Average Steady Flows Driven by Sources of Random Strength in Heterogeneous Aquifers with Application to Partially-Penetrating Wells

Average steady source flow in heterogeneous porous formations is modelled by regarding the hydraulic conductivity K as a stationary random space function (RSF). As a consequence, the flow variables become RSFs as well, and we are interested into calculating their moments. This problem has been intensively studied in the case of a Neumann type boundary condition at the source. However, there are many applications (such as well-type flows) for which the required boundary condition is that of Dirichlet. In order to fulfill such a requirement the strength of the source must be proportional to K(x), and therefore the source itself results a RSF. To solve flows driven by sources whose strength is spatially variable, we have used a perturbation procedure similar to that developed by Indelman and Abramovich (1994) to analyze flows generated by sources of deterministic strength. Due to the linearity of the mathematical problem, we have focused on the explicit derivation of the mean head distribution G generated by a unit pulse. Such a distribution represents the fundamental solution to the average flow equations, and it is termed as mean Green function. The general results are subsequently used to investigate flow toward a partially-penetrating well in a semi-infinite domain. Indeed, we construct a σ²-order approximation to the mean as well as variance of the head by replacing the well with a singular segment. It is shown how the well-length combined with the medium heterogeneity affects the head distribution. We have introduced the concept of equivalent conductivity.