Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

Abstract

SUMMARYHigher-resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full-tensor Darcy-flux approximations. A sequence of non-uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher-order schemes are constructed using non-uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full-tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher-order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts.

A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

“…Frequently, this condition follows from (20). However, if the edge mid-point x e lies outside the convex hull of points x k 2R T the reconstructed function may be negative at this point.…”

“…For instance, our attempts to use only edge mid-points x e in (20) or drop out (22) resulted in numerical instabilities in the nonlinear iterative method which is introduced in Section 4. Due to (20), we get thatg T 0 in local minima and maxima. Fig.…”

A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids

“…2(b), the local flux coefficients result from the constraint Eqs. (21), (22), (23) and (25) and are functions of cluster subcell geometry, permeability tensor coefficients and quadrature point. However, while the discrete matrix is not generally symmetric positive definite, conditions for a positive definite discrete matrix are deduced from the local cluster contribution to inner product of potential with the fully discrete operator.…”

Higher Order Multidimensional Upwind Convection Schemes for Flow in Porous Media on Structured and Unstructured Quadrilateral Grids

“…A higher order unstructured grid approximation is now presented with respect to the saturation variables, following [14,23]. (For the remainder of this section, superfix n is omitted, and it is understood that all saturations are computed at level n.) The scheme is expressed as a two-step process.…”