Equivalent transmissivity of heterogeneous leaky aquifers for steady state radial flow

The purpose of this study is to estimate the equivalent transmissivity for two-dimensional steady state radial flow toward a fully penetrating well in a heterogeneous leaky aquifer. The log transmissivity of the pumped aquifer is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram, while the aquitard conductance is assumed to be spatially uniform. An approximate expression for the upscaled or block transmissivity, defined as the transmissivity of an equivalent homogeneous leaky aquifer system with the same pumping, is proposed. The proposed relation expresses the upscaled transmissivity as a weighted average of the log transforms of the point transmissivity values. Monte Carlo simulations are used to numerically validate the proposed upscaling expression for wide ranges of aquifer parameters. The relation of the weighing function to the statistical parameters of the transmissivity field is discussed. The first and second moments of the upscaled transmissivity are also estimated as a function of the problem parameters. The expected value of the equivalent transmissivity is shown to increase from the geometric mean to the arithmetic mean as the aquitard conductance increases from zero to infinity.