Two-point Flux Approximation Research Articles

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod’s representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.

A positivity-preserving RD-FV scheme for diffusion problems on triangular meshes

In this paper, we propose a positivity-preserving residual distribution based finite volume scheme (RD-FV) for steady diffusion problems on unstructured triangular meshes. In this scheme, we use the RD method to obtain the auxiliary values of vertex, then adopt the positivity-preserving finite volume scheme to update the cell-centered values for diffusion problems, in which a nonlinear two-point flux approximation is utilized to obtain nonnegative solution. Thus a positivity-preserving conservative property for the RD-FV scheme is guaranteed. Several numerical experiments show the optimal convergence rates for the numerical solution, that is, approximate second-order accuracy for the solution and first-order accuracy for the flux, and demonstrate the positivity-preserving property of our new scheme.

Poro-Mechanical Coupling for Flow Diagnostics

Accounting for poro-mechanical effects in full-field reservoir simulation studies and uncertainty quantification workflows using complex reservoir models is challenging, mainly because of the high computational cost. We hence introduce an alternative approach that couples hydrodynamics through existing flow diagnostics simulations with poro-mechanics to screen the impact of coupled poro-mechanical processes on reservoir performance without significantly increasing computational overheads. In flow diagnostics, time-of-flight distributions and influence regions can be used to characterise the flow field in the reservoir, which depends on the distribution of petrophysical properties that are altered due to production-induced changes in pore pressure and effective stress. These extended flow diagnostics calculations hence enable us to quickly screen how the dynamics in the reservoirs (e.g. reservoir connectivity, displacement efficiency, and well allocation factors) are affected by the complex interactions between poro-mechanics and hydrodynamics. Our poro-mechanically informed flow diagnostics account for steady-state and single-phase flow conditions based on the poro-elastic theory and assume that the reservoir does not contain fractures. Fluid flow and rock deformation calculations are coupled sequentially. The equations are discretised using a finite-volume method with two-point flux-approximation and the virtual element method, respectively. The solution of the coupled system considers stress-dependent permeabilities. Due to the steady-state nature of the calculations and the effective proposed coupling strategy, these calculations remain computationally efficient while providing first-order approximations of the interplay between poro-mechanics and hydrodynamics, as we demonstrate through a series of case studies. The extended flow diagnostic approach hence provides an efficient complement to traditional reservoir simulation and uncertainty quantification workflows and enable us to assess a broader range of reservoir uncertainties.

Accurate Two-Phase Flow Simulation in Faulted Reservoirs by Combining Two-Point Flux Approximation and Mimetic Finite Difference Methods

Summary Faulted reservoirs are commonly modeled by corner-point grids (CPGs). Because the two-point flux approximation (TPFA) method is not consistent on non-K-orthogonal grids, multi-phase flow simulation using TPFA on CPGs may have significant discretization errors if grids are not K-orthogonal. We developed a novel method to improve the simulation accuracy where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular K-orthogonal grid is first constructed, and then cells are divided where fault planes are present. Most cells remain K-orthogonal while irregular non-K-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD, and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to the part of the domain with severe grid non-K-orthogonality, while the TPFA method is applied to the rest of the domain. We compare flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid, while the same problem is solved by TPFA and MFD methods on a grid with non-K-orthogonal cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault, while fluxes computed using MFD are still as accurate as the reference solution. We also compare saturation accuracy for two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when the pressure equation is solved by the TPFA method. Because the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions agree with reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, we observe that the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method, while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.

On several numerical strategies to solve Richards’ equation in heterogeneous media with finite volumes

We propose several numerical approaches building on upstream mobility two-point flux approximation finite volumes to solve Richards’ equation in domains made of several rock-types. Our study encompasses four different schemes corresponding to different ways to approximate the nonlinear transmission condition systems arising at the interface between different rocks consisting in the continuity of the pressure and the fluxes, as well as different resolution strategies based on Newton’s method with variable switch. We benchmark the robustness and accuracy of the different methods on filling and drainage test cases with standard nonlinearities of Brooks-Corey and van Genuchten-Mualem type, as well as with challenging steep nonlinearities.

A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. We focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell-centered Lagrange multipliers are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. This formulation is consistent, but is not uniformly inf-sup bounded and requires a stabilization. For the resulting 3×3 block Jacobian matrix, robust and efficient solution methods are not available, so we aim at designing new scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a benchmark problem. The objective of the analysis is to identify the most promising solution strategy.

Load-balanced parallel simulations for embedded discrete fracture model on non-conforming staggered 3D unstructured grids

Recently, embedded discrete fracture model (EDFM) using non-conforming staggered 3D unstructured grids was presented for the numerical simulation of fluid-flows in large-scale complex fractured porous media. In this paper, we implement load-balanced parallel simulations for this approach on distributed memory systems. A load-balanced domain decomposition algorithm is presented to partition matrix grids and embedded fracture grids uniformly. We classify all grid connections on the boundary of subdomains to five types and illustrate corresponding inter-processor connections. A packaged inter-processor communication approach is implemented based on message passing interface (MPI) to improve communication efficiency for this simulation method. A two-phase parallel simulator is developed, and block coupled method is adopted to assemble the equations discretized in matrix and grid domains using two-point flux approximation (TPFA) finite volume method. The load-balanced algorithm is tested by nonuniformly distributed fracture networks. The performance and scalability are also tested on 1024 CPU cores by two complex fracture network models with up to 25 million matrix grids and 11 million embedded fracture grids. The results show satisfactory parallel efficiency and scalability.

The VEM-based positivity-preserving conservative scheme for radiation diffusion problems on generalized polyhedral meshes

In this paper, for the 3D radiation diffusion equation on generalized polyhedral meshes, we construct a positivity-preserving conservative scheme based on the virtual element method (VEM). Firstly, the equation is discretized by finite volume, and the one-sided flux on the cell-face is discretized by using the fixed stencil of all vertices. Then, the nonlinear two-point flux approximation is used to obtain the equations about cell-centered unknowns and vertex unknowns; the lowest order VEM solves the vertex values of the mesh. Finally, the conservative equations containing only cell-centered unknowns are solved. This scheme ensures the high accuracy of VEM and embodies the local conservation and positivity-preserving. Compared with the existing nonlinear positivity-preserving finite volume schemes, the new scheme does not need nonlinear iteration for linear problems. For nonlinear problems, it only needs to use the VEM to solve the nonlinear system. Numerical results show that the new scheme can achieve the optimal convergence order and maintain the nonnegativity of the solution for generalized polyhedral meshes and arbitrary diffusion tensors.

A Positivity-Preserving Finite Volume Scheme for Nonequilibrium Radiation Diffusion Equations on Distorted Meshes.

In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme.

Fast Screening Tool for Assessing the Impact of Poro-Mechanics on Fractured Reservoirs Using Dual-Porosity Flow Diagnostics

Summary Accounting for poro-mechanical effects in full-field reservoir simulation studies and uncertainty quantification workflows is still limited, mainly because of their high computational cost. We introduce a new approach that couples hydrodynamics and poro-mechanics with dual-porosity flow diagnostics to analyze how poro-mechanics could affect reservoir dynamics in naturally fractured reservoirs without significantly increasing computational overhead. Our new poro-mechanically informed dual-porosity flow diagnostics account for steady-state and single-phase flow conditions in the fractured medium while the fracture-matrix fluid exchange is approximated using a physics-based transfer rate coefficient, which models two-phase flow using an analytical solution for spontaneous imbibition or gravity drainage. The deformation of the system is described by the dual-porosity poro-elastic theory, which is based on mixture theory and micromechanics to compute the effective stresses and strains of the rock matrix and fractures. The solutions to the fluid flow and rock deformation equations are coupled sequentially. The governing equations for fluid flow are discretized using a finite-volume method with two-point flux-approximation while the governing equations for poro-mechanics are discretized using the virtual element method. The solution of the coupled system considers stress-dependent permeabilities for fractures and matrix. Our framework is implemented in the open-source MATLAB Reservoir Simulation Toolbox (MRST). We present a case study using a fractured carbonate reservoir analog to illustrate the integration of poro-mechanics within the dual-porosity flow diagnostics framework. The extended flow diagnostics calculations enable us to quickly screen how the dynamics in fractured reservoirs (e.g., reservoir connectivity, sweep efficiency, and fracture-matrix transfer rates) are affected by the complex interactions between poro-mechanics and fluid flow where changes in pore pressure and effective stress modify petrophysical properties and hence affect reservoir dynamics. Because of the steady-state nature of the calculations and the effective coupling strategy, these calculations do not incur significant computational overheads. They provide an efficient complement to traditional reservoir simulation and uncertainty quantification workflows because they enable us to assess a broader range of reservoir uncertainties (e.g., geological, petrophysical, and hydromechanical uncertainties). The capability of studying a much broader range of uncertainties allows the comparison and ranking from a large ensemble of reservoir models and select individual candidates for more detailed full-physics reservoir simulation studies without compromising on assessing the range of uncertainties inherent to fractured reservoirs.

A convergent finite volume scheme for dissipation driven models with volume filling constraint

In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications.

A physics-informed deep convolutional neural network for simulating and predicting transient Darcy flows in heterogeneous reservoirs without labeled data

The physics-informed neural network (PINN) is a general deep learning framework for simulating physical processes and surrogate modeling without labeled data. The basic idea is to formulate the loss function according to the governing PDEs such that the neural network (NN) can be trained to minimize the PDE residual along with other misfits such as initial and boundary conditions. Following PINN, various networks have been developed for simulating steady and transient flows with or without labeled data. However, according to literature review, it is still not clear how to use NNs to simulate transient Darcy flows in highly heterogeneous reservoir models with source/sink terms in the absence of labeled data. In the current study, a physics-informed deep convolutional neural network (PIDCNN) architecture for simulating and predicting such flows is presented. Convolutional neural network is found to be more efficient than fully-connected neural network since 2D variables can be regarded as images. The finite volume discretization scheme is adopted to build the loss function to approximate the PDE residual such that flux continuity between neighboring cells of different properties can be implemented conveniently using the two-point flux approximation. Test cases are used to show that PIDCNN can accurately simulate transient Darcy flows in homogeneous and heterogeneous reservoirs. Further, it is demonstrated that PIDCNN can be trained as a surrogate to predict the transient flow fields of reservoir models not included in training. In addition, the CNN structure in the current study can be trained as a surrogate with labels for a particular output for better accuracy. A workflow is presented to demonstrate that CNN can be trained as an accurate surrogate for production rates using labels generated by the PIDCNN-based solver such that the entire workflow is external-label-free.

Energy-stable discretization of two-phase flows in deformable porous media with frictional contact at matrix–fracture interfaces

We address the discretization of two-phase Darcy flows in a fractured and deformable porous medium, including frictional contact between the matrix–fracture interfaces. Fractures are described as a network of planar surfaces leading to the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. Phase pressures are supposed to be discontinuous at matrix–fracture interfaces, as they provide a better accuracy than continuous pressure models even for high fracture permeabilities.The general gradient discretization framework [31] is employed for the numerical analysis, allowing for a generic stability analysis and including several conforming and nonconforming discretizations. We establish energy estimates for the discretization, and prove existence of a solution. To simulate the coupled model, we employ a Two-Point Flux Approximation (TPFA) finite volume scheme for the flow and second-order (P2) finite elements for the mechanical displacement coupled with face-wise constant (P0) Lagrange multipliers on fractures, representing normal and tangential stresses, to discretize the frictional contact conditions. This choice allows to circumvent possible singularities at tips, corners, and intersections between fractures, and provides a local expression of the contact conditions. We present numerical simulations of two benchmark examples and one realistic test case based on a drying model in a radioactive waste geological storage structure.

A Cut-Cell Polyhedral Finite Element Model for Coupled Fluid Flow and Mechanics in Fractured Reservoirs

Summary Mechanical deformation induced by injection and withdrawal of fluids from the subsurface can significantly alter the flow paths in naturally fractured reservoirs. Modeling coupled fluid flow and mechanical deformation in fractured reservoirs relies on either sophisticated gridding techniques or enhancing the variables (degrees of freedom) that represent the physics to describe the behavior of fractured formation accurately. The objective of this study is to develop a spatial discretization scheme that cuts the “matrix” grid with fracture planes and utilizes traditional formulations for fluid flow and geomechanics. The flow model uses the standard low-order finite volume method with the compartmental embedded discrete fracture model (cEDFM). Due to the presence of nonstandard polyhedra in the grid after cutting/splitting, we use numerical harmonic shape functions within a polyhedral finite element (PFE) formulation for mechanical deformation. To enforce fracture-contact constraints, we use a penalty approach. We provide a series of comparisons between the approach that uses conforming unstructured grids and an unstructured discrete fracture model (uDFM) with the new cut-cell PFE formulation. The manuscript validates and compares both methods for linear elastic, single-fracture slip, and Mandel’s problems with tetrahedral, Cartesian, and perpendicular-bisectional (PBI) grids. Finally, the paper presents a fully coupled 3D simulation with multiple inclined intersecting faults activated in shear by fluid injection, which caused an increase in effective reservoir permeability. Our approach allows for great reduction in the complexity of the (gridded) model construction while retaining the solution accuracy together with great savings in the computational cost compared with uDFM. The flexibility of our model with respect to the types of grid polyhedra allows us to eliminate mesh artifacts in the solution of the transport equations typically observed when using tetrahedral grids and two-point flux approximation.

A Flux Based Approximation to Simulate Coupled Hydromechanical Problems for Mines with Heterogeneous Rock Types Using the Material Point Method

Advances in numerical simulation techniques play an important role in helping mining engineers understand those parts of the rock mass that cannot be readily observed. The Material Point Method (MPM) is an example of such a tool that is gaining popularity for studying geotechnical problems. In recent years, the original formulation of MPM has been extended to not only account for simulating the mechanical behaviour of rock under different loading conditions, but also to describe the coupled interaction of pore water and solid phases in materials. These methods assume that the permeability of mediums is homogeneous, and we show that these MPM techniques fail to accurately capture the correct behaviour of the fluid phase if the permeability of the material is heterogeneous. In this work, we propose a novel implementation of the coupled MPM to address this problem. We employ an approach commonly used in coupled Finite Volume Methods, known as the Two Point Flux Approximation (TPFA). Our new method is benchmarked against two well-known analytical expressions (a one-dimensional geostatic consolidation and the so-called Mandel-Cryer effect). Its performance is compared to existing coupled MPM approaches for homogeneous materials. In order to gauge the possible effectiveness of our technique in the field, we apply our method to a case study relating to a mine known to experience severe problems with pore water.

Sequential implicit vertex approximate gradient discretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

This work deals with sequential implicit schemes for incompressible and immiscible two-phase Darcy flows which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhedral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequential implicit schemes are built from the total velocity formulation of the two-phase flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequential implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes. The numerical investigation of the sequential schemes is also extended to the widely used finite volume Two-Point Flux Approximation spatial discretization.

A New Nonlinear Two-Point Flux Approximation Method for Solving the Anisotropic Diffusion Equation with Reduced Violations of the Discrete Maximum/Minimum Principle

Summary A class of monotone cell-centered nonlinear finite-volume methods has been proposed in the past decade to solve the anisotropic diffusion equation. The nonlinear two-point flux approximation (TPFA) (NTPFA) method preserves the nonnegativity of the solution values but can violate the discrete maximum/minimum principle (DMP). To enforce DMP, the nonlinear multipoint flux approximation (NMPFA) method ought to be used. In this work, we propose a novel NTPFA method that can reduce the severity of DMP violations significantly compared with the standard NTPFA method. The new formulation uses conormal decomposition for the construction of the one-sided fluxes. To define the unique flux through a connection between two cells, we choose a convex combination of the two one-sided fluxes such that the absolute differences of the magnitudes of the two transmissibility terms associated with the two neighboring cells are minimized, thus bringing the discrete coefficient matrix closer to having the zero row-sum property. Numerical experiments are conducted to test the performance of the new NTPFA method. The results demonstrate that the new scheme has comparable convergence order for both the solution and the flux compared with the standard NTPFA method or the classical multi-point flux approximation (MPFA-O) method. Moreover, the new NTPFA formulation shows marked improvements over the standard NTPFA in terms of reducing DMP violations. However, depending on the specific problem configuration, our new NTPFA formulation can lead to a system of nonlinear equations that is more difficult to solve.

Semi-Implicit Finite Volume Procedure for Compositional Subsurface Flow Simulation in Highly Anisotropic Porous Media

Subsurface multiphase flow in porous media simulation is extensively used in many disciplines. Large meshes with non-orthogonalities (e.g., corner point geometries) and full tensor highly anisotropy ratios are usually required for subsurface flow applications. Nonetheless, simulations using two-point flux approximations (TPFA) fail to accurately calculate fluxes in these types of meshes. Several simulators account for non-orthogonal meshes, but their discretization method is usually non-conservative. In this work, we propose a semi-implicit procedure for general compositional flow simulation in highly anisotropic porous media as an extension of TPFA. This procedure accounts for non-orthogonalities by adding corrections to residual in the Newton-Raphson method. Our semi-implicit formulation poses the guideline for FlowTraM (Flow and Transport Modeller) implementation for research and industry subsurface purposes. We validated FlowTraM with a non-orthogonal variation of the Third SPE Comparative Solution Project case. Our model is used to successfully simulating a real Colombian oil field.

A nonlinear correction scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

In this paper, a nonlinear positivity-preserving finite volume scheme for the heterogeneous and anisotropic diffusion problems is proposed. Firstly, a linear diamond finite volume scheme is introduced. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as auxiliary ones and are expressed as linear combinations of the surrounding cell-centered unknowns, which reduces the scheme to a pure cell-centered one. Two new interpolation algorithms are suggested through the linearity-preserving approach and a novel discretization of diffusion coefficient. The new vertex interpolation algorithms improve a great deal the numerical performance of the linear scheme on distorted meshes with strongly anisotropy. Secondly, a new positivity-preserving cell-centered scheme is constructed via the aforementioned linear scheme and a nonlinear correction technique proposed by [28]. This nonlinear scheme is different from most existing nonlinear two-point flux approximation schemes in the way that it is not based on the nonlinear two-point flux approximation and it doesn't require the positivity-preserving vertex interpolation. Its implementation is also very easy and simple. Numerical experiments conducted in this paper indicate that the new positivity-preserving scheme is efficient and has approximately second order accuracy in most extreme cases.

Error estimate of the non-intrusive reduced basis method with finite volume schemes

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non-Intrusive Reduced Basis (NIRB) method has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meanings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).