Successive Linear Estimator Research Articles

Comparison of Hydraulic Travel Time and Attenuation Inversions, Thermal Tracer Tomography and Geostatistical Inversion for Aquifer Characterization: A Numerical Study

For the characterization of heterogeneous aquifers, transient hydraulic tomography (THT) was proposed as a promising method to obtain the distribution of hydraulic parameters with satisfying spatial resolution using different approaches. These include hydraulic travel time, attenuation inversions, thermal tracer tomography, and geostatistical inversion with successive linear estimator (SLE). For the same hydrogeological test, different inversion methods tend to use different sub-data sets to obtain different hydraulic parameters. Up to now, however, few studies have focused on revealing the respective characteristics of these inversion methods and attempted to improve the accuracy of aquifer characterization by bridging the shortcomings of the inversion methods. The main objective of this study was to evaluate the utility of multiple inversion techniques on aquifer heterogeneity characterization. A series of warm water injection tests were first simulated in a fluvial aquifer analogue outcrop. The calculated head and temperature datasets from these tests were fully utilized to reveal the aquifer heterogeneity by using all of the four above-mentioned inversion methods. The results show that the thermal tracer tomography, hydraulic travel time, and attenuation tomography characterized the high permeability zones more accurately within the well area, whereas the geological statistical method tended to depict the overall distribution of K values for a larger area. By comparison analysis and combinations of the individual inversion results, the scientific and economic complementarity can be studied and some valuable advice for the choice of different inversion methods can be recommended for future practices.

A Short‐Term Pumping Strategy for Hydraulic Tomography Based on the Successive Linear Estimator

AbstractIn this study, the random finite element method, a finite element method with random field generation techniques, was applied to investigate the cross correlations between the observed head and hydraulic conductivity and specific storage at different locations and different times in pumping tests. The results show that the two cross correlations between the pumping well and the observation well reach their maximums before pumping reaches a steady state. Specifically, the cross correlation between the observed head and hydraulic conductivity is the greatest when the temporal derivative of the observed head does not change significantly, and that between the observed head and specific storage is the greatest when the temporal derivative (the rate) of the observed head is maximum. Based on the results of cross‐correlation analysis, a short‐term pumping strategy for hydraulic tomography is proposed to obtain the spatial distribution of hydraulic conductivity and specific storage using the successive linear estimator. Furthermore, this strategy was validated by Monte Carlo simulations. This paper points out that the sensitivity and cross‐correlation analyses report the ensemble (averaged) behaviors of any heterogeneous aquifers, which is not necessarily suitable for one realization. Furthermore, Monte Carlo simulation is suggested for validating any groundwater inverse modeling result.

Mapping conduits in two-dimensional heterogeneous karst aquifers using hydraulic tomography

Hydraulic tomography (HT) is a well-established approach to yield the spatial distribution of hydraulic conductivity of an aquifer. This work explores the potential of HT for the characterization of the distribution and connectivity of conduits in a two-dimensional sandbox and its corresponding synthetic aquifer. Two inversion techniques were implemented and compared: the geostatistics-based inversion which uses the simultaneous successive linear estimator (SimSLE) algorithm to conduct stochastic inversions on the transient hydraulic heads, and the travel time-based inversion which employs the simultaneous iterative reconstruction technique (SIRT) algorithm on the hydraulic travel times for tomography reconstructions. Four artificial karst conduits of different geometries were placed in an aquifer of layer with different hydraulic conductivities. We conducted 6 pumping tests at 6 different locations, and the resultant pressure responses were recorded at 42 observation points in both the sandbox and the corresponding synthetic aquifer. The measured data were then used for the inversion of hydraulic diffusivity using the SimSLE and SIRT algorithms. Our results show that both algorithms were able to approximately identify the embedded karst conduits and yield similar hydraulic diffusivity distribution. Statistically, the travel time-based inversion re-constructed high-contrast diffusivities which clearly differentiate the karst structures from the surrounding matrix. The geostatistics-based SimSLE algorithm yields a better agreement on the positions and the shapes of the embedded karst structures, compared to those obtained by the travel time-based SIRT algorithm. Uncertainties and limitations of our results are also discussed in this work, followed by recommendations on hydraulic tests in karst aquifers.

Exploration of Ordovician limestone aquifer heterogeneity with tomographic water releasing tests

Water inrush is a typical geological disaster in the process of underground mining. It is crucial to depict potential water inrush areas for mining safety. In this study, mining activity in Dongtan coal mine is under the water inrush threat from deep Ordovician limestone confined aquifers. Tomographic water releasing tests and simultaneous successive linear estimator (SimSLE) were conducted to characterize aquifer heterogeneity by stimulating hundreds of square kilometers. An artificial injection well was firstly introduced to alleviate the influence from identified recharge channels. The estimated parameter distributions indicated that the study area was divided into two separately hydrogeological compartments. Transmissivity (T) and storativity (S) in the north were relatively higher due to water-conducting structures, and more attention should be paid to this region which may develop into water inrush incidents. The southern area is occupied with lower T and S, which indicated relatively worse flow conditions. Auxiliary drawdown recovery data, borehole chemistry variations, drilling fluid losses, and existing geological features further confirmed the estimated parameter distributions. In the end, we concluded that Hydraulic Tomography technique (HT) could obtain aquifer parameter information in large scales and provide supports to formulate reasonably effective water prevention and control schemes before mining deep coal seams.

Numerical investigation of hydraulic tomography for mapping karst conduits and its connectivity

Hydraulic tomography (HT) is a well-established approach to successfully describe subsurface hydraulic heterogeneity. This work explored the potential of HT and the simultaneous successive linear estimator algorithm to map the distribution and connectivity of karst conduits and their hydraulic parameters at the engineering scale. Three generalized synthetic karst conduit models, auxiliary conduit, cave and waterfall, were established. A series of pumping tests were conducted at different locations as forward simulations to collect hydraulic head responses. These responses were then used for the HT inversion analysis to estimate hydraulic parameter fields of karst aquifers under steady-state and transient conditions. Our results show that high-density observation wells substantially assist with interpreting the reliable spatial distribution of hydraulic parameters. With the increase in the number of pumping wells in the HT inversion, the estimates of hydraulic conductivity and specific storage became more accurate for the auxiliary conduit model, but for the cave and waterfall models, there was no significant improvement. The hydraulic conductivity (K) tomograms from transient hydraulic tomography (THT) better indicated the heterogeneity of the three karst models than those from steady-state hydraulic tomography (SSHT) because a large number of hydraulic head records were introduced into the THT inversion process. Moreover, when reliable geological data are available, the accuracy of the estimated hydraulic parameter maps can be greatly improved even if only infrequent hydraulic head measurements can be obtained, implying that a priori geological information may be extremely valuable in mapping karst conduit heterogeneity. Finally, the robustness of HT in mapping karst conduits is verified through the simulation of independent cross-hole pumping tests. This study proved that hydraulic tomography utilizing groundwater responses from a series of cross-hole pumping tests is an efficient and cost-effective way and provided a feasible method for accurate identification of hydraulic heterogeneity of karst aquifers.

Conditional mean, effective, and realizations of hydraulic conductivity fields

This study first discusses the conditional mean, realizations, and effective hydraulic conductivity in a theoretical framework. It then introduces Monte Carlo simulation (MCS) algorithms for constraining the outcome by either hydraulic conductivity (K) samples or hydraulic head (h) measurements from the hydraulic tomographic survey (HT). It demonstrates that kriging using K measurements leads to a conditional mean K field, while inverse modeling using successive linear estimator (SLE) with head measurements of HT yields the conditional effective K field. The effects of conditioning using K measurements are different from those using heads. Besides, the conditional effective K leads to the unbiased prediction of the head that honors the observed head at measurement locations. More importantly, the study reveals that the harmonic and geometric means of conditional realizations of K fields of MCS, given head measurements, are equivalent to the conditional effective K in one- and two-dimensional flows, respectively. The first-order approximation in the SLE results in a conditional covariance similar to that from MCS with smaller magnitudes. Despite the difference, all approaches predict unbiased conditional mean head behaviors.

Influence of correlation scale errors on aquifer hydraulic conductivity inversion precision

In order to investigate the influence of correlation scale error on the inversion precision of the hydraulic conductivity of the aquifer, the successive linear estimator (SLE) was used to invert the hydraulic conductivity field of a heterogeneous aquifer based on synthetic experiments. By increasing the numbers of observation wells and pumping tests, we analyzed the difference between the estimated and true values of hydraulic conductivity with different correlation scale errors. The relationships between the observation well number and the error in inversion results, and between the pumping test number and the error in inversion results were investigated. The results show that, if the amount of observed head data is insufficient, there will be errors in inversion results with changing correlation scale. Due to the existence of correlation scale error, the improvement of inversion precision gradually slows down with the increase of the amount of observed head data, which indicates that too much observed head data causes data redundancy. Therefore, for the synthetic experiments described in this paper, the observation well number should be less than 41, the pumping test number should be less than 17, and a more suitable method should be selected according to the precision requirements of specific situations in practical engineering.

Identification of groundwater basin shape and boundary using hydraulic tomography

Shapes and boundary types of a groundwater basin play essential roles in the analysis of groundwater management and contaminant migration. Hydraulic tomography (HT), a recently developed new approach for high-resolution characterization of aquifers, is not only an inverse method but a logical strategy for collecting non-redundant hydraulic information. In this study, HT was applied to synthetic 2-D aquifers to investigate its feasibility to map the irregular shapes and types of the aquifer boundaries. We first used the forward model of VSAFT2 to simulate hydraulic responses due to HT surveys in the aquifer with irregular geometry and predetermined constant head conditions at some boundaries, and no-flow conditions at others. The SimSLE (Simultaneous Successive Linear Estimator) inverse model in VSAFT2 was then used to interpret the simulated HT data to estimate the spatial distribution of hydraulic properties of the aquifer using a domain with a wrong geometry surrounded by boundaries of a constant head condition. The inverse modeling experiment used steady-state and transient-states data from the HT forward simulations, and it used the same monitoring network as in the aquifer with irregular geometry to assess the ability of HT for detecting types and shapes of the boundary as well as heterogeneity in the aquifer. Results of the experiment show that no-flow boundaries, which were incorrectly treated as constant head boundaries in inverse models, were portrayed as low permeable zones of the aquifer near the boundaries. Overall, the results show that HT could delineate not only the irregular shape of the aquifer in general but also heterogeneity in the aquifer. Improvements of the estimation with prior information of transmissivity and storage coefficient was also investigated. The study shows that using homogeneous initial guess parameters resulted in a slightly better estimate than others. Moreover, this study employs Monte Carlo simulations to ensure statistically meaningful conclusions.

Mapping soil layers using electrical resistivity tomography and validation: Sandbox experiments

Knowledge of the geologic structure at a field site is a useful piece of information for hydrologic modeling since it can serve as more site-specific prior information about the hydraulic parameter patterns at the site than generic spatial statistics. Widely accepted electrical resistivity tomography (ERT) survey for mapping subsurface anomalies could be a viable tool for acquiring this information. However, their ability to delineate geologic structures has not been thoroughly investigated. In this study, two-dimensional ERT numerical experiments were first conducted to study the effects of boundary conditions on the dipole-dipole and pole-pole array configurations. An ERT setup subsequently was implemented in a sandbox consisting of complex layers of different sands. A continuous copper wire was installed along the sides of the sandbox to impose potential boundaries. Using data collected with the pole-pole array in the sandbox under different degrees of drainage and the Successive Linear Estimator (SLE) algorithm, we show that ERT yields electrical conductivity estimates of complex layer structures with small uncertainties. In addition, using SLE with physically meaningful correlation scales as prior information can lead to an electrical conductivity field that is consistent with visually observed layer structures. The correlation scale concept also was demonstrated to provide guidance to the design of the electrode spacing in the surveys. Moreover, the estimated field was validated by predicting electrical potential fields from two independent ERT surveys using electrodes at different locations. Results of this study suggest that the combination of ERT and SLE is a viable geophysical survey tool for mapping geologic layer structures. Research SignificanceThis study develops a method to implement prescribed potential boundaries for enhancing the pole-pole ERT survey, illustrates the importance of correlation scales and develops an approach for validating ERT results.

Fusion of Hydraulic Tomography and Displacement Back Analysis for Underground Cavern Stability Investigation

AbstractThis paper first elucidates the importance of hydraulic heterogeneity of a rock mass on the design of an underground water‐sealed storage cavern. It then introduces an information fusion approach that integrates hydraulic tomography, displacement back analysis, and a stochastic successive linear estimator to map spatially varying hydraulic conductivity (Ks), Young's modulus (E′), cohesion (c′), and internal friction angle (ϕ′). This algorithm yields statistically unbiased estimate of the Ks, E′, c′, and ϕ′ fields and their uncertainty due to insufficient samples of their heterogeneity. Afterward, this uncertainty is translated into the probability of failure distribution around the cavern for the risk assessment. Results show that heterogeneity of Ks plays an important role in the deformation and failure of the cavern. They also show that the proposed approach can reveal more detailed parameter distributions than kriging approach, based on a large number of samples of the parameters before excavations. Further, the predicted probability of failure distribution based on the proposed approach can effectively reflect the actual failure locations.

A Reduced‐Order Successive Linear Estimator for Geostatistical Inversion and its Application in Hydraulic Tomography

AbstractHydraulic tomography (HT) is a recently developed technology for characterizing high‐resolution, site‐specific heterogeneity using hydraulic data (nd) from a series of cross‐hole pumping tests. To properly account for the subsurface heterogeneity and to flexibly incorporate additional information, geostatistical inverse models, which permit a large number of spatially correlated unknowns (ny), are frequently used to interpret the collected data. However, the memory storage requirements for the covariance of the unknowns (ny × ny) in these models are prodigious for large‐scale 3‐D problems. Moreover, the sensitivity evaluation is often computationally intensive using traditional difference method (ny forward runs). Although employment of the adjoint method can reduce the cost to nd forward runs, the adjoint model requires intrusive coding effort. In order to resolve these issues, this paper presents a Reduced‐Order Successive Linear Estimator (ROSLE) for analyzing HT data. This new estimator approximates the covariance of the unknowns using Karhunen‐Loeve Expansion (KLE) truncated to nkl order, and it calculates the directional sensitivities (in the directions of nkl eigenvectors) to form the covariance and cross‐covariance used in the Successive Linear Estimator (SLE). In addition, the covariance of unknowns is updated every iteration by updating the eigenvalues and eigenfunctions. The computational advantages of the proposed algorithm are demonstrated through numerical experiments and a 3‐D transient HT analysis of data from a highly heterogeneous field site.

Estimation of permeability of naturally fractured reservoirs by pressure transient analysis: An innovative reservoir characterization and flow simulation

Fluid flow processes in naturally fractured reservoirs are strongly controlled by fracture intensity, their orientation, and interconnectivity. Therefore, knowledge of fracture properties plays a critical role in reservoir management. Developing a detailed description of subsurface fracture map is challenging for geothermal and petroleum industry due to a number of reasons: It is often difficult to obtain sufficient hard data such as borehole images and core description. Also, there is a general lack of quality of interpretation of soft data, such as well logs, seismic attributes, and tectonic history etc. for fracture interpretation. To overcome the short comings the industry has been relying heavily on geo-statistical analysis of both hard and soft data.In this paper we used well test data (dynamic data) to reduce the level of uncertainty of the existing methods for generating fracture map through an innovative inversion technique. The major inversion techniques include simulated annealing, sequential successive linear estimator, and gradient and streamline based method. In this paper we used gradient based method which utilises adjoint equation. The inversion is carried out in a number of steps. First we analyse hard and soft data to generate characteristic fracture properties such as fracture density and fractal dimension. Next we use geo-statistical technique to spatially distribute fractures. Then with use of gradient based technique, we optimise the fracture attributes through different realization. Next we use an innovative simulation of fluid flow through discrete fractures in 3D fractured porous media and estimated pressure and pressure derivatives. The pressure and pressure derivatives are compared with well test data taken from a typical fractured basement reservoir located in offshore Vietnam to determine percentage error.The results showed that the simulated pressure change and pressure derivatives match well with well test data. This has allowed us to capture the complex subsurface fracture pattern. This vital information can help the operator design an effective reservoir management plan.

Validation of hydraulic tomography in an unconfined aquifer: A controlled sandbox study

AbstractIn this study, we demonstrate the effectiveness of hydraulic tomography (HT) that considers variably saturated flow processes in mapping the heterogeneity of both the saturated and unsaturated zones in a laboratory unconfined aquifer. The successive linear estimator (SLE) developed by Mao et al. (2013c) for interpreting HT in unconfined aquifers is utilized to obtain tomograms of hydraulic conductivity (K), specific storage (Ss), and the unsaturated zone parameters (pore size parameter (α) and saturated water content (θs)) for the Gardner‐Russo's model. The estimated tomograms are first evaluated by visually comparing them with stratigraphy visible in the sandbox. Results reveal that the HT analysis is able to accurately capture the location and extent of heterogeneity including high and low K layers within the saturated and unsaturated zones, as well as reasonable distribution patterns of α and θs for the Gardner‐Russo's model. We then validate the estimated tomograms through predictions of drawdown responses of pumping tests not used during the inverse modeling effort. The strong agreement between simulated and observed drawdown curves obtained by pressure transducers and tensiometers demonstrates the robust performance of HT that considers variably saturated flow processes in unconfined aquifers and the unsaturated zone above it. In addition, compared to the case using the homogeneous assumption, HT results, as expected, yield significantly better predictions of drawdowns in both the saturated and unsaturated zones. This comparison further substantiates the unbiased and minimal variance of HT analysis with the SLE algorithm.

A stochastic approach for extended partial blockage detection in viscoelastic pipelines: numerical and laboratory experiments

Research Article| February 02 2015 A stochastic approach for extended partial blockage detection in viscoelastic pipelines: numerical and laboratory experiments C. Massari; C. Massari 1Research Institute for Geo-Hydrological Protection, National Research Council CNR, Via Madonna Alta, 126–06128 Perugia, Italy E-mail: christian.massari@irpi.cnr.it Search for other works by this author on: This Site PubMed Google Scholar T.-C. J. Yeh; T.-C. J. Yeh 2Department of Hydrology and Water Resources, The University of Arizona, John Harshbarger Bldg. 1133E J. E. Rogers Way, Tucson, AZ 85721, USA Search for other works by this author on: This Site PubMed Google Scholar M. Ferrante; M. Ferrante 3Dipartimento di Ingegneria Civile ed Ambientale, The University of Perugia, Via Duranti 93–06125 Perugia, Italy Search for other works by this author on: This Site PubMed Google Scholar B. Brunone; B. Brunone 3Dipartimento di Ingegneria Civile ed Ambientale, The University of Perugia, Via Duranti 93–06125 Perugia, Italy Search for other works by this author on: This Site PubMed Google Scholar S. Meniconi S. Meniconi 3Dipartimento di Ingegneria Civile ed Ambientale, The University of Perugia, Via Duranti 93–06125 Perugia, Italy Search for other works by this author on: This Site PubMed Google Scholar Journal of Water Supply: Research and Technology-Aqua (2015) 64 (5): 583–595. https://doi.org/10.2166/aqua.2015.034 Article history Received: March 22 2014 Accepted: December 09 2014 Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Cite Icon Cite Permissions Search Site Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll JournalsThis Journal Search Advanced Search Citation C. Massari, T.-C. J. Yeh, M. Ferrante, B. Brunone, S. Meniconi; A stochastic approach for extended partial blockage detection in viscoelastic pipelines: numerical and laboratory experiments. Journal of Water Supply: Research and Technology-Aqua 1 August 2015; 64 (5): 583–595. doi: https://doi.org/10.2166/aqua.2015.034 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex Partial blockages in water pipe networks may contribute to large energy dissipation throughout the system and reduce service effectiveness for customers. In this paper, a recently developed stochastic model using transients for detecting partial blockages in water pipelines is tested and numerical and experimental case studies are presented. The model is a stochastic successive linear estimator previously used in groundwater hydrology for detecting the heterogeneity pattern of the subsurface. The model estimates the distribution of diameters within a pipe with partial blockages and quantifies the uncertainty associated with these estimates. Results show that a first good estimate of the extent and size of the blockages can be obtained by a single test generated by the fast closure of a valve. partial blockages, pipe system diagnosis, stochastic estimator, transients, viscoelastic © IWA Publishing 2015 You do not currently have access to this content.

Fast iterative implementation of large-scale nonlinear geostatistical inverse modeling

[1] In nonlinear geostatistical inverse problems, it often takes a significant amount of computational cost to form linear geostatistical inversion systems by linearizing the forward model. More specifically, the storage cost associated with the sensitivity matrix H (m × n, where m and n are the numbers of measurements and unknowns, respectively) is high, especially when both m and n are large in for instance, 3-D tomography problems. In this research, instead of explicitly forming and directly solving the linear geostatistical inversion system, we use MINRES, a Krylov subspace method, to solve it iteratively. During each iteration in MINRES, we only compute the products Hx and HTx for any appropriately sized vectors x, for which we solve the forward problem twice. As a result, we reduce the memory requirement from O(mn) to O(m)+O(n). This iterative methodology is combined with the Bayesian inverse method in Kitanidis (1996) to solve large-scale inversion problems. The computational advantages of our methodology are demonstrated using a large-scale 3-D numerical hydraulic tomography problem with transient pressure measurements (250,000 unknowns and ∼100,000 measurements). In this case, ∼200 GB of memory would otherwise be required to fully compute and store the sensitivity matrix H at each Newton step during optimization. The CPU cost can also be significantly reduced in terms of the total number of forward simulations. In the end, we discuss potential extension of the methodology to other geostatistical methods such as the Successive Linear Estimator.

A Stochastic Tool for Determining the Presence of Partial Blockages in Viscoelastic Pipelines: First Experimental Results

Partial blockages in water pipe networks may contribute to large energy dissipation throughout the system and reduce the service effectiveness for the customers. In this paper, a recently developed stochastic model using transient head measurements for detecting partial blockages in water pipelines is tested with two experimental case studies. The model is a stochastic Successive Linear Estimator (SLE, Yeh et al., 1996) previously used in groundwater hydrology for detecting the heterogeneity pattern of the subsurface. The model provides statistically the best unbiased estimate of diameter distribution due to partial blockages and quantifies the uncertainty associated with these estimates. Tests were carried out at the Water Engineering Laboratory of the University of Perugia (WEL) on two high density polyethylene pipe systems. In the experiments the partial blockages were simulated by placing a smaller diameter pipe of two different lengths between the main pipe. Results show that a first good estimate of the length and the size of the blockage can be obtained by a single short duration transient test executed by maneuvering a ball valve located far away from the blockage position.

Diagnosis of Pipe Systems by means of a Stochastic Successive Linear Estimator

Environmental concerns and legislation of the water industry have recently drawn the attention of many researchers to the management, calibration and power cost reduction of water pipeline systems. Effective water system management rests upon the knowledge of the current state of a water pipeline system. For example, leakages, unknown status of in-line valves, and partial blockages are often a source of management costs. These anomalies must be detected and corrected as early as possible. In this paper a diagnosis tool is presented able to detect leaks, partially closed in-line valves and partial blockages by means of transient head measurements. The algorithm introduced is a Stochastic successive Linear Estimator [SLE] (Yeh et al. Water Resour Res 32: 2757–2766, 1996a) which provides statistically best unbiased estimate of these anomalies and quantifies the uncertainty associated with these estimates via assimilation of available information. Therefore, the information from common diagnosis techniques e.g., inspection methods, and head measurements available at different locations of the system, obtained by different transient tests are fused to provide the most accurate and reliable diagnosis.

Diagnosis of pipe systems by the SLE: first results

In this study, an algorithm is presented for pipe system diagnosis. The algorithm is the Successive Linear Estimator (SLE) previously used in groundwater hydrology to estimate the hydraulic proprieties of soils. The SLE is an iterative cokriging-like technique which provides statistically the best unbiased estimates and quantifies the associated uncertainty. In this paper the algorithm has been extended to the pipe system diagnosis to infer leak areas, status of valves and partial blockage location and extent. We first present the theoretical formulation of the algorithm and then test it by a numerical case study.

Detection and sizing of extended partial blockages in pipelines by means of a stochastic successive linear estimator

Effective water system management depends upon knowledge of the current state of a water pipeline system network. For example, in many cases, partial blockages in a water pipeline system are a source of inefficiencies, and result in an increase of pumping costs. These anomalies must be detected and corrected as early as possible. In this study, an algorithm is developed for detecting blockages by means of pressure transient measurements and estimating the diameter distribution resulting from their formation. The algorithm is a stochastic successive linear estimator that provides statistically the best unbiased estimate of diameter distribution due to partial blockages and quantifies the uncertainty associated with these estimates. We first present the theoretical formulation of the algorithm and then test it with a numerical case study.

Field Study of Subsurface Heterogeneity with Steady‐State Hydraulic Tomography

Remediation of subsurface contamination requires an understanding of the contaminant (history, source location, plume extent and concentration, etc.), and, knowledge of the spatial distribution of hydraulic conductivity (K) that governs groundwater flow and solute transport. Many methods exist for characterizing K heterogeneity, but most if not all methods require the collection of a large number of small-scale data and its interpolation. In this study, we conduct a hydraulic tomography survey at a highly heterogeneous glaciofluvial deposit at the North Campus Research Site (NCRS) located at the University of Waterloo, Waterloo, Ontario, Canada to sequentially interpret four pumping tests using the steady-state form of the Sequential Successive Linear Estimator (SSLE) (Yeh and Liu 2000). The resulting three-dimensional (3D) K distribution (or K-tomogram) is compared against: (1) K distributions obtained through the inverse modeling of individual pumping tests using SSLE, and (2) effective hydraulic conductivity (K(eff) ) estimates obtained by automatically calibrating a groundwater flow model while treating the medium to be homogeneous. Such a K(eff) is often used for designing remediation operations, and thus is used as the basis for comparison with the K-tomogram. Our results clearly show that hydraulic tomography is superior to the inversions of single pumping tests or K(eff) estimates. This is particularly significant for contaminated sites where an accurate representation of the flow field is critical for simulating contaminant transport and injection of chemical and biological agents used for active remediation of contaminant source zones and plumes.