Radial flow in a bounded randomly heterogeneous aquifer with recharge

We present analytical expressions for leading statistical moments of vertically averaged hydraulic head under steady state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer, in the presence of a random recharge. The natural logarithm Y = ln T of aquifer transmissivity T is modelled as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. 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. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness.