Guillaume Caumon
email: guil-laume.caumon@univ-lorraine.fr

Geological objects and physical parameter fields in the subsurface: a review

Geologists and geophysicists often approach the study of the Earth using different and complementary perspectives. To simplify, geologists like to define and study objects and make hypotheses about their origin, whereas geophysicists often see the earth as a large, mostly unknown multivariate parameter field controlling complex physical processes. This chapter discusses some strategies to combine both approaches. In particular, I review some practical and theoretical frameworks associating petrophysical heterogeneities to the geometry and the history of geological objects. These frameworks open interesting perspectives to define prior parameter space in geophysical inverse problems, which can be consequential in under-constrained cases.

Introduction

The earth is three-dimensional, heterogeneous and, for its major part, inaccessible to direct observations. A consequence is that the static and dynamic parameters governing physical processes below the earth surface are generally poorly known. A recurrent challenge for geoscientists and engineers is, therefore, to predict the likely nature or behavior of the subsurface from limited data. In all fields of geophysics sensu lato, these forecasts may use physically and mathematicallybased data processing such as upward continuation of potential fields, seismic processing, classical processing of ground penetrating radar [START_REF] Nobakht | New type curves for analyzing horizontal well with multiple fractures in shale gas reservoirs[END_REF]), reservoir production decline curves [START_REF] Davis | GROUND-PENETRATING RADAR FOR HIGH-RESOLUTION MAPPING OF SOIL AND ROCK STRATIGRAPHY1[END_REF]Fetkovich 1980, Fig. 28.1A) or the resolution of an inverse problem that explicitly uses physical models computing observations from some earth parameters and physical parameters (Fig. 28.1B-D,F-H). In geology, forecasts (e.g., about the location and volume of a specific formation or resource) and geological scenarios involve direct observations and geophysical images [START_REF] Jackson | 3D seismic analysis of the structure and evolution of a salt-influenced normal fault zone: A test of competing fault growth models[END_REF]; [START_REF] Perrouty | 3D modeling of the Ashanti Belt, southwest Ghana: Evidence for a litho-stratigraphic control on gold occurrences within the Birimian Sefwi Group[END_REF]; Fig. 28.1E). In this process, the loop may not always close: in the end, the interpretations are not guaranteed to be compatible with the initial geophysical observations. This may or may not be a problem, depending on the purpose of this interpretation. For example, a qualitative match between reflection seismic data and structural interpretations is probably sufficient to discuss fault growth models [START_REF] Jackson | 3D seismic analysis of the structure and evolution of a salt-influenced normal fault zone: A test of competing fault growth models[END_REF], whereas such mismatch can be problematic in other tasks such as natural resource assessment [START_REF] Caumon | Towards stochastic time-varying geological modeling[END_REF][START_REF] Jessell | Next generation three-dimensional geologic modeling and inversion[END_REF]. Another practical problem is the interpretation and fusion of several independent data sets corresponding to different physical or geological observations [START_REF] Corbel | Framework for multiple hypothesis testing improves the use of legacy data in structural geological modeling[END_REF]Paasche 2016). Geostatistics [START_REF] Chiles | Geostatistics: modeling spatial uncertainty[END_REF][START_REF] Goovaerts | Geostatistics for natural resources evaluation[END_REF]) was historically developed with these problems in mind, and is an attractive theoretical framework to recombine point and volume data coming from geophysical images consistently with spatial statistics. However geological reasoning and statistical reasoning are of different nature [START_REF] Frodeman | Geological reasoning: Geology as an interpretive and historical science[END_REF], so honoring some spatial statistics is very useful but not always sufficient to represent geological concepts. Therefore, several methodologies have been introduced to explicitly incorporate geological knowledge in subsurface interpretation, all of them explicitly considering geological objects (Fig. 28.1F-H).

The main focus of this chapter is to review the main frameworks by which geological concepts can be represented in earth models and inverse methods addressing several types of physics. Thus, it aims at complementing the existing reviews and discussions of [START_REF] Linde | Geological realism in hydrogeological and geophysical inverse modeling: A review[END_REF] and [START_REF] Jessell | Next generation three-dimensional geologic modeling and inversion[END_REF], who address this problem with similar objectives but different perspectives. As the topic is very vast, the reader is also referred to previous review papers related to this topic [START_REF] Farmer | Geological Modelling and Reservoir Simulation[END_REF][START_REF] Lelièvre | Integrated Imaging for Mineral Exploration[END_REF][START_REF] Linde | Geological realism in hydrogeological and geophysical inverse modeling: A review[END_REF][START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF]Mosegaard and Hansen 2016;[START_REF] Oliver | Recent progress on reservoir history matching: a review[END_REF][START_REF] Pyrcz | Stratigraphic rule-based reservoir modeling[END_REF]Zhou et al. 2014a). Several books also present complementary perspectives and more complete descriptions and details [START_REF] Agterberg | Theoretical Foundations, Applications and Future Developments[END_REF][START_REF] Caers | Modeling uncertainty in the earth sciences[END_REF][START_REF] Mallet | Elements of mathematical sedimentary geology: The GeoChron model[END_REF][START_REF] Mallet | [END_REF][START_REF] Perrin | Shared Earth Modeling: Knowledge Driven Solutions for Building and Managing Subsurface 3D Geological Models[END_REF][START_REF] Pyrcz | Geostatistical reservoir modeling[END_REF]). Section 28.2 provides further motivations for considering geology in geophysical models, and tries to define what "geology" means in that sense. Then, Section 28.3 briefly describes the type of parameterizations classically used in computational physics. We discuss some links between these physical parameterizations and the frameworks used to represent geological domains in Section 28.4.

Motivations for explicit geological parameterizations

A wealth of perspectives is essential and complementary to make progresses in the understanding of our planet and its resources. This is exemplified by the various disciplines involved in natural resource characterization, see for instance [START_REF] Ringrose | Reservoir Model Design[END_REF]. Feedbacks and interactions between the various approaches generate many types of possible workflows to integrate geological data and produce forecasts, as illustrated in Fig. 28.1. For example, geophysical processing and inverse methods that use minimal geological prior information (Fig. 28.1A-D) are typically considered as data for geological interpretations (Fig. 28.1E). Whereas these "minimal prior" approaches are not this chapter's focus, they are very useful and are always used to some extent in practical studies, because they provide at least a useful first-order view of the geological domain. This is illustrated in particular in deterministic workflows of Fig. 28.1H that strive for fit-for purpose, simplest as possible, subsurface models [START_REF] Elrafie | The Event Solution--A New Approach for Fully Integrated Studies Covering Uncertainty Analysis and Risk Assessment[END_REF][START_REF] Ringrose | Reservoir Model Design[END_REF][START_REF] Williams | Top-Down Reservoir Modelling[END_REF]. They are also conceptually satisfying in the sense that they produce images or forecasts that depend on the physics, hence can be claimed to be parsimonious and objective. As a consequence of this parsimony and of the non-linear nature of most involved physical processes, these models make it difficult to evaluate uncertainty [START_REF] Watson | Parameter and predictive outcomes of model simplification: Outcomes of Model Simplification[END_REF]). The term "objective" is also relative, as some choices are always made in these methods. In data processing methods these subjective choices relate to the underlying model assumptions (e.g., sub-horizontal layers). In inverse methods, choices must also be made about the parameterization, and a statistical model (e.g., the multi-Gaussian model) or a particular regularization (e.g., [START_REF] Mosegaard | Quest for consistency, symmetry, and simplicity -The legacy of Albert Tarantola[END_REF]).

Among the approaches that try to get the most out of the physics with minimal assumptions, recent and most promising developments use several types of data and petrophysical models to constrain local anisotropy, see for instance [START_REF] Clapp | Incorporating geologic information into reflection tomography[END_REF][START_REF] Ma | Image-guided sparse-model full waveform inversion[END_REF][START_REF] Sava | High definition cross-well electrical resistivity imaging using seismoelectric focusing and image-guided inversion[END_REF][START_REF] Zhou | Image-guided inversion of electrical resistivity data[END_REF] and recent reviews in geophysical imaging [START_REF] Meju | Structural Coupling Approaches in Integrated Geophysical Imaging[END_REF], reservoir seismology [START_REF] Bosch | Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: A review[END_REF], hydrogeophysics [START_REF] Linde | Joint Inversion in Hydrogeophysics and Near-Surface Geophysics[END_REF], mineral exploration [START_REF] Lelièvre | Integrated Imaging for Mineral Exploration[END_REF], petroleum exploration [START_REF] Moorkamp | Joint Inversion in Hydrocarbon Exploration[END_REF]. Two main ideas underlie these approaches. First, some local structural orientations are inferred from borehole data or other geophysical data to constrain the covariance function used during inversion. Second, a petrophysical model that associates the various physical parameters is used to exploit the existing correlation between the physical parameters. As these correlations generally depend on the rock type, the model often includes discrete variables that estimate or sample the rock type at a given location. This notion of rock type is close to the notion of lithofacies, so it is a way to integrate geological reasoning into inverse methods.

In the field of reservoir engineering and hydrogeology, methods incorporating prior geological knowledge in flow and transport models have also been developed very early on, as discussed in several review papers [START_REF] Farmer | Geological Modelling and Reservoir Simulation[END_REF][START_REF] Linde | Geological realism in hydrogeological and geophysical inverse modeling: A review[END_REF][START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF][START_REF] Oliver | Recent progress on reservoir history matching: a review[END_REF]Zhou et al. 2014a). One fundamental reason is that flow and transport processes can be highly non-linear while pressure and concentration measurements are generally quite sparse as compared to the number of potential factors influencing fluid transfers in porous and fractured media. The same observation holds in potential field inversion, where geological prior information can significantly help addressing the ill-posedness of the inverse problem [START_REF] Lelièvre | Integrated Imaging for Mineral Exploration[END_REF]. But what does "geological prior" exactly mean?

As noted in particular by [START_REF] Frodeman | Geological reasoning: Geology as an interpretive and historical science[END_REF], geology is an interpretive science which includes a significant component of historical thinking. One aim of geology is to describe the earth in historical terms by identifying the main geological processes and their impact. In terms of scientific philosophy, it is interesting to highlight that geology generally produces refutable scenarios, whereas mathematics are concerned with formal and irrefutable proofs (given some hypotheses). The encounter of these two scientific methods is deeply written in the DNA of Mathematical Geosciences. Advanced methods in physically-based modeling have been developed to quantitatively model geological processes. Some very interesting inverse methods that use such models have been developed recently to quantitatively integrate spatial observations [START_REF] Charvin | A Bayesian approach to inverse modelling of stratigraphy, part 1: method[END_REF][START_REF] Cross | Construction and application of a stratigraphic inverse model[END_REF][START_REF] Gallagher | Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models, model resolution and model choice for Earth Science problems[END_REF]). These methods are ideal in the sense that they could in principle unify geology and geophysics rigorously. However, the interplay of multiple coupled physical and chemical processes at geological time scales remains extremely challenging to model on a computer. The use of such models in an inverse framework is also very challenging, as the number of unknown or poorly known parameters makes the inverse problem highly ill-posed and computationally intractable. This empty space problem is very general and applies to most inverse problems in geosciences, but it is critical when an explicit time dimension is considered because the density of information in time-space is very small (e.g., only a few points typically constrain pressure and temperature in basin studies). This explains why most of the methods in Fig. 28.1-E-H do not explicitly consider geological time and instead use an object-based approach, a statistics-based approach or a combination of both to represent the geological prior information and make forecasts in the 3D physical space.

Classically, the object-based strategy is essential to the geological approach. For example, geological mapping typically decomposes a complex reality into discrete and interconnected tectonic, igneous, metamorphic, diagenetic, stratigraphic and sedimentological objects. These object definitions do integrate historical and process-based considerations. For instance, time is explicitly considered in the definition of the remarkable surfaces that sequence stratigraphers use to interpret geoscience data. The characterization of these objects in mathematical and computational terms has been a significant focus of the IAMG for that last 50 years. The statistics-based approach, another clear focus of the IAMG, is clearly complementary to the object-based approach. Indeed, objects are heterogeneous, boundaries between objects may be difficult to define and objects can be difficult to map from available observations. Statistical reasoning is key to address these problems. In this chapter, we will try to explain a few manners by which the object-based and statistics-based methods interact in the frame of geo-data and physical modeling integration. For this, we will start from the perspective of what physical modeling needs. [START_REF] Sambridge | Transdimensional inference in the geosciences[END_REF], among others, give a very crisp and generic summary of the parameterizations used in most numerical physical modelling methods. In this view, a model is defined at any point of the physical space by a set of basis functions:

Parameterizations for physical models

.

(1)

For example, in the finite element method with linear triangular elements, a basis function is defined for each mesh vertex : is equal to 1, is equal to 0 and linearly decreases in the mesh elements adjacent to . The values are the parameter values (e.g., thermal conductivity) associated to the mesh vertices.

The general formulation (1) allow to compute or approximate differential operators to solve partial differential equations describing physical processes. Many recent advances in computational physics consist of particular choices of basis functions. For instance, in the extended finite element method, the use of Heaviside basis functions to represent internal discontinuities in a mesh was a step change in the computation of fracture growth [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets-Part I: Mechanical model: NON-PLANAR 3D CRACK GROWTH-PART I[END_REF]. Another very active research field concerns the combination of basis functions at several scales (e.g., [START_REF] Efendiev | Generalized multiscale finite element methods (GMsFEM)[END_REF]). These methods have been applied for instance in finite volume modeling of flow in porous media to solve the flow and transport equations at two distinct and interacting scales [START_REF] Tchelepi | Multi-scale finite-volume method for elliptic problems in subsurface flow simulation[END_REF][START_REF] Møyner | The Multiscale Finite-Volume Method on Stratigraphic Grids[END_REF].

Equation ( 1) is also compatible with the theory of spatial random fields. At point scale, the values are seldom known below the Earth surface. Geostatistics offers many ways to estimate or simulate such values [START_REF] Chiles | Geostatistics: modeling spatial uncertainty[END_REF][START_REF] Goovaerts | Geostatistics for natural resources evaluation[END_REF]) using statistical parameters inferred from subsurface data. One of these parameters is the variogram, which models the statistical correlation between two variables as a function of the distance. In dual kriging, Eq. ( 1) is also used, as the unknown value is estimated as a linear combination of covariance functions centered on the data points. The use of point-based parameterizations is also much studied in computational physics under the term "meshless methods", see for instance [START_REF] Liu | An introduction to meshfree methods and their programming[END_REF]. In the practice of geostatistical methods, the values are generally modeled on a Cartesian grid, but recent papers also discuss about the application of geostatistics on unstructured grids [START_REF] Gross | Geostatistics on Unstructured Grid-Coordinate Systems, Connections and Volumes[END_REF][START_REF] Manchuk | Direct geostatistical simulation on unstructured grids[END_REF][START_REF] Zaytsev | Change-of-Support Models on Irregular Grids for Geostatistical Simulation[END_REF], or directly on points [START_REF] Zagayevskiy | Multivariate grid-free geostatistical simulation with point or block scale secondary data[END_REF]. A major interest of these methods is to estimate or simulate values directly on the physical modeling support, and also to use adaptive resolution depending on the local information density and on the sensitivity between the model parameters and the physical process.

Last, but not least, Equation ( 1) is compatible with a new breed of inverse methods in which the number of parameters is variable, see [START_REF] Sambridge | Transdimensional inference in the geosciences[END_REF] and references therein. These transdimensional inverse methods show much promise to address some of the challenges highlighted in this chapter.

The beauty of equation ( 1) lies in its potential to unify object-based geological descriptions and mathematical descriptions. In a sense, the goal of the various workflows described in Fig. 1 and in the associated references can be seen as a quest to find "geological basis functions" to model the earth. The purpose of Section 28.4 is to try to establish a more explicit correspondence between geological concepts and existing mathematical and computational models for representing geological domains in three-dimensional space. In doing so, we keep in mind that these 3D models will eventually need to be expressed by Eq. ( 1) in physical models.

Geological parameterizations

As discussed in Section 28.2, geologists apply the divide and conquer principle to analyze the earth. Hundreds of years of geological reasoning have essentially led to identify multiple geological features at various scales, depending on their origin:

-Tectonic objects: Faults, joints, folds, cleavages.

-Sedimentary objects: stratigraphic units, horizons and unconformities, sedimentary bodies, facies, bedding structures.

-Intrusive and effusive objects: salt diapirs, salt sheets, shale diapirs, shale mounds, sills or dykes, lava flows, etc.

-Epigenetic objects (originating from chemical and mineralogical processes after rock formation): Metamorphic units, hydrothermalized facies, dissolved rocks (karsts).

These features typically exist at kilometric to micrometric scales (from plates to minerals and fluid inclusions). It is not useful (and not possible) for a model to explicitly represent all objects across these scales. Rather, most modeling approaches hierarchically subdivide the domain to represent a few nested scales [START_REF] Pyrcz | Geostatistical reservoir modeling[END_REF][START_REF] Ringrose | Reservoir Model Design[END_REF].

Two main complementary mathematical and numerical frameworks exist to represent these geological features: spatial random fields and object-based methods. The choice of which framework is most appropriate (or whether and how these frameworks should be combined) depends on the size of the features with re-gard to the density of observations and on the likely impact of the features for the question at hand. Whereas the object size can be objectively discussed and characterized, the impact of features is often based on rules of thumb derived from experience [START_REF] Ringrose | Reservoir Model Design[END_REF]. This may be a source of biases in forecasts. In practical studies, choices may also be constrained by very practical reasons, as some methods are implemented only in commercial software or in distinct software which are not interoperable. These problems and the need for better and abstract knowledge integration are also discussed by [START_REF] Perrin | Shared Earth Modeling: Knowledge Driven Solutions for Building and Managing Subsurface 3D Geological Models[END_REF].

Spatial random fields

As geological processes are not random and result from many physical processes, the resulting spatial fields are generally correlated in space. The characterization of the correlation structure by statistical inference is an essential aspect of geostatistics [START_REF] Chiles | Geostatistics: modeling spatial uncertainty[END_REF][START_REF] Goovaerts | Geostatistics for natural resources evaluation[END_REF]). Indeed, trust can be gained when data are numerous enough to provide robust statistics -even though the modeling assumptions themselves may remain questionable [START_REF] Journel | Beyond covariance: the advent of multiple-point geostatistics[END_REF]. In inverse modeling of flow and transport in porous media, this has led to many approaches that perturb parameters on a grid while preserving variogram or spatial covariance models [START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF][START_REF] Oliver | Recent progress on reservoir history matching: a review[END_REF]Zhou et al. 2014a).

In geostatistics, a result of the divide-and-conquer strategy used in geology is the definition of many types of discrete categories to describe the physical world. These categories can be localized in space in the form of a geological map (or, in three dimensions, a 3D geological model). From a geostatistical standpoint, categories can be modeled with indicator variables. This has led to significant advances, in particular in the field of multiple-point geostatistics (MPS), to represent discrete facies from sparse data and analog training images. Since the seminal work of [START_REF] Guardiano | Multivariate geostatistics: beyond bivariate moments[END_REF], a vast community of mathematical geoscientists has embraced this field and made essential advances, see [START_REF] Hu | Multiple-point geostatistics for modeling subsurface heterogeneity: A comprehensive review: REVIEW OF MULTIPLE POI[END_REF], [START_REF] Mariethoz | Multiple-point geostatistics: stochastic modeling with training images[END_REF]. In particular, MPS have opened concrete and effective ways to using complex (and deliberately subjective) geological priors models in inversion [START_REF] Linde | Geological realism in hydrogeological and geophysical inverse modeling: A review[END_REF][START_REF] Melnikova | History Matching Through a Smooth Formulation of Multiple-Point Statistics[END_REF]. MPS have shown, in a number of instances, the impact of applying analog reasoning and scenarios to find sensible sets of solutions to inverse problems and to assess uncertainties. They also make up an interesting formalism to analyze complex geological systems [START_REF] Scheidt | Quantifying natural delta variability using a multiple-point geostatistics prior uncertainty model: DELTA VARIABILITY AND GEOSTATISTICS[END_REF]).

However, even though progresses can still be made (see for instance Renard et al. 2011), a recurrent challenge with the indicator geostatistical approaches is to ensure that some categories are always connected or adjacent to other categories. This is why, to echo a friendly discussion we had with Andre Journel in 2005, I persist considering that there is more to geological realism than MPS (in its spatial understanding). The Truncated Gaussian method and the Pluri-Gaussian methods [START_REF] Armstrong | Plurigaussian simulations in geosciences[END_REF], even though they rely on multi-Gaussian assumptions, enforce continuity conditions that approach geological reasoning in a very interesting way. For instance, they can produce consecutive successions of facies from shallow marine to offshore environments. This type of method is appropriate when the discrete geological categories originate from an underlying continuous variable (in the previous example, this variable can be assimilated to bathymetry, all facies being defined between consecutive threshold values). In the Pluri-Gaussian approach, the application of Boolean operations on simulated random fields is also a way to emulate the succession of geological events (e.g. simulation of late diagenetic facies overprinting the depositional facies).

In general, spatial random field methods are implemented on grids of fixed resolution. As a result, the discontinuities that may exist in the medium are sampled at that particular resolution. However, some important features such as fractures or shale lenses may be much smaller than the grid resolution, hence cannot be explicitly represented in the grid. Under some hypotheses, this can be addressed by directly modeling a field of equivalent properties assumed representative of the block scale (e.g., equivalent dual porosity and dual permeability fields in fractured media). However, this can be a source of bias in a number of cases [START_REF] Jackson | Surface-based reservoir modelling for flow simulation[END_REF]. The explicit consideration of these objects generally relies on fewer assumptions and provides a way to deal with more complex geometries and with spatial observations, as will be discussed in Section 28.4.2. Note that these two approaches are not mutually exclusive and a combination of both equivalent and explicit approaches are, in general, relevant [START_REF] Bourbiaux | An integrated workflow to account for multi-scale fractures in reservoir simulation models: implementation and benefits[END_REF][START_REF] Maier | An MPFA-based Dual Continuum-Discrete Feature Model for Simulation of Flow in Fractured Reservoirs[END_REF].

Another important aspect of geological reality is that the orientation and the magnitude of spatial correlation can vary in space. This can be modelled with random fields using locally varying anisotropy [START_REF] Boisvert | Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances[END_REF][START_REF] Stroet Cbm Te | Mapping Curvilinear Structures with Local Anisotropy Kriging[END_REF][START_REF] Xu | Conditional curvilinear stochastic simulation using pixel-based algorithms[END_REF]). In geophysics, the use of local anisotropy is illustrated for instance by [START_REF] Clapp | Incorporating geologic information into reflection tomography[END_REF] and by the image-guided inversion methods mentioned in Section 28.2 and Fig. 28.1D. In the absence of exhaustive data to constrain these orientations, one should estimate or simulate the orientations away from local observations [START_REF] Gumiaux | Geostatistics applied to best-fit interpolation of orientation data[END_REF][START_REF] Stroet Cbm Te | Mapping Curvilinear Structures with Local Anisotropy Kriging[END_REF][START_REF] Xu | Conditional curvilinear stochastic simulation using pixel-based algorithms[END_REF]. A practical challenge in the presence of locally varying anisotropy is the inference of geostatistical parameters, as the domain is non-stationary. Object approaches offer another way of dealing with locally varying anisotropy, as will be discussed in the next section.

Object models

In a general sense, object models directly represent the tectonic, sedimentological, intrusive and epigenetic features listed at the beginning of Sec-tion 28.4. As geological objects originate from distinct geological processes at different periods of time, they often correspond to contrasts or discontinuities of the physical parameters of interest. This explains why, beyond pure cartographic goals, so much effort is dedicated to object modeling in geosciences.

Geometry and topology

As discussed by [START_REF] Mallet | [END_REF] and [START_REF] Perrin | Shared Earth Modeling: Knowledge Driven Solutions for Building and Managing Subsurface 3D Geological Models[END_REF], geological objects can be represented in geometrical and topological terms. Topology refers to essential characteristics: the dimension of objects (line, surface or volume), whether objects have inclusions or holes, and if they are connected to other objects. Depending on the type of geological objects, some topological configurations are impossible [START_REF] Caumon | Building and editing a sealed geological model[END_REF]. For instance, a chronostratigraphic horizon must be an open surface and may include internal holes due to faults or intrusions. More generally, the continuity of objects can have a relation to the genesis of the object, hence is a way to constrain geological models. Knowing what is topologically possible and what is not gives precious insights to design modeling methods and to test the validity of geological models [START_REF] Pellerin | RINGMesh: A programming library for developing mesh-based geomodeling applications[END_REF][START_REF] Wellmann | Validating 3-D Structural Models with Geological Knowledge for Improved Uncertainty Evaluations[END_REF]. Topological analysis also provides interesting metrics to characterize and understand geological objects such as karsts [START_REF] Collon | Statistical metrics for the characterization of karst network geometry and topology[END_REF], fracture networks [START_REF] Sanderson | The use of topology in fracture network characterization[END_REF] and structural models [START_REF] Lindsay | Geodiversity: Exploration of 3D geological model space[END_REF][START_REF] Pellerin | Elements for measuring the complexity of 3D structural models: Connectivity and geometry[END_REF]Thiele et al. 2016a;Thiele et al. 2016b). Last, but not least, topology is very important for flow modeling, as it directly relates to the connectivity of permeability conduits and barriers. The links between connectivity and effective flow properties has been much studied at multiple scales in the frame of percolation theory [START_REF] Berkowitz | Percolation theory and its application to groundwater hydrology[END_REF][START_REF] King | Predicting oil recovery using percolation theory[END_REF]). In the cases where geological considerations are not sufficient to fully characterize the topology of the medium, specific methods have been proposed to find possible object geometry honoring some prescribed connectivity [START_REF] Borghi | A pseudo-genetic stochastic model to generate karstic networks[END_REF][START_REF] Collon-Drouaillet | An algorithm for 3D simulation of branchwork karst networks using Horton parameters and A*: Application to a synthetic case[END_REF][START_REF] Henrion | ODSIM: An Object-Distance Simulation Method for Conditioning Complex Natural Structures[END_REF].

Geometry concerns the embedding of the topological objects in 3D space, and is typically described either analytically (e.g., ellipses for fractures) or numerically (using a mesh). Meshes provide much flexibility to discretize the geometry of rock volumes (geological bodies), surfaces (geological boundaries) and lines (contacts between boundaries). All these geometric components are linked by topological relationships [START_REF] Pellerin | RINGMesh: A programming library for developing mesh-based geomodeling applications[END_REF]). More fundamentally, meshes are a way to define basis functions approximating the geometry of the true object. For example, one can define mathematically a triangulated surface as a set of a "hat" basis functions centered on each surface node (taking the value 1 at each node and linearly decreasing it to zero at the node's neighbors), as in Eq. ( 1). This description is very powerful to devise advanced geometry processing algorithms and reduce the dimensionality of complex geometrical shapes [START_REF] Vallet | Spectral geometry processing with manifold harmonics[END_REF]. In the frame of inverse modeling, several inverse methods use the meshed model geome-try as an unknown parameter [START_REF] Fullagar | Drilling-constrained 3D gravity interpretation[END_REF][START_REF] Gjøystdal | COMPUTER REPRESENTATION OF COMPLEX 3-D GEOLOGICAL STRUCTURES USING A NEW "SOLID MODELING" TECHNIQUE*[END_REF][START_REF] Mondal | Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods[END_REF].

Over the past decade, computational advances have also made it possible to consider implicit surfaces to represent geological boundaries. In these approaches, the surfaces are considered as level sets of some three-dimensional scalar field [START_REF] Calcagno | Geological modelling from field data and geological knowledge: Part I. Modelling method coupling 3D potentialfield interpolation and geological rules[END_REF][START_REF] Cowan | Practical implicit geological modelling[END_REF][START_REF] Frank | 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data[END_REF][START_REF] Henrion | ODSIM: An Object-Distance Simulation Method for Conditioning Complex Natural Structures[END_REF]). These methods share the same principles as the Truncated Gaussian and Pluri-Gaussian methods [START_REF] Mannseth | Relation Between Level Set and Truncated Pluri-Gaussian Methodologies for Facies Representation[END_REF]), but the underlying random function model is not necessarily Gaussian, and their focus is set on the geometry of object boundaries. These level set methods are very powerful to automate geometric modeling tasks such as interpolation and extrapolation. In particular, they have shown much interest in stratigraphic modeling as one single scalar field can represent a conformable stratigraphic series at once, which opens new possibilities in structural data interpolation [START_REF] Calcagno | Geological modelling from field data and geological knowledge: Part I. Modelling method coupling 3D potentialfield interpolation and geological rules[END_REF][START_REF] Caumon | Three-Dimensional Implicit Stratigraphic Model Building From Remote Sensing Data on Tetrahedral Meshes: Theory and Application to a Regional Model of La Popa Basin, NE Mexico[END_REF][START_REF] Hillier | Three-Dimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial Basis Functions[END_REF][START_REF] Laurent | Implicit modeling of folds and overprinting deformation[END_REF]. Implicit surfaces also offer very nice ways to consider geometric model perturbations needed to address inverse problems in geosciences [START_REF] Cardiff | Bayesian inversion for facies detection: An extensible level set framework: LEVEL SETS FOR FACIES DETECTION[END_REF][START_REF] Caumon | Elements for stochastic structural perturbation of stratigraphic models[END_REF][START_REF] Noetinger | An Explicit Formula for Computing the Sensitivity of the Effective Conductivity of Heterogeneous Composite Materials to Local Inclusion Transport Properties and Geometry[END_REF][START_REF] Zheglova | 2-D reconstruction of boundaries with level set inversion of traveltimes[END_REF]. A major distinction between explicit and implicit surface models is about topological control: the surface topology has to be chosen before interpolation in explicit methods, whereas it emerges from the interpolation in implicit models, see also [START_REF] Collon | 3D modeling from outcrop data in a salt tectonic context: Example from the Inceyol minibasin, Sivas Basin, Turkey[END_REF] for more discussions.

As in Pluri-Gaussian simulation, it is possible to indirectly account for geological time in object models using the truncation between implicit or explicit objects [START_REF] Calcagno | Geological modelling from field data and geological knowledge: Part I. Modelling method coupling 3D potentialfield interpolation and geological rules[END_REF][START_REF] Caumon | Surface-based 3D modeling of geological structures[END_REF][START_REF] Gjøystdal | COMPUTER REPRESENTATION OF COMPLEX 3-D GEOLOGICAL STRUCTURES USING A NEW "SOLID MODELING" TECHNIQUE*[END_REF]. Boolean operations also provide ways to obtain sharp features in object geometry using constructive solid geometry principles [START_REF] Rongier | Simulation of 3D karst conduits with an object-distance based method integrating geological knowledge[END_REF][START_REF] Ruiu | Modeling Channel Forms and Related Sedimentary Objects Using a Boundary Representation Based on Non-uniform Rational B-Splines[END_REF]. In terms of Eq. ( 1), Boolean operations between implicit objects can be described as indicator (or Heaviside) basis functions [START_REF] Mannseth | Relation Between Level Set and Truncated Pluri-Gaussian Methodologies for Facies Representation[END_REF][START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets-Part I: Mechanical model: NON-PLANAR 3D CRACK GROWTH-PART I[END_REF]): these functions are equal to zero on one side of the interface and equal to 1 on the other side. The representation of faults is a major challenge which is specific to geosciences. Indeed faults are not just discontinuities or sharp geometric features: they result from sliding of rocks that were previously connected. Several authors have proposed mathematical or numerical solutions to address this problem by considering directly or indirectly the displacement between either sides of a fault [START_REF] Calcagno | Geological modelling from field data and geological knowledge: Part I. Modelling method coupling 3D potentialfield interpolation and geological rules[END_REF][START_REF] Georgsen | Fault displacement modelling using 3D vector fields[END_REF][START_REF] Hale | Methods to compute fault images, extract fault surfaces, and estimate fault throws from 3D seismic images[END_REF][START_REF] Holden | Stochastic structural modeling[END_REF][START_REF] Jessell | Structural geophysics: integrated structural and geophysical modelling[END_REF][START_REF] Laurent | A parametric method to model 3D displacements around faults with volumetric vector fields[END_REF][START_REF] Mallet | Elements of mathematical sedimentary geology: The GeoChron model[END_REF][START_REF] Mallet | [END_REF].

From objects to physical parameters

Generally, geological object geometry cannot be described analytically and determining the associated physical parameter fields is not straightforward. In most cases, objects are first discretized in space with a mesh that will support the nu-merical resolution of the physical equations [START_REF] Kolditz | OpenGeoSys: an open-source initiative for numerical simulation of thermo-hydromechanical/chemical (THM/C) processes in porous media[END_REF][START_REF] Pellerin | RINGMesh: A programming library for developing mesh-based geomodeling applications[END_REF]. This mesh is a numerical translation of Eq. ( 1) discretizing the space in elementary volumes deemed representative of some effective physical properties (the values in Eq. ( 1)).

A possible working assumption is to consider a constant (or analytically defined) parameter value associated to each type of geological object. This principle is used for simplicity in a number of numerical models [START_REF] Gjøystdal | COMPUTER REPRESENTATION OF COMPLEX 3-D GEOLOGICAL STRUCTURES USING A NEW "SOLID MODELING" TECHNIQUE*[END_REF][START_REF] Jackson | Reservoir Modeling for Flow Simulation by Use of Surfaces, Adaptive Unstructured Meshes, and an Overlapping-Control-Volume Finite-Element Method[END_REF]. However, as discussed above, heterogeneity exists at many different scales and can have an impact on the physical process below the scale of the objects that are explicitly represented in a numerical model. For example, it is well known in stochastic hydrogeology and reservoir engineering that petrophysical heterogeneity exists within layers or sedimentary facies and impacts flow and transport (see for instance de Marsily et al. 2005 for a review). In many cases, the orientation of heterogeneities within a geological object depends on the object geometry (e.g., crystal orientations in a dyke may be preferentially aligned along the dyke boundaries; sedimentary heterogeneities tend to be more continuous along layers than orthogonally to layers). This can be addressed in modeling by explicitly using locally variable directions of anisotropy [START_REF] Boisvert | Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances[END_REF] or by considering a geometric transform between two spaces [START_REF] Mallet | Elements of mathematical sedimentary geology: The GeoChron model[END_REF][START_REF] Shtuka | Petrophysical Simulation Within an Object-Based Reservoir Model[END_REF]. This last option is very promising as it provides a way to simplify geostatistical modeling, and as it allows to define some useful geological variables such as the apparent sedimentation rate [START_REF] Kedzierski | Combining Stratigraphic and Sedimentological Information for Realistic Facies Simulations[END_REF][START_REF] Mallet | Elements of mathematical sedimentary geology: The GeoChron model[END_REF][START_REF] Massonnat | Breaking of a paradigm: geology can provide 3D complex probability fields for stochastic facies modelling[END_REF]. Such use of indirect geological parameters is an essential and powerful way to introduce geological principles in earth models.

Nonetheless, one should not neglect that object geometry affects model predictions at the two main stages of geostatistical models: (1) geostatistical inference (distributions of continuous variables within each subdomain, multivariate relationships between different variables, trends, spatial variability) and (2) geostatistical modeling (interpolation or simulation). The separation of integrated modeling into an object-modeling phase and a petrophysical modeling phase are, therefore, relatively easy in the classical case where objects are known, when a clear separation of scales exist between representative elementary volume (REV) properties and object geometry, and when objects do not affect geostatistical parameters. However, uncertainty about object geometry and topology can have a significant impact on statistical parameters [START_REF] Lallier | Uncertainty assessment in the stratigraphic well correlation of a carbonate ramp: Method and application to the Beausset Basin, SE France[END_REF], which can be a significant source of complexity in practical studies. More generally, finding at what scale explicit objects properties and REV effective properties can be separated is a fundamental problem in modeling. Therefore, more research is clearly needed to capture the interactions between object geometric (and topological) parameters and random field parameters.

Object uncertainty

Geometric uncertainty can be sampled by adding geometric perturbations to an existing reference model [START_REF] Caumon | Elements for stochastic structural perturbation of stratigraphic models[END_REF][START_REF] Corre | Integrated Uncertainty Assessment For Project Evaluation and Risk Analysis[END_REF][START_REF] Lecour | Modelling of stochastic faults and fault networks in a structural uncertainty study[END_REF] or creating several models after perturbing data [START_REF] Lindsay | Geodiversity: Exploration of 3D geological model space[END_REF][START_REF] Wellmann | Towards incorporating uncertainty of structural data in 3D geological inversion[END_REF]. As the very existence of some objects is also uncertain in many cases, it is also useful to consider object-based stochastic simulation. In random set theory, geometric objects are placed randomly and independently in the domain by combining the simulation of points (Poisson Point Process) and the simulation of objects shapes around these points (see Chiles and Delfiner 2012 and references therein;[START_REF] Lantuéjoul | Boolean model[END_REF]. Classically, objects are geometric primitives defined analytically, whose shape, orientation and size parameters are simulated from some input distribution. Random set theory places a lot of emphasis on the statistical aspects of this process and on conditioning to spatial data, see in particular [START_REF] Lantuéjoul | Boolean model[END_REF] and [START_REF] Allard | Conditional Simulation of Multi-Type Non Stationary Markov Object Models Respecting Specified Proportions[END_REF]. These models, in particular the Boolean Model, have been used to simulate many types of geological objects such as fractures [START_REF] Chiles | Fractal and geostatistical methods for modeling of a fracture network[END_REF], shale lenses [START_REF] Haldorsen | A New Approach to Shale Management in Field-Scale Models[END_REF] or sedimentary channels [START_REF] Deutsch | Hierarchical object-based stochastic modeling of fluvial reservoirs[END_REF][START_REF] Holden | Modeling of fluvial reservoirs with object models[END_REF]. Extensions of the Boolean Model have also been proposed to introduce interactions between objects such as attraction or repulsion between fractures to reproduce their mechanical interactions [START_REF] Aydin | Quantifying structural uncertainty on fault networks using a marked point process within a Bayesian framework[END_REF][START_REF] Bonneau | Impact of a Stochastic Sequential Initiation of Fractures on the Spatial Correlations and Connectivity of Discrete Fracture Networks[END_REF][START_REF] Chiles | Fractal and geostatistical methods for modeling of a fracture network[END_REF][START_REF] Hollund | Havana-a fault modeling tool[END_REF].

From a random set perspective, a deterministic object model is a particular realization of some underlying random set process. In this case, the relatively large data density allows one to consider mainly the data conditioning problem rather than focusing on the number of objects and on their spatial density. Another focus of deterministic object modeling approaches relates to the expert-guided definition of the interactions between objects using interactive editing tools to ensure that the connectivity between objects is compatible with the geological history of the domain (e.g., how faults branch one onto another and how faults displace horizons).

Yet, more and more complex geometric object parameterizations have recently been introduced in object-based simulation methods. For instance, several authors propose to anchor sedimentary channels on discrete polygonal curves (Mariethoz et al. 2014;[START_REF] Pyrcz | ALLUVSIM: A program for event-based stochastic modeling of fluvial depositional systems[END_REF][START_REF] Rongier | A geostatistical approach to the simulation of stacked channels[END_REF][START_REF] Ruiu | Modeling Channel Forms and Related Sedimentary Objects Using a Boundary Representation Based on Non-uniform Rational B-Splines[END_REF][START_REF] Viseur | Caracterisation de reservoirs turbiditiques: simulations stochastiques basees-objet de chenaux meandriformes (10 fig.)[END_REF]). Other variants consider the bounding surfaces of stratigraphic deposits together with some rules to mimic depositional processes [START_REF] Graham | Three-dimensional modeling of clinoforms in shallow-marine reservoirs: Part 1. Concepts and application[END_REF][START_REF] Labourdette | LOSCS' Lateral Offset Stacked Channel Simulations: Towards geometrical modelling of turbidite elementary channels: Lateral offset stacked channels simulations[END_REF][START_REF] Michael | Combining geologic-process models and geostatistics for conditional simulation of 3-D subsurface heterogeneity[END_REF][START_REF] Pyrcz | Stratigraphic rule-based reservoir modeling[END_REF][START_REF] Pyrcz | Stochastic surface-based modeling of turbidite lobes[END_REF][START_REF] Rongier | A geostatistical approach to the simulation of stacked channels[END_REF][START_REF] Ruiu | Modeling Channel Forms and Related Sedimentary Objects Using a Boundary Representation Based on Non-uniform Rational B-Splines[END_REF][START_REF] Sech | Three-dimensional modeling of a shorefaceshelf parasequence reservoir analog: Part 1. Surface-based modeling to capture high-resolution facies architecture[END_REF]. As argued in the review of [START_REF] Pyrcz | Stratigraphic rule-based reservoir modeling[END_REF], these models make it possible to consider genetic principles such as erosion, progradation and aggradation of sedimentary deposits in an automatic way. Similarly, pseudo-process-based models have also been proposed in the area of fracture modeling to approximate mechanical interactions and truncations that occur during fracture growth [START_REF] Bonneau | A methodology for pseudo-genetic stochastic modeling of discrete fracture networks[END_REF][START_REF] Davy | A model of fracture nucleation, growth and arrest, and consequences for fracture density and scaling: A DISCRETE FRACTURE NETWORK MODEL[END_REF][START_REF] Srivastava | Geostatistical simulation of fracture networks[END_REF]. At a larger scale, a recent trend has been to simulate possible stochastic geometries where the number and the connectivity of faults is variable [START_REF] Aydin | Quantifying structural uncertainty on fault networks using a marked point process within a Bayesian framework[END_REF][START_REF] Cherpeau | Method for Stochastic Inverse Modeling of Fault Geometry and Connectivity Using Flow Data[END_REF][START_REF] Cherpeau | Stochastic simulations of fault networks in 3D structural modeling[END_REF][START_REF] Cherpeau | Stochastic structural modelling in sparse data situations[END_REF][START_REF] Holden | Stochastic structural modeling[END_REF]Julio et al. 2015a). In all these approaches, the use of rules is often a means to generate realistic objects and to produce likely connectivities and spatial features without being constrained by some input grid resolution. However, conditioning to dense spatial data sets remains challenging with these approaches. A possible way forward is to consider parameter-rich object-models and to consider process-based rules backward in time [START_REF] Parquer | Conditioning channel backward migration modeling to seismic data[END_REF][START_REF] Ruiu | Semiautomatic interpretation of 3D sedimentological structures on geologic images: An object-based approach[END_REF]. In all cases, expert control of model realism is also difficult and may call for additional "geological likelihood" functions to scrutinize the realizations [START_REF] Jessell | Towards an integrated inversion of geoscientific data: What price of geology?[END_REF].

Interestingly, the use of continuous functions around the Poisson points used in object-based simulation (Random Function Model [START_REF] Jeulin | MODELLING RANDOM MEDIA[END_REF]), is a possible way to relate random sets to Eq. ( 1). However, formalizing the link between object models and basis functions used in physical models is not easy and relies on the assumption that values are analytically defined on each object, and that objects have stationary statistics [START_REF] Jeulin | Morphology and effective properties of multi-scale random sets: A review[END_REF][START_REF] Oda | An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock masses[END_REF]. Dealing with more realistic geometries and sequential Boolean operations to reproduce the succession of geological events calls for further numerical and mathematical developments. Meanwhile, as statistical properties of random sets are not easily checked in practical cases, the numerical approach to relate objects to physics clearly remains an area of much interest [START_REF] Botella | Indirect unstructured hex-dominant mesh generation using tetrahedra recombination[END_REF][START_REF] Cacace | MeshIt-a software for three dimensional volumetric meshing of complex faulted reservoirs[END_REF][START_REF] Karimi-Fard | A general gridding, discretization, and coarsening methodology for modeling flow in porous formations with discrete geological features[END_REF][START_REF] Merland | Voronoi grids conforming to 3D structural features[END_REF][START_REF] Mustapha | G23FM: a tool for meshing complex geological media[END_REF][START_REF] Pellerin | Toward Mixed-element Meshing based on Restricted Voronoi Diagrams[END_REF][START_REF] Zehner | Workflows for generating tetrahedral meshes for finite element simulations on complex geological structures[END_REF].

Figure 28.2: Summary of the various complementary ways to incorporate geological knowledge in earth models.

Conclusions and challenges

Several complementary ways exist to incorporate geological information in earth models (Fig. 28.2): spatial statistics, geological variables, geometry and topology of geological objects and explicit geological process modeling. Links exist between the random field and object-based frameworks in cases where the canonical random field theory is applicable (e.g., homogeneous and stationary object densities). This forms the rationale for most modeling methods where "small" objects are treated though their (spatially correlated) equivalent properties at the representative elementary volume scale. "Large" objects are modeled explicitly using rules and parameters that incorporate geological principles and may be calibrated from data and analogs.

Although geostatistics has proven an invaluable theoretical framework to rigorously describe geological domains, it needs to be complemented by geological reasoning (sensu [START_REF] Frodeman | Geological reasoning: Geology as an interpretive and historical science[END_REF]. Namely, considering discrete time steps approximating geological history and geological variables cannot be directly measured can significantly help generating more predictive geological models, which may not always have stationary statistical properties. Geometric and topological interactions between objects have a direct connection to geological history and prove a powerful tool to characterize geological domains.

From a physical modeling perspective, geometric object models allow to represent small spatial features which can have a large impact on physical processes [START_REF] Jackson | Reservoir Modeling for Flow Simulation by Use of Surfaces, Adaptive Unstructured Meshes, and an Overlapping-Control-Volume Finite-Element Method[END_REF]Julio et al. 2015b;[START_REF] Matthäi | Numerical simulation of multi-phase fluid flow in structurally complex reservoirs[END_REF]). This calls for specific developments in meshing and physical simulation, for example to better account for object features directly in the numerical code [START_REF] Pichot | A Generalized Mixed Hybrid Mortar Method for Solving Flow in Stochastic Discrete Fracture Networks[END_REF]). In the frame of inverse problems, sensitivity analysis is essential in practical studies. Theoretically, specific methods integrating the probability of existence of objects also need to be considered more widely, such as random vector parameterization [START_REF] Cherpeau | Method for Stochastic Inverse Modeling of Fault Geometry and Connectivity Using Flow Data[END_REF], reversible jump Monte-Carlo Markov Chain simulation [START_REF] Green | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination[END_REF][START_REF] Sambridge | Transdimensional inference in the geosciences[END_REF] or ensemble-based methods [START_REF] Scheidt | Representing Spatial Uncertainty Using Distances and Kernels[END_REF]. Both in forward and inverse physical models, an additional and significant challenge is to better characterize the multi-sale interactions between geometrical and petrophysical parameterizations (basis functions and associated parameters values).

Acknowledgements

The ideas expressed in this chapter owe much to Bruno Lévy, Albert Tarantola, Andre Journel and Jean-Laurent Mallet. Their encouragements, trust and critical remarks have been essential influences. I am also grateful to the graduate students and colleagues of the Research for Integrative Numerical Geology (RING) Team, especially my colleagues Pauline Collon and Paul Cupillard, for their multiple contributions to such a stimulating research environment. Discussions in the frame of the HIWAI ANR project led by Yann Capdeville also fed some of the ideas about scale expressed in this chapter. Last, but not least, at a time where research funding is getting more and more complex, I express my great appreciation to the academic and industrial sponsors of RING-Gocad Consortium for their continued support and to ASGA for its effective Consortium management.

Figure

  Figure 28.1: Examples of approaches and workflows using geological and geophysical data to make forecasts about the