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 (theta) 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 theta), and the Richards Equation (for h). The other two forms can be given for generalized matric flux potential (phi) 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 theta could be useful for further theoretical and practical studies.

Automated summary

The sustainable exploitation of groundwater resources is a complex problem that involves water flow in soils and sediments. Modeling water flow using partial differential equations is an effective approach, which assumes that the change in water content is balanced by the net water flux and that the flux is related to the matric potential. Different equations can be used to express the relationship between these physical quantities, and the choice of boundary conditions is also important.