Upscaling transmissivity under radially convergent flow

When pumping from a single well in a real (heterogeneous) aquifer, the steady state drawdown at the well is proportional to the pumping rate through a value called the equivalent transmissivity, Tea, which, in short, is the value that best suits Thiem's formula. Equivalent transmissivity is not a local value, but some representative value of a certain area surrounding the well. By means of numerical simulations we study the validity of a recent analytical formula (Sânchez-Vila et al., 1999) that provides a value for Teq which is given as a weighted average of the fluctuations in log T throughout the domain. The formula is found to work acceptably well for mildly heterogeneous T fields. I N T R O D U C T I O N Most field methods used to estimate transmissivity values rely on the analysis of drawdown under radially convergent flow conditions. In heterogeneous media, flow towards a pumping well is still convergent, but not uniformly radial. If we consider a single well pumping in a real (heterogeneous) aquifer and under steady state flow conditions, the drawdown at the well, s, is directly related to the pumping rate, Q, through a value called the equivalent or upscaled transmissivity, Teq. This value would be the transmissivity assigned to the well location in a simple interpretation of a steady state pumping test (using Thiem's formula). Equivalent transmissivity is not a local value, but some representative value of a certain area surrounding the well. Recently, Sânchez-Vila et al. (1999) provided an analytical formula for Tcq, based upon an extension of Thiem's equation. The resulting upscaled value is given as a weighted average of the fluctuations in log T throughout the heterogeneous domain. In this paper we test this analytical formula by means of numerical simulations for a number of heterogeneous transmissivity fields. THE A N A L Y T I C A L UPSCALING FORMULA A rigorous definition (although not necessarily the only possible one) of block or equivalent transmissivity, Teq, in a two-dimensional annular domain is the value that suits the following relationship: Q = 2itATeq (1) 142 X. Sânchez-Vila et al. Q being the pumping rate and A = {hehw)l\xi(relrw); hw and he are the measured heads at the inner (rw) and outer (re) radii, respectively. In short, Teq is the value that best suits Thiem's formula. In general it is not possible to derive an exact formula for Teq in a heterogeneous domain. Trying to overcome this problem, Sânchez-Vila et al. (1999) presented an analytical expression for Teq given as a weighted average of the fluctuations in log T throughout the domain. The solution for Teq is based on a series expansion of Q: