Patent Publication Number: US-2021172316-A1

Title: Method of modelling a sedimentary basin using a hex-dominant mesh representation

Description:
FIELD OF THE INVENTION 
     The present invention relates to the field of petroleum reservoir or geological gas storage site exploration and exploitation. 
     Petroleum exploration is the search for hydrocarbon reservoirs within a sedimentary basin. Understanding the principles of hydrocarbon genesis and the connections thereof with the subsurface geological history has allowed to develop methods for assessing the petroleum potential of a sedimentary basin. 
     The general procedure for assessing the petroleum potential of a sedimentary basin comprises shuttles between:
         a prediction of the petroleum potential of the sedimentary basin, from available data relative to the basin studied (outcrops, seismic surveys, drilling data for example). This prediction aims to better understand the architecture and the geological history of the basin studied, and notably to study whether hydrocarbon maturation and migration processes may have developed, to identify subsurface zones where these hydrocarbons may have accumulated, to define which zones have the best economic potential, assessed from the volume and the nature of the hydrocarbons probably trapped (viscosity, rate of mixing with water, chemical composition, etc.), as well as their operating cost (controlled for example by the fluid pressure and depth),   exploratory drilling operations in the various zones having the best potential, in order to confirm or invalidate the previously predicted potential and to acquire new data intended to fuel new and more precise studies.       

     Petroleum exploitation of a reservoir consists, from the data collected during the petroleum exploration phase, in selecting the reservoir zones with the best petroleum potential, in defining optimum exploitation schemes for these zones (using reservoir simulation for example in order to define the numbers and positions of the exploitation wells allowing optimum hydrocarbon recovery), in drilling exploitation wells and, in general terms, in setting up the production infrastructures necessary for reservoir development. 
     A sedimentary basin results from the deposition, over geological times, of sediments within a depression of the Earth&#39;s crust. These soft and water-rich sediments are then subjected, as they are progressively buried in the basin, to pressure and temperature conditions that convert them to compact sedimentary rocks referred to as geological layers. 
     The current architecture of a sedimentary basin notably results from a mechanical deformation of the subsoil over geological times. This deformation comprises, a minima, a compaction of the geological layers due to the gradual burial of these layers in the basin, under the effect of the supply of new sediments. However, a sedimentary basin is also most often subjected to large-scale tectonic movements, generating for example geological layer folding, or faults causing breaks in the geological layers. 
     The nature of the hydrocarbons present in a sedimentary basin notably depends on the type of organic matter present in the deposited sediments, and on the pressure and temperature conditions undergone by the basin over geological times. 
       FIG. 1  schematically shows a sedimentary basin comprising several geological layers (a, c) delimited by sedimentary interfaces (b) traversed by a fault (e), and a hydrocarbon accumulation (d) in one of the geological layers of the basin considered (c). 
     Formation of a sedimentary basin thus involves a large number of complex physical and chemical processes, which may additionally interact with one another. Given such complexity, prediction of the petroleum potential of a sedimentary basin requires computer tools allowing to simulate, as realistically as possible, the physical and chemical phenomena involved in the formation of the basin studied. 
     This type of reconstruction of the formation history of a sedimentary basin, also referred to as “basin modelling”, is most often performed by means of a family of computer tools allowing to simulate in one, two or three dimensions, the sedimentary, tectonic, thermal, hydrodynamic, organic and inorganic chemical processes involved in the formation of a petroleum basin. 
     Basin modelling conventionally comprises three steps:
         a step of constructing a mesh representation of the basin studied, known as geomodelling. This mesh representation (also referred to as mesh) is most often structured in layers, i.e. a group of cells is assigned to each geological layer of the modelled basin. Then, each cell of this mesh is filled with one or more petrophysical properties, such as porosity, facies (clay, sand, etc.) or the organic matter content at the time of sedimentation. The construction of this model is based on data acquired through seismic surveys, measurements in wells, core drilling, etc.;   a step of structural reconstruction of this mesh, representing prior states of the basin architecture. This step can be carried out using a method referred to as “backstripping”, ora method referred to as structural restoration;   a step of numerical simulation of physical phenomena taking place during the basin evolution and contributing to the formation of oil traps. This step, known as “basin simulation”, is based on a discretized representation of space and time for reconstructing the basin formation over geological times. In particular, basin simulation allows to simulate, over geological times, the formation of hydrocarbons from notably the organic matter initially buried with the sediments, the state of the stresses and strains in the basin, and the transport of these hydrocarbons, from the rocks in which they are formed to those where they are trapped. Basin simulation thus provides a map of the subsoil at the current time, showing the probable location of the reservoirs, as well as the proportion, the nature and the pressure of the hydrocarbons trapped therein. An example of such a basin simulator is the TemisFlow® software (IFP Energies nouvelles, France).       

     Thus, this integrated procedure allowing the phenomena that have caused hydrocarbon generation, migration and accumulation in sedimentary basins to be taken into account and analysed allows the success rate to be increased when drilling an exploration well, thus enabling better exploitation of this basin. 
     BACKGROUND OF THE INVENTION 
     The following documents are mentioned in the description:
     Cook, Robert D., Finite Element Modeling for Stress Analysis, John Wiley &amp; Sons, 1995.   Coussy, O., Mécanique des milieux poreux, Editions Technip, 1991, Paris.   Liu, G. R., Nguyen Thoi Trung, Smoothed Finite Element Methods, CRC Press, 2010.   Schneider F., Modelling multi-phase flow of petroleum at the sedimentary basin scale. Journal of Geochemical exploration 78-79 (2003) 693-696).   Steckler, M. S., and A. B. Watts, Subsidence of the Atlantic-type continental margin off New York, Earth Planet. Sci. Lett., 41, 1-13, 1978.   Zienkiewicz, O. C., Taylor, R. L.,  The finite element method—Volume  1 : the basis—Fifth  Edition, Butterworth Heinemann, 2000.   

     Basin simulation tools allowing to numerically simulate the formation of a sedimentary basin are known. Examples thereof are the tools described in patent EP-2,110,686 (U.S. Pat. No. 8,150,669) or in patent applications EP-2,816,377 (US-2014/0,377,872), EP-3,075,947 (US-2016/0,290,107), EP-3,182,176 (US-2017/0,177,764). These tools notably allow to assess the evolution of quantities such as temperature and pressure in the entire sedimentary basin over geological times, and thus to simulate, over geological times, both the transformation of the organic matter present in a source rock of the basin into hydrocarbons and the migration, into a reservoir rock of the basin, of the hydrocarbons thus produced. 
     Conventionally, as described for example in the document (Schneider, 2003), basin simulation softwares assume only vertical variations of the mechanical stresses that affect a sedimentary basin. More precisely, basin simulation softwares only take into account the vertical component of the mechanical stress variations induced by the weight of the successive sediment deposits in the course of time. This is referred to as 1D mechanical effects simulation. 
     However, a sedimentary basin can undergo, throughout its history, mechanical stresses characterized by components in the three dimensions of space, and these stresses may be local or regional, and time variant. These mechanical stresses are on the one hand induced by the sediment deposits themselves. In this case, the mechanical stresses involve a vertical component, related to the weight of the sediments on the already deposited layers, but they also often have horizontal components, as well as shear components as the sediment deposits are generally not invariant laterally. On the other hand, a sedimentary basin undergoes, throughout the formation thereof, mechanical stresses induced by tectonic movements related to the geodynamics of the Earth, such as extension movements (causing opening of the basin with, for example, rift formation) or compression movements (causing folds, overlaps, fractures within the basin, etc.). These tectonic movements most often induce mechanical stress variations in the three dimensions of space. It is noted that an already deposited layer undergoes stress variations induced by the tectonic movements undergone by a sedimentary basin throughout the formation thereof. 
     Thus, in the case of such basins, high-precision modelling of the stress field is required, and the vertical model is no longer satisfactory. Document EP-3,182,176 (U.S. Pat. No. 10,296,679) describes a coupling between a conventional basin simulator (i.e. with 1D simulation of the mechanical effects) and a mechanical simulator allowing to determine and to take into account the displacement and stress field in 3 dimensions in a basin modelling simulation. The coupling described in this document allows to carry out the thermal, hydrodynamic and mechanical computations with a single mesh. 
     In general terms, the meshes used in basin simulation must be hexahedral-dominant (or hex-dominant) and consistent with the boundaries between stratigraphic layers so that hydrodynamic modelling is as precise as possible. Now, sometimes, some basins comprise a succession of very fine geological layers (a few meters, whereas the conventional dimensions of a basin model are of the order of a hundred km in the two horizontal directions and of the order of 10 to 20 km in the vertical direction) having very heterogeneous mechanical and hydraulic properties and/or pinched-out stratigraphic layers, and these characteristics need to be taken into account in the mesh representation of the basin for a precise numerical simulation. 
     Fine layers and pinchouts are two geometric objects that make the generation of the mesh to be used for basin modelling very complex, notably for mechanical behaviour simulation. Conventionally, simulation of the mechanical behaviour is performed with the finite-element method (FEM). The quality of the solution that can be obtained with this method depends on the size of the mesh cells (accuracy increases with mesh refinement) and on the shape thereof. Ideally, the geometry of the cell must be regular and its aspect ratio (ratio between the lengths of the smallest edge and the greatest edge thereof) should be 1: a hexahedral cell should ideally be a cube, a tetrahedral cell should ideally be a regular tetrahedron, etc. In the presence of fine layers and pinchouts, compliance with these conditions would inevitably lead to the generation of meshes with a very large number of cells, thus making the algorithms using these meshes unusable due to the excessive increase in memory and computation time demand. In the case of pinchouts, it is even impossible to comply with the condition on the cell geometry, even when significantly increasing the mesh refinement: the presence of very low angles at the end of the pinchout shows flattened or distended cells, regardless of the refinement level. In some cases, the unfavourable geometry of these cells generates numerical problems that prevent successful simulation. 
     In practice, in case of fine layers or stratigraphic pinchouts, in order to limit the number of cells to be used for numerical simulation, cells having a lateral extension of one to several kilometers are conventionally used, whereas the extension is only a few meters in the vertical direction. Besides, the presence of stratigraphic pinchouts further requires being able to process cells whose faces are connected at a very low angle, of the order of 1° for example. Thus, some practical application cases of numerical basin simulation do not enable generation of a mesh where the conditions relative to the cell shape are verified, which results in that the quality of the numerical basin simulation is not guaranteed. In some cases, it may even occur that the numerical method cannot find a solution. 
     In order to overcome these drawbacks, it is possible to use the FS-FEM method (Face-based Smoothed Finite-Element Method). The FS-FEM method is a variation of the S-FEM method (Smoothed Finite-Element Method), which is itself a variant of the FEM method. Each variation of the S-FEM method has its convergence and accuracy properties. The FS-FEM variant is known as the most accurate variation of the S-FEM method. 
     However, as shown hereafter, it appears that application of the FS-FEM method to solve the balance equation of poromechanics is unstable in the case of a hex-dominant mesh having flattened cells (ratio of the smallest vertical edge to the greatest vertical edge 1/5 at most, preferably 1/10). Now, in practice, in order not to disproportionately increase the number of cells of the numerical basin simulation, the fine layers and the stratigraphic pinchouts are often represented by flattened cells. 
     The present invention aims to overcome these drawbacks. In particular, the present invention allows to solve numerically, in a stable manner that guarantees the quality of the solution, the equations involved in a numerical basin simulation, in particular the balance equation of poromechanics by means of a face-based smoothed finite-element method, even in case of a hex-dominant mesh representing fine layers and/or stratigraphic pinchouts. 
     SUMMARY OF THE INVENTION 
     The present invention relates to a computer-implemented method of modelling a sedimentary basin, said sedimentary basin having undergone a plurality of geological events defining a sequence of states of said basin, by means of a computer-executed numerical basin simulation, said numerical basin simulation solving at least a balance equation of poromechanics according to a face-based smoothed finite-element method for determining at least a stress field and a strain field. The method according to the invention comprises at least the following steps: 
     A. performing physical quantity measurements relative to said basin by means of sensors and constructing a first mesh representative of said basin for each of said states of said basin, said first meshes of said basin for each of said states predominantly consisting of hexahedral cells. 
     B. by means of said numerical basin simulation and said first meshes for each of said states, determining at least a strain field and a stress field for each of said states by carrying out at least the following steps for each of said first meshes of said states: 
     a) determining a second mesh representative of said state by subdividing each of said hexahedral cells of said first mesh representative of said state into six pyramidal cells, said non-hexahedral cells of said first mesh being unchanged in said second mesh, 
     b) modelling said basin by determining at least said displacement field and said stress field for said state by means of said numerical simulation applied to said second mesh representative of said state. 
     According to an implementation of the invention, a hexahedral cell of said first mesh can be subdivided into six pyramidal cells by connecting, for each face of said hexahedral cell, each node of said face to an additional node located at the barycenter of said hexahedral cell. 
     According to an implementation of the invention, it is possible, in step b), for each face of each of said cells of said second mesh representative of said state, to determine a smoothing domain relative to said face-based smoothed finite-element method by connecting each node of said face to at least one additional node located at the barycenter of said at least one cell to which said face belongs. 
     According to an implementation of the invention, said smoothing domain for a face belonging to at least one pyramidal cell of said second mesh can correspond to at least a tetrahedron or a pyramid. 
     Furthermore, the invention relates to a computer program product downloadable from a communication network and/or recorded on a computer-readable medium and/or processor executable, comprising program code instructions for implementing the method as described above, when said program is executed on a computer. 
     The invention also relates to a method for exploiting hydrocarbons present in a sedimentary basin, said method comprising at least implementing the method for modelling said basin as described above, and wherein, from at least said modelling of said sedimentary basin, an exploitation scheme is determined for said basin, comprising at least one site for at least one injection well and/or at least one production well, and said hydrocarbons of said basin are exploited at least by drilling said wells of said site and by providing them with exploitation infrastructures. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       Other features and advantages of the method according to the invention will be clear from reading the description hereafter of embodiments, given by way of non-limitative example, with reference to the accompanying figures wherein: 
         FIG. 1  shows an illustrative example of a sedimentary basin, 
         FIG. 2  shows an example of a sedimentary basin (left) and an example of a mesh representation (right) of this basin, 
         FIG. 3  shows an example of structural reconstruction of a sedimentary basin according to an embodiment of the invention, represented by three strain states taken in three different geological states, 
         FIG. 4  shows the strain in a beam subjected to bending, calculated with the FEM method and the FS-FEM method, in the case of flattened hexahedral cells, and 
         FIG. 5  shows an example of subdivision of a hexahedral cell into six pyramidal cells. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     According to a first aspect, the invention relates to a computer-implemented method of modelling a sedimentary basin, by means at least of a numerical basin simulation solving at least a balance equation of poromechanics according to a face-based smoothed finite-element method known as FS-FEM method. 
     In general terms, the balance equation of poromechanics is written (see for example document (Coussy, 1991)): 
       ∇· τ +(ρ+ m )   g   =∇·( p B   )
 
     where  σ ′ is the effective stress tensor, ρ the homogenized density of the porous medium, m the fluid mass exchange,  g  the acceleration of gravity, p the fluid pressure and  B  the Biot tensor. In a numerical simulation such as a basin simulation, this equation is numerically solved. In general, this equation is usually discretized in a mesh representation (or mesh) by means of the finite-element method (FEM) or one of its variants such as, for example, the smoothed finite-element method (S-FEM). 
     Different variations of the S-FEM method exist in the literature in the field of mechanical simulation, each with its convergence and accuracy properties. The variation known as FS-FEM (Face-based Smoothed Finite-Element Method) is known as the most accurate variation of the S-FEM method. However, in the literature, the FS-FEM method is described to date only for tetrahedral cells, and all the application cases found in the literature in the field of mechanical simulation use a tetrahedral mesh only. 
     The applicant has observed that the FS-FEM method is in practice unstable in the case of a mesh having very flattened hexahedral cells. It therefore appears that the FS-FEM method cannot be applied as is in the case of basin models whose mesh representation comprises cells representative of thin geological layers (for example of the order of a few meters in the vertical direction) and/or when these layers comprise stratigraphic pinchouts (characterized by an angle less than 1°). 
     The instability of the FS-FEM method in the presence of flattened hexahedral cells (i.e. whose ratio of the greatest horizontal edge to the smallest vertical edge is at least 5, preferably 10) materializes in two different ways in numerical simulations: either the computation stops with a fatal error because the stiffness matrix of the model is singular (zero determinant), which means that it cannot be inverted, or the computed solution is incorrect and it locally exhibits oscillations and/or peaks in the computed displacements, stresses and strains. An example of the latter case is shown in  FIG. 4 , within the context of a beam subjected to bending, the beam being represented by cells consisting of flattened hexahedra (aspect ratio 1:10). More precisely,  FIG. 4 , top, shows the geometry of the deformed beam computed with the FEM method. This deformed shape is correct because it is coherent with the known analytical solution to the problem.  FIG. 4 , bottom, shows the deformed beam geometry obtained with the FS-FEM method: the strain is clearly incorrect and the accordion shape is the typical sign of numerical instability (see also paragraph 4.6 of document (Cook, 1995)). 
     The present invention relates to an improvement to the FS-FEM method in order to make it applicable to the hexahedral cells used in basin models. More specifically, in a particularly shrewd manner, the applicant has shown that improving the FS-FEM method according to the invention allows to stabilize the numerical simulation, including in the case of flattened hexahedral cells. 
     Thus, the numerical basin simulation according to the invention aims to solve numerically at least the balance equation of poromechanics according to the face-based smoothed finite-element method (FS-FEM) in the case of a hex-dominant type mesh. 
     In general terms, discretization of the balance equation of poromechanics leads to write a system of algebraic equations of the following form (see for example Equation 2.23 of the document (Zienkiewicz and Taylor, 2000)): 
         Ka+f=r   (1)
 
     where K is the stiffness matrix of the system, a is the vector of the degrees of freedom of the nodes of the mesh used for discretization (nodal displacements, see Equation 2.1 of the document (Zienkiewicz and Taylor, 2000)), f is a vector of the nodal forces (equivalent nodal forces, see Equation 2.24b of the document (Zienkiewicz and Taylor, 2000)), and r is the vector of the external concentrated forces (see Equation 2.14 of the document (Zienkiewicz and Taylor, 2000)). Details of the solution of this system of equations by means of a face-based smoothed finite-element method specifically suited to the case of hexahedral cells are given in step 2 of the method according to the invention. 
     According to the invention, it is assumed that the sedimentary basin has undergone a plurality of geological events defining a sequence of states of the basin, each of said states extending between two successive geological events. Preferably, the sequence of states can cover a period of time covering at least the production of hydrocarbons, notably by maturation of an organic matter present in a source rock of the basin, deformation of the basin due to 3D mechanical stresses, displacement of these produced hydrocarbons into at least one reservoir rock of the basin over geological times. By way of non-limitative example hereafter, Ai denotes a state of the sequence of states of the basin, i being an integer ranging from 1 to n, An representing the state of the basin at the current time. According to the invention, n is at least 2. In other words, the sequence of states according to the invention comprises the state of the basin at the current time and at least one state of said basin at an earlier geological time. 
     According to a second aspect, the invention relates to a method of exploiting hydrocarbons present in a sedimentary basin, the method according to the second aspect comprising implementing the method of modelling a sedimentary basin according to the first aspect of the invention. 
     The method according to the first aspect of the invention comprises at least steps 1) and 2) described hereafter. 
     The method according to the second aspect of the invention comprises at least steps 1) to 3) described hereafter. 
     1) Constructing Mesh Representations for Each of the States 
     This step consists in measuring physical quantities relative to the basin by means of sensors and in constructing a mesh representative of the basin at the current time, then meshes for each state of the basin. According to the invention, the mesh representations of the basin for each of said states predominantly consist of hexahedral cells. This is conventionally known as hex-dominant meshing. 
     In this step, at least the three following substeps are carried out: 
     1.1) Measuring Physical Quantities Relative to the Basin 
     This substep consists in acquiring physical quantity measurements relative to the basin studied, by means of sensors. 
     By way of non-limitative example, the sensors can be logging tools, seismic sources and receivers, fluid samplers and analyzers, etc. 
     Thus, the measurements according to the invention can consist of outcrop surveys, seismic acquisition surveys, measurements in wells (logging for example), petrophysical and/or geochemical analyses of core samples taken in situ. 
     These measurements allow to deduce petrophysical properties associated with the basin studied, such as facies (lithology), porosity, permeability, or the organic matter content at measuring points of the basin. Information relative to the properties of the fluids present in the basin can also be obtained, such as saturation values of the various fluids present in the basin. Temperatures can also be measured at different points of the basin (notably bottomhole temperatures). 
     1.2) Constructing a Mesh Representative of the Basin at the Current Time 
     This substep consists in constructing a hex-dominant mesh representative of the basin at the current time, from the physical quantity measurements performed in the previous substep. 
     More precisely, construction of a mesh representation of a basin consists in discretizing in three dimensions the architecture of the basin and in assigning properties to each of the cells of this mesh. The physical quantity measurements performed at various points of the basin as described above are therefore notably exploited, extrapolated and/or interpolated, in the various cells of the mesh, according to more or less restrictive hypotheses. 
     Most often, the spatial discretization of a sedimentary basin is organized in cell layers representing each the various geological layers of the basin studied.  FIG. 2  illustrates, on the left side, an example of a sedimentary basin at the present time and, on the right side, an example of a mesh of this basin. 
     According to an implementation of the invention, the mesh constructed for the current state An of the basin studied notably comprises in each cell information on the lithology, a porosity value, a permeability value, an organic matter content, and properties relative to the fluids present in the cell, such as saturation. 
     According to an implementation of the invention, the mesh constructed for the current state An of the basin predominantly consists of hexahedral cells. The OpenFlow® software (IFP Energies nouvelles, France) or the GOCAD® software (Emerson-Paradigm, USA) can be used. 
     1.3) Structural Reconstruction of the Basin Architecture for the Various States 
     This substep consists in reconstructing the past architectures of the basin for the various states Ai, with i ranging from 1 to n−1. The mesh constructed in the previous substeps, which represents the basin at the current time, is therefore deformed in order to represent the anti-chronological evolution of the subsoil architecture over geological times, and for the various states Ai. At the end of this substep, a mesh is available for each state Ai, with i ranging from 1 to n. 
     According to a first embodiment of the present invention, structural reconstruction can be particularly simple if it is based on the assumption that its deformation only results from a combination of vertical movements by compaction of the sediment or by uplift or downwarping of its basement. This technique, known as backstripping, is described in (Steckler and Watts, 1978) for example. 
     According to another embodiment of the present invention, in the case of basins whose tectonic history is complex, notably in the cases of basins with faults, less restrictive hypotheses, such as structural restoration, need to be used. Such a structural restoration is described for example in document FR-2,930,350 A1 (US-2009/0,265,152 A1). Structural restoration consists in computing the successive deformations undergone by the basin by integrating the deformations due to compaction and those resulting from tectonic forces. 
     In the example of  FIG. 3 , three states are used to represent the subsoil deformation over geological times. The mesh on the left represents the current state, where a slip interface (a fault here) can be observed. The mesh on the right represents the same sedimentary basin for a state Ai, prior to the current state. For this state Ai, the sedimentary layers are not fractured yet. The central mesh represents an intermediate state, i.e. the sedimentary basin in a state Ai′ between state Ai and the current state. It is observed that the slip along the fault has started to modify the basin architecture. 
     2) Determining a Strain and Stress Field by Numerical Basin Simulation for Each State 
     In this step, by means of numerical basin simulation and of the meshes determined for each of the states in the previous step, at least one strain field and one stress field are determined for each state. 
     The “first mesh representation” of a state is understood to be hereafter a mesh as constructed in step 1 for said state. Step 2 according to the invention comprises at least applying substeps 2.1 and 2.2 described below to each of the states. In substep 2.1, which is applied for a given state, a second mesh representative of this state is constructed from the first mesh representation of this state. In substep 2.2, which is applied for a given state, the numerical basin simulation according to the invention is applied to the second mesh representative of the state considered, thus solving at least a balance equation of poromechanics according to the face-based smoothed finite-element method. Substeps 2.1 and 2.2 are repeated for each of the states. 
     2.1) Determining a Second Mesh Representative of a State 
     This substep consists in constructing a second mesh representative of a state by subdividing each of the hexahedral cells of the first mesh representation of this state into six pyramidal cells (pyramids whose base is a quadrilateral, i.e. pyramids having five faces and five apices). It is clear that the non-hexahedral cells of the first mesh considered are not subdivided as described in this substep. The non-hexahedral cells of the first mesh considered are simply transferred, with all their dimension and position characteristics, to the second mesh. In other words, the non-hexahedral cells of the first mesh considered are unchanged in the second mesh. 
     According to a preferred embodiment of the invention, each hexahedral cell of the first mesh can be subdivided into six pyramidal cells as follows: for each face of the hexahedral cell considered, each node of this face is connected to an additional node located at the barycenter of the hexahedral cell considered. 
       FIG. 5 , left, shows in grey a pyramidal cell determined according to this preferred embodiment of the invention, formed from the face ABCD of a hexahedral cell of the first mesh and barycenter G of this hexahedral cell.  FIG. 5 , right, shows the six pyramidal cells formed from each of the faces of a hexahedral cell of the first mesh according to this preferred embodiment of the invention, the thin dotted lines representing the connections between the nodes of each face of the hexahedral cell with the node located at barycenter G of this hexahedral cell. 
     This substep is repeated for each hexahedral cell of the first mesh representative of a state. The non-hexahedral cells of the first mesh are unchanged in the second mesh. 
     2.2) Determining the Displacement Field and the Stress Field for a Mesh Representative of a State 
     This substep, which is applied for a given state, consists in determining the displacement field and the stress field by means of said numerical basin simulation according to the invention solving at least the balance equation of poromechanics according to the FS-FEM method applied to the second mesh representative of the state considered and determined in the previous substep. 
     In other words, the face-based smoothed finite-element method as described in the literature, and notably in document (Liu and Nguyen Thoi Trung, 2010), is applied to the second mesh representative of a state. 
     We however describe hereafter the main steps, denoted by 2.2.1 to 2.2.4, to be carried out for applying the face-based smoothed finite-element method as described in the literature to the second mesh representative of a state. 
     2.2.1) Determining a Smoothing Domain for Each Face of Each Cell of the Second Mesh Representative of a State 
     In this substep of the FS-FEM method applied to the second mesh according to the invention, for each face of each of the cells of the second mesh determined in the previous substep, a smoothing domain relative to the face-based smoothed finite-element method is determined. 
     In general terms, the face-based smoothed finite-element method (FS-FEM) and, more generally, the smoothed finite-element method (S-FEM) use, in the mechanical balance formulation, a smoothed strain calculated by a weighted average of the strain according to an equation of the type (see for example Equation 4.19 of document (Liu and Nguyen Thoi Trung, 2010)): 
     
       
         
           
             
               
                 
                   
                     
                       
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     where  ε  is the smoothed strain of the S-FEM method or of the FS-FEM method, is the strain (used by the FEM method) and A k   s  is the volume of a smoothing domain Ω k   s  on which the strain is smoothed. 
     According to these methods, whose description can be found in document (Liu and Nguyen Thoi Trung, 2010), Equation (2) above is calculated in each smoothing domain. The smoothing domains for the S-FEM method or the FS-FEM method are determined in such a way that:
         the smoothing domains cover the mesh entirely,   there is no overlap between the domains.       

     Conventionally for the S-FEM method or the FS-FEM method, a smoothing domain relative to a face belonging to at least any cell is formed by connecting the nodes of the face considered to the point(s) corresponding to the barycenters of the cells to which the face considered belongs. The smoothing domains associated with each face of the cells of the second mesh are created according to the same general principle. 
     By way of example, a smoothing domain relative to a face belonging to a single pyramidal cell (in other words, a free cell, on the border of the mesh) is determined by forming at least one tetrahedron or a pyramid from the nodes of said face and a point at the barycenter of the pyramidal cell considered. It is clear that the smoothing domain of a face belonging to a single pyramidal cell corresponds to a tetrahedron when the face considered is one of the four triangles of the pyramid, and that the smoothing domain of a face belonging to a single cell corresponds to a pyramid when the face considered is the quadrangle (i.e. the base) of the pyramid. It is also clear that, for a face belonging to two neighbouring pyramidal cells, the smoothing domain consists of two tetrahedra or two pyramids because the nodes of a face are then connected to the barycenters of each neighbouring cell. 
     Determination of a smoothing domain according to these principles is repeated for each face of the cells of the second mesh representative of a state. 
     2.2.2) Determining a Strain-Displacement Relation for Each Smoothing Domain 
     In this substep of the FS-FEM method applied to the second mesh according to the invention, for each of the smoothing domains determined as described above, a strain-displacement relation is determined according to the face-based smoothed finite-element method applied to the smoothing domains as determined above. “Applied to the smoothing domains” means that the face-based smoothed finite-element method is applied as described in general terms in the literature. 
     According to an implementation of the invention, said strain-displacement relation takes the form of a matrix relating the displacement vector to a strain vector. This matrix is also referred to as strain-displacement transformation matrix. 
     According to an implementation of the invention, using Equation (2) above for the definition of the smoothed strain leads to the following expression for the strain-displacement transformation matrix, denoted by B (see also Equations 4.29, 4.30 and 4.31 in document (Liu and Nguyen Thoi Trung, 2010)): 
         B =[ B   1   B   2    . . . B   n-1   B   n ]  (3)
         with       

     
       
         
           
             
               
                 
                   
                     
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     where B 1  . . . B n  are the matrices respectively associated with nodes i=1 . . . n of the cell(s) to which smoothing domain Ω k   s  belongs, A k   s  is the volume of smoothing domain Ω k   s , ∂Ω k   s  is the boundary of the smoothing domain, n is the outward normal of smoothing domain Ω k   s  and N i  is the shape function (see paragraph 2.2.1 of document (Zienkiewicz and Taylor, 2000)) associated with node i. 
     2.2.3) Determining a Stiffness and Nodal Forces for Each Smoothing Domain 
     In this substep of the FS-FEM method applied to the second mesh according to the invention, a stiffness and nodal forces are determined for each of the smoothing domains determined as described above, from at least the strain-displacement relation determined for this smoothing domain as described above. 
     According to an implementation of the invention, said stiffness and said nodal forces can take the form of a stiffness matrix and of a nodal force vector respectively. 
     According to an implementation of the invention, stiffness matrix K Ω  and nodal force vector f Ω  relative to a smoothing domain constructed as described above can be determined by means of Equations 2.24a, 2.24b of document (Zienkiewicz and Taylor, 2000) and using strain-displacement transformation matrix B as determined above. 
     In general, in the FS-FEM method, for a smoothing domain denoted by V, stiffness matrix K is calculated according to Equation 2.24a of document (Zienkiewicz and Taylor, 2000): 
         K=∫   V   B   T   DBDV   (5)
 
     where B is the matrix as described above, D is a matrix representative of the material stiffness (see Equation 2.5 of document (Zienkiewicz and Taylor, 2000)) calculated with the material constitutive law connecting stresses and strains. 
     In general, in the FS-FEM method, for a smoothing domain denoted by V, nodal force vector f is calculated according to Equation 2.24b of document (Zienkiewicz and Taylor, 2000): 
         f=−∫   V   N   T   b  dV−∫ A   N   T   t  dA−∫ V   B   T   Dε   0  dV+∫ V   B   T σ 0  dV  (6)
 
     where N is the shape function matrix of the finite element of the parent cell as described above, b is the body force vector,  t  the surface force vector (distributed external loading), ε 0  the initial strain (see paragraph 2.2.3 of document (Zienkiewicz and Taylor, 2000)) and σ 0  the initial residual stress (see paragraph 2.2.3 of document (Zienkiewicz and Taylor, 2000)). 
     2.2.4) Determining a Stiffness and Nodal Forces for the Second Mesh Representative of a State 
     This substep of the FS-FEM method applied to the second mesh consists in determining a stiffness and nodal forces relative to the second mesh representative of the state considered from at least the stiffness and the nodal forces determined for each smoothing domain as described above. 
     According to an implementation of the invention wherein the stiffness and/or the nodal forces can take the form of α stiffness matrix and a nodal force vector respectively, stiffness matrices K Ω  and vectors f Ω  determined for each smoothing domain Ω in the previous substep are assembled by means of a standard assembly used in the finite-element method (see paragraph 1.3 of document (Zienkiewicz and Taylor, 2000)), which can be written as follows: 
         K=Σ   t=1   n   K   Ω   (7)
 
         f=Σ   t=1   n   f   Ω   (8)
 
     where n is the total number of smoothing domains Ω. 
     Then, according to the invention, the sedimentary basin studied is modelled by determining at least the displacement field and the stress field for the mesh considered, by means of the numerical basin simulation according to the invention and of at least the stiffness and the nodal forces determined for the second mesh considered and determined as described above. 
     According to an implementation of the invention, the discretized problem defined by Equation (1) above, implemented in the basin simulator according to the invention and used with stiffness matrix K determined as described above (and computed with Equation (7) above for example) and nodal force vector f determined as described above (and computed with Equation (8) above for example), is solved. Determining vector r of Equation (2), which is the boundary conditions vector, can be done using the conventional FEM method, as described for example in paragraph 1.3 of document (Zienkiewicz and Taylor, 2000). 
     According to the invention, at least substeps 2.1 and 2.2 are repeated for each first mesh representative of each state of the basin. 
     3) Exploiting the Hydrocarbons of the Basin 
     This step is carried out within the context of the second aspect of the invention, which concerns a method of exploiting the hydrocarbons present in a sedimentary basin. 
     After carrying out the previous steps, basin simulation results are available. The basin simulation according to the invention allows at least to determine the basin stress and strain field for various states of the basin, in a stable manner in the case of a hex-dominant mesh some cells of which have a poor aspect ratio. 
     Implicitly, and as is conventional in basin simulation, the amount of hydrocarbons present in each cell of the mesh representative of the basin for the past and current times is also known. 
     Furthermore, depending on the basin simulator used for implementing the invention, it is possible for example to have information on:
         i. the development of sedimentary layers   ii. their compaction under the effect of the weight of the overlying sediments   iii. their temperature evolution during burial   iv. the changes in fluid pressure resulting from this burial   iv. the thermogenic formation of hydrocarbons.       

     From such information, the specialist can then determine cells of the mesh representative of the basin at the current time comprising hydrocarbons, as well as the proportion, the nature and the pressure of the hydrocarbons trapped therein. The zones of the basin studied having the best petroleum potential can then be selected. These zones are then identified as reservoirs (or hydrocarbon reservoirs) of the sedimentary basin studied. 
     This step consists in determining at least one scheme for exploiting the hydrocarbons contained in the sedimentary basin studied. Generally, an exploitation scheme comprises a number, a geometry and a site (position and spacing) for injection and production wells to be drilled in the basin. An exploitation scheme can further comprise a type of enhanced recovery for the hydrocarbons contained in the reservoir(s) of the basin, such as enhanced recovery through injection of a solution containing one or more polymers, CO 2  foam, etc. A hydrocarbon reservoir exploitation scheme must for example enable a high rate of recovery of the hydrocarbons trapped in this reservoir, over a long exploitation time, and requiring a limited number of wells. In other words, the specialist predefines evaluation criteria according to which a scheme for exploiting the hydrocarbons present in a reservoir of a sedimentary basin is considered sufficiently efficient to be implemented. 
     According to an embodiment of the invention, a plurality of exploitation schemes is defined for the hydrocarbons contained in one or more geological reservoirs of the basin studied, and at least one evaluation criterion is assessed for these exploitation schemes, by means of a reservoir simulator (such as the PumaFlow® software (IFP Energies nouvelles, France)). These evaluation criteria can comprise the amount of hydrocarbons produced for each of the various exploitation schemes, the curve representative of the production evolution over time for each well considered, the gas-oil ratio (GOR) for each well considered, etc. The scheme according to which the hydrocarbons contained in the reservoir(s) of the basin studied are really exploited can then correspond to the one meeting at least one of the evaluation criteria of the various exploitation schemes. 
     Then, once an exploitation scheme determined, the hydrocarbons trapped in the petroleum reservoir(s) of the sedimentary basin studied are exploited according to this exploitation scheme, notably at least by drilling the injection and production wells of the exploitation scheme thus determined, and by installing the production infrastructures necessary to the development of this or these reservoirs. Moreover, in cases where the exploitation scheme has been determined by estimating the production of a reservoir associated with different enhanced recovery types, the selected additive type(s) (polymers, surfactants, CO 2  foam) are injected into the injection well. 
     It is understood that a scheme for exploiting hydrocarbons in a basin can evolve during the exploitation of the hydrocarbons of this basin, for example according to additional basin-related knowledge acquired during this exploitation and to improvements in the various technical fields involved in the exploitation of a hydrocarbon reservoir (advancements in the field of drilling, of enhanced oil recovery for example). 
     Equipment and Computer Program Product 
     The method according to the invention is implemented by means of an equipment (a computer workstation for example) comprising data processing means (a processor) and data storage means (a memory, in particular a hard drive), as well as an input/output interface for data input and method results output. 
     The data processing means are configured for carrying out in particular step 2 described above. 
     Furthermore, the invention concerns a computer program product downloadable from a communication network and/or recorded on a computer-readable medium and/or processor executable, comprising program code instructions for implementing the method as described above, and notably step 2), when said program is executed on a computer. 
     Thus, the method according to the first aspect of the invention allows to stabilize a numerical basin simulation solving at least one balance equation of poromechanics according to a face-based smoothed finite-element method, in the case of a hex-dominant mesh, even in the case of fine and/or pinched-out geological layers in the sedimentary basin. 
     Besides, the method according to the second aspect of the invention allows to predict the petroleum potential of complex sedimentary basins, which may have undergone for example complex tectonic movements and exhibit fine layers and/or stratigraphic pinchouts, which contributes to improving the exploitation of hydrocarbons in this type of sedimentary basins.