DeepAI AI Chat
Log In Sign Up

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

by   N. K. Mapakshi, et al.
University of Houston

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.


page 1

page 19

page 20

page 22

page 23

page 28

page 29

page 33


Numerical approximation of singular-degenerate parabolic stochastic PDEs

We study a general class of singular degenerate parabolic stochastic par...

Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow

We propose two different discrete formulations for the weak imposition o...

A Variational Finite Element Discretization of Compressible Flow

We present a finite element variational integrator for compressible flow...

Benchmark computations of dynamic poroelasticity

We present benchmark computations of dynamic poroelasticity modeling flu...

Adaptive and Pressure-Robust Discretization of Incompressible Pressure-Driven Phase-Field Fracture

In this work, we consider pressurized phase-field fracture problems in n...

On Formulations for Modeling Pressurized Cracks Within Phase-Field Methods for Fracture

Over the past few decades, the phase-field method for fracture has seen ...

Free surface flow through rigid porous media – An overview and comparison of formulations

In many applications free surface flow through rigid porous media has to...