I. Aavatsmark

An Introduction to Multipoint Flux Approximations for Quadrilateral Grids

Control-volume discretizations using multipoint flux approximations (MPFA) were developed in the last decade. This paper gives an introduction to these methods for quadrilateral grids in two and three dimensions. The introduction is kept on a basic level, and a brief summary to more advanced results is given. Only the O-method with surface midpoints as continuity points is discussed. Flux expressions are derived both in physical and in curvilinear space. Equations for calculation of the transmissibility coefficients are given, and an explicit solution is shown for constant coefficients. K-orthogonality, stability and monotonicity are discussed, and an iterative solution technique is presented. Two numerical examples close the paper.

Control-Volume Discretization Method for Quadrilateral Grids with Faults and Local Refinements

This paper extends the multipoint flux-approximation (MPFA) control-volume method to quadrilateral grids for which the adjacent cells do not necessarily share corners. Examples are grids with faults and locally refined grids. This paper gives a derivation of the method for such grids. The difference between two-point flux-approximation (TPFA) results and MPFA results for faults and local grid refinements is demonstrated for synthetic problems. Further, the results are compared with results from uniform fine-grid simulations. The effect of repeated fault patterns as well as anisotropy is investigated. Large errors may be found for the TPFA method for flow through a series of faults in an anisotropic medium. Finally, a comparison is done for a reservoir field application.

Symmetry and M-Matrix Issues for the O-Method on an Unstructured Grid

More sophisticated discretization methods than the traditional control-volume finite-difference methods, have been proposed by Aavatsmark et al. in recent papers for solving the mass balance equations for porous media flow. These methods are based on a local representation of fluxes across cell-edges of control volumes (CVs). This paper will focus on mathematical properties of the discrete operator that arises when an elliptic term of the form −∇⋅(K∇p) is discretized based on these discretization principles.

MPFA methods applied to irregular grids and faults

Recent contributions to discretising the mass balance equations for the fluid phases in porous media flow, have been presented by Aavatsmark et. al. The methods are termed multipoint flux approximation methods, and are derived from a control volume formulation of the governing equations. The methods are locally mass conservative, and allow for flexible geometry, heterogeneity and/or anisotropy. We will here emphasize the discretisation principles on irregular grids and on grids for which adjacent control volumes do not share corners. Such cases may occur for locally refined grids and faulted media.

Discretization on general grids for general media

Discretization methods are presented for control-volume formulations on quadrilateral, triangular and polygonal grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy’s law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy.