Viscous Two-Phase Flow in Porous Media Driven by Source Terms: Analysis and Numerics

In this work we consider the initial-boundary value problem for a generalized, compressible, porous media two-phase model. The formulation is motivated by principles from mixture theory and involves interstitial fluid velocity 𝐮𝑖 instead of Darcy type velocity 𝐔$_𝑖$ =𝜙⁢𝑠$_𝑖$⁢𝐮$_𝑖$, where 𝜙 is porosity and 𝑠$_𝑖$ is saturation (volume fraction) of phase 𝑖. The momentum balance equations account for fluid-rock resistance forces as well as fluid-fluid interaction effects, in addition to internal viscosity through a Brinkmann type viscous term. We explore a one-dimensional version of this model where we account for two-phase dynamics driven by injection and production of fluids through realistic source terms and where physically relevant capillary pressure is accounted for. Our investigations are twofold: (i) Various a priori estimates are derived that give rise to an existence result subject to a constraint on the magnitude of the viscous terms and the strength of the injection and production rate. (ii) A numerical finite difference scheme, designed for dealing effectively with the strong nonlinear coupling between the mass and momentum equations, is then used to demonstrate a variety of two-phase injection-production scenarios for a realistic reservoir setting. ... mehrDifferent physical effects are highlighted such as the difference between compressible and incompressible flow, balance between gravity and pressure-driven flow, and effect of viscous coupling.