A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity

TL;DR

This paper introduces a scalable variational inequality-based numerical method for modeling flow through porous media with pressure-dependent viscosity, ensuring discrete maximum principles are satisfied even in complex, anisotropic conditions.

Contribution

The paper develops a novel VI-based formulation that enforces DMP in nonlinear porous media flow models, compatible with various discretizations, and demonstrates its scalability and efficiency.

Findings

VI formulation enforces DMP in pressure-dependent viscosity models

Method is scalable and compatible with multiple discretizations

Parallel and static-scaling studies confirm efficiency

Abstract

Mathematical models for flow through porous media typically enjoy the so-called maximum principles, which place bounds on the pressure field. It is highly desirable to preserve these bounds on the pressure field in predictive numerical simulations, that is, one needs to satisfy discrete maximum principles (DMP). Unfortunately, many of the existing formulations for flow through porous media models do not satisfy DMP. This paper presents a robust, scalable numerical formulation based on variational inequalities (VI), to model non-linear flows through heterogeneous, anisotropic porous media without violating DMP. VI is an optimization technique that places bounds on the numerical solutions of partial differential equations. To crystallize the ideas, a modification to Darcy equations by taking into account pressure-dependent viscosity will be discretized using the lowest-order Raviart-Thomas (RT0) and Variational Multi-scale (VMS) finite element formulations. It will be shown that these formulations violate DMP, and, in fact, these violations increase with an increase in anisotropy. It will be shown that the proposed VI-based formulation provides a viable route to enforce DMP. Moreover, it will be shown that the proposed formulation is scalable, and can work with any numerical discretization and weak form. Parallel scalability on modern computational platforms will be illustrated through strong-scaling studies, which will prove the efficiency of the proposed formulation in a parallel setting. Algorithmic scalability as the problem size is scaled up will be demonstrated through novel static-scaling studies. The performed static-scaling studies can serve as a guide for users to be able to select an appropriate discretization for a given problem size.