## 1. Introduction

[2] Predicting the fate of contaminants in heterogeneous aquifers requires careful characterization of hydraulic parameters, such as hydraulic conductivity and specific storage. Traditionally, pumping tests have been used for this purpose. However, pumping test analysis provides effective hydraulic parameters of large volumes, such that much of the spatial variability on a scale of relevance for transport remains unresolved. Unfortunately, existing tests with small support volume, such as slug tests and laboratory analysis on core samplers, cannot fill this gap of knowledge because of their limited spatial coverage, which makes data fusion a difficult, and often elusive, goal [see, e.g., *Rubin et al.*, 1998; *Beckie and Harvey*, 2002]. If spatial variability is poorly resolved, as actually occurs in most applications, relevant subsurface features, such as interconnected high or low conductivity zones inducing preferential flow paths, may be missed, thus limiting our capability to predict early (and late) arrival times in the successive modeling efforts [e.g., *Knudby and Carrera*, 2005]. This bears negative consequences on decisions taken by trusting the outcomes of flow and transport models, as for example in risk analysis.

[3] Several techniques have been proposed in order to better characterize the spatial variability of hydraulic parameters at the local scale, including tracer tests, geophysical techniques and a new way of conducting and analyzing hydraulic tests, which is called hydraulic tomography [see, e.g., *Gottlieb and Dietrich*, 1995; *Yeh and Liu*, 2000; *Bohling et al.*, 2007, and references therein]. Hydraulic tomography, in particular, has been proposed as a viable alternative to traditional pumping tests because it can be implemented with a spatial resolution that allows resolving the most relevant scales of variability for transport.

[4] The general hydraulic tomography concept, as explained above, can be exploited in several ways, from the interpretation of constant discharge multiwell pumping tests by type-curve analysis [*Neuman et al.*, 2004], or inversion [*Yeh and Liu*, 2000; *Indelman*, 2003; *Wu et al.*, 2005; *Firmani et al.*, 2006; *Li et al.*, 2005, 2007], to multiple pulse multiple receiver (MPMR) transient hydraulic tests in a cross-well configuration, as suggested by *Butler et al.* [1999] and *Bohling et al.* [2007]. So far, field applications of the MPMR configuration have been hampered by the complexity of the experimental setup, but recent advances in tubing, pulse emission equipments, and monitoring devices opened interesting opportunities for its practical implementation [*Butler et al.*, 1999; *Bohling et al.*, 2007; *Liu et al.*, 2008]. Therefore in the present work we focus on the MPMR configuration because it showed the potential for resolving intermediate to small-scale variability with only a moderate increase in complexity of the experimental setup with respect to traditional pumping tests.

[5] In a typical MPMR test, a localized flow or pressure pulse is applied sequentially at several depths within a borehole while the response of the aquifer to this perturbation is monitored at several locations in one or more additional boreholes (for this purpose the boreholes are subdivided in intervals, for example by using packers). The recorded head data are then utilized in an inversion procedure to infer spatially variable hydraulic parameters.

[6] The idea of inverting head data in order to obtain the hydraulic parameters is not a new one. An extensive literature exists on this subject, mainly referring to head data collected under natural and forced flow conditions [e.g., *Yeh*, 1986; *Kuiper*, 1986; *Carrera*, 1987; *Ginn and Cushman*, 1990; *Sun*, 1994], and Bayesian methods have been suggested to alleviate the often overarching difficulties encountered in solving a mathematical problem that is inherently ill posed [e.g., *Dagan*, 1985; *Carrera and Neuman*, 1986; *McLaughlin and Townley*, 1996; *Tarantola*, 2005]. In this context, synthetic studies showed that inversion is facilitated if redundant measurements are available, as for example when multiple well testing is utilized. Redundancy allows a better separation of noise from signal, although it leads to a larger computational burden [*Giudici et al.*, 1995; *McLaughlin and Townley*, 1996; *Tarantola*, 2005]. The MPMR configuration matches perfectly this philosophy because the aquifer is explored with many pulses following different pathways to the receivers.

[7] In an attempt to alleviate the otherwise high computational burden, *Bohling et al.* [2002, 2007] utilized the steady-shape approach for inversion of transient hydraulic data, a methodology envisioned by *Butler* [1988] following an early work of *Wenzel* [1942], whose justification lies in the observation that a steady-shape configuration with nearly constant (in time) hydraulic gradients is established very rapidly in many field settings [*Butler*, 1988; *Kruseman and de Ridder*, 1990; *Bohling et al.*, 2002]. In such a situation the spatial gradients are stationary (time independent) and can be approximated by the space derivatives of the heads obtained by solving the steady state flow equation. An important advantage of this methodology, in comparison with the classical analysis of steady state drawdown data, is a much shorter duration of the hydraulic test, which comes at the expense of inability to determine the specific storage. Furthermore, the need to approximate space derivatives with differences, often computed by using data from a few measurement locations, introduces an additional source of uncertainty that, to the best knowledge of the authors, has not been investigated so far. On the same line of thought, i.e., avoiding replicate solutions of the computationally expensive transient flow equation, is the inversion of drawdown temporal moments [*Li et al.*, 2005, 2007; *Zhu and Yeh*, 2006].

[8] A compelling difficulty in using drawdown data for inversion, or their transformations such as temporal moments, is that the inverse model is underdetermined with many more unknowns (i.e., the hydraulic parameters at the computational cells) than measurements. An effective way to overcome this difficulty is by coupling hydraulic tomography with Bayesian inversion. Firstly, the MPMR configuration reduces the ratio parameters to observations by shifting the volume interrogated at each pulse test, thereby making the problem less underdetermined. Secondly, the Bayesian inversion provides a powerful framework to sort out the space of admissible solutions and retain only those compatible with the observational data [*Fienen et al.*, 2008]. However, a further difficulty emerges because the application of the Bayes' theorem to invert drawdown data leads to a nonlinear relationship between the observational data and the hydraulic parameters, which makes the solution of the inverse problem a difficult task to achieve. In order to facilitate the solution of the problem and reduce the computational burden this relationship is typically linearized by using either the Sequential Successive Linear Estimator (SSLE), developed by *Yeh et al.* [1996] and *Yeh and Liu* [2000], and applied in subsequent works [see, e.g., *Zhu and Yeh*, 2005], or the quasilinear method, developed by *Kitanidis* [1995] and used in several two and three-dimensional applications [see, e.g., *Li et al.*, 2005, 2007; *Fienen et al.*, 2008].

[9] In this work, we use an alternative approach based on particle swarming, an efficient genetic algorithm, which avoids linearization of the underlying optimization problem, and thus the computation of the sensitivity matrix, which is the most demanding step in the previous approaches. Furthermore, instead of solving directly for the conditional mean, we seek an ensemble of independent realizations conditional to the measurements of the hydraulic properties and the head data, all representing possible realizations of the actual (true) variability of the hydraulic parameters. Inversion is performed by using a Bayesian Maximum a Posteriori (MAP) approach, already shown to be successful with hydrological inversion [see, e.g., *Kowalsky et al.*, 2005] when the geostatistical model of spatial variability is known. Guidelines for obtaining the parameters of the spatial variability model from pumping and tracer test experiments have been provided in a number of studies [see, e.g., *Wilson and Rubin*, 2002; *Bellin and Rubin*, 2004; *Firmani et al.*, 2006], typically under ergodic conditions. However, a common assumption in existing works dealing with the inversion of hydraulic tomography data is that they are totally or partially known in advance [*Kowalsky et al.*, 2004, 2005; *Nowak and Cirpka*, 2004]. In this work we remove this hypothesis and assume that only the type of geostatistical model of spatial variability is known in advance, while structural parameters such as mean, variance, and integral scales of the hydraulic conductivity, are estimated as part of the inversion procedure. A similar approach has been adopted by *Li et al.* [2007], who inverted head data collected at the Krauthausen (Germany) test site by using the quasilinear method developed by *Kitanidis* [1995] with the structural parameters that were progressively updated from their prior estimates. However, since their exercise was based on real data, they could not assess the accuracy of their estimates. This is another question we seek to answer with this paper.

[10] Therefore the main objective of the present work is to assess how much spatial variability is granted by hydraulic tomography data collected in a MPMR configuration. Its basic configuration is composed of two boreholes subdivided into intervals, which are used to apply the pulse and to measure the resulting perturbations of the hydraulic head.

[11] In section 2, we discuss the theoretical framework, followed in section 3 by the inversion methodology. In section 4, we discuss a few synthetic examples in a two-dimensional setup to explore the potentiality of the inverse methodology; in section 5, we discuss an application in a three-dimensional setup and finally we present our conclusions in section 6.