Variational formulations for a parabolic partial differential equation describing variably saturated flow

The variably saturated flow in a porous medium is described by a parabolic equation with hydraulic conductivity and soil moisture capacity as the flow characteristics. Variational formulations have been developed to solve the governing parabolic partial differential equation describing variably saturated flow in a porous medium. The finite element method has been used to solve the partial differential equation for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop the numerical algorithms for solving the partial differential equation to generate time-varying hydraulic heads in the subsurface. A computer program has been written to solve a symmetric positive definite system obtained from the variational formulation for the finite element method. The system of equations is solved using the conjugate gradient method. Analytical solutions have been developed for the simplified governing equation. The finite element solutions were compared with the analytical solutions. A reasonable agreement between finite element and analytical solutions was observed. A variational formulation was also developed using boundary element method. The matrix characteristics and the solution algorithm using boundary element method were investigated.