Upscaling transmissivity under radially convergent flow in heterogeneous media

Abstract

Most field methods used to estimate transmissivity values rely on the analysis of drawdown under convergent flow conditions. For a single well in a homogeneous and isotropic aquifer and under steady state flow conditions, drawdown $s$ 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 $T_{eq}$. In this context, $T_{eq}$ is defined as the value that best fits Thiem's equation and would, for example, be the transmissivity assigned to the well location in the classical interpretation of a steady state pumping test. This equivalent or upscaled transmissivity is clearly not a local value but is some representative value of a certain area surrounding the well. In this paper we present an analytical 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 literature, and differences are pointed out. $T_{eq}$ depends upon an infinite series which may be expressed in terms of coefficients of the finite Fourier transform of the log transmissivity function. Sufficient conditions for convergence of this series are examined. Finally, we show that our solution agrees with existing analytical ones to second order and test the solution with a numerical example

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