Upscaling transmissivity under radially convergent flow in heterogeneous media

Most field methods used to estimate transmissivity values rely on the analy sis of drawdown under convergent flow conditions. For a single well in a ho mogeneous and isotropic aquifer and under steady state flow conditions, dra wdown s is directly related to the pumping rate Q through transmissivity T. In real, nonhomogeneous aquifers, s and Q are still directly related, now through a value called equivalent transmissivity T In this context, T-eq is defined as the value thar best fits Thiem's equation and would, for exampl e, be the transmissivity assigned to the well location in the classical int erpretation of a steady state pumping test. This equivalent or upscaled tra nsmissivity is clearly not a local value but is some representative value o f a certain area surrounding the well. In this paper we present an analytic al solution for upscaling transmissivities under radially convergent steady state flow conditions produced by constant pumping from a well of radius r (W) in a heterogeneous aquifer based upon an extension of Thiem's equation. Using a perturbation expansion, we derive a second-order expression for T- eq given as a weighted average of the fluctuations in log T throughout the domain. This expression is compared to other averaging formulae from the li terature, and differences are pointed out. T depends upon an infinite serie s which may be expressed in terms of coefficients of the finite Fourier tra nsform of the log transmissivity function. Sufficient conditions for conver gence of this series are examined. Finally, we show that our solution agree s with existing analytical ones to second order and test the solution with a numerical example.