Multipoint Flux Approximation Research Articles

Monotonicity of the cell-centred triangular MPFA method for saturated and unsaturated flow in heterogeneous porous media

Summary In this work, the monotonicity of the symmetric cell-centred Multi-Point Flux Approximation (MPFA) method on triangular grids is studied for saturated and unsaturated flow in heterogeneous porous media. For the linear saturated flow, we derive necessary conditions on the triangulation to satisfy the Positive Transmissibility (PT) condition which ensures monotonicity of the discrete scheme. For the nonlinear unsaturated flow, an upwind MPFA scheme is developed using the upstream weighting of the hydraulic conductivity. We show analytically that this scheme cannot satisfy the PT condition whatever the kind of the triangulation. In spite of that, numerical experiments in 2D heterogeneous domains and unstructured meshes show that the upwind MPFA scheme avoids spurious oscillations in the numerical solution.

A multi-resolution multiscale finite volume method for simulation of fluid flows in heterogeneous porous media

This paper presents an extension of the multiscale finite volume (MsFV) method to multi-resolution coarse grid solvers for single phase incompressible flows. To achieve this, a grid one level coarser than the coarse grids used in the MsFV method is constructed and the local problems are redefined to compute the basis and correction functions associated with this new grid. To construct the coarse-scale pressure equations, the coarse-scale transmissibility coefficients are calculated using a new multi-point flux approximation (MPFA) method. The estimated coarse-scale pressures are utilized to compute the multiscale pressure solution. Finally a reconstruction step is performed to produce a conservative velocity field which is used to solve the transport equations. The computational cost of the proposed method is compared with that of the MsFV method and the relevant time complexity formulas are given. Several two-dimensional test cases with permeability fields ranging from two-scale to multi-scale problems are solved. The performance of the proposed method in handling problems with shale layers or discrete fractures is assessed. Also, a number of layers from the tenth SPE comparative study problem are used to examine general abilities of the method when facing realistic reservoir problems. The results are compared with fine-scale reference solutions to assess the accuracy of the proposed method.

A Multipoint Flux Approximation of the Steady-State Heat Conduction Equation in Anisotropic Media

In this work, we introduce multipoint flux (MF) approximation method to the problem of conduction heat transfer in anisotropic media. In such media, the heat flux vector is no longer coincident with the temperature gradient vector. In this case, thermal conductivity is described as a second order tensor that usually requires, at least, six quantities to be fully defined in general three-dimensional problems. The two-point flux finite differences approximation may not handle such anisotropy and essentially more points need to be involved to describe the heat flux vector. In the framework of mixed finite element method (MFE), the MFMFE methods are locally conservative with continuous normal fluxes. We consider the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method with a special quadrature rule that allows for nodal velocity elimination resulting in a cell-centered system for the temperature. We show comparisons with some analytical solution of the problem of conduction heat transfer in anisotropic long strip. We also consider the problem of heat conduction in a bounded, rectangular domain with different anisotropy scenarios. It is noticed that the temperature field is significantly affected by such anisotropy scenarios. Also, the technique used in this work has shown that it is possible to use the finite difference settings to handle heat transfer in anisotropic media. In this case, heat flux vectors, for the case of rectangular mesh, generally require six points to be described.

MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS

In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

Multidimensional upstream weighting for multiphase transport on general grids

The governing equations for multiphase flow in porous media have a mixed character, with both nearly elliptic and nearly hyperbolic variables. The flux for each phase can be decomposed into two parts: (1) a geometry- and rock-dependent term that resembles a single-phase flux; and (2) a mobility term representing fluid properties and rock–fluid interactions. The first term is commonly discretized by two- or multipoint flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single-point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, SPU suffers from grid orientation effects. An important example of this, which will be considered in this work, is the displacement of a heavy oil by water. For these adverse mobility ratio flows, the governing equations are unstable at the modeling scale, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results. In this work, we present a framework for multidimensional upstream weighting for multiphase flow with buoyancy on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multidimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for time integration of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multidimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multidimensional upstream weighting is used.

Upscaling permeability to unstructured grids using the multipoint flux approximation

Grids used for flow simulation are often at a much coarser scale than that of grids for geological modelling due to computational demand. Unstructured grids offer increased flexibility for the flow grid design; however, solving the flow equations and upscaling from high resolution geological grids to the coarse flow grid is more complex than using coarse regular grids. The multipoint flux approximation (MPFA) is one technique applied to discretize the flow equations on unstructured grids. This paper develops an upscaling technique that uses the MPFA method to solve the flow equations on the fine- and coarse-scale grids. Unlike most cases where the fine-scale grid is regular or structured, this work utilizes a high resolution triangular grid that conforms to the coarse-scale grid. The triangular grid is generated using the coarse-scale interaction regions as constraints. Upscaling leads to transmissibility matrices of the coarse-scale interaction regions. Two different types of local boundary conditions for the MPFA upscaling approach are developed, including linear varying pressures and pressures computed by solving the flow equations around the element boundary. The method is tested using flow simulation on several cases. Results are comparable with flow using a high resolution regular grid.

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture–Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

Convergence of the Multipoint Flux Approximation L-Method on General Grids

We prove first order convergence of the flux and the pressure for the multipoint flux approximation (MPFA) method known as the MPFA L-method. The proof is based on that presented for the mimetic finite difference paper by Brezzi, Lipnikov, and Shashkov in 2005. Particular to this procedure is that no velocity interpolation is needed inside the element. Hence the volume integration of the velocity field is substituted with a quadrature involving only the fluxes. The proof is valid on polyhedra but limits the possibility of the MPFA L-method to choose a stencil to each and every half-face, as is the usual procedure. The assumption needed for convergence is local and poses limits on grid deformation and the anisotropy of the permeability tensor. This assumption is then used to compare the convergence properties of the MPFA L-method with those of the MPFA O-method and the mimetic finite difference method.

Modeling 2D Multispecies Reactive Transport in Saturated/Unsaturated Porous Media with the Eulerian–Lagrangian Localized Adjoint Method

In the present paper, the Eulerian–Lagrangian localized adjoint method (ELLAM) formulation developed by Younes et al. (Advances in Water Resources 29:1056–1074, 2006) is combined with the sequential noniterative approach to accurately simulate 2D multicomponent reactive transport in saturated/unsaturated porous media. The performance and accuracy of the developed model, named ELLAM_REACT, are compared against those of an existing numerical model based on a combination of discontinuous Galerkin and multipoint flux approximation methods (DGMPFA_REACT). Three studied test cases, dealing with reactive transport in saturated and unsaturated porous media and involving chemical reactions with only aqueous species or both fixed and aqueous species, show the superiority of the ELLAM_REACT model compared to the DGMPFA_REACT model.

A family of multi‐point flux approximation schemes for general element types in two and three dimensions with convergence performance

SUMMARYA family of flux‐continuous, locally conservative, control‐volume‐distributed multi‐point flux approximation (CVD‐MPFA) schemes has been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids. These schemes are applicable to the full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is characterized by a quadrature parameterization. Improved numerical convergence for the family of CVD‐MPFA schemes using the quadrature parameterization has been observed for structured and unstructured grids in two dimensions.The CVD‐MPFA family cell‐vertex formulation is extended to classical general element types in 3‐D including prisms, pyramids, hexahedra and tetrahedra. A numerical convergence study of the CVD‐MPFA schemes on general unstructured grids comprising of triangular elements in 2‐D and prismatic, pyramidal, hexahedral and tetrahedral shape elements in 3‐D is presented. Copyright © 2011 John Wiley & Sons, Ltd.

Multiscale Mimetic Solvers for Efficient Streamline Simulation of Fractured Reservoirs

Summary Advances in reservoir characterization and modeling have given the industry improved ability to build detailed geological models of petroleum reservoirs. These models are characterized by complex shapes and structures with discontinuous material properties that span many orders of magnitude. Models that represent fractures explicitly as volumetric objects pose a particular challenge to standard simulation technology with regard to accuracy and computational efficiency. We present a new simulation approach based on streamlines in combination with a new multiscale mimetic pressure solver with improved capabilities for complex fractured reservoirs. The multiscale solver approximates the flux as a linear combination of numerically computed basis functions defined over a coarsened simulation grid consisting of collections of cells from the geological model. Here, we use a mimetic multipoint-flux approximation to compute the multiscale-basis functions. This method has limited sensitivity to grid distortions. The multiscale technology is very robust with respect to fine-scale models containing geological objects such as fractures and fracture corridors. The methodology is very flexible in the choice of the coarse grids introduced to reduce the computational cost of each pressure solve. This can have a large impact on iterative modeling workflows.

Modelling variable density flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods

Modelling variable density flow problems under heterogeneous porous media conditions requires very long computation time and high performance equipments. In this work, the DASPK solver for temporal resolution is combined with advanced spatial discretization schemes in order to improve the computational efficiency while maintaining accuracy. The spatial discretization is based on a combination of Mixed Finite Element (MFE), Discontinuous Galerkin (DG) and Multi-point Flux Approximation methods (MPFA). The obtained non-linear ODE/DAE system is solved with the Method of Lines (MOL) using the DASPK time solver. DASPK uses the preconditioned Krylov iterative method to solve linear systems arising at each time step. Precise laboratory-scale 2D experiments were conducted in a heterogeneously packed porous medium flow tank and the measured concentration contour lines are used to evaluate the numerical model. Simulations show the high efficiency and accuracy of the code and the sensitivity analysis confirms the density dependence of dispersion.

Convergence of the multipoint flux approximation L‐method for homogeneous media on uniform grids

AbstractIn this article, we show the convergence of the multipoint flux approximation (MPFA) L‐method. To do so, we introduce a boundary modification of the original MPFA L‐method. On basis of this local modification, we obtain the equivalence between the MPFA L‐method and a conforming finite element approach with modified right hand side. The influence of the modified right hand side can be easily analyzed within the abstract framework of variational crimes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems

The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart–Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.

Efficient algorithms for multiscale modeling in porous media

AbstractWe describe multiscale mortar mixed finite element discretizations for second‐order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. We discuss the construction of multiscale mortar basis and extend this concept to nonlinear interface operators. We present a multiscale preconditioning strategy to minimize the computational cost associated with construction of the multiscale mortar basis. We also discuss the use of appropriate quadrature rules and approximation spaces to reduce the saddle point system to a cell‐centered pressure scheme. In particular, we focus on multiscale mortar multipoint flux approximation method for general hexahedral grids and full tensor permeabilities. Numerical results are presented to verify the accuracy and efficiency of these approaches. Copyright © 2010 John Wiley & Sons, Ltd.

Monotonicity for MPFA methods on triangular grids

Flow in a porous medium can be described by a set of non-linear partial differential equations. The pressure variable satisfies a maximum principle, which guarantees that the solution will have no oscillations. A discretisation of the pressure equation should preserve this monotonicity property. Whether a numerical method is monotone will depend both on the medium and on the grid. We study monotonicity of Multi-point Flux Approximation methods on triangular grids. We derive necessary conditions for monotonicity on uniform grids. Further, we study the robustness of the methods on rough grids, and quantify the violations of the maximum principle. These investigations are done for single phase flow, however, they are supported by two-phase simulations.

Finite Volume Methods with Multi-Point Flux Approximation with Unstructured Grids for Diffusion Problems

This paper addresses the key issue of calculating fluxes at the control-volume interfaces when finite-volumes are employed for the solution of partial differential equations. This calculation becomes even more significant when unstructured grids are used, since the flux approximation involving only two grid points is no longer correct. Two finite volume methods with the ability in dealing with unstructured grids, the EbFVM-Element-based Finite Volume Method and the MPFA-Multi-Point Flux Approximation are presented, pointing out the way the fluxes are numerically evaluated. The methods are applied to a porous media flow with full permeability tensors and non-orthogonal grids and the results are compared with analytical solutions. The results can be extended to any diffusion operator, like heat and mass diffusion, in anisotropic media.

First-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods

In this paper we show first-order convergence of a multi-point flux approximation control volume method (MPFA) on unstructured triangular grids. In this approach the flux approximation is derived directly in the physical space. In order to do this, we introduce a perturbed mixed finite element method that is equivalent to the MPFA scheme and prove the first-order convergence of this approach. Moreover, we carefully compare the computational performance properties of the MPFA method with those of a lowest order Raviart–Thomas and Brezzi–Douglas–Marini mixed finite element approximation.

Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes

In this paper, we employ the so-called linearity preserving method, which requires that a difference scheme should be exact on linear solutions, to derive a nine-point difference scheme for the numerical solution of diffusion equation on the structured quadrilateral meshes. This scheme uses firstly both cell-centered unknowns and vertex unknowns, and then the vertex unknowns are treated as a linear combination of the surrounding cell-centered unknowns, which reduces the scheme to a cell-centered one. The weights in the linear combination are derived through the linearity preserving approach and can be obtained by solving a local linear system whose solvability is rigorously discussed. Moreover, the relations between our linearity preserving scheme and some existing schemes are also discussed, by which a generalized multipoint flux approximation scheme based on the linearity preserving criterion is suggested. Numerical experiments show that the linearity preserving schemes in this paper have nearly second order accuracy on many highly skewed and highly distorted structured quadrilateral meshes.

A Multipoint Flux Mixed Finite Element Method on Hexahedra

We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell-centered finite differences. The paper is an extension of our earlier paper for quadrilateral and simplicial grids [M. F. Wheeler and I. Yotov, SIAM J. Numer. Anal., 44 (2006), pp. 2082-2106]. The construction is motivated by the multipoint flux approximation method, and it is based on an enhancement of the lowest order Brezzi-Douglas-Duran-Fortin (BDDF) mixed finite element spaces on hexahedra. In particular, there are four fluxes per face, one associated with each vertex. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate first-order convergence for pressures and subface fluxes on sufficiently regular grids, as well as second-order convergence for pressures at the cell centers. Second-order convergence for face fluxes is also observed computationally.