Solution of Liquid–Gas–Solid Coupled Equations for Porous Media Considering Dynamics and Hysteretic Retention Behavior

Abstract

This paper presents a solution procedure based on the finite element method for the three-constituent coupled problem representing the behavior of unsaturated porous media. A formulation is derived from the theory of porous media considering the dynamic behavior, the balance of momenta of porous liquid, porous gas, and solid porous matrix. The hysteresis of the liquid retention behavior is also modeled. A retention model able to smoothly represent the transition from maximum saturation to partially saturated states, and vice versa, is considered. With these features, the resulting system of equations is fairly complex; however an efficient, stable, and accurate numerical method is developed for its solution. The derivation of consistent tangent operators of first and second order to obtain quadratic convergence is presented in addition to all details to properly implement the numerical solver, e.g., consistent treatment of initial conditions. Finally, a number of numerical experiments are studied in order to illustrate the model capabilities in particular with regard to the influence of gas pressure on the unsaturated porous media behavior.