Green element simulations of the transient nonlinear unsaturated flow equation

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This paper presents numerical solutions to transient one-dimensional (1-D) and 2-D nonlinear flows in unsaturated media which are based on the Green element method (GEM). GEM is a novel numerical scheme based essentially on the singular integral theory of the boundary element method (BEM) but which implements the theory in an element-by-element fashion so that when the discrete equations from all the elements are assembled the resultant global coefficient matrix is banded and easier to invert, in contrast to the fully populated global coefficient matrix commonly associated with conventional BEM. Because GEM implements the singular integral theory of BEM within each element, the elemental integrals of the theory are evaluated analytically, thereby enhancing the accuracy of the method, and medium properties that vary spatially and temporally as a result of their dependence on the flow are readily incorporated into the theory. The transient nonlinear flow problem in unsaturated porous medium with soil constitutive relations that are of practical interest is highly nonlinear and had not been amenable to the boundary element theory, but here, for the first time, such problems are successfully solved by that theory using the Green element approach which incorporates a Picard nonlinear solution algorithm. Two Green element models for transient 1-D and 2-D unsaturated flows are developed and successfully tested on four examples of infiltration flow problems whose soil constitutive relations cover most of those reported in the literature.