A Scalable Variational Inequality-Based Formulation That Preserves Maximum Principles for Darcy Flow with Pressure-Dependent Viscosity
The overarching goal of this thesis is to present a robust and scalable finite element computational framework based on Variational inequalities (VI) which models nonlinear flow through heterogeneous and anisotropic porous media without violating discrete maximum principles (DMP) for pressure. VI is an optimization technique that places bounds on the numerical solutions of partial differential equations, and previous studies have shown that the VI approach can enforce DMP for linear and semi linear subsurface flow and transport problems. Herein, the same VI framework is extended to the nonlinear Modified Darcy flow (MDF) model which incorporates pressure dependent viscosity. Although it can be proven that the MDF model satisfies maximum principles, most finite element formulations, including the classical Galerkin formulation with Raviart-Thomas elements and the variational multi-scale formulation, will not adequately enforce the DMP if strong levels of anisotropy are present. Several representative reservoir problems with realistic parameters are presented, and both the algorithmic and parallel scalability of the proposed computational framework are studied.