Flux-continuous Schemes Research Articles

Numerical convergence of the family of flux-continuous schemes with variable quadrature (qu1,qu2) for single phase flow in porous media

Finite-volume schemes, which honor pressure and flux-continuity conditions, is developed using double quadrature (qu1,qu2), referred as double family scheme. The scheme is applicable to solve the elliptic pressure equation used in reservoir simulation. Schemes are applicable on both regular cartesian and unstructured triangular meshes. The scheme is defined over a control-volume distributed formulation. The scheme can be applied to both diagonal and full permeability tensor elliptic pressure equation with discontinuous coefficients. The scheme removes the first order errors, which are introduced by standard reservoir simulation schemes when applied to full tensor flow. The scheme is quantified with help of a quadrature rule. When the scheme is applied to highly heterogeneous and anisotropic porous media it does not honor maximum principle resulting in unstable solution with oscillatory behavior. The numerical solution is termed non-monotonicity for high anisotropy ratios with results showing oscillations in the numerical pressure solution. In this paper a double (qu1,qu2) quadrature flux continuous schemes is presented, which with specific choice of quadrature (qu1,qu2) helps in improved stability of the numerical solutions. Numerical convergence of the scheme is also demonstrated with help of a number of numerical test cases and schemes impact on monotonicity behavior is also demonstrated with numerical examples.

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.

Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids

This paper focuses on flux-continuous pressure equation approximation for strongly anisotropic media. Previous work on families of flux-continuous schemes for solving the general geometry–permeability tensor pressure equation has focused on single-parameter families. These schemes have been shown to remove the O(1) errors introduced by standard two-point flux reservoir simulation schemes when applied to full-tensor flow approximation. Improved convergence of the schemes has also been established for specific quadrature points. However these schemes have conditional M-matrices depending on the strength of the off-diagonal tensor coefficients. When applied to cases involving full-tensors arising from strongly anisotropic media, the point-wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution. New double-family flux-continuous locally conservative schemes are presented for the general geometry–permeability tensor pressure equation. The new double-family formulation is shown to expand on the current single-parameter range of existing conditional M-matrix schemes. The conditional M-matrix bounds on a double-family formulation are identified for both quadrilateral and triangle cell grids. A quasi-positive QM-matrix analysis is presented that classifies the behaviour of the new schemes with respect to double-family quadrature in regions beyond the M-matrix bounds. The extension to double-family quadrature is shown to be beneficial, resulting in novel optimal anisotropic quadrature schemes. The new methods are applied to strongly anisotropic full-tensor field problems and yield results with sharp resolution, with only minor or practically zero spurious oscillations.

A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support

A new family of flux-continuous, locally conservative, finite-volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media. The new schemes have full pressure continuity imposed across control-volume faces. Previous families of flux-continuous schemes are point-wise continuous in pressure and flux. When applying the earlier point-wise flux-continuous schemes to strongly anisotropic full-tensor fields their failure to satisfy a maximum principle (as with other FEM and finite-volume methods) can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution. An M-matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9-point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field. The full pressure continuity schemes are shown to possess a larger range of flux-continuous schemes, than the previous point-wise counter parts. For strongly anisotropic full-tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points (previously out of range) that are shown to minimize spurious oscillations in discrete pressure solutions. The new formulation leads to a more robust quasi-positive family of flux-continuous schemes applicable to general discontinuous full-tensor fields.

Unstructured, Control-Volume Distributed, Full-Tensor Finite Volume Schemes With Flow Based Grids

Locally conservative flux-continuous, full-tensor, discretization schemes are presented for general unstructured grids. The schemes are control-volume distributed, where flow variables and rock properties are assigned to the polygonal control-volumes derived from the primal grid. A relationship between these finite volume schemes and the mixed finite element method is established. An extension for unstructured grids is described that leads to a general symmetric positive definite discretization matrix for both quadrilateral and triangular grids. A novel flow based gridding approach for unstructured mesh generation is also proposed for heterogeneous reservoir domains. Results computed with the flux continuous schemes on unstructured flow-based grids demonstrate the advantages of the methods.