Combining Glimm’s Scheme and Operator Splitting for Simulating Constrained Flows in Porous Media

Abstract

This paper studies constrained Newtonian fluid flows through porous media, accounting for the drag effect on the fluid, modeled using a Mixture Theory perspective and a constitutive relation for the pressure—namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix supersaturation (that is controlled) and the transition from unsaturated to saturated flow (and vice versa). The mathematical model gives rise to a nonlinear, non-homogeneous hyperbolic system. Its numerical simulation combines Glimm’s method with an operator-splitting strategy to account for the Darcy and Forchheimer terms that cause the system’s non-homogeneity. Despite the Glimm method’s proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operator-splitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm’s method with operator-splitting to account for linear and nonlinear drag effects. Glimm’s scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure—an increasing function of the saturation, with the first derivative also increasing, convex, and positive, enables us to obtain explicit expressions for the Riemann invariants. The results show the influence of the Darcy and Forchheimer drag terms on the flow.

Keywords

• flow through unsaturated porous media
• hyperbolic description
• Glimm’s scheme
• operator splitting
• Riemann problem