Cell-Centered Nonlinear Finite-Volume Methods With Improved Robustness

Abstract

Summary We present a nonlinear finite-volume method (NFVM) that is either positivity-preserving or extremum-preserving with improved robustness. The key ingredient of the method is the construction of one-sided fluxes, which involves decomposition of conormal vectors by introducing harmonic-averaging points as auxiliary points. The original NFVM using harmonic-averaging points is not robust in the sense that decomposition of conormal vectors with nonnegative coefficients can easily run into difficulties for heterogeneous and anisotropic permeability tensors on general nonorthogonal meshes. To improve NFVM robustness, we first present an alternative derivation of harmonic-averaging points and give a different formula that shows more clearly a point's location. On the basis of the derivation of the new formula, a correction algorithm is proposed to make modifications to those problematic harmonic-averaging points so that all the conormal vectors can be decomposed with nonnegative coefficients successfully. As a result, the resulting NFVM can be applied to more-challenging problems when conormal decomposition with nonnegative coefficients is not possible without correction. The correction algorithm is a compromise between robustness and accuracy. While it improves the robustness of the resulting NFVM, results of numerical convergence tests show that the effect of our correction algorithm on accuracy is problem-dependent. Optimal order of convergence is still maintained for some problems, and the convergence rate is reduced for others. Monotonicity and extremum-preserving properties are verified by numerical experiments. Finally, a field test case is used to demonstrate that the NFVM combined with our correction algorithm can be applied to simulate real-life reservoirs of industry-standard complexity.