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

Abstract

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.