Modeling Water Flow in Variably Saturated Porous Soils and Alluvial Sediments

The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial differential equation, which translates the mass conservation principle into mathematical terms. Such an equation assumes that the variation of the volumetric water content ( θ ) in the medium is balanced by the net flux of water flow, i.e., the divergence of specific discharge, if source/sink terms are negligible. Specific discharge is in turn related to the matric potential ( h ), through the non-linear Darcy–Buckingham law. The resulting equation can be rewritten in different ways, in order to express it as a partial differential equation where a single physical quantity is considered to be a dependent variable. Namely, the most common instances are the Fokker–Planck Equation (for θ ), and the Richards Equation (for h ). The other two forms can be given for generalized matric flux potential ( Φ ) and for hydraulic conductivity ( K ). The latter two cases are shown to limit the non-linearity to multiplicative terms for an exponential K -to- h relationship. Different types of boundary conditions are examined for the four different formalisms. Moreover, remarks given on the physico-mathematical properties of the relationships between K , h , and θ could be useful for further theoretical and practical studies.