Computing the Electrokinetic Response with Simple Models via Eigenvalue Decomposition

The efficient solution of coupled hydrogeophysical problems both numerically and analytically is important to their use in parameter estimation. We present a general approach for decoupling the governing equations for groundwater flow and the associated electrokinetic problem. The approach can use either a symbolic or numerical eigenvector decomposition of the matrix that arises when writing the two equations in vector form. The two coupled problems, once uncoupled, can then be solved using any existing approaches for the simple non-coupled component problems. Solutions can be either analytic or numerical in nature with the effective parameters being computed in the decomposition. The final solution, in terms of the physical potentials of interest, is computed through a simple matrix multiplication. We solve the fully coupled electrokinetic problem (water flow driving electrical flow and electrical flow driving water flow) for a single layer using the Theis solution, and for multilayer problems using MODFLOW. The approach is quite general, with the main limitation being a required symmetry between the coupled processes in their differential equation (e.g., both processes must be governed by the diffusion equation). The solution obtained with this approach is shown to agree with that obtained by Malama et al. (2009). Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000