## 1. Introduction

[2] Pumping and slug tests are the most commonly used field techniques for characterizing the transmissive properties of an aquifer. Unfortunately, when the focus of an investigation is on contaminant transport, the information provided by these techniques is often of relatively limited value. Conventional pumping tests yield hydraulic conductivity (K) estimates that represent an average over a relatively large volume of an aquifer [*Butler*, 1990; *Butler and Liu*, 1993; *Meier et al.*, 1998]. Contaminant transport, however, is sensitive to the manner in which hydraulic conductivity varies in space and, particularly, to the spatial arrangement of laterally continuous regions of extreme values. Slug tests can provide information about spatial variations in K at a scale of relevance for transport investigations [*Yeh et al.*, 1995], but the parameter estimates are subject to significant bias as a result of incomplete well development [*Butler*, 1998; *Butler and Healey*, 1998]. Furthermore, conventional slug tests, which only use measurements in the test well, are not capable of yielding information about the continuity of layers between wells. Clearly, new approaches are needed if hydraulic tests are to provide information of more significance for transport investigations.

[3] Over the last decade, several research groups have begun work on a new hydraulic test method, known as hydraulic tomography, that has the potential to yield a detailed description of spatial variations in hydraulic conductivity between wells [*Bohling*, 1993; *Tosaka et al.*, 1993; *Gottlieb and Dietrich*, 1995; *Butler et al.*, 1999; *Yeh and Liu*, 2000]. As shown in Figure 1, this approach essentially consists of a series of short-term pumping tests in which the position of the stressed interval in the pumping well, isolated with packers, is varied between tests. In each test, the aquifer is “probed” by a streamline configuration that converges on a different pumping interval. The sequence of tests produces a pattern of crossing streamlines in the region between the pumping and observation wells similar to the pattern of crossed ray paths used in seismic or radar tomography [*Peterson et al.*, 1985; *Vasco et al.*, 1997]. Although a large number of drawdown measurements is required to delineate the numerous streamlines produced by the test sequence, new methods for drawdown measurement have been developed that can provide the requisite density of data in a practically feasible manner [*Butler et al.*, 1999].

[4] The primary objective of this paper is to present an efficient and robust approach for the analysis of drawdown data from hydraulic tomography experiments. The field methods presented by *Butler et al.* [1999] do reduce the time and expense involved with the field implementation of the tomography procedure. These methods, however, are of practical utility only if they can be coupled with efficient means for analysis of the drawdown data. Most previously proposed methods for drawdown analysis have assumed steady state conditions [*Bohling*, 1993; *Yeh and Liu*, 2000], but reliance on steady state conditions severely limits the utility of such methods. For example, attainment of steady state conditions often requires many hours, if not days, of pumping, a duration that would not be feasible for practical applications. Moreover, a steady state analysis requires assumptions about the boundary conditions controlling the transition of the flow system to steady state, an issue that is usually the source of considerable uncertainty. The propagation of that uncertainty will often have a significant impact on K estimates.

[5] *Vasco et al.* [2000] propose an approach that has potential to greatly increase the efficiency of the analysis of data from pumping tests in tomographic format. This technique involves identifying and matching the arrival times of the peak slope of the drawdown-versus-time curve at different locations. However, the approach does have some significant limitations. First, arrival times are primarily controlled by the aquifer diffusivity (ratio of hydraulic conductivity to specific storage), making it difficult to separate the effects of conductivity and storage. Second, identifying peak slope arrival times requires use of early-time drawdown data that can be affected by a variety of nonideal mechanisms (e.g., inertia-induced mechanisms in highly permeable systems [*Shapiro*, 1989]) that have an insignificant impact on drawdown data at larger times.

[6] The analysis approach introduced in this paper avoids the limitations of these previous methods by exploiting the simplifications possible under a flow regime known as steady shape, unsteady state [*Heath and Trainer*, 1968; *Butler*, 1990]. This steady shape concept, which has its origins in the early work of *Theis* [1940] on the nature of pumping-induced drawdown, enables transient drawdown data to be analyzed with a steady state model. Transient data can be processed with the computational efficiency of a steady state model, while avoiding the errors introduced by inappropriate assumptions about the boundary conditions driving the system to steady state or the nonidealities associated with early-time data. When combined with the field methods described by *Butler et al.* [1999], the new analysis approach should greatly enhance the practical viability of the hydraulic tomography methodology.

[7] This paper will describe the steady shape analysis method and demonstrate its efficacy in a series of numerical experiments using a synthetic, imperfectly layered aquifer. The paper begins with an overview of the steady shape concept followed by a description of the numerical approaches used to generate and analyze drawdown from a series of simulated tests. The computed drawdowns are analyzed with transient, steady state, and steady shape inversion techniques. The K estimates from these analyses are compared to the hydraulic conductivity values used as inputs for the test simulations to assess the viability of the various approaches. The dependence of the steady state and steady shape approaches on the assumed mechanism responsible for steady state conditions is also examined. The paper concludes with an assessment of the sensitivity of the steady shape approach to the number of drawdown measurements and to the magnitude of measurement error.