Numerical Simulation of Constrained Flows through Porous Media Employing Glimm’s Scheme

Abstract

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem’s complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm’s semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm’s scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic.

Keywords

• flow through unsaturated porous media
• hyperbolic description
• transition unsaturated–saturated flow
• Glimm’s scheme
• Riemann problem