Statistics of hydraulic head in randomly heterogeneous well fields with truncated multiscale variogram

We consider two-dimensional steady state flow toward a well that fully penetrates a randomly heterogeneous aquifer. A constant pumping rate is prescribed deterministically at the well while a constant head is maintained at a circular outer boundary of radius L. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous, and isotropic random fields (modes) of natural log transmissivity, Y(r), each of which is associated with a Gaussian variogram. Here we consider only a lower cut-off of the hierarchy. We develop an analytical solution for the mean and variance of hydraulic head based on the nonlocal theory first proposed for steady state flows in bounded, randomly heterogeneous media by Neuman and Orr [1993] and Guadagnini and Neuman [1999a]. In particular, we develop and solve analytically recursive closure approximations of the governing nonlocal moment equations to second order in Y. Analytical solutions are evaluated by Gaussian quadratures. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. Potential uses include the analysis of pumping tests and tracer test, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.