Transient Hydraulic Tomography Research Articles

HT2DINV: A 2D forward and inverse code for steady-state and transient hydraulic tomography problems

Hydraulic tomography is a technique used to characterize the spatial heterogeneities of storativity and transmissivity fields. The responses of an aquifer to a source of hydraulic stimulations are used to recover the features of the estimated fields using inverse techniques. We developed a 2D free source Matlab package for performing hydraulic tomography analysis in steady state and transient regimes. The package uses the finite elements method to solve the ground water flow equation for simple or complex geometries accounting for the anisotropy of the material properties. The inverse problem is based on implementing the geostatistical quasi-linear approach of Kitanidis combined with the adjoint-state method to compute the required sensitivity matrices. For undetermined inverse problems, the adjoint-state method provides a faster and more accurate approach for the evaluation of sensitivity matrices compared with the finite differences method. Our methodology is organized in a way that permits the end-user to activate parallel computing in order to reduce the computational burden. Three case studies are investigated demonstrating the robustness and efficiency of our approach for inverting hydraulic parameters.

A fast algorithm for parabolic PDE-based inverse problems based on Laplace transforms and flexible Krylov solvers

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method.

Hydraulic Tomography: Continuity and Discontinuity of High-K and Low-K Zones.

Hydraulic tomography is an emerging field and modeling method that provides a continuous hydraulic conductivity (K) distribution for an investigated region. Characterization approaches that rely on interpolation between one-dimensional (1D) profiles have limited ability to accurately identify high-K channels, juxtapositions of lenses with high K contrast, and breaches in layers or channels between such profiles. However, locating these features is especially important for groundwater flow and transport modeling, and for design and operation of in situ remediation in complex hydrogeologic environments. We use transient hydraulic tomography to estimate 3D K in a volume of 15-m diameter by 20-m saturated thickness in a highly heterogeneous unconfined alluvial (clay to sand-and-gravel) aquifer with a K range of approximately seven orders of magnitude at an active industrial site in Assemini, Sardinia, Italy. A modified Levenberg-Marquardt algorithm was used for geostatistical inversion to deal with the nonlinear nature of the highly heterogeneous system. The imaging results are validated with pumping tests not used in the tomographic inversion. These tests were conducted from three of five clusters of continuous multichannel tubing (CMTs) installed for observation in the tomographic testing. Locations of high-K continuity and discontinuity, juxtaposition of very high-K and very low-K lenses, and low-K "plugs" are evident in regions of the investigated volume where they likely would not have been identified with interpolation from 1D profiles at the positions of the pumping well and five CMT clusters. Quality assessment methods identified a suspect high-K feature between the tested volume and a lateral boundary of the model.

Restarted inverse Born series for the Schrödinger problem with discrete internal measurements

Convergence and stability results for the inverse Born series (Moskow and Schotland 2008 Inverse Problems 24 065005) are generalized to mappings between Banach spaces. We show that by restarting the inverse Born series one obtains a class of iterative methods containing the Gauss–Newton and Chebyshev–Halley methods. We use the generalized inverse Born series results to show convergence of the inverse Born series for the Schrödinger problem with discrete internal measurements. In this problem, the Schrödinger potential is to be recovered from a few measurements of solutions to the Schrödinger equation resulting from a few different source terms. An application of this method to a problem related to transient hydraulic tomography is given, where the source terms model injection and measurement wells.

Hydraulic conductivity imaging from 3‐D transient hydraulic tomography at several pumping/observation densities

3‐D Hydraulic tomography (3‐D HT) is a method for aquifer characterization whereby the 3‐D spatial distribution of aquifer flow parameters (primarily hydraulic conductivity, K) is estimated by joint inversion of head change data from multiple partially penetrating pumping tests. While performance of 3‐D HT has been studied extensively in numerical experiments, few field studies have demonstrated the real‐world performance of 3‐D HT. Here we report on a 3‐D transient hydraulic tomography (3‐D THT) field experiment at the Boise Hydrogeophysical Research Site which is different from prior approaches in that it represents a “baseline” analysis of 3‐D THT performance using only a single arrangement of a central pumping well and five observation wells with nearly complete pumping and observation coverage at 1 m intervals. We jointly analyze all pumping tests using a geostatistical approach based on the quasi‐linear estimator of Kitanidis (1995). We reanalyze the system after progressively removing pumping and/or observation intervals; significant progressive loss of information about heterogeneity is quantified as reduced variance of the K field overall, reduced correlation with slug test K estimates at wells, and reduced ability to accurately predict independent pumping tests. We verify that imaging accuracy is strongly improved by pumping and observational densities comparable to the aquifer heterogeneity geostatistical correlation lengths. Discrepancies between K profiles at wells, as obtained from HT and slug tests, are greatest at the tops and bottoms of wells where HT observation coverage was lacking.

Transient hydraulic tomography in a fractured dolostone: Laboratory rock block experiments

The accurate characterization of fractured geologic medium, imaging of fracture patterns and their connectivity have been a challenge for decades. Recently, hydraulic tomography has been proposed as a new method for imaging the hydraulic conductivity (K) and specific storage (Ss) distributions of fractured geologic media. While encouraging results have been obtained in the field, the method has not been rigorously assessed in a controlled laboratory setting. In this study, we assess the performance of transient hydraulic tomography (THT) in a fractured dolomitic rock block. The block is characterized through flow‐through tests and multiple pumping tests. The pumping test data were then analyzed with the THT code of Zhu and Yeh (2005) to image the fracture patterns and their connectivity through the delineation of K and Ss distributions (or tomograms). Results show that the THT analysis of pumping tests yields high‐K and low‐Ss zones that capture the fracture pattern and their connectivity quite well and those patterns become more vivid as additional pumping test data are added to the inverse model. The performance of the estimated K and Ss tomograms are then assessed by: (1) comparing the tomograms obtained from synthetic to real data; (2) comparing the tomograms from two different pumping configurations; (3) comparing the estimated geometric mean of the hydraulic conductivity (KG) from the K tomogram to the effective hydraulic conductivity (Keff) estimated from the flow‐through tests; and (4) predicting five independent pumping tests not used in the construction of the K and Ss tomograms. The performance assessment of the K and Ss tomograms reveals that THT is able to image high‐K and low‐Ss zones that correspond to fracture locations in the fractured rock block and that the tomograms can be used to predict drawdowns from pumping tests not used in the construction of the tomograms with reasonable fidelity.

Improved predictions of saturated and unsaturated zone drawdowns in a heterogeneous unconfined aquifer via transient hydraulic tomography: Laboratory sandbox experiments

► Uniform parameters lead to biased drawdown predictions in the (un)saturated zones. ► Accounting for aquifer heterogeneity yields significant improvements in predictions. ► Considering unsaturated zone (UZ) heterogeneity produced a minor improvement. ► UZ heterogeneity plays a minor role in forming the S shape drawdown curve. ► Hydraulic tomography can be used to characterize unconfined aquifers. Interpretation of pumping tests in unconfined aquifers has largely been based on analytical solutions that disregard aquifer heterogeneity. In this study, we investigate whether the prediction of drawdown responses in a heterogeneous unconfined aquifer and the unsaturated zone above it with a variably saturated groundwater flow model can be improved by including information on hydraulic conductivity ( K ) and specific storage ( S s ) from transient hydraulic tomography (THT). We also investigate whether these predictions are affected by the use of unsaturated flow parameters estimated through laboratory hanging column experiments or calibration of in situ drainage curves. To investigate these issues, we designed and conducted laboratory sandbox experiments to characterize the saturated and unsaturated properties of a heterogeneous unconfined aquifer. Specifically, we conducted pumping tests under fully saturated conditions and interpreted the drawdown responses by treating the medium to be either homogeneous or heterogeneous. We then conducted another pumping test and allowed the water table to drop, similar to a pumping test in an unconfined aquifer. Simulations conducted using a variably saturated flow model revealed: (1) homogeneous parameters in the saturated and unsaturated zones have a difficult time predicting the responses of the heterogeneous unconfined aquifer; (2) heterogeneous saturated hydraulic parameter distributions obtained via THT yielded significantly improved drawdown predictions in the saturated zone of the unconfined aquifer; and (3) considering heterogeneity of unsaturated zone parameters produced a minor improvement in predictions in the unsaturated zone, but not the saturated zone. These results seem to support the finding by Mao et al. (2011) that spatial variability in the unsaturated zone plays a minor role in the formation of the S-shape drawdown-time curve observed during pumping in an unconfined aquifer.

A field proof‐of‐concept of aquifer imaging using 3‐D transient hydraulic tomography with modular, temporarily‐emplaced equipment

Hydraulic tomography is a field scale aquifer characterization method capable of estimating 3‐D heterogeneous parameter distributions, and is directly sensitive to hydraulic conductivity (K), thus providing a useful data source for improving flow and transport models. We present results from a proof‐of‐concept field and modeling study in which we apply 3‐D transient hydraulic tomography (3DTHT) to the relatively high‐K and moderately heterogeneous unconfined aquifer at the Boise Hydrogeophysical Research Site. Short‐duration (20 min) partially penetrating pumping tests, for which observed responses do not reach steady state, are used as the aquifer stimulation. To collect field data, we utilize a system of temporarily emplaced packer equipment to isolate multiple discrete intervals in boreholes. To analyze the data, we utilize MODFLOW combined with geostatistical inversion code based on the quasilinear approach of Kitanidis (1995). This combination of practical software allows inversion of large datasets (>250 drawdown curves, and almost 1000 individual data points) and estimation of K at >100,000 locations; reasonable runtimes are obtained using a single multicore computer with 12 GB of RAM. The K heterogeneity results from 3DTHT are cross‐validated against K characterization from a large set of partially penetrating slug tests, and found to be quite consistent. The use of portable, modular equipment for field implementation means that 3DTHT data collection can be performed (including mobilization/demobilization) within a matter of days. Likewise, use of a practical, efficient and scalable numerical modeling and inversion strategy means that computational effort is drastically reduced, such that 3‐D aquifer property distributions can be estimated quickly.

3‐D transient hydraulic tomography in unconfined aquifers with fast drainage response

We investigate, through numerical experiments, the viability of three‐dimensional transient hydraulic tomography (3DTHT) for identifying the spatial distribution of groundwater flow parameters (primarily, hydraulic conductivity K) in permeable, unconfined aquifers. To invert the large amount of transient data collected from 3DTHT surveys, we utilize an iterative geostatistical inversion strategy in which outer iterations progressively increase the number of data points fitted and inner iterations solve the quasi‐linear geostatistical formulas of Kitanidis. In order to base our numerical experiments around realistic scenarios, we utilize pumping rates, geometries, and test lengths similar to those attainable during 3DTHT field campaigns performed at the Boise Hydrogeophysical Research Site (BHRS). We also utilize hydrologic parameters that are similar to those observed at the BHRS and in other unconsolidated, unconfined fluvial aquifers. In addition to estimating K, we test the ability of 3DTHT to estimate both average storage values (specific storage Ss and specific yield Sy) as well as spatial variability in storage coefficients. The effects of model conceptualization errors during unconfined 3DTHT are investigated including: (1) assuming constant storage coefficients during inversion and (2) assuming stationary geostatistical parameter variability. Overall, our findings indicate that estimation of K is slightly degraded if storage parameters must be jointly estimated, but that this effect is quite small compared with the degradation of estimates due to violation of “structural” geostatistical assumptions. Practically, we find for our scenarios that assuming constant storage values during inversion does not appear to have a significant effect on K estimates or uncertainty bounds.

Three‐dimensional transient hydraulic tomography in a highly heterogeneous glaciofluvial aquifer‐aquitard system

Hydraulic tomography has been proposed as an alternative site characterization method, however, relatively few field scale studies have been attempted. In this paper, we characterize the highly heterogeneous glaciofluvial aquifer‐aquitard system at the North Campus Research Site, located at the University of Waterloo, Waterloo, Ontario, Canada using transient hydraulic tomography (THT). In particular, we performed 9 pumping tests in a network of wells to image the hydraulic conductivity (K) and specific storage (Ss) distributions (or tomograms) as well as their uncertainties in three‐dimensions using the THT code of J. Zhu and T.‐C. J. Yeh (2005). We first performed stochastic inverse modeling of the 9 pumping tests individually to gain insight into the level of detail that can be imaged. Then, we sequentially included 4 of the pumping tests in a THT analysis. The resulting K and Ss tomograms were then validated visually by comparing them to stratigraphy and permeameter K estimates. The K and Ss tomograms were also rigorously assessed through the simulation of all 9 pumping tests and comparing the simulated and observed drawdowns. We find that performing the inversion with multiple pumping tests (i.e., hydraulic tomography) yields improved results when compared to the analysis of individual pumping tests.

Capturing aquifer heterogeneity: Comparison of approaches through controlled sandbox experiments

Groundwater modeling has become a vital component to water supply and contaminant transport investigations. These models require representative hydraulic conductivity (K) and specific storage (Ss) estimates, or a set of estimates representing subsurface heterogeneity. Currently, there are a number of approaches for characterizing and modeling K and Ss heterogeneity in varying degrees of detail, but there is a lack of consensus for an approach that results in the most robust groundwater models with the best predictive ability. The main goal of this study is to compare different heterogeneity modeling approaches (e.g., effective parameters, geostatistics, geological models, and hydraulic tomography) when input into a forward groundwater model and used to predict 16 independent cross‐hole pumping tests. We first characterize a sandbox aquifer through single‐ and cross‐hole pumping tests, and then use these data to construct forward groundwater models of various complexities (both homogeneous and heterogeneous distributions). Two effective parameter models are constructed: (1) by taking the geometric mean of single‐hole test K and Ss estimates and (2) calibrating effective K and Ss estimates by simultaneously matching the response at all ports during a cross‐hole test. Heterogeneous models consist of spatially variable K and Ss fields obtained via (1) kriging single‐hole data; (2) calibrating a geological model; and (3) conducting transient hydraulic tomography (Zhu and Yeh, 2005). The performance of these parameter fields are then tested through the simulation of 16 independent cross‐hole pumping tests. Our results convincingly show that transient hydraulic tomography produces the smallest discrepancy between observed and simulated drawdowns.

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

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.

Hydraulic tomography in fractured granite: Mizunami Underground Research site, Japan

Two large‐scale cross‐hole pumping tests were conducted at depths of 191–226 m and 662–706 m in deep boreholes at the Mizunami Underground Research Laboratory (MIU) construction site in central Japan. During these two tests, induced groundwater responses were monitored at many observation intervals at various depths in different boreholes at the site. We analyze the two cross‐hole pumping tests using transient hydraulic tomography (THT) based on an efficient sequential successive linear estimator to compute the hydraulic conductivity (K) and specific storage (Ss) tomograms, as well as their uncertainties in three dimensions. The equivalent K and Ss estimates obtained using asymptotic analysis treating the medium to be homogeneous served as the mean parameter estimates for the 3‐D stochastic inverse modeling effort. Results show several, distinct, high K and low Ss zones that are continuous over hundreds of meters, which appear to delineate fault zones and their connectivity. The THT analysis of the tests also identified a low K zone which corresponds with a known fault zone trending NNW and has been found to compartmentalize groundwater flow at the site. These results corroborate well with observed water level records, available fault information, and coseismic groundwater level responses during several large earthquakes. The successful application of THT to cross‐hole pumping tests conducted in fractured granite at this site suggests that THT is a promising approach to delineate large‐scale K and Ss heterogeneities, fracture connectivity, and to quantify uncertainty of the estimated fields.

Laboratory sandbox validation of transient hydraulic tomography

Hydraulic tomography is a method that images the hydraulic heterogeneity of the subsurface through the inversion of multiple pumping or cross‐hole hydraulic test data. Transient hydraulic tomography is different from steady state hydraulic tomography in that it utilizes transient hydraulic head records to yield the distribution of hydraulic conductivity (K) as well as specific storage (Ss) of an aquifer. In this paper we demonstrate the robustness of transient hydraulic tomography through the use of hydraulic head data obtained from multiple cross‐hole pumping tests conducted in a laboratory sandbox with deterministic heterogeneity. We utilize the algorithm developed by Zhu and Yeh (2005) to conduct the transient inversions and validate the K and Ss tomograms using a multimethod and multiscale validation approach previously proposed by Illman et al. (2006). Validation data consist of cross‐hole tests not used in the inversion as well as other hydraulic tests that provided local (core, single‐hole tests) as well as large‐scale (unidirectional flow‐through tests) estimates of hydraulic parameters. Results show that the algorithm is able to yield consistent estimates that agree with independently collected local as well as large‐scale hydraulic parameter data. In addition, we find that the transient hydraulic tomography requires a fewer number of pumping tests to estimate a similar quality K tomogram when compared with steady state hydraulic tomography, as the former approach utilizes more data from each pumping test. Overall, we find that transient hydraulic tomography is a robust subsurface characterization technique that can delineate the subsurface heterogeneity in both K and Ss from multiple pumping or cross‐hole hydraulic tests.

Analysis of hydraulic tomography using temporal moments of drawdown recovery data

Transient hydraulic tomography (THT) is a potentially cost‐effective and high‐resolution technique for mapping spatial distributions of the hydraulic conductivity and specific storage in aquifers. Interpretation of abundant well hydrographs of a THT survey, however, is a computational challenge. We take on this challenge by developing an estimation approach that utilizes the zeroth and first temporal moments of well hydrographs, instead of drawdown itself. 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 evolution. Likewise, the adjoint equations for evaluating sensitivities of the moments for parameter estimation also take the same forms. Therefore a temporal moment approach is expected to expedite the interpretation of THT surveys. On the basis of this premise we extend our sequential successive linear estimator to use the zeroth moment and characteristic time of the drawdown‐recovery data generated by THT surveys. We subsequently investigate computational efficiency and accuracy of the moment approach. Results of the investigation show that the temporal moment approach yields results similar to those from the approach that uses transient heads but at significantly less computational costs. Limitations using temporal moments are discussed subsequently.

Characterization of aquifer heterogeneity using transient hydraulic tomography

Hydraulic tomography is a cost‐effective technique for characterizing the heterogeneity of hydraulic parameters in the subsurface. During hydraulic tomography surveys a large number of hydraulic heads (i.e., aquifer responses) are collected from a series of pumping or injection tests in an aquifer. These responses are then used to interpret the spatial distribution of hydraulic parameters of the aquifer using inverse modeling. In this study, we developed an efficient sequential successive linear estimator (SSLE) for interpreting data from transient hydraulic tomography to estimate three‐dimensional hydraulic conductivity and specific storage fields of aquifers. We first explored this estimator for transient hydraulic tomography in a hypothetical one‐dimensional aquifer. Results show that during a pumping test, transient heads are highly correlated with specific storage at early time but with hydraulic conductivity at late time. Therefore reliable estimates of both hydraulic conductivity and specific storage must exploit the head data at both early and late times. Our study also shows that the transient heads are highly correlated over time, implying only infrequent head measurements are needed during the estimation. Applying this sampling strategy to a well‐posed problem, we show that our SSLE can produce accurate estimates of both hydraulic conductivity and specific storage fields. The benefit of hydraulic tomography for ill‐posed problems is then demonstrated. Finally, to affirm the robustness of our SSLE approach, we apply the SSLE approach to a hypothetical three‐dimensional heterogeneous aquifer.