Seepage analysis in multi-domain general anisotropic media by three-dimensional boundary elements

A three-dimensional boundary element solution for the seepage analysis in multi-domain general anisotropic media has been developed based on the transformation approach. Using analytical eigenvalues and eigenvectors of the hydraulic conductivity tensor, a closed-form coordinate transformation matrix has been provided to transform the quadratic form of governing equation of seepage for the general anisotropic media to the Laplace equation. This transformation allows the analysis to be carried out using any standard BEM codes for the potential theory on the transformed space by adding small pre- and post-processing routines. With this transformation, any physical quantity like the total head remains unchanged at corresponding nodes on the physical and transformed space, and the normal gradient across the domain boundaries should also be transformed. In multi-domain problems, compatibility equations (equality of the potential on corresponding nodes on the interface) and equilibrium equations (conservation of the flux across the interface boundaries of adjacent domains) on the corresponding nodes of interface between two neighbor domains are needed for boundary element method. In the transformed space, the compatibility equation remains unchanged. However, due to the distortion of boundaries in the mapped space and therefore misalignment of the unit outward normal vectors along the inter-domain boundaries, the equilibrium of the normal fluxes have to be transformed accordingly. Based on the proposed transformation, the normal to boundary flux boundary conditions in the mapped space and the transformed equilibrium equation for interface of adjacent zones have been given in this paper. Examples have been solved with the proposed scheme and the results were verified with the finite element method. Excellent agreement of the results shows the veracity of the proposed transformation and the formulas given for transformation of equilibrium equation for multi-domain general anisotropic media.