## 1. Introduction

[2] Pumping tests are common techniques for hydrogeological site investigation. During pumping tests, water is injected or extracted from a production well and the changes of water level are monitored in adjacent observation wells as well as in the production well itself. Conventional pumping tests are restricted to a single pumping well. Analysis of these tests provides hydraulic properties over a large influence zone, essentially an ellipse, between the production and observation wells [*Butler and Liu*, 1993; *Gottlieb and Dietrich*, 1995]. The obtained transmissivity is a weighted average and does not provide detailed spatial information [*Yeh and Liu*, 2000].

[3] To overcome the limitations of conventional pumping tests, the method of hydraulic tomography has been proposed [*Neuman*, 1987; *Butler and Liu*, 1993; *Gottlieb and Dietrich*, 1995]. In hydraulic tomography, we inject or extract water in multiple wells and monitor the changes of heads or drawdowns at other multiple monitoring wells [*Butler and Liu*, 1993; *Gottlieb and Dietrich*, 1995; *Yeh and Liu*, 2000], obtaining multiple sets of head data. In dry or at least unsaturated formations, one may inject or withdraw gas instead of water [*Vesselinov et al.*, 2001a]. These data sets need to be jointly analyzed by methods similar to those used in electrical resistivity tomography [*Henderson and Webster*, 1978; *Lytle and Dines*, 1978]. Numerical investigations [*Bohling*, 1993; *Gottlieb and Dietrich*, 1995; *Vesselinov et al.*, 2001a, 2001b; *Zhu and Yeh*, 2006] and sandbox experiments [*Yeh and Liu*, 2000] have demonstrated that hydraulic tomography may produce a significantly improved description of spatially variable hydraulic parameters.

[4] In this paper, we analyze pumping test data obtained at the test site in Krauthausen of the Jülich Research Center, Germany [*Vanderborght and Vereecken*, 2001; *Vereecken et al.*, 1999, 2000]. The research center has conducted a sequence of aquifer tests; that is, water was pumped at different wells and head changes were monitored at adjacent wells. Although the way of conducting these tests differs from the suggested three-dimensional setups for hydraulic tomography in the references given above, it still follows the same philosophy, namely giving stress to the aquifer at different locations and observing the response at other locations. It can be viewed as the special case of two-dimensional hydraulic tomography.

[5] Type-curve methods are the basis of conventional analysis of hydraulic aquifer tests [e.g., *Meier et al.*, 1998]. The conventional approaches are straightforward and easy to implement, but they are based on the assumption of a homogeneous isotropic formation and an infinite domain, which may lead to biased estimates of hydraulic parameters. The results of the conventional approaches are apparent uniform values related to particular stress/observation points. In contrast to the conventional methods, the geostatistical inverse approach is based on the assumption that the parameter fields are spatially correlated random functions, which may be more consistent with the heterogeneous nature of aquifers [e.g., *Hoeksema and Kitanidis*, 1984; *Rubin and Dagan*, 1987]. The disadvantages of these inverse approaches are that they are more difficult to implement and require significantly higher computational effort than the type-curve methods.

[6] Several studies have shown that the conventional analysis of pumping tests can provide valuable information of real heterogeneous media in spite of the assumption of homogeneity. *Meier et al.* [1998] and *Sánchez-Vila et al.* [1999] assessed the applicability of Jacob's approach in heterogeneous aquifers. They simulated a conventional pumping test in a virtual two-dimensional aquifer where heterogeneous transmissivity and homogeneous storativity fields were used. Jacob's approach was applied to the late-time response of drawdown curves. The estimated transmissivity closely agreed with the real effective transmissivity. The estimated storativity values varied strongly from one pumping test to the other, although the real storativity was uniform. Based on Theis' method, *Leven and Dietrich* [2006] estimated the hydraulic parameters for different pumping test configurations in a virtual aquifer. In their study, only the pumping wells were monitored. The estimated transmissivity from multiple pumping tests showed a close agreement with the actual distribution. The estimated transmissivity from the conventional pumping tests showed low variation and approached the effective transmissivity of the virtual aquifer. Like *Meier et al.* [1998] and *Sánchez-Vila et al.* [1999], *Leven and Dietrich* [2006] observed similar behavior in estimating storativity, namely a strong variation in the estimate, despite the fact that the actual field of storativity was uniform. *Schad and Teutsch* [1994] applied the conventional analysis of pumping tests in a real alluvial aquifer and successfully estimated the effective length scale of the heterogeneous structures. *Neuman et al.* [2004] developed a type-curve approach to estimate the variance and integral scale of log transmissivity in real media. This type-curve method can be used to estimate the variance and the integral scale if a sufficient number of pumping wells are available.

[7] In the literature, a number of studies on radial flow toward a well in heterogeneous aquifers have been reported [*Sánchez-Vila et al.*, 1999; *Copty and Findikakis*, 2004]. *Dagan* [1982] and *Guadagnini et al.* [2003] derived analytical relationships between the effective transmissivity and drawdown. These analytical solutions do not assume homogeneity of aquifers and have the potential of analyzing pumping tests to obtain the hydraulic parameters of a formation.

[8] The purpose of this paper is to test whether transmissivity and storativity estimates, obtained by conventional type-curve analysis, can be viewed as local measurements of the hydraulic parameters themselves. If the latter assumption was valid, a continuous image of the field could be obtained by geostatistical interpolation, i.e., kriging. We will compare the kriged results of conventional pumping test analysis with the best estimate from the geostatistical inverse approach, which is conceptually more consistent. We perform the estimates based on the two-dimensional hydraulic tomography data from the test site in Krauthausen. Unlike *Leven and Dietrich* [2006], who considered only head measurements in multiple pumping wells, we will include the drawdown information also from adjacent wells.

[9] As geostatistical inverse approach, we apply the quasi-linear method of *Kitanidis* [1995] to invert temporal moments of drawdown [*Li et al.*, 2005]. In this approach, we estimate a spatial variable, but smooth parameter field, maximizing the posterior probability density of the parameters, linearized about the estimate itself. We develop a strategy to analyze data from multiple two-dimensional tomographic pumping tests. Unlike *Zhu and Yeh* [2006], we apply the method to field data, where the true parameters are not known and also the geostatistical parameters are uncertain.

[10] In case that the functional relationship between measurements and parameters is linear, the most likely value of our inverse approach is identical to the mean of conditional realizations. When the nonlinearity of the functional relationship is pronounced, the most likely value obtained by our method differs from the conditional mean. To overcome this shortcoming and obtain the unbiased best estimate, one may generate multiple realizations meeting the measurements via Monte Carlo simulations [*Sahuquillo et al.*, 1992; *Gutjahr et al.*, 1994] which may be computationally demanding. An alternative method of computing the conditional mean with high-order accuracy is based on conditional nonlocal ensemble moment equations [*Guadagnini and Neuman*, 1999a, 1999b; *Hernandez et al.*, 2006]. This method relies neither on multiple realizations nor on linearizations. It comes, however, with significantly higher computational costs than the quasi-linear approach used in our study. In order to solve the underlying integrodifferential equations, the conditional covariance matrix of the parameters must be computed explicitly and stored in each iteration, which may become rather demanding for large-scale problems discretized by hundreds of thousands of nodes.

[11] Both kriging and the quasi-linear geostatistical approach of inversion require the knowledge of structural parameters, such as the variance and correlation length. We estimate these parameters from the data using the restricted maximum likelihood approach [*Kitanidis and Vomvoris*, 1983], and discuss the influence of the measurement error on the identifiability of the structural parameters. The advantages of this approach are threefold. First, it avoids biased results of conventional experimental variogram analysis. Second, it can infer the structural parameters also from related secondary information, such as hydraulic heads. Third, the hydraulic parameter fields and the structural parameters are estimated jointly.