## 1. Introduction

[2] The transmissivity (*T*) and storage coefficient (*S*) are two important properties that control groundwater flow in aquifers and are of practical importance for water resources development and management. Traditionally, these aquifer properties are determined by collecting drawdown time data of the aquifer induced by pumping, and then matching the data with analytical solutions, which assume homogeneity of the aquifer. Theis' solution [*Theis*, 1935] is one of the commonly used analytical solutions in aquifer tests. It is derived from the equation of unsteady radial, horizontal, groundwater flow in a confined aquifer with constant *T* and *S*. Although the Theis solution is strictly applicable only to such idealized flow and aquifer conditions, it has been widely used in the field to estimate aquifer properties given drawdown time data from an observation well during an aquifer test.

[3] Radial flow in heterogeneous aquifers has been studied by many researchers in the past [see *Meier et al.*, 1998]. In particular, *Butler and Liu* [1993] derived an analytical solution for the case of transient, pumping-induced drawdown in a uniform aquifer into which a disk-shaped inclusion of anomalous properties (different *T* and *S*) has been placed. They found that changes in drawdown are sensitive to the hydraulic properties of a discrete portion of an aquifer for a time of limited duration. After that time, it is virtually impossible to gain further information about those properties. They concluded that constant rate pumping tests are not an effective tool for characterizing lateral variation in flow properties. *Oliver* [1993] derived the Fréchet derivatives and kernels to study the effect of areal variations in *T* and *S* on drawdown at an observation well. He concluded that small-scale variation in *T* near the well bore can influence the late time drawdown at distant observations depending on the location of the nonuniformity. Interpretation of a drawdown anomaly might be difficult because the effect on the drawdown derivative of a spatially small near-well nonuniformity is similar to the effect of a spatially large nonuniformity located farther from the well bore. *Meier et al.* [1998] conducted numerical simulations of pumping tests in two-dimensional horizontal aquifers with spatially varying *T* and a constant *S*. Analyzing the simulated drawdown at observation wells at various distances from the pumping well, they found that the estimated *T* from late time drawdown data using the Cooper-Jacob method [*Cooper and Jacob*, 1946] is very close to the effective *T* of the medium for uniform flows, practically independent of the location of the observation point. *Sánchez-Vila et al.* [1999] conducted an analytical study of drawdown under flow toward a well in heterogeneous aquifers of spatially varying *T* with a constant *S*. Using Jacob's method, they showed that estimated *T* values for different observation points tend to converge to the effective *T* derived under parallel flow conditions. Estimated *S* values, however, displayed higher variability but the geometric mean of the estimated S values could be used as an unbiased estimator of the actual S.

[4] Using an analytical stochastic approach, *Indelman* [2003] investigated the unsteady well flow in heterogeneous aquifers by modeling the hydraulic conductivity (*K*) as a three-dimensional stationary random function of axisymmetric anisotropy and Gaussian correlations. He assumed that the aquifer thickness is uniform and much greater than the vertical correlation scale of *K*, and specific storage *S*_{s} is a deterministic constant. Then, closed-form approximations of the ensemble mean drawdown were derived. He showed that the *T* estimated based on the ensemble mean drawdown, using the Cooper-Jacob asymptotic, is precisely the effective conductivity for uniform horizontal flow.

[5] These studies in general have suggested that the conventional Cooper-Jacob method is viable for estimating mean parameter values in heterogeneous aquifer from late time data, or a long duration of pumping. These studies, however, have not investigated effects of the variability of *S* on the *T* and *S* estimates, nor do they examine the behaviors of *T* and *S* estimates at early times. The behaviors of *T* and *S* estimates at early times can be important because an extended pumping could include effects of large-scale heterogeneity, as well as boundary effects. More importantly, few studies have examined the meaning of estimated *S* for heterogeneous aquifers. Even if they have, they have assumed that aquifers are made of spatially variable *T* and a spatially uniform *S*. Since the storage coefficient is the key parameter for evaluating groundwater availability in a basin, knowing the real meanings of the estimate of *S* in aquifers with heterogeneous *S* is of critical importance to groundwater resource management.

[6] Therefore previous numerical and theoretical analyses are incomplete. A practical but important question remains: What kind of estimates of the properties do we obtain from either early or late time drawdown data from an individual observation well in a heterogeneous aquifer? Also, do the estimates reflect the local properties near the observation well, some averaged properties between the pumping well and observation well, or none of the above?

[7] To answer these questions, this paper develops two theoretically consistent methods (i.e., distance drawdown and spatial moments) to estimate the effective transmissivity (*T*_{eff}) and storage coefficient (*S*_{eff}) values for radial flow in a given aquifer, as opposed to an ensemble of aquifers. Using numerical simulations and cross-correlation analysis, we investigate effects of heterogeneity in both *T* and *S* on the analysis of traditional aquifer tests using the Theis analytic solution.