STOCHASTIC DIAGRAMMATIC ANALYSIS OF GROUNDWATER-FLOW IN HETEROGENEOUSPOROUS-MEDIA

The diagrammatic approach is an alternative to standard analytical met hods for solving stochastic differential equations governing groundwat er flow with spatially variable hydraulic conductivity. This approach uses diagrams instead of abstract symbols to visualize complex multifo ld integrals that appear in the perturbative expansion of the stochast ic 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 con ventional first-order series approximations, which are based on assump tions such as constant mean hydraulic gradient, infinite flow domain, and neglect of cross correlation terms. This formulation includes simp le schemes, like finite-order diagrammatic perturbations that account for mean gradient trends and boundary condition effects, as well as mo re advanced schemes, like diagrammatic porous media description operat ors which contain infinite-order correlations. In other words, diagram matic analysis covers not only the cases where low-order diagrams lead to good approximations of flow, but also those situations where low-o rder perturbation is insufficient and a more sophisticated analysis is needed. Diagrams lead to a nonlocal equation for the mean hydraulic g radient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity. Three-dimensional fl ow in an isotropic bounded domain with Dirichlet boundary conditions i s considered, and an integral equation for the mean hydraulic head is derived by means of diagrams. This formulation provides an explicit ex pression for the boundary effects within the three-dimensional flow do main. In addition to these theoretical results, the numerical performa nce of the diagrammatic approach is tested, and useful insight; is obt ained by means of one-dimensional flow examples where the exact stocha stic Solutions are available.