analysis of solute transport in fractured porous media[END_REF][START_REF] Kang | Impact of velocity correlation and distribution on thansport in fractured media: Field evidence and theoretical model[END_REF]. It is a probabilistic approach, which uses the random walk framework to mimic particle transport in heterogeneous media. It describes a path as a series of successive uncorrelated random steps with no direct link to any geometrical or physical support. Continuous time was first introduced by Montroll and Weiss [1965] for modeling diffusion on lattice.

The CTRW framework aims at modeling anomalous transport in heterogeneous media. It is a random walk approach in which both space and time increments are random variables. Formally, the particle transitions depends on a joint probability (pdf) of the transition times, where s denotes the position and t the time, but it is classically assumed that for hydrodynamic transport the decoupled form = ( ) , where is the probability rate for a transition time t between two sites. Detailed overview of the CTRW theory and mathematical framework is given in [START_REF] Berkowitz | Modeling non Fickian transport in geological formations as a continuous time random walk[END_REF] and here we will present only the most important features.

Using the decoupled form presented above, the CTRW transport equation can be written in Laplace space:

(2. [START_REF] Blunt | Pore scale imaging and modelling[END_REF] where the sign denotes the Laplace transform, is the Laplace variable, and are the apparent velocity vector and dispersion tensor respectively (both being respectively different from the average velocity and Fickian dispersion), and is a memory function which depends on and some transition time characteristics [Berkowitz et al., 2006 and references therein]. Note that in equation (2.21), c denotes the total concentration while the equation (2.18) written for the MRMT model gives the solution for the mobile concentration . The common form for is [e.g., Bijeljic et al., 2011a]:

(2.22)
with a single characteristic transition time. Truncated power law for is often used to model anomalous dispersion but other expressions for and can be investigated (see section 5.3). Note that for , i.e. an exponential form of , the Fickian dispersion model (ADE Equation (2.8)) is retrieved. By performing inverse Laplace transform of Equation 22one can obtain time domain transport equation:

(2.23)
As mentioned earlier, the transport behavior in the CTRW framework is controlled by the functional shape of the pdf. It depends on the probability rate of the transition times. For instance, a power law distribution of transition times may be used to model power law tailed breakthrough curves. In this case, CTRW formulation is mathematically equivalent to the MRMT approach [START_REF] Dentz | Transport behavior of a passive solute in continuous time random walks and multirate mass transfer[END_REF][START_REF] Schumer | Fractal mobile/immobile solute transport[END_REF]. The numerical implementation of the 1D CTRW for modeling flow velocity heterogeneity and mobile immobile mass transfers are detailed in section 5.3.

Tracer experiments

Introduction and objectives

Experiments enable to observe natural behaviors in more or less controlled environments. Large scale experiments provide irreplaceable information source on transport as it happens in nature. However, they are long lasting, expensive, and often difficult to interpret because of the superposition of natural flow and the poor knowledge of the true reservoir heterogeneity. On the other hand laboratory experiments provide valuable insight into natural process with an ability to isolate different effects contributing the dispersion and usage of the well known and investigated media. Laboratory experiments can in theory be performed under strictly regulated conditions, eliminating side effects that may lead to the misinterpretation of the results. Nevertheless, these objectives are often difficult to reach in practice and require quite sophisticated measurement equipment and protocols to minimize dispersion due to the experimental setup itself for instance. Also, it is important to perform tracer test over several order of magnitude in concentration to obtain a pertinent characterization of the non Fickian dispersion behavior that is contained in the BTC tailing (i.e. at long time and for low concentration).

In this chapter we present series of 1D like experiments performed on glass beads columns and Berea sandstone cores. The aim is to investigate tracer (pre asymptotic) dispersion through relatively simple and well described porous media. Variations in the porous media structure, flow rate and experiment type (flow through or push pull) allows to distinguish contributions of different natural processes on the hydrodynamic dispersion.

Experiment setup

The experimental circuit is designed to support both flow through and push pull experiments without any changes in the circuit when changing experiment type. Actually the only part that needs to be changed between experiments is the sample and the protective cell around it when changing porous media type.

Experimental circuit layout is presented in Figure 3.1 and all components are detailed described in following subsections. Water is pumped into the circuit with two piston pumps, passes through tracer injection point, sample core and out of the system. Arrows indicate the direction of flow for different experiment stages. Blue arrows indicate flow direction in part of the circuit where flow direction is always the same, green arrows indicate flow direction during flow through experiment and pull phase of push pull experiment, while red arrows indicate flow direction during push phase of push pull experiment. TELog is a high resolution optical sensor presented in details in subsection 3.2.5. It is installed below the experimental cell directly at the outlet from porous media. Data recording and TELog management is performed by personal computer which communicates with TELog through USB controlled signal converter.

In flow through experiments water arrives at the top of the sample. Here we inject tracer at the injection point using micro syringe to precisely dose trace volume and time of injection. It is important not to inject too fast in order not to significantly disturb flow field. Solute then passes through porous media and outlet concentration is measured by TELog. For first (push) phase of push pull experiment flow direction valve is placed in position to direct flow at the bottom of cell. There is another injection point just between the TELog and the sample. Tracer is pushed for a period of time depending on flow rate, permeability and length of sample in order to investigate as much as possible of the porous media without any tracer leaving from porous media on the other side. At the end of the push period, the control valve is switched and water starts to flow from the top of the sample like in flow through experiment. In order to emphasize dispersion due to diffusion in some experiments, we turned off the pumps for several minutes between push and pull phase.

Pumps

The main demand imposed for the pumps was to have constant flow rate without any fluctuations of chosen flow rate over several weeks. Two R2N piston pumps equipped with solenoid valves were chosen. Normally only one pump is used to inject water in the system while other one is waiting full or being recharged. When a pump is almost empty it starts to slow down the piston and decrease flow rate, at the same moment the other pump starts to move its piston maintaining combined flow rate from both pumps equal to chosen flow rate. The ratio in cumulated flow rate is being gradually increased in favor of the second pump until the first pump is completely stopped and the second pump takes over entire flow rate. First pump is being recharged and put in waiting for process to repeat.

Pumps are operated by acquisition program developed in LabVIEW by Richard Leprovost, mechanical engineer in the TMP team. Program monitors and controls pump status (on/off), position and speed of piston (controlling flow rate and recharge of pumps), status of the valves (open/closed). All relevant data are displayed on computer in real time.

Both pumps are exactly the same with internal diameter of 35,685 mm (cross section area 1000.14 mm 2 ) and operating range from 1 to 500 cm 3 /h. Pumps calibration was performed at the beginning of the experiment to ensure that the actual flow rate is exactly the same as the imposed one.

Experimental materials and procedures

In this work we performed tracer experiments on both artificial (glass beads packing) and natural (Berea sandstone) porous media. Sample cell is designed to perform both flow through and push pull experiment and as well that changes between samples can be done fast and simple. As it can be seen in Figure 3.3 a) cell is placed vertical with injection points on the top and on the bottom. Measuring unit (TELog -details in section 3.2.5) is located directly at the bottom inlet/outlet of the sample.

Glass beads used in these experiments are originally intended for industrial application. Therefore, distribution of their diameter is relatively wide. To narrow glass beads diameter distribution we performed sorting process. The sorting process is performed on the vibrating sorting machine (Figure 3.2 b)), where sieves are positioned one above the other in order that upper sieve always has larger openings in the mesh. Glass beads are poured on the top sieve and beads smaller than openings in the mesh drop to lower sieve until they reach sieve with openings smaller than their diameter. Vibrations of the machine assist in the sorting process. At the end of the sorting process remaining beads on each sieve have diameter smaller than openings on the previous (top) sieve and smaller than openings on the sieve they are found on. Chosen glass beads for these experiments have diameters between 355 and 400 m. Column filling with glass beads is made underwater with regular vibrations to ensure optimal homogeneous packing, as low as possible void content and to avoid air pockets (Figure 3.2 c)). First experiments to validate experimental procedure and bead packing were performed with the transparent coating (Figure 3.2 d)). Transparent coating in combination with high concentration tracer allowed monitoring tracer plume movement through the packed beads. During these experiments no side flow or other anomalous behavior was observed. However, the experiments for which results are presented in the section 3.3 were performed on the column with non transparent coating to prevent light pollution of measurement and tracer decay due to light. Non transparent cell used in the experiment is made of plastic material and by design is similar to one used for Berea sandstone cores (Figure 3.3 a)) except it does not have opening to apply external pressure. Both sides of cell are closed with thin highly permeable seals made of glass material.

Berea sandstone core was extracted from a block that is apparently homogeneous at centimeter scale. Core has 18 mm diameter and is 15 cm long (Figure 3.3 d)). Before placing core into the cell it has been coated with thermo plastic material to prevent side flow. Procedure consists of placing core into "raw" thermo plastic material which is later heated with heat gun. During heating process thermo plastic material shrinks and adheres to a core, final result is presented in Figure 3.3 d). Prepared core is placed into metal coating and installed vertically on top of the TELog. A confining pressure (about 3 bar) is applied to the thermo plastic jacket in order to prevent any leakage at the rock jacket interface. Measurement of TELOG can be impaired by several factors such as light pollution or changes in chemical composition of the water. The design of experimental cells and TELog provide good protection from light pollution and in order to ensure that water properties stayed the same during the whole experiment we used bottled water. We used "Eau de Source de montagne d'Auvergne" available in French supermarkets. This water (pH is 7,3 and dry residual at 180 0 C is 52,2 mg/l) has a composition that prevent any chemical reaction with the studied material and specifically the cement material that are present in the Berea sandstone.

Tracer injection point

Injection of tracer is designed in order to minimize dispersion before tracer reaches the sample and in the same time to prevent disturbance in the flow. Figure 3.4 presents the cross section through the injection part installed at the top of the cell. The injection is made with a micro syringe at injection point placed between tube 1 and diffuser. Needle (diameter 0.7 mm) penetrates through tube 2 which has an internal diameter exactly the same as the external diameters of tube 1 and diffuser. It is made of very dense rubber foam which self closes the micro hole after the needle is removed to prevent leakage from the system. Diffuser spreads just before the top of the sample ensuring that tracer can penetrate on the whole surface of the sample. At the cell bottom there is a similar device (with same working principle) for performing the injection during push pull experiments.

Fluorescein

Fluorescein is a synthetic organic compound frequently used in hydrogeological experiments. Usually we can find in a form of disodium salt which is dark red powder also known as "uranin" or D&C Yellow no.8. Even small amount of fluorescein dissolved in the water gives a green color to the solution, while solutions with high fluorescein concentration are colored in orange or red (Figure 3.5). Although organics, sunlight and some chemicals can impair intensity of the dye it is one of the best non toxic conservative tracer used for tracing of water pathways. A reason of its popularity and wide application is that its presence can be detected up to 0.001 g/L [START_REF] Gaspar | Modern Trends in Tracer Hydrology[END_REF]. From chemical point of view fluorescein has only negative functional groups which mean that there is almost no sorption on negatively charged surfaces such as silica sand and sandstones [START_REF] Kasnavia | Flourescent Dye and Media Properties Affecting Sorption and Tracer Selection[END_REF].

Measurement of fluorescein is based on intensity and wavelength distribution of emission spectrum after excitation by a certain spectrum of light [START_REF] So | Fluorescence Spectrophotometry[END_REF]. In water, fluorescein has an absorption maximum at 494 nm and emission maximum of 521 nm. The molecular diffusion coefficient of fluorescein in water is 6.4×10 10 m 2 /s [START_REF] Galambos | Micro Fluidic Diffusion Coefficient Measurement[END_REF]]. For our experiment we can consider fluorescein as an ideal tracer since it has been proven that sorption can be neglected on positively charged surfaces, both glass beads and sandstones are positively charged. Density effect will be investigated in detail during the experiments.

The TELog sensor

TELog is a high resolution optical sensor primarily developed for field usage [START_REF] Gouze | CoFIS and TELog: New downhole tools for characterizing dispersion processes in aquifers by single well injection withdrawal tracer tests[END_REF]. It is in fact a special kind of fluorometer designed for continuous measurement of very low fluorescein concentration. Ability to measure at low concentrations is essential when studying dispersion of a passive tracer in order to avoid density effect and still be able to measure concentration over several orders of magnitude.

Measurement is based on the specific wave lengths absorbed and re emitted by fluorescein. Figure 3.6 displays a schematic representation of the TELog head. Blue led is emitting short light pulses (less than 3 ms) every 10 ms, this light pulses passes the filter that filters all light with different wave length than 485 ± 10 nm. Light beam then passes through the glass tube where the solute is flowing at the outlet from the sample. Fluorescein in the solute absorbs light end re emits it. Re emitted light passes through a second filter that eliminates all light with different wave length than 530 ± 10 nm. Filtered light is detected by photomultiplier and the signal is processed by a computer. The TELog is operated by an acquisition software developed using LabVIEW by Olivier Maudens, electrical engineer it the TMP team. The sensitivity of the photomultiplier is defined by 8 pre programmed scaling factors (gains). Starting from the lower gain value applied to the photomultiplier, the LabVIEW code determines incrementally the optimal gain to obtain the best sensitivity of the photomultiplier; the principle is explained in details in the section 3.2.6. Experiment stability and reproducibility is a very important issue. The use of bottled water, the stability of light source and the accuracy of the flow rate eliminates most of external factors that could cause false readings in final data. Furthermore the black signal (i.e. measure of the photomultiplier response without being illuminated) is used to monitor and correct if necessary light pollution coming from external light source and for the value of the dark signal of the photomultiplier.

TELog calibration and data readings

The photomultiplier installed in TELog detects light that passes through the tube containing fluid at the outlet of the sample. According to the light intensity and wavelength the photomultiplier produces different output voltage. To establish correlation between output voltage and fluorescein concentration we performed calibration. During the calibration process we injected solutions with known fluorescein concentrations and measured the output voltage.

The first step of the calibration process was to produce a set of solutions with known fluorescein concentrations. Solution concentrations are chosen in order that they cover the whole range of values that are expected to be measured (from 1ppm to 1ppt) and that they give enough information to enable interpolation values in between while keeping reasonable number of solutes. First solution is made by adding fluorescein in powder into the pure water. Solute concentration ( ) is calculated by equation:

(3.1)
where denotes the mass of the fluorescein and the mass of the water. The first solution had concentration of 1000 ppm and from this we produced first master solution C 1 . Second master solution C 2 is produced from C 1 ; all other solutions are produced from these two masters (as illustrated on Figure 3.7). Since new solutions are obtained by dilution from existing solutions rather than by adding fluorescein into the pure water, their concentration is calculated by equation:

(3.2)
where denotes the concentration of resulting solute, is the mass of fluorescein in the master solute and is the total mass of the master solute. In order to facilitate calculations, in equation (3.2) the fluorescein mass is substituted by solute concentration, this gives equation (3.3)

(3.3)
where denotes the concentration of master solution. Advantage of the approach with few master solutions to the approach where each solution is obtained by dilution from the first stronger solution is in minimizing cumulative errors and avoiding that the whole series of solutions have wrong concentration due to one mistake. However it requires more attention and very precise scales since resulting concentrations are up to 3 orders of magnitude lower than master concentrations. In order to meet these requirements we used two digital scales with different maximal load and precision. First one is Precise LX220a with the maximal load up to 220g and the precision of 0.001g. Second one covering all weights higher than 220g is Kern EG 2200 2NM with the maximal load up to 2200g and the precision of 0.01g. Made solutions are kept closed in bottles impermeable to light to prevent fluorescein decay due to light.

The calibration procedure consisted in injecting solutions directly into TELog and then flushing it with pure water and drying a tube with air. Solutions were injected starting from the lowest concentration towards highest ones. Each solution is injected three times with 7 minutes of the resting period in between, resulting in a total time of 21 minutes during which measurements were recorded. In the calibration mode TELog records a value every 3 seconds at different gain (i.e. different sensibility of photomultiplier). Recorded value is an average of 30 measurements made each 0.1s at the same gain and then TELog shifts to another gain. Injections every 7 minutes were made to prevent decay of the fluorescein concentration due to light effect, however no decay was observed even for the lowest concentrations. After each injection the system was flushed with pure water for 35 minutes, to remove all fluorescein from the system including some that may be trapped in the dead zones. After flushing, the entire system is dried with compressed air. Drying phase is important since the injected volumes were relatively small and residual drops of water could significantly dilute the injected solution. While in calibration mode tension is measured for all gains, during the experiment TELog was working in standalone mode that was always trying to achieve optimal photomultiplier sensitivity (optimal gain). Optimal photomultiplier sensitivity is achieved between 0.3 and 1.5 V (dashed red lines in Figure 3.8). In this range there is a power law relation between fluorescein concentration and output voltage with relatively small oscillations. A measurement begins with the lowest gain 1 (lowest photomultiplier sensibility) and switches to higher gain if output voltage is lower than 0.3V. Only exception is the highest gain 8 in which there is no lower limit. If during the experiment at any gain (except gain 1) output voltage exceeds 1.5V measurement is switched to gain with lower sensibility. Unlike calibration mode, in the standalone mode output is given as recorded values immediately registered every 0.1 s at given gain, without any additional treatment. where denotes the concentration, refers to gain value (i.e. photomultiplier sensitivity level), is measured voltage output, and are coefficients of the calibration curves related to specific gain. Comparison of experimental results with ADE mode (e.g. Figure 1.1) is based on the CXTFIT model [START_REF] Toride | The CXTFIT Code for Estimating Transport Parameters from Laboratory or Field Tracer Experiments[END_REF] incorporated into STANMOD software package.

Results and discussion

In this section we present results of the laboratory experiments performed on glass beads columns and Berea sandstone cores. All results are presented as breakthrough curves (BTC) showing the tracer concentration at the outlet of the sample.

After each experiment, retrieval rate is calculated as an area below BTC. Retrieval rate varies between 90 and 105% of tracer injected in the system. This difference (as well positive) is explained by the fact that injected volumes were relatively small (2 to 50 L), so there might be small difference (uncertainties on the injected volume) between the planned and the actual injected volume.

Glass beads flow through experiments

We performed series of experiment on the glass beads column. The column (200 mm long and 18 mm in diameter) was filled with glass beads with diameter between 355 and 400 m. Flow rate was 50 cm 3 /h for all experiments. At this flow rate, in the porous medium, there is a laminar flow regime with Reynolds number of 0.55 and the Peclet number is 853 according to equations (2.11) and (4.7) respectively. Porosity of packed beads is estimated at 0.38 [START_REF] Dullien | Porous Media Fluid Transport and Pore Structure[END_REF], characteristic length is taken as median bead diameter and is equal to 380 m, water density is 1000 kg/m 3 , fluorescein molecular diffusivity is 6.4×10 10 m 2 /s [START_REF] Galambos | Micro Fluidic Diffusion Coefficient Measurement[END_REF]] and water dynamic viscosity at 20 0 C is 1.002×10 3 Pa s. Section 3.2.4 shows that in certain experimental conditions, fluorescein can be considered as an ideal tracer. Since both glass beads and sandstones are positively charged surfaces, sorption of the tracer can be neglected, so the only thing that could impair experimental precision is density effect. Several authors studied dispersion behavior in cases when there is significant density difference between the resident fluid and tracer solution, such as e.g. contact interface between salt and fresh water [START_REF] Hassanizadeh | A non linear theory of high concentration gradient dispersion in porous media[END_REF][START_REF] Konz | Variable density flow in heterogenous porou media Laboratory experiments and numerical simulations[END_REF]. However density effect resulting from the weak density contrast (e.g. dissolved tracer in the same fluid as resident fluid) is often neglected. To our knowledge only Tenchine and Gouze [2005] investigated density effect for the weak density contrast. They observed that the density effect can influence the transport even if there is only 0.35 grams of the fluorescein sodium per liter. Figure 3.9 presents the study of the possible density effect during the flow through experiments performed on the packed glass beads column. Four different solutions ranging from 10 3 (1000 ppm) to 10 6 (1 ppm) grams of the fluorescein sodium per liter of the water are tested. All BTCs for different injection concentrations have similar shapes displaying non Fickian behavior with emphasized tailing. Analysis of the peak arrival time (Table 3.1) reveals the influence of the density effect. Due to density effect higher concentrations have shorter peak arrival time than lower ones. The observed difference in the peak arrival time between 1000 ppm and 1 ppm is around 6%, and between 500 ppm and 1ppm is around 3%. It can be noted that the peak arrival time for the injected concentration of 50 ppm is slightly larger than the one for the 1ppm; however, the difference is in the range of the experimental error. We conclude that the differences in the peak arrival times are clear indications of density effect the tracer transport, for experimental setup where flow and gravity are oriented in the same direction. Differences in the late time behavior are not so pronounced. In case of 1000 ppm, a density effect in very permeable and relatively short porous medium causes irregular tailing, tails observed for 500 ppm and 50 ppm at late times becomes parallel one to another with a slope of t 10.6 . While slope for 1 ppm (t 13.4 ) is steeper than ones observed at late time for higher concentrations it is in good agreement with ones occurring just after the peak. From this one can conclude that in case of 1 ppm, resolution of the measurements is sufficient only to capture early times after the peak.

Concentration [ppm]

Peak arrival time The irregular late time behavior observed in the experiment with strongest concentration (1000 ppm) of the injected solution (Figure 3.9) requires further investigations of the result reproducibility. In Figure 3.10 one can observe that for the same flow conditions and the same injected concentrations late time behavior can significantly differ. These behaviors appear to be unpredictable and are presumably due to occurrence of variable tracer localization in preferential channels induced by density effects. For all the four experiments peak arrival time differences are inside the experimental error range. These effects do not occur at low value of the injected tracer concentration. For instance Figure 3.11 presents BTCs for two experiments with tracer concentration of 50 ppm and with different tracer volumes of 2 and 10 L, presented with blue and red curve respectively. Results for the two experiments are very consistent. The two BTCs are remarkable similar both in terms of early and late time behavior; the tail of both curves displays a characteristic local slope of t 10.6 . The only differences are a) a wider concentration distribution around the peak and b) a shift in the peak arrival time; the peak arrives later in the experiment with the larger volume injected. This can be explained by the longer injection period for the 10 L injection. Indeed, we observed (see section 3.3.2, Figure 3.14) that experimental results are sensitive to the solute injection rate and therefore slow injection rate (i.e. long injection periods) is used so that disturbance in the flow around injection point is low and at least similar in both cases (using the same tracer injection rate).

Glass beads push pull experiments

Push pull experiments on the glass beads packing are made on the same sample and with the same flow rate as the flow through experiments described earlier. The only difference is that tracer is injected from the bottom of the sample (push phase) and then after a certain time flow direction is inversed and directed from the bottom (pull phase) of the sample. For some experiments between push and pull phase pumps are stopped to create no flow conditions. This period without flow hereafter will be referred to as the resting period. Desired flow rate is achieved in less than 10 s after turning on the pumps, while turning off is instantaneous. Therefore it can be considered that pump manipulations do not have effect on the results. Push period duration is estimated from the flow through experiment with the aim to make it as long as possible, without causing any tracer leakage at the other side of the sample. It is crucial that there is no tracer leakage from the porous media since this could cause misinterpretation of experimental data. Longer push phase allows tracer to explore larger portions of the porous media. Push pull experiments are performed in order to distinguish difference between reversible (spreading) and irreversible (mixing) parts of dispersion. Investigation of the density effect similar to one made for the flow through experiment (section 3.3.1) is presented in Figure 3.12. However, for push pull test only the two lowest concentrations are tested with push period lasting 22 minutes. The BTC curves for both tracer concentrations have similar peak arrival time and behavior just before and after the peak. This confirms conclusions from flow through experiments that there is no visible density effect if tracer concentration is 50 ppm or less. At late times for the experiment with higher tracer concentration, one can observe occurrence of the tailing that is not visible in the experiment with the lower tracer concentration because of the resolution limit of the TeLoG sensor as already observed for the flow through experiments. Therefore, further experiments will be made with a tracer concentration of 50 ppm. Figure 3.13 presents push pull experiments with different push phase and resting period duration. The aim of this comparison is to investigate solute transport behavior in relation with different time spent in the porous media and different distances that it explores. All experiments are performed at the same conditions (flow rate 50cm 3 /h, injected concentration solution 50 ppm) and on the same sample. As expected, difference in the peak arrival time is equal to the differences in the sum of the push and resting period durations. The peak concentration is lower in the experiment when peak arrival time is longer; this is due to the diffusion that has more time to influence on the plumb center. It can be noticed that longer push and resting period does not affect BTC's shape tail since all three BTCs display similar late time slopes of t 6.2 . Experiments with longer duration of push and stagnant phases have tendency to faster converge between two slopes. This means that difference in the time spent in the system does not affect the late time behavior.

Comparing results to similar flow through experiment (Figure 3.11), one can observe that in push pull there are two very distinct behaviors. Immediately after the peak, one can observe a faster decrease in the concentration which is explained by the fact that the reversible part (spreading) of hydrodynamic dispersion is cancelled. At later times, the second slope appears which is controlled by the irreversible dispersion (e.g. controlled by diffusion in slow flowing zones), whereas, in flow through experiments, the concentration decrease rate is rather constant and its value is somewhere in between the two slopes that characterize the push pull experiment. Clearly it is reasonable to expect that the concentration decrease in the flow through experiment immediately after the peak is slower than in the push pull experiment due to spreading of solute plume. However, it is also expected that BTC slopes at late time for both experiments have similar values; since late time behavior is supposed to be controlled by the diffusion transfer between fast and stagnant (no) flowing zones. One can observe that the slope in the flow through experiment is significantly steeper than the second slope captured in the push pull experiment. The reason for this difference is in the fact that in case of the flow through experiments late time behavior is not captured and that the slope is still controlled by overlapping effects of flow field heterogeneity and mobile immobile transfer.

Figure 3.14 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, injected concentration solution 50 ppm, push duration 22min. Magenta curve -fast injection, Green curve -slow injection

As for flow through experiments, repeatability is very important issue. Since the sample is relatively small, it is reasonable to assume that manipulations should be very precise to ensure good repeatability and ability to compare different experiments. Figure 3.14 presents BTC for two experiments performed with same injection concentration, injection volume, push period and flow rate. Difference is made in the injection procedure. Since tracer injection is made by human with a syringe it is impossible to expect that there will be two injections with exactly the same conditions. Therefore we compared two "extreme" injection cases. Magenta curve presents experiment where whole volume of the tracer is injected in 1 or 2 seconds (i.e. fast injection), while green curve represents slower injection in which tracer was injected during approximately 10 seconds. The BTC from the experiment with faster injection has earlier peak arrival time and narrower peak. Also concentration decreases faster immediately after the peak, however one can observe that late time behavior in both cases is the same, i.e. is not dependent on the solute injection uncertainties.

Despite precautions (e.g. bottled water, elimination of the light pollution…) taken during the designing and performing experiment we encountered unresolvable problem with measurement resolution. For unexplained reasons measurement equipment would decrease its resolution. Example can be seen on Figure 3.15 presenting two exactly identical push pull experiments performed in two consecutive days. Experiment performed on day before (blue line) had significantly lower resolution than second experiment. There were no differences in the water, room temperature and light conditions between two experiments. Only possible explanation can be in the fluctuations of the TELog sensibility due to possible changes in the voltage or the temperature. However, this investigation has shown remarkable consistency at higher concentrations and excellent repeatability of the experiment results.

Berea sandstone flow through experiments

In this section we present results of the flow through experiment performed on the Berea sandstone core (length 15 cm, diameter 18 mm). We performed experiments at two different flow rates, 10 and 2 cm 3 /h. Calculated Peclet numbers (2.11) are 100 and 20, with Reynolds number (4.7) ranging between 0.06 and 0.01 for given flow rates. To calculate Peclet and Reynolds numbers characteristic length ( 110 m) is taken from calculations made on segmented images presented in section 4.2.4. Porosity is 0.19; it was measured during mercury porosimetry test. Other parameters were taken from the literature; water density is 1000 kg/m 3 , fluorescein molecular diffusivity is 6.4×10 10 m 2 /s [START_REF] Galambos | Micro Fluidic Diffusion Coefficient Measurement[END_REF]] and water dynamic viscosity at 20 0 C is 1.002×10 3 . Experimental results are presented in Figure 3.16. Red curve represents BTC for the experiment with higher flow rate and blue one BTC obtained at the lower flow rate. Both BTC displays signatures of anomalous transport with elongated tails. BTC resulted from the experiment with higher flow rate is characterized at the late time by constant slope that scales in time as a power law with a slope of 2.8. On the other hand on the BTC obtained at lower flow rate the slope cannot be determined. BTCs tails scaling as a power law in time are commonly recorded in the conditions with higher Peclet number and late time behavior is more pronounced. However from the Figure 3.16 it is clear that the concentration decrease for slower flow rate is significantly slower than one noticed in at higher flow rate. This behavior is contrary to our expectations. In theory the late time behavior is controlled by matrix diffusion and therefore is function of the porous medium properties and independent on the flow rate.

To understand this observed a priori incoherent behavior, first we have to explain a problem that occurred during the experiment with higher flow rate. Due to relatively high pressure at the injection points there was a leakage during the injection. Rubber foam tube (tube 2, see section 0 and Figure 3.4) started to leak at the position where needle had penetrated it. Yet, although the hole was closed in couple of seconds with another needle a small volume of tracer leaked and therefore was not injected in the water flow. Note that the needle used to close the hole was clean and inserted just in the rubber tube in order to not disturb the flow. We believe that this small leakage did not influence the experimental results and BTCs shape; however, it is clear that the amount of tracer actually injected in the core sample was smaller than expected. It is difficult to have a precise appraisal of the amount of tracer that was lost. Nevertheless, we can make some rough estimation by noting that we approximately recover 14 % of the tracer in the higher rate experiment compared to the lower rate experiment (Figure 3.16). Although it seems quite large, since leakage was almost immediately stopped (one small drop leaked), it is worth reminding that the total injected volume is small as well (50 L) and that excellent retrieval rate was measured in previous experiments. Figure 3.17 a) presents the scaled BTC according to the estimated injected volume. To explain the observed behavior it is compared with modeling results (modified Figure 11 from section 5.3). Figure 3.17 b) presents results of the transport simulations with different Peclet numbers ( =1000 pink symbols, =100 green symbols) performed on the segmented images of the same Berea sandstone rock as used in the experiment (more details are in section 5.3). Connected symbols denotes simulations where both influences of the flow field heterogeneity and mass transfer to the microporous phase are considered, while unconnected symbols denotes simulations where mass transfer to microporous phase is neglected. Qualitative comparison between Figure 3.17 a) and b) reveals similarities in the observed behavior. The rectangular window (Figure 3.17 b)) represents the limits of the experiment resolution, so experimental BTCs are compared only with part of numerically obtained BTCs inside the limits of this window. BTC for the simulation with higher have more pronounced slope after the peak and that behavior influenced by matrix diffusion starts later than in the case of the lower . The experimental BTC for higher displays the same behavior, except that influence of the matrix diffusion is not visible due to resolution limitations. For lower slowing of the concentration decrease rate observed in the experiments is visible also in the numerical simulations, but only ones that include matrix diffusion. This comparison reveals significant limitations in the experiment resolution, despite its ability to measure over more than 5 orders of magnitude. Due to these limitations it is not possible to study BTC behavior influenced only by the matrix diffusion (i.e. late time behavior) and they can mislead to the conclusion that the heterogeneities in the flow field are exclusively responsible for non Fickian behavior. However experimental results with lower emphasize importance of inclusion matrix diffusion into transport calculations, which cannot be observed at higher .

Summary and conclusion

In this section we presented an experimental set up that allows studying dispersion of the non reactive tracer. Test bench design allowed continuous measurement of the tracer concentration over several orders of magnitude. Improvements in the injection procedure minimized tracer dispersion in the system before entering the porous media. However it also caused leakage in experiments where a high pressure difference was imposed.

Series of experiments with different tracer concentrations performed on the glass beads column emphasize the importance of the density effect. The density effect is usually neglected, especially in the large scale experiments. However, we proved that density effect can impair experimental results even if the concentration is 500 ppm. Although, concentration (and consequently density) of the tracer decreases during the experiment, final results may be misinterpreted due to density effect in vicinity of the injection point. In this scope we demonstrated influence of different injection times (rates) on the BTCs shape.

Comparison between push pull and flow through experiments showed limitations of the experimental equipment when trying to capture late time behavior. These limitations were confirmed on the flow through experiments performed on the Berea sandstone. Also, qualitative comparison between experimentally and numerically obtained BTCs on the Berea sandstone emphasized the importance of inclusion both matrix diffusion and flow field heterogeneity in the pore scale transport modeling.

Finally, one must admit that despite a tremendous effort for obtaining state of the art core tracer test, the results cannot be used with confidence for parameterizing non Fickian model. This conclusion is the results of a large set of experiments for testing 1) the effects of tracer concentration, injection procedure and optimization of the equipment and 2) the result reproducibility. Albeit there is still space for improvement of the equipment and protocol, we believe that tracer tests on cores must be analyzed with care because 1) a small experimental bias can produce large effects on the BTC and 2) technological barriers limit our possibilities of tackling part of the BTC tails. This is specifically obvious for low Peclet experiments corresponding to the most general case in natural geological situations. It follows that an apparently power law behavior defined over less than one order of magnitude in concentration can be just a transitional behavior between, for instance velocity heterogeneity effects and MRMT effects.

Pore scale numerical simulations of the flow field

Introduction and framework

Recent development of pore scale imaging techniques, computational fluid dynamics solvers and more powerful computers has allowed investigating origin of anomalous transport at pore scale. The common approach is to process X ray microtomography (XRMT) images in order to distinguish the pore from the solid, then simulate the steady state flow for incompressible fluid and finally solve the diffusion advection in order to obtain the distribution of the tracer concentration according to specified boundary conditions [START_REF] Ovaysi | Pore scale modeling of dispersion in disordered porous media[END_REF][START_REF] Blunt | Pore scale imaging and modelling[END_REF].

XRMT acquisition and image processing is described in section 4.2. These images are then used to calculate flow field. Flow field is calculated using Navier Stokes equation pre programmed in computational fluid dynamics software OpenFOAM. Detailed procedure is described in section 4.3. Results of the flow simulations with emphasis on comparison between different mesh resolutions are shown in section 4.4. Transport results are presented in the first article in Chapter 5.

Rock structure imaging

X ray microtomography (XRMT)

Internal geometry of the rock is the main input parameter needed for flow and transport simulations at pores scale. XRMT is a non invasive imagine technique used to obtain realistic presentation of the pore space geometry. Image resolution can vary from few decimeters to the tenths of micrometers.

Figure 4.1 illustrates procedure of obtaining real 3D model of pore space from a rock sample using XRMT approach. X ray beam is emitted from the source passing through the sample and captured by CCD camera placed behind sample. The image consists of the projection of the sample in terms of X ray intensity. Projections of samples are taken from different angles ranging from 0 0 to 180 0 (or 360 0 ) and the step change between projections does not exceed half degree. Recorded projections are transferred from the camera to the processor where they are filtered and reconstructed. Result is a set of 2D images representing sample slices in terms of X ray attenuation per pixel. X ray attenuation depends on chemical compound and density. For mono crystalline rocks, XRMT images are actually maps of rock density (porosity), since there is no variation in chemical compound. Sets of 2D images are arranged and together form a 3D image of pore space. This 3D image is suitable for physical parameter analysis and pore space geometry reconstruction.

For this work the high resolution XRMT image of the Berea sandstone core (10mm length and 6mm diameter) was acquired using the BM5 beamline at the European Synchrotron Radiation Facility (Grenoble, France). The 3D volume was reconstructed from 3495 projections acquired at an energy of 110 keV using a GGG:Eu 100 microns scintillator and a SCMOS based 2048 x 2048 pixels detectors (model PCO edge; http://www.pco.de). The reconstruction was performed using the single distance phase retrieval algorithm described by [START_REF] Paganin | Simltaneous phase and amplitude extraction from a single defocused image of homogeneous object[END_REF] applying an unsharp filter before reconstruction. Details on the reconstruction algorithm and performances can be found in [START_REF] Iassonov | Segmentation of x ray computed tomography images of porous materials: A crucial step for characterization and quantitative analysis of pore structures[END_REF]. The final 3D image is formed by 4667 × 2130 × 2099 voxels of characteristic size 3.16 m (3.16 10 6 m). The 16 bits encoded value associated to each of the voxels denotes the X ray absorption integrated over a volume of 31.5 m 3 .

Segmentation

Analysis of the gray level histogram from raw XRMT images pointed out existence of three types of materials, denoted hereafter phases. The three phases correspond respectively to the macroporosity (i.e. the void space filled with air or liquid), the microporosity (voxels in which pores are smaller than image resolution) and the solid rock. Furthermore, existence of pores smaller than image resolution is confirmed by the Mercury Intrusion Porosimetry (MIP) test. MIP is a widely used technique for characterizing the pore size distribution in porous media. It is based on the premise that a non wetting fluid; in our case mercury, will not intrude capillaries of a given diameter unless a given pressure is applied [START_REF] Abell | Mercury Intrusion Porosimetry and Image Analysis of Cement Based Materials[END_REF]. Relationship between the capillary pressure ( ) and the pore radius ( ) is given by Washburn equation (4.1) [START_REF] Washburn | The Dynamics of Capillary Flow[END_REF] where is the surface tension and is the angle of contact.

Throat size distribution is determined from volume of mercury intruded at each step. One of the most common approaches to image segmentation is applying global threshold. This simple approach consists in deciding a grey scale value that marks separation point in X ray attenuation between two phases on the basis of image histogram analysis. However, by definition, the microporous material can display a large range of gray level values denote pixels sampling both solid and voids. Consequently it is a priori not pertinent to use the global thresholding method for porous materials in which the existence of microporosity is known. Several alternative methods have been developed in order to improve the accuracy of the segmentation. A comparative study of different approaches has been presented by [START_REF] Iassonov | Segmentation of x ray computed tomography images of porous materials: A crucial step for characterization and quantitative analysis of pore structures[END_REF].

For our study we used the so called "region growing" method based on the assumption that all the voxels belonging to a given phase cluster are connected and they are similar in term of gray level [START_REF] Siprkovska | A Summary of Image Segmentation Techniques[END_REF][START_REF] Noiriel | Hydraulic properties and microgeometry evolution in the course of limestone dissolution by CO2 enriched water[END_REF]. A significant difficulty of growing methods is the need for carefully defining the grey level thresholds that defines boundaries between regions. This is typically done manually from analyzing the histogram. However, the image histogram does not contain enough information to properly identify the different phases and relatively small differences in thresholds may lead to noticeable errors on the determination of the phase fraction and structure of pore space. For this reason in the present work we applied method proposed by [START_REF] Mangane | Permeability impairment of a limestone reservoir trigered by heterogeneous dissolution and particles migration during CO2 rich injection[END_REF]. It consists in conditioning the determination of the threshold values by the value of the total porosity measured by laboratory techniques.

Delimitation of 3 phases is done in two steps. In the first step we perform standard segmentation in which 2 thresholds are applied. All values that are smaller than lower threshold belong to first phase. Values larger than higher threshold are assigned to second phase, while the algorithm decides on assigning pixels with values between threshold limits to one of two phases. For 3 phase segmentation we have to perform this procedure two times to obtain two images, consequently we need 4 thresholds. Two images differ in arrangement of segmented phases. Phase one at first image is denoting void, while microporosity and solid are combined in second phase. On the second image void and microporosity are combined together in first phase while second phase contains only solid. Second step is overlapping and addition of two images. Areas that are marked as first phase on both images are considered as void, if the area is marked by second phase on first image and by first phase on second image it is considered as microporosity and finally all area that are marked with second phase on both images are considered as rock.

Described procedure is applied to perform 3 phase segmentation on Berea sandstone. From gray scale level distribution of raw image presented in Figure 4.3, 4 thresholds were defined. After finished segmentation the total porosity ( ) of the segmented image is calculated by equation:

(4.2)
with and denoting fraction of the void and of the microporous phase respectively, and denoting the intrinsic porosity. The convergence toward a satisfactory agreement between the total porosity of segmented images and the connected porosity obtained by MIP is obtained by an iterative procedure. Threshold values were adjusted until the total porosity was in good agreement with the experimentally obtained porosity. Final threshold values are presented on Figure 4.3. Blue dashed lines are denoting thresholds used to delimitate between void and microporosity, while red dashed lines denotes thresholds delimitating microporosity and rock. Procedure was applied assuming that the mineralogical composition of the microporous cement is homogeneous and second that the intrinsic porosity of microporous phase range from 0.01 to 60 %. Assumption of maximal intrinsic porosity is based on the fact that above 60 %, material cannot be cohesive. Whole procedure is based on comparing the experimental porosity to the total porosity (4.2). Certainly XRMT images may contain pixels belonging to the unconnected porosity which cannot be measured by MIP test. However, we performed measurement of this unconnected porosity a posteriori (details in section 4.2.3) and found that share of unconnected porosity is lower than 0.8% for all extracted sub volumes. Accordingly one can consider that this value represents the error in the segmentation.

On XRMT images there are some voxels located directly on the interface between void and rock phases. Gray level of these pixels is similar to gray level of microporosity and during segmentation they are assigned to microporous phase. In order to avoid the appearance of microporous layer with thickness of one voxel on all void rock interfaces we performed one layer abrasion during segmentation process. This abrasion is applied during first step on the image where microporosity and rock are place together in the same phase.

With the procedure described in this section we obtained realistic pore space geometry representations of the Berea sandstone (Figure 4.4). The fraction of the different phases are 79.45, 18.15 and 2.4 for solid, void (macroporous) and microporous phase respectively. The intrinsic porosity of microporous phase is heterogeneously distributed between 0.01 and 60%. Distribution is obtained directly from XRMT images as explained in section 4.2.1 .

Percolation clusters

The first step of segmented data processing consist in eliminating unconnected pores since they do not participate in solute transport through porous medium. Network of interconnected pores that spans through the whole sample volume in the flow direction connecting boundaries is called percolation cluster. In this work percolation cluster were defined by cluster labeling method proposed by Hoshen and Kopelman [1976].

Connected clusters were determined with two different assumptions, first that microporosity is percolating and second that it is not. Figure 4.5 illustrates differences between two approaches. White color represents macroporous zone that is percolating in both cases and this zone will be referred as mobile domain. Immobile domain consists of microporous zone denoted in orange and macroporosity that is connected only through microporosity that is colored in blue. In the mobile domain flow field can be calculated with standard approach using Stokes equation and transport is driven by both velocity and diffusion. Flow calculations in the immobile domain are feasible [e.g., [START_REF] Brinkman | A calculation of the viscous force exerted by a flowing fluid on a dense swarn of particles[END_REF], but not very common on pore scale. However, even if flow field is calculated, fluid velocities are significantly lower than in mobile phase and transport is dominated by diffusion. Inclusion of the immobile domain increases porosity on average by 5.3 % in the segmented sub volumes of Berea sandstone; while macroporosity connected only through microporosity contributes in this increase with around 40 %.

Characteristic length

The characteristic length (L) is the property of porous media necessary to calculate Reynolds (4.7) and Peclet (2.11) numbers, which is usually expressed as average grain (experiments with unconsolidated porous media) or pore diameter. At pore scale it is usually associated with the average pore diameter in the relation with the definition of the Peclet number in a capillary [START_REF] Taylor | Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube[END_REF].

For heterogeneous porous medium the characteristic length can be defined from the variogram of the mobile domain porosity. The variogram for Berea sandstone is presented in Figure 4.6 where dashed red line marks distance ( 110 m) at which one can consider that the variogram reached asymptotic value. This is considered as the characteristic length. [START_REF] Mostaghimi | Simulation of Flow and Dispersion on Pore Space Images[END_REF] proposed alternative way to calculate the characteristic length based on the cubic packing of regular spheres. For this idealized system, the characteristic length ( ) can be associated with the sphere diameter

(4.3)
where is the bulk volume of the porous medium (void and solid) and is the interface area between the void and solid. Applying this method on Berea sandstone we obtain characteristic length equal to 124 m. We note that characteristic lengths calculated with both methods are in good agreement with characteristic lengths calculated on Berea sandstone in other relevant studies: 100 m [START_REF] Bijeljic | Pore scale modeling of longitudinal dispersion[END_REF], 131.13 m [START_REF] Mostaghimi | Simulation of Flow and Dispersion on Pore Space Images[END_REF] and 150 m [START_REF] Berkowitz | Modeling non Fickian transport in geological formations as a continuous time random walk[END_REF].

Obtaining flow field

Representative elementary volume

To obtain a realistic velocity field of heterogeneous medium it is necessary to satisfy two opposite conditions. First one is to have a high resolution in order to capture all geometrical properties and the second one is to have a domain that is large enough to capture all heterogeneities of porous media. Increase in resolution is resulting in an increase of computational cell number and therefore reduces the size of the media that can be simulated, thus it is essential to achieve a good balance between resolution and domain size.

Properties of a heterogeneous medium vary in space and therefore the volume on which simulations are performed should be large enough to the capture average properties of medium. Volume that satisfies this minimal requirement is usually called the representative elementary volume (REV). One of the most popular methods to define REV is through studying the scale dependence of the porosity [START_REF] Bear | Dynamics of fluids in porous media[END_REF]. To evaluate REV of Berea sandstone images, six points were chosen completely randomly in the whole image. From these starting points, cubes of different volumes were extracted. For each cube, connected porosity is calculated and divided by the total porosity of the whole image. Similarly volume of each cube was divided by volume of the cube with side dimensions of 300x300x300 pixels since this size is chosen for numerical simulations presented in this work. Figure 4.8 presents results of investigation of the REV in terms of porosity. We observe that the porosity starts to become consistent with the total porosity at volumes that are twice smaller than the chosen volume and continuous this trend for larger volumes. Dashed red line presents volume equal to 300x300x300 block (0.85 mm 3 ). However porosity defined REV size can differ in terms of the hydrodynamic properties of the porous media that is being observed. Therefore we decided to investigate REV in terms of permeability. We randomly choose one series of blocks originating from same point used in porosity comparison. On each block flow simulations are performed and permeability [m 2 ] is calculated from Darcy's Law by equation is area [m 2 ] of cross section perpendicular to flow direction. Figure 4.9 presents results of permeability calculation for cubes with different volumes. Sample volumes are normalized to the volume of 300x300x300 pixel (0.85 mm 3 ) cube, which is represented by dashed red line. Permeability is normalized to the permeability of the referent cube and we observe that it starts to become uniform from cubes that are 40% smaller than referent. From both of these investigations we can conclude that cubes with side length of 0.95 mm used for the simulations are large enough to have good representation of Berea sandstone sample. Estimated value is in a good agreement with the REV investigations performed by other authors on the same rock, who estimated that cubes with side length larger than 1.

Mesh generator

Solving partial differential equations using CFD simulators requires that the domain is divided in smaller sub domains. Inside these sub domains the governing equations are discretized and solved. The sub domains are usually called cells and collection of cells is called a mesh or a grid. There are different types of meshes which are usually classified as structured and unstructured meshes. Structured meshes are the ones made with always the same elements that have the same size, while in unstructured meshes cells varies in size and shape.

Mesh generation is quite complex and time consuming task which has direct influence on the simulation convergence as well as accuracy of the final result. Therefore it is essential to obtain a mesh with a good quality for given problem. Good quality mesh captures the system/problem of interest in a discrete manner with enough resolution to enable capturing enough information while retaining reasonable number of cells.

In this work we use structured mesh made of cubes (regular hexahedra). Such a decision is based on a fact that cubic mesh easily captures pore structure from segmented XRMT images and it is suitable for post processing to obtain necessary input data for the transport simulations. Pore scale simulations on XRMT image commonly use image resolution as mesh resolution, which means that cells are the same size as voxels. However a recent study of flow properties by Guibert et al. [2015] has shown that mesh resolution should be smaller than XRMT image resolution.

Mesh generation utility provided with OpenFOAM is called snappyHexMesh. This utility enables to generate structured and unstructured 3D meshes automatically from triangulated surface geometries saved in Stereolithography (.stl) format. To produce *.stl file we used the commercial image processing software Avizo, however this software treats each voxel in XRMT images as a point with coordinates but without a real volume. This causes changes in the pore structure between original image and resulting *.stl file. Also this approach is very memory consuming, thus limiting maximal size of domain that can be meshed.

To resolve this problem we developed our own mesh generator that reads XRMT images and produces mesh with the same geometry as original image. One of the features of this mesh generator is that each voxel can be divided by any integer in all three directions creating a structured mesh with several times higher resolution than the original image. Comparison of mesh generators is presented on the Figure 4.10. On the left side (Figure 4.10 a)) is the mesh made with snappyHexMesh. Black area is representing solid phase, white parts are void spaces, green line is surface made by Avizo (*.stl file), blue lines are internal mesh boundaries between neighboring cells and red lines are external boundaries between void and solid phase. It can be observed that there are some void spaces that are not included in the mesh. On the right there is a mesh made with our software. We can see that mesh external boundaries (red line) are at the same position as boundaries between void and solid in the XRMT image. Comparison on the same geometry and in the case when each voxel is divided into 27 equal cells shows that approximately 7% of void phase is not included in the approach when using snappyHexMesh. Figure 4.11 illustrates impact of the mesh resolution to the flow simulation results on an example of a throat with diameter equal to one voxel in two dimensions. In laminar flow, velocity distribution is not uniform due to friction with side walls. According to the Poiseuille's Law velocity has a parabolic distribution [START_REF] Adler | Porous Media: Geometry and Transport[END_REF] as illustrated on Figure 4.11 a) with fast moving fluid in the middle of the throat and almost stagnant. Figure 4.11 b) illustrates a case in which cell size is equal to the XRMT image size. Parabolic velocity distribution is substituted with only one value and in this case velocity distribution in the throat is neglected. On Figure 4.11 c) there is presented a case in which the mesh resolution is three times smaller than image resolution. In this case we see that velocity in the middle cell is larger than in the ones next to the boundary. It is possible to capture part of the throat velocity distribution. In ideal case mesh resolution would be small enough to capture full parabolic velocity distribution. However this would mean a very big mesh and consequently would make simulations extremely long and demanding in terms of computer power usage. Therefore it is necessary to find a mesh resolution that captures the most important flow field properties while creating a mesh that can be handled by available computer resources. Investigation to determine optimal mesh resolution is made on the 150x150x150 voxel image extracted from the one of the subsamples used in the numerical simulations and with approximate porosity as the whole image. We made 7 simulations with different cell size, every time voxels were divided by an integer in all three directions. After each simulation permeability is calculated (4.4) and results are presented in Table 4.1. Pixel divided column denotes integer used to divide voxel, e.g. by 3 denotes that voxel is divided by 3 in each direction and that each voxel contains 27 cells. We observe that permeability decreases with increase in the mesh resolution which is consistent with similar studies [Guibert et The control of the refinement level on the computed permeability value is larger for coarser mesh and it decreases as mesh is refined. Permeability difference between coarsest and finest mesh is more than 22%, while difference between two finest meshes is less than 0.5%. We decided to use mesh that is refined by 3 from the original image and relative difference of less than 5% from the finest mesh resolution. Maximal sample volume for each refinement level in Table 4.1 shows that refinement by 3 gives a good compromise between simulation results quality and sufficiently large volumes of observed medium.

The boundary conditions at the edges of the computational domain are set to mimic an experimental permeameter: on the inlet and outlet sides of the sample a pressure difference is imposed, and no flow conditions are specified on the other faces of the sample. No slip condition is set at the solid void interface. Twenty layers (21 m) of void voxels have been added to the porous volume at the inlet and outlet boundaries in order to minimize the boundary effects [Guibert et al., 2015]. Figure 4.12 shows the final mesh of pore space structure for Berea sandstone including inflow and outflow manifolds. OpenFOAM solver is based on a finite volume method which enables solving system of non linear equations. The pressure and velocity are solved iteratively by a steady state solver based on Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF]. SIMPLE algorithm belongs to Pressure Based Method techniques to deal with pressure velocity coupling in numerical simulations of the fluid flow. Detailed procedure has been described in [START_REF] Patankar | A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows[END_REF], [START_REF] Caretto | Two Calculation Procedures for Steady, Three Dimensional Flows with Recirculation[END_REF] and [START_REF] Patankar | Numerical Predictions in Three Dimensional Flow[END_REF].

The iterative procedure to couple Navier Stokes equations can be summed up as follows: In our case simulations are pressure driven, which means that pressure is imposed on inlet and outlet of the system as mentioned above. Since simulations are steady state, pressure value at the boundaries is constant and flow is driven only by pressure difference ( ) between inlet and outlet. Example of flow field is given in Figure 4.13. Blue color denotes regions with low local velocity while red color is used for regions with high local velocities.

When

Stokes flow occurs and Navier Stokes equation can be simplified to Stokes equation. However comparison between built in solvers in OpenFOAM shows that the solver for full Navier Stokes equation based on SIMPLE algorithm is much more efficient than comparable one for Stokes equation.

Convergence criterion

A convergence criterion in steady state numerical simulations is always an arbitrary issue. Since every problem that needs to be simulated is different, there is no straight forward answer what is the optimal setup. For flow simulations in this work we decided to use two criterions to estimate when simulations are fully converged.

First criterion is based on mass balance and we monitored inlet and outlet flux from the domain. Standard utilities to calculate flux at the boundaries built in the OpenFOAM are not appropriate for running in the same time with simulations, therefore we used a plug in swak4Foam that prints inlet and outlet flux in real time.

Second criterion is based on initial residual values. Initial residual value presents relative differences in flow field parameters (e.g. velocity, pressure...) between the current iteration and the previous one. To determine initial residual values below which we have fully converged simulations, we run simulation till initial residuals for pressure and all three velocity components were lower than 10 9 . Permeability is calculated by equation (4.4) every time the residual is lower or equal to threshold. Figure 4.14 present the evolution of permeability with decreasing values of the initial residual. Permeability is stabilized when initial residual level is lower that 10 4 . To be sure that we use adequate convergence criteria that is still reasonable in terms of computational time all simulations in this thesis are stopped when initial residuals are below 10 5 .

Figure 4.14 Normalized permeability in function of initial residual, red dashed line denotes initial residual level chosen as convergence criterion of all OpenFOAM simulations

As expected, there is globally a weak positive correlation between porosity and permeability (Figure 4.16). It can be noted that adding flow calculation in the microporosity does not have a significant influence on the permeability while it induces a noticeable change of the porosity. This is reasonable since velocities in the microporous zone have one or several orders of magnitude lower values than in macroporous zone and there is only very few zones were the connectivity is controlled by microporous clusters.

Mesh comparison

Mesh resolution has significant effect on the flow simulation results. Table 4.1 presents influence of seven different mesh resolutions on the permeability of the small sub volume of Berea sandstone. Here we present a more detailed investigation focused on coarse and fine mesh resolution. Table 4.2 shows differences in the permeability between two mesh resolutions for six different sub volumes of Berea sandstone. Permeability calculated from the coarse mesh is always higher than the one calculated from the fine mesh by 14 to 18.3%. These results are in good agreement with previous investigation (Table 4.1) and relevant studies [Guibert et al., 2015].

Figures from 4.17 to 4.22 show average (a), maximal (b) and minimum (c) velocities per slice perpendicular to the flow direction for different sub volumes and different mesh resolution. Blue line is always denoting fine mesh; orange line is for coarse mesh and dashed black line represents local porosity. There is clear negative correlation between porosity and average velocity, however there is no correlation between porosity and minimal and maximal velocities.

When comparing averages velocities per slice there is a clear offset and average velocities are always higher in the coarser mesh. There is no significant variation in differences between average velocities. Observed offset of around 10 15% is in accordance with permeability differences and also cross plot between coarse grained average velocity from fine mesh and corresponding velocity in coarse mesh (section 5.3, Figure 9).

Comparison between maximal and minimal velocities (Figures 4.17 andc)) shows that in all situations maximal velocity is higher in the coarse mesh. Similar minimal velocity per slice is lower in the coarse mesh; however this difference is even less highlighted. Although it is clear that coarser mesh causes greater differences in velocity distribution, deviation is relatively small. To gain even better understanding of the mesh resolution impact on the velocity field distribution we created 3D artificial structure simulating a synthetic throat between two pores (Figure 4.23). The structure is completely symmetric and it gradually decreases in area perpendicular to the flow direction until it reaches the size of one pixel. The coarsest mesh has same resolution as the original structure, and all other meshes are gradually finer. Resolution is increased by principle explained in the section 4.3.2. Permeability and maximal velocity for different mesh resolutions are summarized in Table 4.3. Relative differences are calculated regarding to finest mesh results. Again one can observe a gradual decrease in the absolute permeability that is in accordance with relevant studies and previous investigations presented in this thesis (Table 4.1). However, differences in this case are more pronounced. This is probably due to the structure geometry which is designed to investigate case where largest differences are expected. From the table it can be noticed that maximal velocities in the meshes where pixels were divided by even number are lower than comparable meshes where pixels are divided by odd number. This is due to the fact that meshes where pixels are divided by even integer fail to capture the maximal velocity in the middle of the throat. 4.27 presents flow field results for finest mesh (each pixel divided by 7 in all 3 directions), it is considered as the best approximation of the real flow field. One can observe that if mesh has same resolution as image (Figure 4. [START_REF] Gaspar | Modern Trends in Tracer Hydrology[END_REF], it completely fails to reproduce realistic pressure and velocity distribution. Even more worrying is that maximal velocities occur in cross sections before and after the narrowest one. Therefore, mass conservation is questionable, although fluxes at inlet and outlet of the system at the end of simulation are the same. We repeated the simulation several times with different pressure gradients and increasing convergence criterion, however results are always the same. Detailed investigation of the relative cell divergence revealed that divergence in the cross section before throat is equal to 7×10 4 , while divergence in finer meshes never reaches 6×10 6 . However this does not fully explain the value of 9 % lower local velocity in the throat compared to neighboring cross sections. Relative cell ( ) divergence (

) is calculated by equation

(4.8)
where is the sum of the fluxes at all six faces of the cell and is magnitude of the velocity in the cell centre.

Pressure distribution for the second mesh (pixels divided by 2) is significantly better (Figure 4.25 a)). However the velocity distributions map (Figure 4.25 b)) shows that at this resolution there are still significant misinterpretations of velocities in the throat. Contrary to the coarser mesh, in case when mesh has a three times higher resolution than the original image, pressure and velocity distributions (Figure 4.26) are in good agreement with the finest mesh.

Summary and conclusion

In this section the methodology used for solving flow from the XRMT images is presented. Starting from image acquisition, followed by the segmentation and removal of non connected porosity, to the Navier Stokes solver implemented in the OpenFOAM software package, which allowed flow field calculations. Also different ways for calculating the characteristic length and representative elementary volume are presented.

Development of mesh generator allowed transferring pore geometry directly from the segmented images to the numerical solver. Also it allowed investigating the mesh resolution influence on the accuracy of the flow simulations. This issue was recently accentuated by Guibert et al. [2015] and in this work we expand their investigation by investigating further refinements of the mesh size and discretization.

Similar to the other authors it has been observed a decrease in the permeability with the mesh refinement (Table 4.1). We showed that common practice where one image voxel is modeled by one mesh cell induces significant deviations from the results using meshes with higher discretization of the same structure. Also in some cases (e.g. modeling one pixel sized throat) flow field representation is completely unrealistic. It has been shown that the optimal mesh resolution is if each voxel is represented by 27 mesh cells in the case of our Berea sandstone image Similar scaling of the permeability to mesh resolution observed in our investigation is expected in other porous media displaying dissimilar complexity and pore geometry characteristic. However, the mesh refinement factor necessary to obtain satisfactory results may differ significantly due to the image resolution and porous medium properties. As a general rule for porous medium with very narrow throats it is advisable to use odd mesh refinement factor (e.g. 3,5,7..) in order to achieve more realistic flow field approximations.

Pore scale modeling of anomalous transport

Introduction

This section presents transport simulations performed directly on the X ray microtomography images focusing on the dispersion of ideal tracer in a porous medium. Image processing and flow simulations are presented in the previous chapter (Chapter 4). Selected sub volumes (300 3 voxels) were post processed after flow simulations and used as an input data for transport simulations.

The hydrodiffusive transport of a passive tracer is modeled by random walk particle tracking method in which solute concentration field is represented by the ensemble average of particles. Actually these particles are random walkers, which are considered to be without mass, infinitesimally small and that they do not interact between each other or with the porous medium. In the random walk methods a large number of particle moves through the medium taking random steps that may vary in distance, direction and time need to make a step. Although path taken by each particle is not representative; the ensemble average of the motion of a large number of particles represents the solute transport behavior for given boundary conditions.

There are numerous different random walk methods and in this work we decided to use the time domain random walk (TDRW). TDRW proved to be very efficient tool since it is based on the particle movements with constant length, while stochasticity arises from the direction of the step and time between two consecutive steps. Constant step size makes TDRW very suitable to use in the discretized media with complex geometry; where other methods with stochastic distribution of step size would cause numerous missteps and thus have negative influence on the duration of simulations.

Main results, discussion and conclusions are presented in form of an article (see section 5.3) accepted in Water Resources Research. Expanded theory and assumptions taken into account during modeling transport in microporous domain are presented hereafter (section 5.2).

Immobile region properties

Percolation threshold of microporosity

Similar as discussed in section 0 for macroporosity, there is also a fraction of unconnected porosity in the microporous phase which is inaccessible for solute transport. Contrary to macroporosity, unconnected microporosity cannot be detected from XRMT images, since pores are smaller than image resolution. In terms of continuous model, intrinsic porosity of microporous phase has to be higher than percolation threshold ( ) in order that microporosity is accessible to flow and transport. Figure 5.1 presents an illustration of percolation threshold effect. In this illustration it is assumed that solute is diffused into immobile domain from mobile domain (blue) at the top. Figure 5.1 a) shows distributed intrinsic porosity of microporous phase without percolation threshold applied. Gray level denotes value of intrinsic porosity, where darker colors represent areas with lower intrinsic porosity while lighter colors are used for areas with lower intrinsic porosity. Influence of low percolation threshold is shown on Figure 5.1 b) where one can see yellow areas that are considered to be inaccessible for diffusion since intrinsic porosity is lower than percolation threshold. Here we observe that inaccessible microporosity is scattered in microporous phase and lacunarity is increased. As a consequence of lacunarity (i.e. heterogeneity of microporous phase), macroscopic tortuosity in microporous phase is increased as presented on the Figure 5.1 c). Red arrows denote flow paths taken by solute in microporosity. This increase of heterogeneity in microporous phase means that transient time that solute experience when diffuse into microporous phase is increased. Consequently tails in break through curve is elongated. Finally when percolation threshold is increased so that most of microporosity is inaccessible to diffusion (Figure 5 In bulk solution the rate of molecular diffusion is defined by the molecular properties of the fluid and the tracer as well as the temperature and pressure in the system. However under same conditions, the spreading of a passive tracer in a porous medium is significantly slower than in free fluid due to the inhibiting effects of the complex structure that slow down spreading in all directions. The diffusional tortuosity can be modeled with a random walk method; however in case of the microporosity defined in our images it is not possible since pores are smaller than the image resolution, and therefore we must define the effective diffusion coefficient from a model.

Effective diffusion in the microporosity

Diffusive mass transfer rate in microporosity varies spatially due to structure complexity of microporosity. Complexity arises from different intrinsic porosity and tortuosity structure at different locations and thus causing spatial variations of effective diffusion coefficient . Diffusive mass transfer into microporosity domain is described by diffusion equation:

(5.3)
where initial concentration in microporosity is 0 ( and boundary condition at the interface between macro and microporous domain . Diffusion coefficient depends on the local properties that are the intrinsic porosity and the tortuosity . It is defined by equation:

(5.4)

The tortuosity of the microporous phase cannot be determined from XRMT images. Many authors [START_REF] Avellandea | Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media[END_REF][START_REF] Clennel | Tortuosity: a guide through the maze[END_REF][START_REF] Dias | Tortuosity variation in a low density binary particulate bed[END_REF] suggested that the inverse of tortuosity scales as a power law of porosity. This approximation is confirmed for porosities ranging from 0.4 to 0.5. To the best of our knowledge, this approach has not been validated for lower porosities. However, due to lack any other information or means to calculate tortuosity in the microporous phase, we consider the above mentioned relation to be valid even for low intrinsic porosity. Therefore, in equation (5.4) tortuosity term can be expressed as power law of intrinsic porosity , and accordingly the effective diffusion can be rewritten:

(5.5)

In section 5.2.1 we discussed the regions of microporosity inaccessible to mass transfer because they display low porosity values so low that they do not permit percolation. It follows that the formulation for the effective diffusion is:

(5.6)
where

. In this work we will perform sensitivity analysis with different values of from n = 1 to n = 4., since we have no information on the exact value, following the methodology proposed by Gouze et al. [2008b]. The effect of the exponent in terms of intrinsic porosity distribution is illustrated on Figure 5.3. As one can see, increasing the value of widens the distribution and decreases effective intrinsic porosity.

Summary and conclusion

In this chapter we presented transport simulations performed directly on the XRMT images. Image processing and flow field calculations necessary to calculate transport are explained earlier in Chapter 4. The transport is simulated with a stochastic particle tracking method based on the random walk approach; more precisely we used time domain random walk (TDRW) [START_REF] Dentz | Diffusion and trapping in heterogeneous media: An ihomogeneous continiuous time random walk approach[END_REF]].In the TDRW distance that particle makes in each step is constant and stochasticity arises from different time needed for the step and the direction. The constant step length makes it suitable to use in a discretized system with complex structure.

Once again, mesh resolution proved to be an important factor. In transport simulations inadequate mesh resolution lead to overestimating the anomalous transport and the error in the results appears to increase with decreasing Peclet number values. Comparison between transport simulations including only flow field heterogeneity and ones including both flow field heterogeneity and mobile immobile mass transfer emphasizes their concomitant control in the intermediary domain of the BTC tail while, as expected, the mobile immobile mass transfer control the late time behavior which is, as discussed in section 3.3.3, difficult to tackle experimentally.

From XRMT images we cannot extract a lot of information regarding the microporosity (e.g. tortuosity, percolation threshold). Actually only thing that can be estimated is its intrinsic porosity: the quality of this estimation depends on the quality of the 3 phase segmentation procedure. Therefore we performed a parametric analysis of the percolation threshold and effective diffusion on the BTC shape. Increase of the percolation threshold at the lower values increases the particle resident time in the immobile domain, while at higher values it significantly decreases the fraction of immobile domain accessible to the particles which results in shorter resident time and decreased tailing of the BTC. Conversely, the decrease of the effective diffusion causes increase in the particle residence time in the immobile domain increasing the BTC tailing.

An interesting point evidenced by the result analysis is that if one assumes that the properties of the microporous zone are positively correlated with intrinsic porosity, heterogeneous distribution of the microporous zone can be substituted with its equivalent homogeneous distribution. In this case the single diffusion coefficient for the microporosity must be equal to the harmonic mean of the heterogeneously distributed microporosity.

Finally we proposed an upscaled 1D model based on the continuous time domain random walk approach able to capture behavior caused by both flow field heterogeneity and mobile immobile mass transfer. This is achieved by having two different power law distributions for representing transit times in mobile and immobile domain.

Simulations of flow in both the macroporosity and the microporosity: the Darcy Brinkman approach

Introduction

Numerical calculations of a flow field at pore scale are usually based on solving the Navier Stokes equation (see 4.3) on two phase segmented images. However in Chapters 4 and 5, we already showed that the microporosity cannot be neglected and demonstrated its influence on the anomalous transport behavior. However, in the first article (section 5.3), advection in the microporous phase was neglected, assuming that transport in this zone is completely driven by diffusion; this simplification was based on assumption that the velocities in this region is an order or two lower than in the macroporous zone.

The emphasis of this chapter is on the flow calculation in the microporous zone. There is no question that flow in the microporous zone has significant effect in the porous medium with high microporosity content (e.g. carbonates...). However, the idea is to prove importance of inclusion of the microporous phase into the flow calculations even if the microporous domain is only a fraction of the total medium volume. Problem of the flow calculations in the microporous domain could be solved by using high resolution X ray microtomography images which can capture pore geometry even in such low pore size regions. Performing flow simulations with such high spatial resolution for volume corresponding to the macroporosity REV far beyond the actual computational possibilities, but it could be used for characterizing the "macroscopic" diffusivity for small volumes of the microporous phase. Yet, we believe that anyway it would be difficult (highly costly in term of computational effort) to evaluate the variability of the microporous phase at the scale of the macroporosity REV. Accordingly, we use here the same Berea sandstone images which were segmented by three phase segmentation (see section 4.2.2). The percolation clusters were determined as in section 4.2.3 with the assumption that microporosity is always percolating (i.e. percolating threshold in the microporous zone is 0). Note that this assumption can be easily avoided, but in this preliminary work the objective is to evaluate the maximum difference in terms of permeability and flow distribution between the two models.

In the present chapter, the used approach is based on the Darcy Brinkman equation that allows solving Stokes equation in the macroporosity and an equivalent to Darcy flow in the microporous zone. In this approach, a further assumption has to be made which consists in using an heuristic model relating the permeability to the porosity in the microporous material. The Darcy Brinkman solver compatible with OpenFOAM software package was developed by Cyprien Soulaine from Stanford University. Results are presented in form of a paper which is currently in preparation for submission.

Article 2 -Effect of microporosity on pore scale simulation of flow using

X ray microtomography images

Results -comparison with Navier Stokes equation

In addition to the results presented in the previous section, Table 6.1 presents the comparison of the permeability values between simulations performed on the exactly same (300 3 voxel) blocks with Navier Stokes and Darcy Brinkman equations based approaches. For Navier Stokes simulations the microporous phase is added to the solid phase, to obtain 2 phase images. Permeability and porosity are calculated by equations (4.4) and (4.2) respectively.

Peremability [mD]

Porosity Results reveal that the inclusion of the microporous phase into flow calculations leads to an increase in permeability ranging from 4 to 7 %, while we expect errors from the flow simulations to be lower than 1 %.

Summary and conclusion

Starting from 3 phase segmented XRMT images (see section 4.2.2) we performed 3D pore scale flow simulations both in the macro and microporosity. The new solver developed in the OpenFOAM framework has proved to be appropriate for simultaneously solving the flow in the macroporosity and the microporous domain. The approach is based on the Darcy Brinkman equation, which uses single equation to calculate fluid velocity in both Stokes (i.e. macroporosity) and Darcy (i.e. microporosity) domain. It consists of adding Darcy resistance term in Stokes equation, which represents the friction between solid and void in the macroporous domain. In the macroporosity domain the Darcy resistance term is null and we solve standard Stokes equation. The most important part of defining the Darcy resistance term is in the determination of the permeability in the microporous domain. Since we have no real data on the microporosity properties (see sections 4.2 and 5.2), the permeability is estimated by Kozeny Carman equation.

The aim of this work was to study the importance of the additional flow in the microporosity even in rocks with low microporosity content such as Berea sandstone (2 %). The increase in permeability due to the microporosity is around 5 % which is relatively small but still noticeable and we can anticipate that the contribution of the flow in the microporosity should become a major process in rocks with higher fraction of microporosity such as altered sandstone and carbonates. Nevertheless it is worth recalling that the Darcy Brinkman simulation results hang on the porosity permeability model used in the microporous phase (here Kozeny Carman relationship). The implications of using different models or model parameterization (e.g. using power law k laws such as proposed by [START_REF] Gouze | On the characterization of porosity permeability relationships and reastive surface areas during heterogeneous dissolution induced by C02 injection in limestone reservoir[END_REF]) on the permeability and further on the solute transport will have to be tested in futures works. Also, as mentioned in section 6.1, high resolution microtomography (with pixel size smaller than 500 nm for instance) can be used to help determining appropriate k laws for the microporosity. the microporosity is compulsory for explaining the observations. Furthermore, we identified the methodological requirements and necessary technologic upgrading that seem to be essential for improving the experimental setup. Specifically, modifications must be made 1) to allow higher Peclet flow rate in order to better investigate the flow velocity heterogeneity on the anomalous transport and 2) to even increase the concentration measurement accuracy and resolution.

Conversely, we believe that further developments should focus on a better characterization of the microporosity and velocity distribution in order to relate the CTRW model parameters to measurable properties. While the directions of research for characterizing the microporosity can be easily drawn (e.g., measuring microporosity cluster volume, interface area, …), improving the characterization of the flow field with the objective of parameterizing CTRW is more challenging and will require probably to analyze the velocity correlations in a Lagrangian framework.

Introduction

Les milieux poreux sont tout autour de nous et de notre vie quotidienne. Comprendre les mécanismes de transferts de masse et d'énergie qui se produisent dans ces milieux poreux est un objectif scientifique depuis longtemps. L'un des processus très important est le transport de substance dissoute, qui intervient dans de nombreux processus naturels, en particulier en géosciences (dispersion des polluants, exploitation des ressources souterraines, récupération assistée du pétrole, …).

La théorie traditionnelle du transport de solutés dans les milieux poreux par diffusion et advection est basée sur les travaux de, par exemples [START_REF] Saffman | A theory of dispersion in porous media[END_REF] ) montrent des courbes de restitution de traceurs très asymétriques, avec la signature d'un transport anormal. L'EDA ne parvient pas à prédire les comportements observés expérimentalement car il ne peut pas reproduire les mécanismes réels qui contrôlent la dispersion pour les durées et / ou les échelles d'observation. La non Fickianité du transport de solutés en milieux poreux est généralement expliqué par l'hétérogénéité de champ d'écoulement qui existe même dans les milieux macroscopiquement apparemment homogènes [START_REF] Datta | Spatial Fluctuations of Fluid Velocities in Flow through a Three Dimensional Porous Medium[END_REF]. La dispersion non Fickienne est un régime de pré asymptotique; la dispersion devient Fickienne aux temps longs si l'échelle de l'hétérogénéité est finie, à savoir lorsque le traceur a visité l'hétérogénéité à toutes les échelles. Pourtant, dans les milieux naturels présentant une hétérogénéité à pratiquement toutes les échelles, le régime effectif de dispersion est en général non Fickian.

Les propriétés de dispersion non fickienne, leur origine et leurs relations avec l'hétérogénéité structurale de la porosité sont encore débattues. Pendant les quelques dernières décennies, de nombreux auteurs ont développé avec plus ou moins de succès différents modèles pour simuler et prévoir un transport non Fickéen. Beaucoup de ces modèles reposent largement sur la formulation de l'ADE et essayent d'adapter les résultats expérimentaux par les variations spatiales ou temporelles du coefficient de dispersivité. Cependant, les modèles alternatifs tels que le Continuous Time Random Walk (CTRW) ou Multi Rate Mass transfer (MRMT) décrivant le comportement de transport anormal basé sur l'hypothèse que la dispersion non Fickiéenne ont été développés. I sont basés sur la prose en compte de la distribution des temps de transition que le soluté rencontre dans les milieux poreux et se sont avérées très efficaces pour expliquer les comportements observés.

Les développements récents en ressources informatiques nous ont donné l'occasion d'étudier les mécanismes qui contrôlent le transport à l'échelle du pore. Par exemple, des techniques fiables pour l'imagerie tridimensionnelle des structures et de flux dans l'espace des poral sont maintenant utilisables. La résonance magnétique nucléaire (RMN) [par exemple, [START_REF] Scheven | Pore Scale Mixing and Transverse Dispersivity of Randomly Packed Monodisperse Spheres[END_REF]

Transport en milieu poreux

Les deux modes de base de déplacement de solutés en milieux poreux sont miscibles et non miscibles. Dans ce travail, nous étudions le transport miscible d'un traceur idéal. Il est caractérisé par un seul fluide présent dans le milieu poreux, mais ses propriétés ou la composition peut varier dans l'espace et le temps. Deux mécanismes de transport principaux sont la diffusion moléculaire et l'advection. La diffusion moléculaire est un mécanisme ubiquiste de transport se produisant dans le milieu poreux dès que la concentration du soluté varie dans l'espace. Le traceur se déplace à partir d'une région de haute concentration pour une région de faible concentration. L'advection (ou convection) correspond au déplacement du traceur à la vitesse local de l'écoulement. L'advection provoque la dispersion du traceur en raison des différentes longueurs de trajet et les différences de vitesse. Ce processus est appelé dispersion mécanique. Cependant, la dispersion dans le milieu poreux avec de l'eau qui coule est influencée à la fois par la diffusion et la dispersion mécanique et ils ne peuvent pas être séparés. Par conséquent ces deux influences sont définies ensemble comme dispersion hydrodynamique. La contribution relative à dispersion hydrodynamique de dispersion et de diffusion mécanique est caractérisée par nombre de Peclet (2,11), qui est utilisé pour caractériser régime de dispersion.

Expériences de traçage

Le chapitre 2 présente montage expérimental et les résultats de l'expérience de laboratoire. Des expériences de laboratoire fournissent un aperçu précieux dans les processus naturels avec une capacité d'isoler les différents effets qui contribuent dispersion. Dans ce cas, l'expérience a été conçue pour permettre l'étude de la dispersion de traceur non réactif (fluorescéine) dans des colonnes de billes de verre et des carottes de grès de Berea dans des conditions strictement contrôlées. Le plan du circuit expérimental est présenté dans la figure 3.1. Il se compose de deux pompes à piston qui assurent un débit constant de flux, cellule d'échantillon, l'appareil de mesure (fluorimètre), la soupape de direction de l'écoulement, réservoir d'eau avec de l'eau propre et récipient pour l'eau de sortie. Le milieu poreux est placé dans la cellule qui est positionnée verticalement sur la partie supérieure du fluoromètre. Différentes vannes contrôles la direction de l'écoulement (vers le bas ou vers le haut) à travers la cellule, ce qui permet d'effectuer des expériences push pull. Selon le type d'expérimentation le traceur est injecté dans le bas ou le haut de l'échantillon avec une seringue grâce à un système d'injection à spécialement conçus (Figure 3.4).

Nous avons effectué série d'expériences sur la colonne de billes de verre (L = 200 mm, Ø = 18 mm). Les résultats en termes de courbes de percée (BreakThrough Curves, BTCs) sont présentés sur les figures 3.9 3.15. Nous avons d'abord étudié le possible effet de la densité pour différentes concentrations de traceur. Il a été observé qu'il existe un effet significatif de densité se produit si la concentration du traceur est de 500 ppm ou plus. L'effet de la densité peut conduire à une mauvaise interprétation des résultats expérimentaux et par conséquent, toutes les autres expériences sont réalisées avec une concentration de traceur de 50 ppm. Expériences push pull a révélé l'impact significatif de la durée de l'injection sur la forme des BTCs, mais il n'y a que peu d'effet sur t le comportement aux temps long (tailing). En outre, il a été montré que le temps de résidence du traceur dans le système n'a aucun effet sur la pente des BTCs aux temps longs. En outre, nous avons comparé les résultats expériences en mode push pull et des expériences standard et les résultats indiquent que le comportement aux temps long sont indentiques. Ces limitations ont été confirmés par deux expériences avec différents débits effectués sur la colonne de grès de Berea (l = 150 mm, Ø = 18 mm), i.e. la même roche que pour la modélisation numérique. Nous avons effectué la comparaison qualitative entre l'expérience et résultats de la modélisation (Figure 3.17), qui a souligné l'importance de l'inclusion à la fois la matrice de diffusion et le débit champ hétérogénéité dans la modélisation du transport pores échelle.

Enfin, il faut admettre que, malgré un effort considérable pour améliorer la résolution et le contrôle des expériences de laboratoire, les résultats ne peuvent pas être utilisés avec confiance pour le paramétrage modèle non Fickian. Cette conclusion est le résultat d'un grand nombre d'expériences pour les essais 1) les effets de la concentration du traceur, la procédure d'injection et l'optimisation de l'équipement et 2) la reproductibilité des résultats.

Modélisation numérique du champ d'écoulement à l'échelle du pore

Dans le chapitre 4, nous présentons les procédures nécessaires qui doivent être effectuées afin d'effectuer écoulement et de transport des simulations sur la microtomographie à rayons X (XRMT) images. XRMT est une technique d'imagerie non invasive permettant d'obtenir la géométrie interne de la roche sur la base des faisceaux de rayons X passant à travers l'échantillon sous des angles différents. Le résultat de XRMT est un ensemble de 2D en niveaux de gris, où le niveau d'échelle de gris représente atténuation des rayons X par pixel. Pour les roches mono cristallins tels que grès de Berea atténuation de rayons X ne dépend que de la densité de la roche (porosité). Ces images en niveaux de gris sont ensuite segmentées afin de distinguer les différentes phases (par exemple, rock, void). Processus de segmentation dans ce travail a été effectué en utilisant ce qu'on appelle «région de croissance" algorithme [START_REF] Siprkovska | A Summary of Image Segmentation Techniques[END_REF][START_REF] Noiriel | Hydraulic properties and microgeometry evolution in the course of limestone dissolution by CO2 enriched water[END_REF]; dans lequel nous avons ajusté de manière itérative seuils pour obtenir la même porosité que dans l'essai de porosimétrie par intrusion de mercure effectuée sur la même roche [START_REF] Mangane | Permeability impairment of a limestone reservoir trigered by heterogeneous dissolution and particles migration during CO2 rich injection[END_REF]. La nouveauté dans ce travail est que nous avons utilisé 3 phases segmentation de distinguer vide, rock et microporosité (c. à région avec des pores plus petits que la résolution de l'image); la place de la segmentation (seulement vide et rock) 2 phases habituellement utilisé dans la modélisation pores échelle [START_REF] Ovaysi | Direct pore level modeling of incompressible fluid flow in porous media[END_REF][START_REF] Blunt | Pore scale imaging and modelling[END_REF]. Depuis porosité non connecté ne participe pas au transport, ils ont été éliminés par l'algorithme de marquage de la grappe sur la base de la méthode proposée par Hoshen Kopelman et [1976], qui identifie également des pores interconnectés qui enjambe à travers la totalité de l'échantillon. Sur les images segmentées nous avons calculé longueur caractéristique de deux manières différentes. Propriété définie comme le diamètre de pore moyen utilisé pour définir les régimes d'écoulement et de transport à travers Reynolds (4,7) et Peclet (2,11) respectivement des numéros. Les deux valeurs obtenues par la structure de la roche variogramme ( 110 um, Pour obtenir une représentation réaliste de l'écoulement en milieu poreux, domaine de calcul doit être suffisamment grande pour capturer tous les hétérogénéités. Dans ce travail, le volume élémentaire représentatif a été étudié en fonction de la porosité (Figure 4.8) et la perméabilité (figure 4.9). Il a été extrait montrent que sous volume de 300x300x300 voxels peut être considérée comme représentative de milieu poreux sélectionné. Pour préparer milieu poreux pour les simulations de flux de domaine doit être discrétisée. Dans ce travail, nous utilisons maillage structuré en hexahedars réguliers, qui capte facilement la structure des pores à partir d'images de XRMT segmentés et il est adapté pour le post traitement nécessaire pour les simulations de transport. Champ d'écoulement Pore échelle est obtenue en résolvant conservation de la masse (4.5) et Naiver Stokes (4.6) équations, pour l'écoulement monophasé avec la densité et de la viscosité constante. Ces équations sont construits dans une base de solveur l'état d'équilibre de la méthode semi implicite pour les équations de pression Lié (simple) [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF] algorithme qui fonctionne à l'intérieur du progiciel OpenFOAM. Algorithme simple utilise procédure itérative pour coupler de Navier Stokes équation et obtenir champ d'écoulement pour des conditions aux limites données. Figure 4.16 présente un graphique de corrélation de la perméabilité calculée en fonction de la porosité pour toutes les simulations effectuées dans la portée de ce travail. Reste de la section de résultat est dédié à l'étude de l'influence de la résolution de maillage. Nous avons confirmé les résultats publiés par Guibert et al. [2015] en ce qui concerne la diminution de la perméabilité avec l'augmentation de la résolution de maillage et élargi leur enquête. Nous avons montré tha pratique courante où une image voxel est modélisé par un maille peut induire des écarts importants par les résultats en utilisant des maillages avec discrétisation plus élevé de la même structure. Il a été démontré dans le cas de haute résolution Berea images de grès comme ceux utilisés ici; résolution optimale de maille est si chaque voxel est représenté par 27 mailles. Pour les autres médias ce facteur peut varier considérablement en raison de la résolution de l'image et des propriétés milieu poreux, mais en règle générale, il est conseillé d'utiliser impaire facteur de raffinement de maillage pour les médias poreux de gorges étroites pour atteindre approximations de champ d'écoulement plus réalistes.

Modélisation du transport à l'échelle du pore

Le chapitre 5 présente des simulations de transport effectuées sur les images XRMT focalisation sur la dispersion de traceur idéal. Sous volumes sélectionnés (3003 voxels) ont été post traité après des simulations de flux (chapitre 4) et utilisé comme une entrée pour les simulations de transport. Le transport hydrodiffusive d'une trace passive est modélisé par marche aléatoire méthode de suivi de particule dans laquelle la concentration du soluté est représentée par la moyenne d'ensemble de particules. En réalité, ces particules sont marcheurs aléatoires, qui sont considérés être sans masse, infiniment petit et ils ne interagissent entre eux ou avec le milieu poreux. Dans ce travail, nous utilisons le domaine du temps de marche aléatoire (TDRW). TDRW avérée outil très efficace car il est basé sur les mouvements de particules avec une longueur constante, ce qui est très approprié à utiliser dans les médias discrétisé avec une géométrie complexe. En cas de TDRW stochasticité provient de la direction et pas de temps entre deux étapes consécutives. Principaux résultats, la discussion et les conclusions sont présentées sous forme d'un article (voir rubrique 5.3) accepté dans Water Resources Research. Théorie et hypothèses élargi pris en compte lors du transport de modélisation dans le domaine microporeux sont présentés ci après (section 5.2). Une fois de plus, la résolution de maillage avéré être un facteur important. Dans les simulations de transport insuffisante résolution de maillage conduit à surestimer le transport anormal et l'erreur dans les résultats semble augmenter avec la diminution de Peclet valeurs numériques. Comparaison entre simulations de transport, y compris l'hétérogénéité de champ seulement de flux et ceux dont l'hétérogénéité de champ d'écoulement et de transfert de masse mobile immobile souligne leur contrôle concomitante dans le domaine intermédiaire du BTC queue tandis que, comme prévu, le contrôle de transfert de masse mobile immobile le comportement de temps tardif qui est, comme on le verra dans la section 3.3.3, difficile à aborder expérimentalement.

Des images XRMT, nous ne pouvons pas extraire beaucoup d'informations concernant la microporosité (par exemple, la tortuosité, seuil de percolation). En fait seule chose qui peut être estimée est sa porosité intrinsèque: la qualité de cette estimation dépend de la qualité de la procédure de segmentation 3 phases. Par conséquent nous avons effectué une analyse paramétrique du seuil de percolation et la diffusion efficace de la forme BTC. Augmentation du seuil de percolation, les valeurs inférieures augmente le temps de séjour des particules dans le domaine immobile, tandis que des valeurs plus élevées, il diminue de manière significative la fraction de domaine immobile accessible aux particules qui se traduit par temps de séjour plus court et une diminution de traînée de la CTB. Inversement, la diminution de la diffusion efficace entraîne une augmentation dans le temps de séjour des particules dans le domaine de queue immobile augmentant la CTB.

Un point intéressant en évidence par l'analyse du résultat est que, si l'on suppose que les propriétés de la zone microporeuse sont en corrélation positive avec la porosité intrinsèque, la distribution hétérogène de la zone microporeuse peut être substituée par sa répartition homogène équivalente. Dans ce cas, le coefficient de diffusion unique pour la microporosité doit être égal à la moyenne harmonique de la microporosité distribué de façon hétérogène.

Enfin, nous avons proposé un modèle 1D rehaussée sur la base du domaine temporel continue approche de marche aléatoire en mesure de capturer le comportement causés par l'hétérogénéité de champ d'écoulement et de transfert de masse mobile immobile. Ceci est réalisé en ayant deux distributions en loi de puissance différentes pour représenter les temps de transit dans le domaine mobile et immobile.

Simulations de flux à la fois dans la macroporosité et la microporosité: l'approche de Darcy Brinkman

Dans le chapitre 6, nous nous concentrons sur les simulations d'écoulement pores échelle, qui comprennent à la fois dans le domaine flux micro et macroporosité. Comme indiqué précédemment approche «normalisé» de résoudre débit à pores échelle consiste à résoudre l'équation de Stokes dans macroporosité, tandis que les flux dans la microporosité est négligée. A partir de 3 phases segmenté images XRMT (voir section 4.2.2) nous avons effectué des simulations 3D des pores d'écoulement d'échelle. Le nouveau solveur développé par Cyprien Soulaine de Stanford Univerity dans le cadre de OpenFOAM est avérée appropriée pour résoudre simultanément le débit de la macroporosité et le domaine microporeux. L'approche est basée sur l'équation de Darcy Brinkman, qui utilise une seule équation pour calculer la vitesse du fluide dans les deux Stokes (c. à macroporosité) et Darcy (c. à microporosité) domaine. Elle consiste à ajouter Darcy terme de résistance dans l'équation de Stokes, qui représente la friction entre le solide et non avenue dans le domaine macroporeux. Dans le domaine de la macroporosité le terme de résistance Darcy est nulle et nous résoudre Stokes standard. La partie la plus importante de définir le terme de résistance Darcy est dans la détermination de la perméabilité dans le domaine microporeux. Depuis que nous n'avons pas de données réelles sur les propriétés de microporosité (voir rubriques 4.2 et 5.2), la perméabilité est estimé par l'équation Kozeny Carman.

Le but de ce travail était d'étudier l'importance de la circulation supplémentaire dans la microporosité, même dans les roches à faible teneur en microporosité tels que le grès de Berea (2%). L'augmentation de la perméabilité en raison de la microporosité est d'environ 5% (voir la comparaison dans le tableau 6.1) qui est relativement petite, mais toujours perceptible et nous pouvons anticiper que la contribution de l'écoulement dans la microporosité devrait devenir un processus majeur dans les roches avec une fraction élevée de microporosité comme le grès et les carbonates modifié.

Bien que, dans ce travail, nous n'avons pas réussi à établir un lien formel entre les données expérimentales sur le grès de Berea et les simulations de transport effectuées sur les images de XRMT de la même roche, mais nous avons fait quelques comparaisons qualitatives qui ont prouvé que la mise en oeuvre dans les modèles de le transfert de masse dans la microporosité est obligatoire pour expliquer les observations. En outre, nous avons identifié les exigences méthodologiques et les améliorations nécessaires technologiques qui semblent être essentiels pour améliorer le dispositif expérimental. Plus précisément, les modifications doivent être faites 1) pour permettre à plus de débit Peclet afin de mieux enquêter sur l'hétérogénéité de la vitesse d'écoulement sur le transport anormal et 2) à même d'accroître la précision et la résolution de mesure de la concentration. Inversement, nous croyons que les développements futurs devraient se concentrer sur une meilleure caractérisation de la microporosité et de la distribution de vitesse afin de relier les paramètres du modèle CTRW aux propriétés mesurables. Bien que les directions de recherche pour caractériser la microporosité peuvent être facilement établis (par exemple, la mesure du volume de la microporosité de cluster, zone d'interface, ...), l'amélioration de la caractérisation du champ d'écoulement avec l'objectif de paramétrage CTRW est plus difficile et nécessitera sans doute d'analyser les corrélations de vitesse dans un cadre de Lagrange.

Figure 1 22 Figure 2 . 4 22 Figure 2 . 5 23 Figure 2 . 6 24 Figure 2 . 7 34 Figure 3 . 2 39 Figure 3 . 6 40 Figure 3 . 7 42 Figure 3 . 8 45 Figure 3 . 46 Figure 3 .

 122242225232624273432393640374238453463 Figure 1.1 BTC of fluorescein tracer from laboratory experiment on Berea sandstone core (blue line) compared with the solution of ADE (red line). Experimental setup: flow through experiment, flow rate 10 cm 3 /h; Berea sandstone, core diameter 18mm, core length 15 cm; pulse injection, concentration 50 ppm, tracer volume 50 L ........................................................................................................................... 14 Figure 1.2 Concentration of solute in the porous medium consisting of glass beads. Different permeability zones were created by inserting circular regions filled with glass beads with significantly smaller radius than the surrounding medium. Three experiments are presented; they differ in permeability of low permeability zones. Concentration is represented as fraction of initial concentration, scale is at the bottom of image -reprinted from Zinn et al. [2004] .......................................................... 16 Figure 2.1 Spreading of a solute pulse with time (t), transport only due to diffusion. Solute pulse was injected into the aquifer at time t 0 with an initial concentration C 0 -reprinted from Fetter [1999] . 19 Figure 2.2 Illustration of diffusion a) free in space and b) in the pore space ..................................... 20 Figure 2.3 Illustration of advective transport in the porous medium with time. Solute pulse was injected at x=0 and at the time t 0 . .............................................................................................................. 22 Figure 2.4 Illustration of different trajectories taken by solute traveling through the porous medium .................................................................................................................................................................... 22 Figure 2.5 Illustration of the mechanical dispersion in the porous medium. On the illustration at the top, blue line denotes flow path with lower velocity while red line denotes flow path with faster flowing fluid. On the bottom image there is an illustration of two different paths taken by the solute, both paths have same starting and finishing points but the path length is significantly different. Magnified is example of the parabolic velocity distribution in one pore that causes Taylor dispersion. ....................... 23 Figure 2.6 Illustration of the transport and spreading of the solute pulse in the porous medium with time, according to advection dispersion theory. Solute pulse was injected at x=0+a and at the time t 0 . Transported and spreading is due to advection and dispersion -reprinted from Fetter [1999] .............. 24 Figure 2.7 Illustration of dead end pores in the natural porous medium .......................................... 25 Figure 2.8 Experimental results showing relation between the longitudinal dispersion coefficient ( ) and Peclet number ( ). Dashed lines indicate limits between five different transport regimesreprinted from Sahimi [1993] ..................................................................................................................... 26 Figure 2.9 a) Illustration of "natural" heterogeneous porous medium with solute exchange between different phases indicated by arrows. Left image presents mass transfer between zones with different permeability (slow and fast flowing zones) and right presents mass exchange on the pore scale between porosity with different flow regimes (i.e. diffusion between stagnant and flowing zones), microporosity (e.g. clay pockets) and surface sorption. b) Illustration of different mass exchange processes incorporated in one multi rate mass transfer model designed to simulate different mass exchange principles that happen in nature. -reprinted from Haggerty and Gorelick[1995] .................................... 30

  Figure3.13 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, injected tracer concentration 50 ppm, injected tracer volume 10 L. Differences between BTC's are in duration of push and resting period. Red curve -push duration 900s, resting duration 300s; green curve -push duration 1320s, resting period 0s; blue curve-push duration 1320s, resting period 300s. Times noted in legend are in minutes, SW stands for duration of push and ST for resting period. ............................................... 49 Figure 3.14 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, injected concentration solution 50 ppm, push duration 22min. Magenta curve -fast injection, Green curveslow injection .............................................................................................................................................. 50 Figure 3.15 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50 cm 3 /h, injected concentration solution 50 ppm, injected volume 10 L, duration of push phase 22 min. ........................ 51 Figure 3.16 Experimental breakthrough curve data of tracer concentration at the outlet of the Berea sandstone core (L=15cm Ø=18mm), flow through experiment, injected concentration solution 50 ppm, injected volume 50 L. Comparison between two different flow rates 10 cm 3 /h (red line) and 2 cm 3 /h (blue line). Black line denotes power law function that fits late time behavior of the experiment with higher flow rate. ................................................................................................................................. 52

62 Figure 4 . 7 75 xi 89 Figure 5 . 3

 6247758953 Figure3.17 Scaled BTC according to the estimated tracer volume that is injected (a); red line 10 cm 3 /h, blue line 2 cm 3 /h. Compared with results of transport simulations performed on XRMT images of the same rock (b) at (purple symbols) and (green symbols). Black rectangular limits resolution that can be observed in the experiments. Unconnected symbols denotes simulations in which mass transfer to microporosity is neglected, while connected symbols represents simulations that account for mass transfer into the microporosity -original figure is in section 5.3, Figure11................. 53 Figure 4.1 Image acquisition from X ray microtomography to the final 3D model of heterogeneous media -reprinted from Fernandes et al. 2012 ........................................................................................... 56 Figure 4.2 Mercury Intrusion Porosimetry test results for Berea sandstone (black line). Dashed red line denotes XRMT image resolution (3.16 m), delimitating microporosity and macroporosity, blue line marks average pore size (11.71 m) ........................................................................................................... 57 Figure 4.3 Gray scale level distribution for Berea sandstone (black line), dashed lines marks threshold values used for segmentation. Blue lines are for void micoporosity interface and red lines are for solid microporosity interface ................................................................................................................ 59 Figure 4.4 3 dimensional visualization of pore space network (blue) and connected microporosity (yellow) ....................................................................................................................................................... 60

Figure 1 .

 1 Figure 1.1 BTC of fluorescein tracer from laboratory experiment on Berea sandstone core (blue line) compared with the solution of ADE (red line). Experimental setup: flow through experiment, flow rate 10 cm 3 /h; Berea sandstone, core diameter 18mm, core length 15 cm; pulse injection, concentration 50 ppm, tracer volume 50 L

Figure 1 . 2

 12 Figure 1.2 Concentration of solute in the porous medium consisting of glass beads. Different permeability zones were created by inserting circular regions filled with glass beads with significantly smaller radius than the surrounding medium. Three experiments are presented; they differ in permeability of low permeability zones. Concentration is represented as fraction of initial concentration, scale is at the bottom of image -reprinted from Zinn et al. [2004]

Figure 2 . 1

 21 Figure 2.1 Spreading of a solute pulse with time (t), transport only due to diffusion. Solute pulse was injected into the aquifer at time t 0 with an initial concentration C 0 -reprinted from Fetter [1999]

(2. 1 )Figure 2 . 2

 122 Figure 2.2 Illustration of diffusion a) free in space and b) in the pore space

Figure 2 . 3

 23 Figure 2.3 Illustration of advective transport in the porous medium with time. Solute pulse was injected at x=0 and at the time t 0 .

Figure 2 . 4

 24 Figure 2.4 Illustration of different trajectories taken by solute traveling through the porous medium

Figure 2 . 5

 25 Figure 2.5 Illustration of the mechanical dispersion in the porous medium. On the illustration at the top, blue line denotes flow path with lower velocity while red line denotes flow path with faster flowing fluid. On the bottom image there is an illustration of two different paths taken by the solute, both paths have same starting and finishing points but the path length is significantly different. Magnified is example of the parabolic velocity distribution in one pore that causes Taylor dispersion.

Figure 2 . 6

 26 Figure 2.6 Illustration of the transport and spreading of the solute pulse in the porous medium with time, according to advection dispersion theory. Solute pulse was injected at x=0+a and at the time t 0 . Transported and spreading is due to advection and dispersion -reprinted from Fetter [1999]

  where denotes the velocity of the fluid in pores [m/s], is the distance from the central axis of the tube [m],

Figure 2 . 7

 27 Figure 2.7 Illustration of dead end pores in the natural porous medium

Figure 2 .

 2 Figure 2.9 a) Illustration of "natural" heterogeneous porous medium with solute exchange between different phases indicated by arrows. Left image presents mass transfer between zones with different permeability (slow and fast flowing zones) and right presents mass exchange on the pore scale between porosity with different flow regimes (i.e. diffusion between stagnant and flowing zones), microporosity (e.g. clay pockets) and surface sorption. b) Illustration of different mass exchange processes incorporated in one multi rate mass transfer model designed to simulate different mass exchange principles that happen in nature. -reprinted from Haggerty and Gorelick[1995]

Figure 3 . 1

 31 Figure 3.1 Schematic diagram of the experimental test bench circuit layout with arrows indicating flow direction. Blue arrows -permanent flow direction; green arrows -flow direction for flow through experiment and pull phase of push pull experiment; red arrows -flow direction during push phase of push pull experiment

Figure 3 . 2

 32 Figure 3.2 Different stages of glass bead column preparation for the experiments. a) microscopic image of glass beads; b) sorting machine at the Geosciences Montpellier laboratory and one of the sieves; c)underwater column filling with sorted glass beads; d)experiment on the transparent column with high fluorescein content in the tracer to visual check for side flow and other anomalies

Figure 3 . 3

 33 Figure 3.3 Cross section through experimental cell for laboratory tracer experiment (a) and different stages of preparing core samples; b) Berea sandstone block from which core is extracted; c) coring machine at Geosciences Montpellier laboratory and d) on the left Berea sandstone core and on the rights same core coated with thermo plastic coating

Figure 3 . 4

 34 Figure 3.4 Schematic cross section of injection part at the top of the cell with labeled essential parts

Figure 3 . 5

 35 Figure 3.5 Fluorescein in various forms, on the right is disodium salt which is also known as "uranin" and on the left is dissolution in the water with typical bright green color.

Figure 3 . 6

 36 Figure 3.6 Schematic representation of the TELog measurement head displaying optical path reprinted form Gouze et al. [2009].

Figure 3 . 7

 37 Figure 3.7 Schematic illustration of master concentration and its diluted products with list of all solutes concentrations. Measured tension for different TELog gains at different fluorescein concentrations obtained during calibrations process are presented on Figure 3.8. Symbols present measured values, while lines are fitted functions for each gain used to process experimental results.

Figure 3 . 8

 38 Figure 3.8 TELog calibration results. Symbols present measured tension output for given concentration at different gains (photomultiplier sensitivity), lines present functions used to transform voltage to fluorescein concentration for different gains (line color matches symbol color for same gain), two dashed red lines present limit of function application in standalone mode for all gains except gain 8.

Figure 3 . 9

 39 Figure 3.9 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), flow through experiment, flow rate 50cm 3 /h. Comparison between different solution concentration of injected tracer ranging from 1 ppm to 1000 ppm.

  Figure 3.10 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), flow through experiment, flow rate 50cm 3 /h. Displaying different tail behavior due to density effect in experiments with injection concentration 500ppm and tracer volume 2 L.

Figure 3 .

 3 Figure 3.11 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), flow through experiment, flow rate 50cm 3 /h. Displaying behavior for injected tracer with concentration of 50ppm, with two different tracer volumes 2 L (blue line) and 10 L (red line).

Figure 3 .

 3 Figure 3.12 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, duration of push phase is 22 minutes, injected tracer volume 10 L. Comparison between different concentrations of the injected solutions; 50 ppm (green line) and 1 ppm (red line).

Figure 3 .

 3 Figure 3.13 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50cm 3 /h, injected tracer concentration 50 ppm, injected tracer volume 10 L. Differences between BTC's are in duration of push and resting period. Red curve -push duration 900s, resting duration 300s; green curve -push duration 1320s, resting period 0s; blue curve-push duration 1320s, resting period 300s. Times noted in legend are in minutes, SW stands for duration of push and ST for resting period.

Figure 3 .

 3 Figure 3.15 Experimental breakthrough curve data of tracer concentration at the outlet of the column packed glass beads (L=20cm Ø=18mm), push pull experiment, flow rate 50 cm 3 /h, injected concentration solution 50 ppm, injected volume 10 L, duration of push phase 22 min.

Figure 3 .

 3 Figure 3.16 Experimental breakthrough curve data of tracer concentration at the outlet of the Berea sandstone core (L=15cm Ø=18mm), flow through experiment, injected concentration solution 50 ppm, injected volume 50 L. Comparison between two different flow rates 10 cm 3 /h (red line) and 2 cm 3 /h (blue line). Black line denotes power law function that fits late time behavior of the experiment with higher flow rate.

Figure 3 .

 3 Figure 3.17 Scaled BTC according to the estimated tracer volume that is injected (a); red line 10 cm 3 /h, blue line 2 cm 3 /h. Compared with results of transport simulations performed on XRMT images of the same rock (b) at (purple symbols) and (green symbols). Black rectangular limits resolution that can be observed in the experiments. Unconnected symbols denotes simulations in which mass transfer to microporosity is neglected, while connected symbols represents simulations that account for mass transfer into the microporosity -original figure is in section 5.3, Figure 11

Figure 4 . 1

 41 Figure 4.1 Image acquisition from X ray microtomography to the final 3D model of heterogeneous media -reprinted from Fernandes et al. 2012

  Figure 4.2 shows results of MIP test for Berea sandstone. Red dashes line denotes image resolution (3.16 m) and all pores with radius smaller than this (left from the line) are considered as microporosity. Blue line marks average pore size (11.71 m) for given sample.

Figure 4 . 2

 42 Figure 4.2 Mercury Intrusion Porosimetry test results for Berea sandstone (black line). Dashed red line denotes XRMT image resolution (3.16 m), delimitating microporosity and macroporosity, blue line marks average pore size (11.71 m)

Figure 4 . 3

 43 Figure 4.3 Gray scale level distribution for Berea sandstone (black line), dashed lines marks threshold values used for segmentation. Blue lines are for void micoporosity interface and red lines are for solid microporosity interface

Figure 4 .

 4 Figure 4.4 3 dimensional visualization of pore space network (blue) and connected microporosity (yellow)

Figure 4 . 5

 45 Figure 4.5 Cross section through segmented Berea sandstone displaying pore space structure of connected clusters (i.e. all unconnected porosity is eliminated). Black filled areas denote the solid phase, white colored areas denote connected macroporosity, orange colored areas indicate the microporous phase and blue areas denote macroporosity that is connected with percolation cluster only through microporous phase

Figure 4 . 6

 46 Figure 4.6 Rock structure variogram for Berea sandstone

  Figure 4.7 presents a diagram to determine REV in terms of porosity. Dashed line at U 0 marks when the REV is achieved since porosity values are stabilized.

Figure 4 . 7

 47 Figure 4.7 Definition of the representative elementary volume as a function of porosity -reprinted from Bear [1972]

Figure 4 . 8

 48 Figure 4.8 Normalized porosity in function of the normalized sample volume

(4. 4 )

 4 where is fluid viscosity [Pa s], is total flux [m 3 /s], is length [m] of block in the flow direction, is pressure [Pa] and

  2 mm [Ovaysi and Piri, 2010] or 0.68 mm [Okabe and Oseto, 2006] can be considered as representative.

Figure 4 . 9

 49 Figure 4.9 Normalized permeability in function of normalized sample volume

Figure 4 .

 4 Figure 4.10 Graphical illustration of differences between two mesh generation approaches. Left image (a) presents mesh generated from *.stl file (green line) with snappyHexMesh and right image (b) is presenting mesh made by developed mesh generator. Black -solid phase; white -void phase; green -surface from *.stl file; blue -internal mesh boundaries; red -external mesh boundaries

Figure 4 .

 4 Figure 4.11 Velocity distribution in the pore throat with diameter equal to the XRMT image resolution a) parabolic velocity profile in real conditions; b) velocity distribution when mesh resolution is equal to image resolution; c) velocity distribution when mesh resolution is three times smaller than image resolution

Figure 4 .

 4 Figure 4.12 Mesh generated in pore space of Berea sandstone used for flow simulations, including manifolds on inlet and outlet boundaries

  where denotes superficial velocity [m/s], is the fluid density [kg/m 3 ], is the time [s], is the pressure [Pa] and is dynamic fluid viscosity [Pa s].

1.

  Set the boundary conditions 2. Solve the discretized momentum equation to compute the intermediate velocity field 3. Compute the mass fluxes at the cell faces 4. Solve the pressure equation and apply under relaxation 5. Correct the mass fluxes at the cell faces 6. Correct the velocities on the basis of the new pressure field 7. Update the boundary conditions 8. Repeat until convergence

Figure 4 .

 4 Figure 4.13 Flow field obtained from OpenFOAM, blue color denotes areas with low velocities while red denotes areas with high velocities

(4. 7 )

 7 where denotes the fluid density [kg/m 3 ], is the average pore velocity [m/s], L is the characteristic length [m] and is the dynamic fluid viscosity [Pa s]. It defines ratio of inertial forces to viscous forces. Low Reynolds number indicates dominance of viscous forces, thus occurrence of laminar flow regime which is associated with low velocities in the flow field.

Figure 4 .

 4 Figure 4.16 Cross plot of calculated permeability from pore scale flow simulations in function of the porosity for different sub volumes extracted from Berea sandstone segmented image.

  4.22 b)

Figure 4 .

 4 Figure 4.17 SubV1. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.18 SubV2. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.19 SubV3. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.20 SubV4. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.21 SubV5. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.22 SubV6. Comparison of a) average, b) maximal and c) minimal velocity per cross section profiles perpendicular to the flow direction for coarse (orange line) and fine mesh (blue line) with corresponding local porosity (black dashed line).

Figure 4 .

 4 Figure 4.23 Artificial 3d symmetric structure, designed to investigate flow through smallest possible throat (1 pixel)

Figure 4 .

 4 Figure 4.24 Results of flow simulations on the artificial structure presented as a) pressure and b) velocity fields. Mesh resolution is the same as image resolution

Figure 4 .

 4 Figure 4.27 Results of flow simulations on the artificial structure presented as a) pressure and b) velocity fields. Mesh resolution seven times finer than image resolution (1 image pixels is represented with 343 mesh cells).

Figure 5 . 1

 51 Figure 5.1 Illustration of evolution of percolation threshold in microporous phase. Blue color denotes mobile domain from which solute diffuses in microporosity, microporous phase is represented with gray pixels where gray level represents intrinsic porosity (darker gray lower intrinsic porosity and vice versa) and yellow color denotes areas with intrinsic porosity lower than percolation threshold; a) no percolation threshold applied, b) low percolation threshold level, c) medium percolation threshold level and d) high percolation threshold

Figure 5 . 3

 53 Figure 5.3 Illustration of intrinsic porosity distribution as the function of exponent

  Figure 4.6) et par la méthode proposée par Mostaghimi et al. [2012] (4.3) (124 um) sont en bon accord avec la littérature.

  al., 2015].

	Pixel divided Permeability [mD] Relative Difference	Maximal sample	Characteristic
			[%]	volume [mm 3 ]	length [1]
	by 1	736	+22.3	2.84	25.8
	by 2	657	+9.1	1.42	12.9
	by 3	630	+4.7	0.948	8.6
	by 4	617	+2.5	0.711	6.5
	by 5	610	+1.3	0.569	5.2
	by 6	605	+0.5	0.474	4.3
	by 7	602		0.406	3.7

Table 4 .1 Comparison of permeability to the mesh resolution, relative difference of calculated permeability compared to the finest mesh resolution, maximal sample volume that can be processed with available computational resources and characteristic lengths contained in maximal sample volume

 4

Table 4 .3 Comparison of the permeability and the maximal velocity at different mesh resolution. Simulations are performed on the 3D symmetric artificial structure simulating smallest possible throat in porous media

 4

	Pixel	Permeability	Relative	Max velocity	Relative
	divided	(mD)	difference (%)	(m/s)	difference (%)
	by 1	77.2	+175	1.825 10 5	+19.4
	by 2	44.9	+60.4	1.19 10 5	22.1
	by 3	36.1	+28.9	1.644 10 5	+7.3
	by 4	32.2	+15	1.443 10 5	5.6
	by 5	30.1	+7.5	1.557 10 5	+1.9
	by 6	28.9	+3.2	1.472 10 5	3.6
	by 7	28		1.528 10 5
	Figures 4.24 4.27 shows the pressure (a) and velocity field (b) distribution for different mesh
	resolutions. Figure

Table 6 .1 Comparison in permeability and porosity between simulations performed on 2 phase (Navier Stokes) and 3 phase (Darcy Brinkman) segmented images. All simulations are performed on exactly the same 300 3 voxel blocks, fine mesh.

 6

	[%]

  ; [1960];[START_REF] Scheidegger | General Theory of Dispersion in Porous Media[END_REF]; Ours et Bachmat[1967]. La théorie de dispersion macrocosmique décrit le transport d'un traceur idéal dans un milieu poreux saturé satisfaisant aux conditions d'écoulement de la loi de Darcy et est decrite par l'équation d'advection dispersion (EDA) (1,1). L'ADE représente le transport par un modèle similaire à la lois de Fick (Modèle Fickian) à une échelle à laquelle on suppose un mélange complet du soluté. Cependant, de nombreuse expériences de terrain, par exemple, Mackay et al.,[1986], LeBlanc et al., [1991], Gouze et al, [2008a] et de laboratoire (par exemple, [Levy et Berkowitz, 2002; Gouze et al., 2009; Bijeljic et al., 2011a]

  et la microtomographie à rayons X (XRMT) [par exemple,[START_REF] Fourar | Non Fickian Description of Tracer Transport Through Heterogeneous Porous Media[END_REF][START_REF] Klise | Comparison of laboratory scale solute transport visualization experiments with numerical simulation using cross bedded sandstone[END_REF] ont été utilisés pour visualiser directement les effets du transport dans les médias poreux naturel et artificiel. Toutefois, la contribution de ces méthodes pour caractériser les mécanismes de dispersion multi échelles est plus qualitatif que quantitatif, similaire à d'autres techniques d'imagerie développées pour l'étude de la dispersion dans les milieux poreux transparent. Néanmoins, ces méthodes sont inégalées pour caractériser les modèles moyens de comportement et d'essais macroscopiques.A l'inverse, une étude détaillée des processus de transport est maintenant possible par résolution numérique directe à l'échelle des pores. L'approche standard commence avec l'acquisition d'images à haute résolution de XRMT de la structure de pores. Les images segmentées sont ensuite utilisés pour mailler la porosité et calculer l'écoulement et le transport en utilisant des codes de dynamique des fluides calculée[START_REF] Ovaysi | Pore scale modeling of dispersion in disordered porous media[END_REF][START_REF] Blunt | Pore scale imaging and modelling[END_REF]. Cette approche donne des informations utiles sur les processus qui se déroulent à l'échelle des pores et de ses principales limites sont liées à la qualité et la résolution des images XRMT et les limitations de calcul pour résoudre les flux sur les grands domaines. En outre, les résultats sont très sensibles à la segmentation des images XRMT, la précision des calculs d'écoulement et à la représentativité de l'échantillon par rapport à lobject macroscopique initial[Arns et al, 2005 [Schlüter et al, 2014.]. Guibert et al 2015. Sienne et al., 2015].

(2.5)

Figure 4.26 Results of flow simulations on the artificial structure presented as a) pressure and b) velocity fields. Mesh resolution three times finer than image resolution (1 image pixels is represented with 27 mesh cells).

Acknowledgement

Parallelization performance

Optimal usage of available computer resources reduces time needed to perform numerical simulations. Simulations were performed on a large memory computer using four 24-cores Intel Xeon (2.3 GHz) processors with 1,5 TB of RAM. To optimize our simulations, we performed scalability test on the simulation with same geometry, boundary conditions, convergence criteria and solver setup, only variable that changes is number of nodes used for calculations. All cases used simple method for mesh decomposition provided in OpenFOAM. Figure 4.15 presents results of parallelization performance investigations on mesh similar to ones used in the simulations. We observe a significant speed up when increasing the number of process up to 16, while speed up is negligible when increasing process number up to 24. All the simulations were performed using 18 to 24 process and they converged in less than three days.

Results and discussion

In this section we focus mostly on the flow statistics and properties, since transport results are presented in Chapter 5. Figure 4.16 presents a cross plot between porosity and calculated permeability made on the Berea sandstones sub volumes with different mesh resolutions and using different simulation approaches. Mesh resolution is discussed in details in section 4.3.2 and in this comparison we use three different mesh resolutions. The coarse mesh has same resolution as image and the fine mesh is obtained by dividing each voxel by 3 in all directions.

Regarding simulation approaches we used a conservative approach which implies solving Stokes equation in the mobile zone as presented in the section 4.3.3. These results are then compared with the Darcy Brinkman approach that allows flow computations in the microporous zone as well. Differences between two mesh resolutions are also visible at high velocities in the probability density function (section 5.3, Figure 9), where probability of achieving higher velocity is higher in coarser mesh. That was in conflict with our assumption shown in the Figure 4.11 where one could expect lower maximal velocities in the coarse mesh due to averaging errors.

Tortuosity

In this work we will not calculate the tortuosity explicitly; however since it is an important parameter that strongly influences the transport in the 3 phase segmented porous medium, we will briefly present the theory standing behind. The concept of the tortuosity ( ) was introduced by [START_REF] Carman | Fluid flow through granulat beds[END_REF], who defined it as square of the ratio of the effective average path length (l e ) to the shortest distance (l) as presented on the Figure 5.2 and with equation (5.1). Many authors developed other definitions of tortuosity (e.g.Bear and Bachmat, 1986); however, there is still no universal definition [START_REF] Lorenz | Tortuosity in Porous Media[END_REF][START_REF] Dullien | Porous Media Fluid Transport and Pore Structure[END_REF][START_REF] Clennel | Tortuosity: a guide through the maze[END_REF]. Different definitions of the tortuosity is used by physicists, engineers and geologists to describe the sinuosity and interconnectedness of the pore space that affects transport processes taking place in a porous medium. Some of the most common definitions of the tortuosity are geometrical, electrical, hydraulic and diffusional tortuosity. With the exception of the geometrical tortuosity which can be measured only if the 3D geometry of the pore space is known, the other definitions of tortuosity refer to different mechanisms of mass or energy transport. It follows that relations between these definitions of tortuosity are not simple because they come distinctly different types of measurement or computation [START_REF] Clennel | Tortuosity: a guide through the maze[END_REF].

In this work tortuosity is considered as one of the parameters contributing to the reduction of the molecular diffusivity in the immobile domain (see section 0). Therefore, the most suitable definition for our problem is the diffusional tortuosity It can be defined as a ratio between molecular diffusivity ( and the effective diffusivity

General conclusions and prospective

In this thesis we presented new insights into the flow and transport in porous media obtained from both laboratory experiments and pore scale numerical modeling. The main focus was on the simultaneous study of the flow field heterogeneity and multi rate mass transfer as coexisting mechanisms contributing to the control of anomalous transport.

Newly designed experimental bench allowed highly accurate measurements over several orders of the concentration magnitude. Small experimental scale enabled to notice the significant density effect occurring if the tracer concentration in the solute is 500 ppm or higher. The density effect may lead to the misinterpretation of the experiment results, and it is important to notice that this concentration is well below the ones used in standard tracer tests. Comparison between push pull and flow through experiments performed on the glass beads and Berea sandstone indicated that solute concentration should be measured over more than 5 orders of magnitude to capture late time behavior controlled by the mobile immobile mass transfer.

All together we found that 1) meeting all the requirements for obtaining reliable results (including a good reproducibility) is quite difficult and 2) that the range of Peclet values that can be effectively investigated with a sufficient definition of the BTC tail is quite restricted. Specifically, for the low Peclet number situations generally encountered in natural system, tackling the power law decrease characteristic of the mobile immobile mass transfers seems unrealistic at laboratory scale even with state of the art (optical) sensors such as used in this study, at least for the type of porous media studied here.

Concerning pore scale modeling, we presented some improvements on the understanding of the mesh resolution influence on the local and macroscopic properties. To our knowledge we performed the first transport simulations on segmented X ray microtomography (XRMT) images containing a microporous phase. Transport simulations in the microporous phase are enabled by the implementation of the particle tracking method based on the random walk principles with constant displacement length [START_REF] Dentz | Diffusion and trapping in heterogeneous media: An ihomogeneous continiuous time random walk approach[END_REF]. Results emphasize that taking into account the microporosity in the transport calculations is crucial even in porous media with low microporosity content such as Berea sandstone. Further simulations must be made to conclude on the influence on the overall dispersion of assuming diffusion only in the microporous phase or taking into account advection as well. Nevertheless, we demonstrated the influence of the advection in the microporous phase on the bulk permeability and velocity distribution using a solver based on the Darcy Brinkman equation.

Transport behavior was up scaled within a 1 D model based on the continuous time random walk (CTRW) principles, which incorporates both flow field heterogeneity and mobile immobile mass transfer effects simultaneously. This new model was used to fit pore scale simulations and we showed its predictability in the case of the Berea sandstone. Finally, we did not managed to make a formal link between the experimental data on Berea sandstone and the transport simulations performed on the XRMT images of the same rock, but we made some qualitative comparisons that proved that the implementation in the models of the mass transfer in