Hydraulic tomography using temporal moments of drawdown recovery data: A laboratory sandbox study



[1] Hydraulic tomography (HT) is a new technology that images the hydraulic heterogeneity of the subsurface. Unlike steady state hydraulic tomography (SSHT), which provides a hydraulic conductivity (K) tomogram, transient hydraulic tomography (THT) provides reliable tomograms of both K and specific storage (Ss). Effective as it may be, THT is a computationally demanding technique. To ease the computational burden, a HT which utilizes zeroth and first temporal moments of transient drawdown recovery data (HT-m) has been developed by Zhu and Yeh (2006). This procedure simplifies the governing equation from a single parabolic equation to two Poisson's equations for the zeroth moment and characteristic time defined as the ratio between the first and zeroth moments. The approach was previously tested using synthetic simulations. The numerical experiments tested the feasibility of HT-m under ideal conditions, where measurements and model are assumed to be free of error. In this paper, we further evaluate the performance of HT-m using cross-hole pumping tests conducted in a heterogeneous, synthetic aquifer constructed in a laboratory sandbox in more realistic situations, where the data used in the inversion are not free of experimental errors. Unlike field tests, the laboratory tests were conducted in a synthetic aquifer created with a prescribed heterogeneity pattern and all forcing functions (initial and boundary conditions and source-sink terms) controlled. Results from the HT-m approach were compared to those from THT previously conducted by Liu et al. (2007). Our results show that the estimation of the K tomogram using the HT-m approach is reasonable, but the estimation of the Ss tomogram is unreliable in comparison to the THT approach.

1. Introduction

[2] Characterization of subsurface heterogeneity in aquifer parameters is a topic of great interest to hydrogeologists. There are a number of approaches to capture the spatial variability of parameters such as hydraulic conductivity (K) and specific storage (Ss). One approach which is receiving recent attention is hydraulic tomography (HT). Various inverse methods for HT have been developed which utilize pumping test data simultaneously or sequentially [e.g., Gottlieb and Dietrich, 1995; Yeh and Liu, 2000; Bohling et al., 2002; Brauchler et al., 2003; Zhu and Yeh, 2005, 2006; Li et al., 2005; Fienen et al., 2008]. In particular, Yeh and Liu [2000] developed an algorithm for steady state hydraulic tomography (SSHT) through the use of the sequential successive linear estimator (SSLE). Zhu and Yeh [2005] then extended the SSHT to transient hydraulic tomography (THT).

[3] THT is an effective technique in imaging K and Ss, but is computationally costly. To overcome the computational challenge for THT analysis, Zhu and Yeh [2006], inspired by the moment generating function approach by Harvey and Gorelick [1995], developed an approach that utilizes the zeroth and first temporal moments of drawdown recovery data, instead of drawdown recovery data itself. The method requires the full drawdown recovery curve to calculate the temporal moments [Zhu and Yeh, 2006].

[4] The governing equations for the temporal moments are Poisson's equations. These equations demand less computational resources as opposed to the parabolic equation that governs drawdown recovery evolution. Likewise, the adjoint equations for evaluating sensitivities of the moments for parameter estimation also take the same forms. Therefore a HT which uses the temporal moments of drawdown recovery data (HT-m) expedites the interpretation of THT surveys.

[5] While various algorithms for HT have been developed and some of them have been tested numerically, several sandbox studies have been conducted to evaluate the performance of both SSHT [Liu et al., 2002; Illman et al., 2007, 2008] and THT [Liu et al., 2007]. In the field, THT has also been applied in unconsolidated media [Straface et al., 2007], while a HT based on the steady shape analysis [Bohling et al., 2002] was reported for unconsolidated materials by Bohling et al. [2007]. Most recently, Li et al. [2008] utilized their HT approach to image floodplain deposits.

[6] To date, the evaluation of the HT-m has not been accomplished either in the laboratory or the field setting. A field validation is our ultimate goal, but prior to that, laboratory validations are necessary in which the heterogeneity pattern is prescribed, and all forcing functions and errors can be controlled as opposed to field conditions in which all of these factors are unknown. The objectives of this paper are to invert cross-hole pumping test data obtained in a laboratory aquifer using the HT-m algorithm and to investigate the performance of HT-m in comparison to the THT approach of Zhu and Yeh [2005].

2. Experimental Methods

[7] The cross-hole pumping tests were conducted in a synthetic heterogeneous aquifer built in a sandbox to generate data for testing the HT-m algorithm under a controlled setting. Figure 1 is a computer aided design (CAD) drawing of the sandbox. There were 48 ports each equipped with pressure transducers from which water can be pumped. One additional pressure transducer was installed in each of the two constant head reservoirs on each end of the sandbox to collect data on boundary heads.

Figure 1.

Computer aided design (CAD) drawing of sandbox used for the validation of hydraulic tomography. Numbers next to solid circles indicate port numbers; open circles around numbers indicate the eight ports (2, 5, 14, 17, 32, 35, 44, and 47) used for pumping; the open square around pumping port 46 indicates the pumping location for the ninth cross-hole test used for validation purposes; and dashed rectangles indicate the location of low-K lenses. Types of sand used for packing the sandbox are shown in the legend.

[8] The synthetic aquifer consisted of 4 different commercially sieved sands (20/30 and 4030, U.S. Silica; F-75 and F-85, Unimin Corporation). Eight rectangular sand bodies consisting of lower-K material (4030, F-75, and F-85) were packed within high-K sand (20/30) (Figure 1). After packing the sandbox, horizontal wells were installed and core samples collected at each of the 48 ports locations. Table 1 lists the geometric mean values of K determined from core samples using a constant head permeameter. Further details to the sandbox and data acquisition system can be found in work by Illman et al. [2007].

Table 1. Geometric Mean Values of K Determined From Core Samples Taken From the Sandboxa
Sand TypeManufacturerNumber of SamplesK (cm/s)
20/30U.S. Silica322.60 × 10−1
4030U.S. Silica36.42 × 10−2
F-75Unimin Corp.51.99 × 10−2
F-85Unimin Corp.81.61 × 10−2

[9] Flow through the sandbox can be maintained by two constant-head reservoirs, one at each end of the sandbox. Here, we maintained three constant head boundaries by ponding water over the top of the sand, connecting the two reservoirs. This boundary condition configuration was chosen for all cross-hole tests conducted in the sandbox, as we found the boundary conditions to remain stable during each test.

[10] The cross-hole tests were conducted by pumping at rates ranging from 2.50–3.17 cm3/s at eight separate ports (2, 5, 14, 17, 32, 35, 44, and 47) indicated by open circles in Figure 1. Prior to each cross-hole test, head data were collected in all pressure transducers to establish an initial condition. After establishment of static conditions, we pumped from each port using a peristaltic pump, while taking head measurements in all 50 ports. For each test, pumping continued until the development of steady state flow conditions. The pump was then shut off to collect recovery head data until its full recovery.

3. Inverse Modeling Approach

[11] The HT-m estimation technique [Zhu and Yeh, 2006] is based on the sequential successive linear estimator (SSLE) developed by Zhu and Yeh [2005]. We utilize this algorithm to map the spatial distribution of K and Ss in a real sandbox aquifer. To obtain a K and Ss tomograms from multiple cross-hole pumping tests, we solve an inverse problem for transient flow conditions. The sandbox was discretized into 741 elements and 1600 nodes with element dimensions of 4.1 cm × 10.2 cm × 4.1 cm. This grid was also used previously for SSHT by Illman et al. [2007, 2008] and THT by Liu et al. [2007]. Input data to the inverse model include initial guesses for the K and Ss, estimates of variances and the correlation scales for both parameters, discharge rate (Q) from each pumping test, and zeroth moment and characteristic time computed from the drawdown recovery curves, as well as available point (small-scale, i.e., core, slug, and single-hole tests) measurements of K and Ss. To rigorously test the HT-m code, we elect to not use the point-scale measurements to condition the estimated parameter fields.

[12] We selected 8 pumping tests at ports 2, 5, 14, 17, 32, 35, 44, and 47 and the corresponding drawdown time observations at the rest of 47 ports during each test as data sets for the HT-m analysis. The remaining one test with pumping taking place at port 46 (open square in Figure 1) was reserved for validation purposes.

[13] Prior to the computation of the temporal moments required in the HT-m code, we preprocessed the drawdown recovery data. This is because Illman et al. [2007, 2008] found from the analysis of HT data obtained in a laboratory sandbox aquifer that the signal-to-noise ratio can be critical in inverse modeling of cross-hole pumping test data. To overcome the issue of signal-to-noise ratio on the quality of HT surveys, Xiang [2007] developed a wavelet transform approach based on the work by Zhang et al. [2006] to remove noise from experimental data prior to implementing the data in their inverse code. We utilized the wavelet transform tool implemented in VSAFT2 (http://tian.hwr.arizona.edu/yeh/downloads.html) to denoise all drawdown recovery curves prior to calculating the temporal moments.

[14] The temporal moments of the drawdown are calculated using the moment generation function. The nth temporal moments (Mn(xi)) of drawdown at location xi are given by:

equation image

where t is time, s(xi, t) is drawdown defined as s(xi, t) = H0h(xi, t), H0 is the initial hydraulic head treated to be constant everywhere in the domain, and h(xi, t) is hydraulic head. One can compute the zeroth and first temporal moment using the moment generating function (equation (1)) by setting n = 0 and 1 respectively. The zeroth moment at a particular location represents the area under the drawdown recovery curve. Here, we calculate the zeroth moment of the cross-hole test data using the following approximation:

equation image

Likewise, the first moment of the pumping test data is computed using the following approximation:

equation image

The first moment at a particular location represents the mean arrival time of the pressure disturbance for the drawdown recovery curve. The characteristic time is then given by

equation image

Physically, the characteristic time corresponds to the mean arrival time of the pressure disturbance under the drawdown recovery curve normalized by the area.

4. Results

[15] All inverse modeling runs were executed using 8 of 16 processors on a PC cluster consisting of 1 master and 15 slaves (each with Pentium IV 3.6 GHz with 1 GB of RAM). Previously, the same computational platform was utilized to conduct the THT analysis [Liu et al., 2007] using the same grid. Through the comparison of the THT and HT-m analysis, we found that the runs for the HT-m were approximately a factor of 20 faster than those for the THT for the same problem.

[16] Figures 2a2d are K tomograms obtained by inverting the temporal moment data from 8 pumping tests with pumping taking place at ports 44, 47, 35, 32, 17, 14, 5, 2 (see Figure 1) in that order. The result (Figure 2a) using the first two pumping tests reveals little detail to the heterogeneity pattern throughout the sandbox. As more data are sequentially included to the HT-m algorithm, the heterogeneity structure consisting of low-K blocks emerges (Figures 2b2d), although the contrast between the high- and low-K regions are not very distinct.

Figure 2.

K tomograms computed using HT-m from cross-hole pumping test data included sequentially: (a) ports 44 and 47; (b) ports 44, 47, 35, and 32; (c) ports 44, 47, 35, 32, 17, and 14; and (d) ports 44, 47, 35, 32, 17, 14, 5, and 2. Port numbers (see Figure 1) indicate those used as the pumped well for each cross-hole test. Rectangles indicate locations of low-K blocks.

[17] Figures 3a3d shows the corresponding Ss tomogram that was estimated simultaneously. In contrast to Figures 2a2d, the structure consisting of variable size sand bodies visible in the K tomogram is not visible in the Ss tomograms. We also note that on the basis of Figures 2a2d and 3a3d, K values are not significantly correlated with Ss values in this sandbox.

Figure 3.

Ss tomograms computed using HT-m from cross-hole pumping test data included sequentially: (a) ports 44 and 47; (b) ports 44, 47, 35, and 32; (c) ports 44, 47, 35, 32, 17, and 14; (d) ports 44, 47, 35, 32, 17, 14, 5, and 2. Port numbers (see Figure 1) indicate those used as the pumped well for each cross-hole test. Rectangles indicate locations of low-K blocks.

5. Discussion and Conclusions

5.1. Visual Evaluations of Computed Tomograms

[18] A visual comparison of the K tomogram (Figure 2d) obtained by sequentially including 8 tests into the inversion algorithm with the locations of low-K sand bodies (indicated with rectangles) illustrates the ability of HT-m to image the major low- and high-K features that make up aquifer heterogeneity. In particular, we see in Figure 2d that some of the low-K blocks in the upper portion of the sandbox appear to be captured, but the 2 lowest blocks are not captured very well. In addition, we see a reasonable agreement between the K values of low-K blocks determined from HT-m to those obtained from constant permeameter analysis of core samples (Table 1). The same could also be said of the high-K zones surrounding the low-K blocks.

[19] A direct visual comparison of the Ss tomogram computed using the HT-m algorithm to the low-K blocks shows very little correspondence. This is likely due to the fact that Ss values vary little from one sand type to the next.

5.2. Joint Evaluation of K and Ss Tomograms

[20] Illman et al. [2007, 2008] and Liu et al. [2007] found that a robust approach to validate the tomograms was to test the predictability of head/drawdown measurements from independently conducted cross-hole pumping tests. Here, we adopt this approach by simulating an additional cross-hole pumping test that was not used in the inversion and to examine whether the drawdown at various sampling ports of this independent test can be predicted accurately at various times using a forward model. That is, we utilize the K and Ss tomograms given in Figures 2d and 3d and simulate a cross-hole test with pumping taking place at port 46 (i.e., the heterogeneous case). An additional forward simulation is also conducted using K and Ss fields that are homogeneous (K = 1.726 × 10−1 cm s−1 and Ss = 2.300 × 10−4 cm−1) obtained previously by Liu et al. [2007]. The latter results will be used to assess the performance of the tomograms.

[21] We evaluate the difference between the simulated and observed drawdown values for both the heterogeneous (i.e., the tomograms) and homogeneous cases by computing its mean, variance and correlation coefficient (R). These values are reported for t = 3, 50, and 150 s in Table 2.

Table 2. Summary of the Mean and Variance of the Estimation Errors From the Comparison of Observed to Simulated Drawdownsa
Time (s)Heterogeneous FieldsHomogeneous Fields
Mean (cm)Variance (cm2)CorrelationMean (cm)Variance (cm2)Correlation
  • a

    The K and Ss fields for the heterogeneous case are from the inversion of the laboratory sandbox data using the HT-m algorithm while the homogeneous case represents equivalent K and Ss obtained from Liu et al. [2007].


[22] For the heterogeneous case, the mean, variance, and correlation values all show that the results improve over time when the K and Ss tomograms are used to simulate the independent cross-hole pumping test. In particular, the relatively high correlation values of 0.850 for t = 50 s and 0.855 for t = 150 s suggest the predicted drawdown distribution is close to the observed distribution, at least the drawdowns at late times at the observation ports. This is an encouraging result because it indicates that using the K and Ss fields derived from HT-m approach, one can obtain a reasonably good prediction of the drawdown behavior in the sandbox at late times. However, we note that these results are not as robust as those obtained for the THT approach presented by Liu et al. [2007], who obtained correlation coefficients of 0.995 over the time periods of 3, 10, and 20 s using the same cross-hole test data.

[23] In contrast, for the homogeneous case with effective parameters, Table 2 shows that the means and variances of the difference between the simulated and observed drawdowns increase with time, while the correlation coefficient does not vary significantly. Therefore, Table 2 suggests that using effective parameters, a flow model assuming homogeneity predicts different drawdowns than those observed at the 48 ports in the heterogeneous aquifer.

5.3. Comparison of Results to Those From THT

[24] We next make a direct comparison of the local values from the K and Ss tomograms computed by the HT-m algorithm to those from the THT algorithm [Liu et al., 2007]. Here, the local K and Ss data from the THT algorithm are from Liu et al. [2007, Figures 2f and 3f]. The following four statistics are computed to quantitatively assess the differences (or similarities) between the K and Ss tomograms from the HT-m and THT analyses: (1) the mean absolute error norm L1, (2) the mean square error norm L2, (3) the correlation coefficient (R), and (4) the similarity statistic [Hagen, 2003]. Unlike the norms and the correlation, the similarity statistic was developed to assess the similarity of geographical maps, remote sensing images, and other high-resolution spatial data through the quantitative assessment of fuzziness of feature locations and their values. The similarity value ranges from 0 to 1 and equals 0 for completely dissimilar images and 1 for an identical image.

[25] Table 3 summarizes the results of the statistics computed. It shows that the L1 and L2 statistics are relatively low for the K tomogram, while for Ss tomogram, it is unacceptably high. In terms of the correlation coefficient (R), there is a low correlation (R = 0.43) between the K tomograms from the HT-m and THT approaches. In contrast, there is virtually no correlation (R = 0.09) for the Ss tomograms from the two approaches. The same can be said about the similarity statistic.

Table 3. Comparison of Local K and Ss Values From the HT-m and THT Approachesa
StatisticK TomogramSs Tomogram
L10.78 cm/s5.15 cm−1
L21.00 cm2/s229.20 cm−2

[26] Therefore, on the basis of these results, we conclude that the K tomogram obtained via the HT-m approach is reasonable, but the Ss tomogram is unreliable. This is due to the fact that the HT-m approach relies on temporal moments of the drawdown recovery data, which involves integration. Data integration causes smoothing and perhaps loss of information on aquifer heterogeneity. In particular, the computation of moments and integration could cause significant loss of information on Ss which is contained in the early time data. Previous research by Zhu and Yeh [2006] suggests that this may be the case. Another possible explanation is that the use of the temporal moments, which requires the use of entire drawdown recovery curves, may result in a loss of sensitivity to the estimation of Ss. We also found that the characteristic time was considerably noisier than the zeroth moments contributing to the ineffectiveness in estimating Ss. Therefore, the estimation of Ss tomogram using the HT-m approach was found to be more challenging than the THT approach.

[27] On the basis of the findings from our sandbox study, we recommend the use of the THT over the HT-m algorithm to interpret transient cross-hole test data to obtain more reliable K and Ss tomograms. Computational speed has been more important in the past, but has become less of an issue more recently with the prevalence of more powerful and faster computers.


[28] This research was supported by the Strategic Environmental Research and Development Program (SERDP) under grant ER-1365, by the National Science Foundation (NSF) under grants EAR-0229713, IIS-0431069, and EAR-0450336 and by the Discovery grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. We thank J. Xiang, T.-C. J. Yeh, and J. Zhu for their assistance in using the hydraulic tomography, wavelet, and similarity analyses codes. We also thank the associate editor and the three anonymous reviewers for their constructive suggestions, which significantly improved the manuscript.