Boundary integral solutions to the unsaturated moisture flow equation

Abstract

The boundary integral equation method is formulated for problems concerning unsaturated moisture flow in porous media. The boundary integral formulation presented here is derived using the time‐dependent fundamental solution of the governing partial differential equation. Two boundary integral approaches are presented. The first approach solves the unsaturated moisture flow problem by directly applying the time‐dependent Green's function. The second method is based on the expansion of moisture content into a perturbation series. The governing equation is decomposed into a moisture flow equation without a known gravity term on the right‐hand side. The perturbation equations are then solved in succession by evaluating the gravity term of the unsaturated leachate flow equation from the preceding solution level. The two boundary integral approaches, the direct Green's function and the perturbation Green's function, are closely related because the same time‐dependent fundamental solution is used to formulate the boundary integral expressions. The direct Green's function formulation solves the boundary integral expression in one step. However, the perturbation Green's function approach requires solution of the integral expression for homogeneous boundary conditions. The solutions obtained by both methods are compared with the exact solution. A close agreement of the internal fluxes between boundary integral methods and the exact solution, for an unsaturated portion of a landfill, shows the stability of the boundary integral methods to compute accurate recharge from the unsaturated zone to the saturated leachate mound.