Stochastic analysis of nonstationary subsurface solute transport: 1. Unconditional moments

Abstract

This paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media. Partial differential equations for three unconditional ensemble moments (the concentration mean, concentration covariance, and velocity concentration cross covariance) are derived by applying perturbation techniques to the governing transport equation for a conservative solute. Concentration uncertainty is assumed to be the result of unmodeled small‐scale fluctuations in a steady state velocity field. The moment expressions, which describe how each moment evolves over time and space, resemble the classic deterministic advection‐dispersion equation and can be solved using similar methods. A solution procedure based on a Galerkin finite element algorithm is illustrated with a hypothetical two‐dimensional example. For this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation. The perturbation solution is shown to reproduce the statistics obtained from a Monte Carlo simulation quite well for a natural log conductivity standard deviation of 0.5 and moderately well for a natural log conductivity standard deviation of 1.0. The computational effort required for a perturbation solution is significantly less than that required for a Monte Carlo solution of acceptable accuracy. Sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases.