Boundary Integral Equations for Quasi-Static Unsaturated Porous Media

One of the principal criteria for development of the boundary element method (BEM) in porous media is derivation of the required fundamental solutions in the boundary integral equations (BIE). Furthermore, setting up the governing BIEs based on the governing partial differential equations (PDE) is another challenge in solving a physical phenomenon using BEM. In this regard, the governing BIEs for unsaturated porous media have been developed using the available derived fundamental solutions. In this research, a perturbation type approximation is exploited for developing a system of BIEs for the quasi-static unsaturated porous media with moderate variations in its properties. Nevertheless, the fundamental solutions of the medium with constant properties are applied. The method produces two sets of equations with constant parameters instead of the original equations. Besides, the required boundary conditions have been formulated. This type of BIEs is essential to be used in the BEM for unsaturated porous media as the fundamental solutions for a medium with coordinates dependent properties is not available so far. The resulted introduced BIEs may be used directly in a BEM numerical model for an unsaturated porous media in one, two or three dimensional conditions.