Analysis of a Darcy Flow Model with a Dynamic Pressure Saturation Relation

We consider a new model for groundwater ow. The model diiers from previous models in that the saturation-pressure relation is extended with a dynamic term, namely the time derivative of the saturation. The resulting model equation is of nonlinear degenerate pseudo-parabolic type. We give a rigorous analysis of travelling wave solutions and the local behaviour near fronts. Part of this analysis is then used in a formal asymptotic analysis of solutions with initial support in a half line. As an illustrative special case we also consider Heaviside initial data. Abbreviated Title. Darcy ow with dynamic pressure. We thank C.J. van Duijn for suggesting the mathematical model and encouraging our research in this direction. We gratefully acknowledge the support of the Dutch Organisation for Scientiic Research (NWO) and the British Council. We also thank S.M. Hassanizadeh and A. Leijnse of the National Institute of Public Health and Enviromental Protection (RIVM) in Bilthoven, The Netherlands for stimulating discussions.