STOCHASTIC PERTURBATION ANALYSIS OF GROUNDWATER-FLOW - SPATIALLY-VARIABLE SOILS, SEMIINFINITE DOMAINS AND LARGE FLUCTUATIONS

As is well known, a complete stochastic solution of the stochastic dif ferential equation governing saturated groundwater flow leads to an in finite hierarchy of equations in terms of higher-order moments. Pertur bation techniques are commonly used to close this hierarchy, using pow er-series expansions. These methods are applied by truncating the seri es after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result i s a significant drawback with the application of power series-based pe rturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and c an lead to physically unreasonable and inaccurate solutions of the sto chastic flow equation. To support this argument, one-dimensional, stea dy-state, saturated groundwater flow is examined, for the case of a sp atially distributed hydraulic conductivity field. An ordinary power-se ries perturbation method is used to approximate the mean head, using s econd-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improve d results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such appr oximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the origina l stochastic flow problem to a closed set of equations for the mean an d the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bo unded domains and large fluctuations.