Uncertainty quantification of reactive transport in heterogeneous porous media using a Karhunen-Loeve-based moment equation approach

A new methodology was developed for solving advection-dispersion-reaction (ADR) problems in randomly heterogeneous porous media by combining a Karhunen-Loeve Moment Equation (KLME) technique with an operator splitting algorithm. As an efficient intrusive uncertainty quantification technique, the KLME approach incorporates a Karhunen-Loeve decomposition of the random permeability field and a perturbative and polynomial expansion of the dependent variables (e.g. fluid pressure, contaminant concentration). The resulting model equations include a series of deterministic ADR equations, each of which is similar in form to the original ones. These equations can be solved using an efficient operator splitting method in which advection and dispersion components are computed numerically and the local reaction contributions are evaluated analytically. The mean values and higher spatial moments of the dependent variables (variance, covariance, and cross-variance) can then be calculated readily from the solutions obtained by this scheme, where the higher order moments serve as quantitative measures of uncertainty in the mean solution. As an example, a suite of two-dimensional numerical experiments in a hypothetical heterogeneous system was considered. The model equations included basic first-order reaction, sequential reaction, and parallel reaction terms, and were solved using both the proposed uncertainty quantification method and a conventional Monte Carlo approach as a means to demonstrate the efficiency and accuracy of newer method over the conventional technique.