“Cleopatra’s nose: if it had been shorter, the face of the whole world would have been changed.” Mentioning B. Pascal’s famous passage here aims at emphasizing the power of an image and the important consequences that it may have in real world situations. Mutatis mutandis, the idea behind the use of diagrammatic representations in groundwater flow modelling is based on the concept that the qualitative grasp of form, shape and geometric order may go deeper than the quantitative grasp of abstract mathematical symbol and number. In this spirit, the proposed approach uses topological diagrams which reduce the original stochastic groundwater flow problem to a closed set of equations for the statistical moments. Random integral forms of flow are considered in the light of porous media description operators and graphic Green’s functions. Graphic visualizations of the underlying flow processes allow previously undetected features to be seen and can yield more general and accurate results than traditional methods. Depending upon the choice of a porous medium description operator, the diagrammatic approach can handle both cases of small and large fluctuations and can work at long as well as short range correlation scales.

This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.

As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series 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 is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.

The stochastic modeling of groundwater flow is considered. It is pointed out that in many circumstances analysis in terms of the ordinary perturbation series method may be incapable of representing fundamental characteristics of flow, and may lead to physically unreasonable solutions of the stochastic flow equation. To support this argument, the case of 1-D steady-state flow is examined using ordinary perturbation methods. Then, a more advanced interactive perturbation approach is introduced. This approach goes beyond standard perturbation approximation and can be used in situations where the interactions between flow terms is so significant that the ordinary low-order perturbation approximation will not work. The stochastic flow problem is then analyzed using concepts and techniques from stochastic turbulence and quantum field theory. These well-established techniques yield results similar to those of the interactive perturbation approach, a fact that proves the power of the latter