Fixed domain methods for free and moving boundary flows in porous media

A survey is presented concerning fixed domain methods used to solve mathematical models of free and moving boundary flow problems in porous media. These include the following: variational inequality or quasi-variational inequality formulations; general inequality formulations which have been set and solved in fixed domains; and the residual flow procedure. Finally, some parallel computing methods and mesh adaptation methods are discussed to demonstrate how these fixed domain formulations can be solved with current technology.The fixed domain methods that are referenced herein can be classified into two groups: the variational inequality method and the extended pressure head method. Baiocchi was the first to apply the variational inequality method to free boundary problems of flows through porous media. This method in general also uses an extension of the pressure head but adds an application of an integral transformation (a Baiocchi transformation) to the problem. The method possesses a beautiful mathematical structure for its theory and yields simple numerical solution algorithms. However, application of the method is difficult if not impossible in some cases depending upon the regularity of the seepage domain.The extended pressure head method is based on the concept that the pressure is extended smoothly across the free or moving boundary into the unsaturated region from the flow domain. The extension of the pressure head to the entire porous medium yields an extended coefficient of permeability of the medium which is equal to the saturated coefficient in the seepage region and is equal to zero or some small value (for computational purposes) in the unsaturated region.