Modeling the saturation process of flows through rigid porous media by the solution of a nonlinear hyperbolic system with one constrained unknown

This work proposes a mathematical model to study the filling up of an unsaturated rigid porous medium by a liquid identifying the transition from unsaturated to saturated flow. The mechanical model employs a mixture theory approach – in which the mixture consists of three overlapping continuous constituents, representing the porous matrix (solid constituent), the incompressible fluid (liquid constituent) and an inert gas constituent included to account for its compressibility. The mathematical description gives rise to a nonlinear hyperbolic system in which the fluid fraction must satisfy an inequality – an upper bound – in order to be physically realistic. The model introduced in this work accounts for the physical upper bound of the fluid fraction (and the saturation) that depends on the volume of the pores. The complete solution of a Riemann problem associated to the system of conservation laws satisfying the constraint given by the saturation upper bound is presented.