Stochastic diagrammatic analysis of groundwater flow in heterogeneous porous media

Abstract

The diagrammatic approach is an alternative to standard analytical methods for solving stochastic differential equations governing groundwater flow with spatially variable hydraulic conductivity. This approach uses diagrams instead of abstract symbols to visualize complex multifold integrals that appear in the perturbative expansion of the stochastic flow solution and reduces the original flow problem to a closed set of equations for the mean and the covariance functions. Diagrammatic analysis provides an improved formulation of the flow problem over conventional first-order series approximations, which are based on assumptions such as constant mean hydraulic gradient, infinite flow domain, and neglect of cross correlation terms. This formulation includes simple schemes, like finite-order diagrammatic perturbations that account for mean gradient trends and boundary condition effects, as well as more advanced schemes, like diagrammatic porous media description operators which contain infinite-order correlations. In other words, diagrammatic analysis covers not only the cases where low-order diagrams lead to good approximations of flow, but also those situations where low-order perturbation is insufficient and a more sophisticated analysis is needed. Diagrams lead to a nonlocal equation for the mean hydraulic gradient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity. Three-dimensional flow in an isotropic bounded domain with Dirichlet boundary conditions is considered, and an integral equation for the mean hydraulic head is derived by means of diagrams. This formulation provides an explicit expression for the boundary effects within the three-dimensional flow domain. In addition to these theoretical results, the numerical performance of the diagrammatic approach is tested, and useful insight is obtained by means of one-dimensional flow examples where the exact stochastic solutions are available.