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Richards equation is often used to represent two-phase fluid flow in anunsaturated porous medium when one phase is much heavier and more viscous thanthe other. However, it cannot describe the fully saturated flow due todegeneracy in the capillary pressure term. Mathematically, gravity-drivenvariably saturated flows are interesting because their governing partialdifferential equation switches from hyperbolic in the unsaturated region toelliptic in the saturated region. Moreover, the presence of wetting frontsintroduces strong spatial gradients often leading to numerical instability. Inthis work, we develop a robust, multidimensional mathematical and computationalmodel for such variably saturated flow in the limit of negligible capillaryforces. The elliptic problem for saturated regions is built-in efficiently intoour framework for a reduced system corresponding to the saturated cells, withthe boundary condition of the fixed head at the unsaturated cells. In summary,this coupled hyperbolic-elliptic PDE framework provides an efficient,physics-based extension of the hyperbolic Richards equation to simulate fullysaturated regions. Finally, we provide a suite of easy-to-implement yetchallenging benchmark test problems involving saturated flows in one and twodimensions. These simple problems, accompanied by their correspondinganalytical solutions, can prove to be pivotal for the code verification, modelvalidation (V&V) and performance comparison of such simulators. Our numericalsolutions show an excellent comparison with the analytical results for theproposed problems. The last test problem on two-dimensional infiltration in astratified, heterogeneous soil shows the formation and evolution of multipledisconnected saturated regions.

A hyperbolic-elliptic PDE model and conservative numerical method for gravity-dominated variably-saturated groundwater flow