Abstract. Most pumping tests are interpreted using the classical Theis assumption of large-scale homogeneity with various corrections to account for early time behavior of drawdown curves. When drawdowns are plotted versus log time, late time data often delineate a straight line, which is consistent with Jacob's approximation of Theis' solution but may seem surprising in view of the heterogeneity of natural media. The aim of our work is to show that Jacob's method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data. A review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately, they produce very similar transmissivity T estimates. However, the corresponding estimates for storativity span a broad range. This behavior is verified numerically for several models of formation heterogeneity. A very significant finding of the numerical investigation is that T values estimated using simulated Basic Concepts and Previous WorkFormation transmissivity is the most important parameter to be determined in many hydrogeological problems. Transmissivity is often determined in the field using pumping tests. The analytical solution underlying the Theis and Jacob techniques is based on the assumption of aquifer homogeneity. Other analytical solutions assume that the aquifer can be subdivided into, at most, two or three regions with uniform parameters [e.g., Streltsova, 1988;Butler, 1988Butler, , 1990 Liu, 1991, 1993]. Although such configurations are simplifications of the complex structure of real heterogeneous formations, they can provide insight into the behavior of drawdown in heterogeneous formations. For example, Butler [1990] explains how the Theis and Jacob methods lead to different T values in heterogeneous aquifers because they apply different weightings to different portions of the drawdown curve. As shown in (1), pumping test analysis using Jacob's method is based on the rate of drawdown change. When T only depends on radial distance (axial symmetry), such change reflects aquifer properties only within a ring of influence through which the front of the pressure depression passes within the considered time interval. Therefore estimated transmissivity is independent of the aquifer properties between the inner radius of the ring of influence and the pumping well. However, storage estimates depend on the variations in T between the pumping well and the front of the cone of depression [Butler, 1988]. Butler and Liu [1993] conceptualized the nonuniform aquifer as a uniform matrix into which a disk of anomalous properties has been placed. They found, among other things, that Jacob's method can be used in any laterally nonuniform system to estimate matrix transmissivity if the flow to the pumping well is 1011

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Abstract. Interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity. The applicability of the traditional methods of interpretation in real aquifers can be questioned, since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters. Thus most hydrogeologists tend to look at estimated transmissivities (T) as some average property of the medium, while estimated storativities (S) are disregarded in some cases, particularly when they are obtained from data measured at the pumping well. An analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot. Jacob's method consists of obtaining estimates for T and S from the slope and intercept of this line. We find that even in a heterogeneous field, these estimates provide valuable information about the aquifer. Estimated T values for different observation points tend to converge to a single value, which corresponds to the effective T derived under parallel flow conditions. Estimated storativities, however, display higher variability, but the geometric mean of the S es t values can be used as an unbiased estimator of the actual S. Thus it appears that although Jacob's method was originally derived for homogeneous media, it can provide valuable information in real aquifers. IntroductionThe analysis of constant-rate pumping tests is often carried out using Jacob's method [Cooper and Jacob, 1946]. It consists of plotting drawdown versus log time and fitting a straight line to late time data points. Estimates of the transmissivity (T) and storage coefficient (S) are obtained from the slope and intercept of this line. It is generally accepted that the estimated values (Test and Sest) derived from Jacob's method are representative values for the test area (some area surrounding the pumping and observation wells). In fact, Tes t values calculated from a single pumping test using drawdowns measured at different observation wells tend to be fairly constant. However, the corresponding S es t values display large variability. This seems to somewhat contradict the generally accepted belief that aquifer transmissivity is highly variable. Actually, Meier et al. [1998] and others have shown that these effects in Tes t and Sest are a consequence of the homogeneity assumption used in the interpretation. Thus, in pumping tests performed in heterogeneous media, the variability in the transmissivity field is apparent as a variability in the S estimates, while the estimated T values remain almost constant. In this paper we follow up the numerical work of Meier et al. [1998] with an analytical study. Relationship to Previous Work

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