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Hydraulic tomography has been developed as an alternative to traditional geostatistical methods to delineate heterogeneity patterns in parameters such as hydraulic conductivity (K) and specific storage (S_s). During hydraulic tomography surveys, a large number of hydraulic head data are collected from a series of cross-hole tests in the subsurface. These head data are then used to interpret the spatial distribution of K and S_s using inverse modeling. Here, we use the Sequential Successive Linear Estimator (SSLE) of to interpret synthetic pumping test data created through numerical simulations and real data generated in a laboratory sandbox aquifer to obtain the K tomograms. Here, we define “K tomogram” as an image of K distribution of the subsurface (or the inverse results) obtained via hydraulic tomography. We examine the influence of signal-to-noise ratio and biases on results using inverse modeling of synthetic and real cross-hole pumping test data. To accomplish this, we first show that the pumping rate, which affects the signal-to-noise ratio, and the order of data included into the SSLE algorithm both have large impacts on the quality of the K tomograms. We then examine the role of conditioning on the K tomogram and find that conditioning can improve the quality of the K tomogram, but can also impair it, if the data are of poor quality and conditioning data have a larger support volume than the numerical grid used to conduct the inversion. Overall, these results show that the quality of the K tomogram depends on the design of pumping tests, their conduct, the order in which they are included in the inverse code, and the quality as well as the support volume of additional data that are used in its computation.

Ground water investigations have relied on the determination of aquifer parameters such as hydraulic conductivity (K) and specific storage (S_s). In the past, these were determined by conducting slug and/or pumping tests while using an analytical solution that treats the geological medium to be homogeneous. These solutions and the estimated parameters have been used in a variety of applications despite the fact that the subsurface is heterogeneous at multiple scales, which is the rule rather than the exception. In particular, the knowledge of detailed three-dimensional (3D) distributions of K is critical in prediction of contaminant transport, delineation of well catchment zones, and quantification of ground water fluxes, including surface water/ground water exchange.

However, characterization of subsurface heterogeneity of hydraulic parameters is fraught with difficulties. Information about the spatial variability of flow parameters is most commonly obtained through inference from small-scale measurements of cores, slug/bail tests, and single-hole pressure tests by geostatistical methods, which require numerous measurements. This requires the drilling of numerous boreholes, collection of a large number of samples, and the conduct of multiple measurements within various depth intervals in each of them using sophisticated equipment. The approach is expensive and time consuming, and thus has not been adopted widely in practice.

In addition, the physical meaning of the flow parameter estimates from either traditional pumping or slug tests is considered to be questionable. Furthermore, it is not clear that geostatistical analysis of data collected on relatively small support scales is necessarily indicative of medium properties that impact flow and transport on scales that are much larger. One alternative to traditional geostatistical analyses is hydraulic/pneumatic tomography. Hydraulic tomography is a cost-effective technique for characterizing subsurface heterogeneity of hydraulic parameters. During hydraulic tomography surveys, drawdowns induced by sequential pumping or injection tests at different locations of an aquifer are collected at a large number of subsurface locations. These hydraulic head data are then used to interpret the spatial distribution of hydraulic parameters of the aquifer using an inverse model. Pneumatic tomography is similar in concept to hydraulic tomography, but the well tests are conducted with air in the unsaturated zone particular, developed a sequential geostatistical inverse method, which they refer to as the Sequential Successive Linear Estimator (SSLE). This approach can be applied to hydraulic tomography for the interpretation of cross-hole pumping tests under steady-state conditions. Their approach combines the traditional geostatistical approach and governing flow principles to interpolate and extrapolate at locations where samples are not available. As a consequence, the SSLE as implemented in hydraulic tomography yields more realistic K estimates than kriging, which does not consider principles of flow, and deterministic/zone-based or stochastic inverse modeling approaches that use one pumping or injection data set only. The main advantage of sequentially including pumping tests is its computational efficiency. showed that accurate K distributions can be obtained through data sets from numerically simulated pumping tests in synthetic heterogeneous aquifers. Validation of the steady-state hydraulic tomography in was limited to error-free cases of synthetic simulations.

conducted a laboratory sandbox study to evaluate the performance of hydraulic tomography in characterizing aquifer heterogeneity. This was the first validation study of hydraulic tomography, but the K tomograms were visually compared only to the distribution of sand types and to results from synthetic simulations. The K tomograms were not compared to small-scale estimates of K directly, and the authors explicitly state that the true K distributions were not available for either one of the sandboxes used in the study. The authors mentioned that errors and biases have effects on their K tomograms, but they did not examine the role of errors and biases directly by isolating their causes. Here, we define “K tomogram” as an image of K distribution of the subsurface (or the inverse results) obtained via hydraulic tomography.

further examined the accuracy of the K tomograms obtained from the steady-state hydraulic tomography algorithm developed by They obtained multiple K estimates from core, slug, single-hole and cross-hole tests as well as several unidirectional, flow-through experiments conducted on the sandbox under steady-state conditions. also examined the influence of errors and biases on inversion results using forward and inverse simulations of cross-hole tests. They found out that the pressure transducer offsets, skin effect at the pumped well, among other sources of errors can have a large impact on the quality of the inverse modeling results. Likewise conducted a laboratory validation of the transient hydraulic tomography to estimate both K and S_s tomograms simultaneously, but both and have not examined practical factors such as the effects of signal-to-noise ratio and the role of conditioning on the quality of the inversion results.

There are several issues that need to be further examined. One important issue in applying hydraulic tomography in the field is how much noise a given data set can contain in obtaining an accurate K tomogram without having to apply smoothing and/or signal processing techniques, which may result in the loss of information on the parameters contained in the data set. Another issue is that SSLE incorporates pumping test data sequentially, but no studies have been published to date that examines the role of varying the order of pumping test data included in the SSLE algorithm. Finally, during site characterization, test data other than cross-hole hydraulic test data such as from direct push technologies, core, slug, geophysical, and geochemical data may be collected. In some cases, one could use some of these data to condition the inverse modeling results. However, there are no studies to our knowledge in which the role of conditioning on the estimated K tomogram by means of hydraulic/pneumatic tomography was systematically studied.

To address these practical yet important issues, we continue our study on hydraulic tomography using synthetic data generated through numerical simulations and real data collected in the laboratory. The main objectives of this paper are (1) to further study the validity of steady-state hydraulic tomography through synthetic pumping test data obtained through forward numerical simulations and real data obtained using a laboratory sandbox aquifer; (2) to investigate the effect of varying the pumping rate, which affects the signal-to-noise ratio, and its impact on K tomograms; (3) to investigate the effect of varying the order of test data included into the SSLE inversion algorithm; and (4) to investigate the effect of conditioning on K tomograms.

We first describe the experimental setup for the synthetic and real cases used to conduct our study. Next, we discuss the numerical simulation approach that underlies hydraulic tomography and approaches used to generate synthetic hydraulic test data and corresponding laboratory data sets. We then briefly discuss the reference K tomograms from synthetic and real data generated through laboratory sandbox experiments by. These reference K tomograms will be used to compare against the new results presented in this paper. In all cases, we first compare our results visually to the reference K tomograms obtained as a benchmark result for both the synthetic and real cases. We also assess the validity of the synthetic and real K tomograms by simulating an independently conducted pumping test and comparing the head values obtained from the simulated and observed cases.

We emphasize that the synthetic experiments conducted on the computer are necessary to test the SSLE algorithm under optimal conditions in which the experimental errors are neglected and the forcing functions (boundary condition and source/sink terms) are fully controlled. In addition, through numerical simulations, hydraulic tomography and the K tomograms obtained can be tested rigorously. The laboratory experiments described subsequently are also required to test the SSLE algorithm under controlled conditions, which is a necessary step toward its field applications.