If you mix a fluid such as water with a sediment such as sand, you have a two-phase granular medium. The way the sand piles is influenced by the water, with the sand buoyed and pushed by the movement of the water. Meanwhile, the water can only be where the sand is not. To study this interaction, we combine conservation of mass and Darcy's law for flow through a medium of this type to get a model for the availability of space for water in the mixture, which is called the porosity. This thesis focuses on the visco-elastic case, where the interaction between the grains gives the solid structure a tendency to return to its original shape when any added external pressure is removed. The resulting equation is doubly nonlinear with the nonlinearity influencing the change over both time and space. To cope with the difficulty of the multiple nonlinearities, we separate the problem into several interconnected parts. We use methods of functional analysis, monotonicity and compactness to study these parts separately. We then recombine these parts to arrive at existence results and other qualitative behavior of solutions.