Water Resources Research

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

Authors


Corresponding author: W. A. Illman, Department of Earth and Environmental Sciences, University of Waterloo, Waterloo, ON N2L 3G1, Canada. (willman@uwaterloo.ca)

Abstract

[1] 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.

1. Introduction

[2] Subsurface flow and transport are controlled by the hydraulic properties of the medium (hydraulic conductivity (K), specific storage (Ss)) and their spatial variability, which are critical for the assessment of contaminant transport and other problems. In a fractured geologic medium, the high contrast of these hydraulic properties between the fractures and the matrix along with their spatial variability makes it challenging to characterize the medium accurately. In a fractured medium, fractures usually have higher-K and lower-Ss values than the rock matrix, while the opposite is true for the matrix. Thus, while pumping tests are conducted, pressure propagates rapidly through connected fractures unlike the rock matrix. This large contrast in the hydraulic properties between the fractures and matrix has led to various conceptual models [e.g., Neuman, 1987; National Research Council, 1996; Neuman, 2005; Illman et al., 2009] to describe flow and transport in fractured geologic media.

[3] For example, our inability to characterize the fracture and matrix in detail has led to the creation of the equivalent porous continuum concept [e.g., Bear, 1972; Peters and Klavetter, 1988; Pruess et al., 1990]. The large contrast in fracture and matrix porosities and the slow mixing process between them then prompted the use of dual porosity/mass transfer model [e.g., Bibby, 1981; Moench, 1984; Zimmerman et al., 1993; McKenna et al., 2001; Reimus et al., 2003]. In the dual porosity model, the fracture continuum acts to conduct and store fluids, while the matrix only stores fluids. A dual permeability model [Duguid and Lee, 1977; Wu et al., 2002; McLaren et al., 2000; Illman and Hughson, 2005] is used when both the fracture and matrix continua conduct and store fluids. These models are, however, only suitable for describing or predicting the flow and transport behavior averaged over a large volume of fractured media, which often fail to meet our high-resolution requirements with respect to contaminant transport investigations. The desire for high-resolution predictions, thus promoted the development of discrete fracture network models [e.g., Long et al., 1982; Schwartz et al., 1983; Smith and Schwartz, 1984; Andersson and Dverstorp, 1987; Dershowitz and Einstein, 1988; Dverstorp and Andersson, 1989; Cacas et al., 1990; Dverstorp et al., 1992; Slough et al., 1999; Park et al., 2001a, 2001b, 2003; Darcel et al., 2003; Benke and Painter, 2003; Cvetkovic et al., 2004; Frampton and Cvetkovic, 2010; McLaren et al., 2012].

[4] The discrete fracture approach, however, requires detailed specification of fracture geometries and spatial distributions which are difficult to obtain in the field [Neuman, 1987, 2005]. Uncertainty in characterizing fractures due to our limited characterization technologies then becomes the logic behind the stochastic continuum concept [Neuman, 1987; Tsang et al., 1996; Vesselinov et al., 2001; Ando et al., 2003; Park et al., 2004; Illman and Hughson, 2005; Illman et al., 2009].

[5] Over the past few decades, different hydraulic and pneumatic characterization techniques have been developed to characterize saturated and unsaturated fractured geologic media. For example, Hsieh et al. [1985] conducted cross-hole pumping tests at the Oracle site in Arizona, USA, consisting of fractured granite and obtained the anisotropy of K as well as a value of Ss by treating the fractured rock as a uniform, anisotropic medium. Likewise, numerous single-hole pneumatic injection tests [Guzman et al., 1996] in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS) in central Arizona were interpreted by Illman and Neuman [2000] and Illman [2005] using type curve methods to obtain local scale estimates of air permeability (k) and air-filled porosity (φ). Bulk estimates of k and φ of the tuff at the ALRS were inferred through numerous cross-hole pneumatic injection tests conducted by Illman et al. [1998; see also, Illman, 1999]. In particular, Illman and Neuman [2001] interpreted one of 44 cross-hole pneumatic injection tests at the same site using type curves to obtain k and φ estimates. Due to the highly heterogeneous nature of the fractured tuff at the ALRS, many early time pressure records from various cross-hole pneumatic injection tests deviated significantly from the type curves developed by Illman and Neuman [2001] hence allowing them to analyze only one test among 44. To circumvent the lack of match of transient data to type curves that treat the medium to be homogeneous, Illman and Neuman [2003] analyzed the steady state portion of the pressure records from the other available tests. Because the steady state approach only yields k estimates, an asymptotic approach [Illman and Tartakovsky, 2005a, 2005b] which is analogous to the straight-line method of Cooper and Jacob [1946] but developed for three-dimensional flow, was utilized to interpret the rest of the tests to obtain k and φ estimates. Estimates of hydraulic or pneumatic parameters obtained via type-curve, steady state, and asymptotic analyses all are obtained on the premise that the medium is treated to be uniform. These equivalent parameters are frequently used in various research and practical applications, but Wu et al. [2005] and others have questioned the meaning of these parameters.

[6] Another important issue in site characterization is the quantification of connectivity of high K pathways that may contribute to fast movement of water and contaminants as well as low-K zones that can store contaminants. There are a number of hydraulic, solute transport, and geophysical approaches that have contributed to the understanding of connectivity [Knudby and Carrera, 2005, 2006; Day-Lewis et al., 2003], although the characterization approaches are still under considerable debate. Le Borgne et al. [2006] and Williams and Paillet [2002] used cross-borehole flowmeter pulse tests to characterize subsurface connections between discrete fractures. In this method, hydraulic stress is applied to a borehole through pumping and the propagation of the pressure pulse through the flow system is monitored using a flowmeter. Typically, type curves are utilized to analyze the test data. While flow meters are useful in detecting connections between boreholes, type-curve analysis of flowmeter data cannot yield a map of fractures or their hydraulic parameters that reveals their connectivity. Despite the controversy, Illman [2006] suggested that one possible alternative to imaging the connectivity of hydraulic parameters is hydraulic tomography.

[7] There are a number of inverse algorithms for hydraulic and pneumatic tomography [e.g., Gottlieb and Dietrich, 1995; Vasco et al., 2000; Yeh and Liu, 2000; Vesselinov et al., 2001; Bohling et al., 2002; Brauchler et al., 2003; McDermott et al., 2003; Zhu and Yeh, 2005, 2006; Li et al., 2005, 2008; Ni and Yeh, 2008; Fienen et al., 2008; Castagna and Bellin, 2009; Xiang et al., 2009; Liu and Kitanidis, 2011; Cardiff and Barrash, 2011; Schöniger et al., 2012]. During a hydraulic tomography survey, water is sequentially extracted from or injected into different areas of an aquifer and the corresponding pressure responses are monitored at other intervals to obtain drawdown or buildup data sets. Pneumatic tomography is analogous, but the pumped or injected fluid is air and the investigation takes place in the unsaturated zone. In particular, Yeh and Liu [2000] developed the sequential successive linear estimator (SSLE) to analyze steady state head records from a hydraulic tomography survey. SSLE is an iterative geostatistical inverse method that analyzes available head data from sequential pumping tests to estimate the distribution of hydraulic parameters. In the laboratory, Liu et al. [2002] and Illman et al. [2007, 2008, 2010, 2011] demonstrated the effectiveness of steady state hydraulic tomography (SSHT) using SSLE to estimate the K heterogeneity and its uncertainty. Zhu and Yeh [2005] then extended SSLE for transient hydraulic tomography (THT) to analyze transient drawdown data to estimate both K and Ss heterogeneity simultaneously. Liu et al. [2007] demonstrated encouraging results from the laboratory sandbox experiment for THT. They not only identified the K and Ss distribution in the laboratory sandbox using THT, but also successfully reproduced the observed drawdown as a function of time of an independent aquifer test using estimated K and Ss field. More recently, Berg and Illman [2011a] used laboratory sandbox data to show that THT yields the best predictions of independent pumping tests among several heterogeneity characterization and modeling approaches.

[8] In the field, a number of studies have been published [e.g., Bohling et al., 2007; Straface et al., 2007; Cardiff et al., 2009; Li et al., 2008; Illman et al., 2009; Brauchler et al., 2011; Castagna et al., 2011; Berg and Illman, 2011b; Huang et al., 2011; Cardiff et al., 2012]. In particular, Berg and Illman [2011b] showed that performing the inversion with multiple pumping tests (i.e., hydraulic tomography) yields improved results when compared to the analysis of individual pumping tests at a highly heterogeneous field site consisting of glaciofluvial deposits. While synthetic, laboratory, and field studies on hydraulic tomography in unconsolidated deposits are encouraging, research on the application of hydraulic or pneumatic tomography to fractured rock is limited.

[9] The first study on the pneumatic tomography of fractured rocks was published by Vesselinov et al. [2001]. These authors developed a geostatistical inverse algorithm based on the pilot point method to interpret multiple cross-hole pneumatic injection tests [Illman and Neuman, 2001, 2003] in unsaturated fractured tuffs at the ALRS. The simultaneous inversion of pressure buildup records from three cross-hole pneumatic injection tests amounted to the pneumatic tomography to image the k and φ heterogeneity. The results of the pneumatic tomography were compared to kriged k fields based on single-hole pneumatic injection tests [Chen et al., 2000] and were found to share a similar internal structure. In addition, k estimates obtained through pneumatic tomography were compared to single-hole k estimates along several boreholes yielding a general correspondence between the two estimates.

[10] Brauchler et al. [2003] then developed a hydraulic and pneumatic tomography approach based on the inversion of travel times of the pressure pulse. The algorithm was based on the relation between the peak time of a recorded transient pressure curve and the diffusivity of the investigated system. It was tested in a large diameter cylindrical sample of unsaturated fractured sandstone in the laboratory. The three-dimensional reconstructions of the high diffusivity areas coincided with the location of a vertical fracture.

[11] More recently, Hao et al. [2008] applied the SSLE algorithm to a synthetically generated fractured medium to investigate the feasibility of hydraulic tomography to detect fracture zones and their connectivity. The hypothetical fractured rock aquifer was a 2-D vertical square domain consisting of five orthogonal vertical and two horizontal fracture zones embedded in a rock matrix. They satisfactorily imaged the high K zones from the observation data collected from multiple pumping tests, which reflected the fracture pattern and its connectivity in the synthetic fractured aquifers although estimated values of K and Ss fields were smoother than the true fields. They found that the fracture pattern and connectivity became more vivid and the estimated hydraulic properties approached true values as the number of wells and monitoring ports increased.

[12] Ni and Yeh [2008] extended the SSLE algorithm to pneumatic tomography to delineate fracture permeability, porosity, and connectivity in unsaturated fractured rocks. Their pneumatic tomography algorithm considers compressibility of air and SSLE fully utilizes the cross-correlation between head and pneumatic properties everywhere in a geologic medium. This cross-correlation is ignored by the pilot point approach, as explained by Huang et al. [2011].

[13] Most recently, Illman et al. [2009] interpreted two cross-hole pumping tests at the Mizunami Underground Research Laboratory (MIU) construction site in central Japan and analyzed them using the THT code of Zhu and Yeh [2005] to map the three-dimensional distribution of K and Ss, their connectivity, as well as their uncertainty. They were able to identify two fast flow pathways or conductive fault zones at the site as well as low-K zones, despite the availability of only two cross-hole pumping tests. They assessed the soundness of the estimated fracture K and Ss tomograms using three different approaches: (1) by comparing the calibrated and observed drawdown records as well as predicted the drawdown responses at the monitoring intervals that were not used in the construction of the K and Ss tomograms; (2) by comparing the estimated K and Ss tomograms to previously known fault locations, and (3) by utilizing coseismic groundwater pressure changes recorded during several large earthquakes as a means to evaluate the K and Ss tomograms. While the results were encouraging, there were only two pumping tests available for inverse modeling which precluded Illman et al. [2009] from investigating whether THT could be utilized to map finer details of hydraulic heterogeneity in fractured rocks. The work of Illman et al. [2009] motivates us to conduct a laboratory fractured rock block experiment in which a large number of pumping tests can be conducted and THT can be tested in a controlled setting. Conducting a larger number of pumping test in a controlled environment can help determine whether finer details of K and Ss heterogeneity can be imaged. In addition, pumping tests that are not used for the THT analysis can be utilized to validate the K and Ss tomograms [e.g., Illman et al., 2007; Liu et al., 2007].

[14] Therefore, the main objectives of this study are to investigate the ability of THT to image the K and Ss tomograms of a fractured rock block without the a priori knowledge of fracture locations as well as fracture geometry data and to compare the tomograms to the known fracture locations. We conduct the study using a dolostone rock sample with known fracture locations that is encased in a flow cell. The fractured rock block is initially subjected to flow through tests in order to obtain an estimate of effective hydraulic conductivity (Keff). We then conduct synthetic simulations of pumping tests to design the actual tests for the hydraulic tomography survey of the fractured rock block. With an improved understanding of how the drawdowns propagate through the fractured rock block, we conduct multiple cross-hole pumping tests using multiple observation ports to obtain drawdown data. These drawdown data are then interpreted using the SSLE code developed by Zhu and Yeh [2005] to conduct transient hydraulic tomography, which yields K and Ss tomograms as well as their uncertainty estimates. The obtained K and Ss tomograms are then assessed through a number of methods to test their validity.

2. Experimental Design

2.1. Rock Block Preparation and Flow Cell Design

[15] A dolostone rock sample from the Guelph Formation for the laboratory experiments was obtained from a quarry in Wiarton, Ontario, Canada. The Guelph Formation is a carbonate ramp sequence and mainly composed of tan to dark brown, microcrystalline to fine crystalline, pervasive and massive dolomites [Coniglio et al., 2003, 2004]. The dimensions of the dolostone rock sample are 91.5 cm in length, 60.5 cm in height and 5.0 cm in depth.

[16] Tension fractures were induced by placing a triangular bar underneath the rock sample and forcing it to fracture along it. In order to seal the edges of the fractures, we utilized titanium putty (Devcon, Danvers, Massachusetts, USA) as a sealant. The sealant was applied on the rock surface leaving the fractures open. After the putty dried, we poured resin (Environmental Technology, Inc., California, USA) to cover the front, back, top and bottom surface of the rock in order to prevent any evaporation or leakage, and to make the surfaces smooth. This allowed for water flow to take place only from the left and right surfaces that connect to the constant head reservoirs (Figure 1). After the resin dried and hardened, water was injected at several ports to ensure that the horizontal and vertical fractures did not become sealed.

Figure 1.

Experimental setup of the hydraulic tests on the fractured rock block. Open black circles indicate port locations on the fractures. Black solid circles indicate port locations within the matrix and the constant head reservoirs.

[17] The sealed fractured block was then placed in a flow cell (122 cm in length, 65 cm in height and 6.5 cm thick) constructed of stainless steel with plexiglass used as a front plate (Figure 1). We applied silicon to seal any gaps present between the flow cell and the rock block. The flow cell consisted of constant head reservoirs at the left and right boundaries where it is in contact with the fractured rock block. This allowed for the horizontal fracture to be in contact with the constant head reservoirs allowing for water flow along the length of the rock block. All other boundaries are considered to be “no-flow” boundaries.

2.2. Fractured Rock Block Instrumentation

[18] We installed 31 ports on the fractured rock block to monitor water pressure using a pressure transducer (Figure 1). Each port was connected to a 0–1 pounds per square inch gauge (psig) pressure transducer (model 209, Alpha Controls and Instrumentation, Markham, Ontario, Canada) to record pressure measurements at different locations of the fractured domain during the hydraulic tests. Seventeen pressure transducers were placed on fractures, fourteen were installed on the matrix and two were utilized to record pressure in the constant head reservoirs. Ports placed on fractures were also utilized for water extraction during the pumping tests.

[19] The data acquisition system used for recording pressure measurements consisted of a 64-channel data acquisition board from National Instruments. A hub that separates excitation and output currents for the transducers was assembled. A dedicated PC with National Instruments LabVIEW software also was part of the automated data acquisition system.

3. Description and Interpretation of Flow-Through Tests on the Fractured Rock Block

3.1. Description of Flow-Through Tests

[20] After the fractured block was enclosed in a flow cell, we filled the rock block with water from the bottom and in the constant head reservoirs to allow for the fractured rock block to saturate over several weeks. To minimize air entrapment, we allowed air to escape from the top before we sealed the top of the block. We then conducted four flow-through tests following the work of Illman et al. [2007, 2010] to estimate the bulk or effective hydraulic conductivity (Keff) and the hydraulic aperture of the fractured block. Flow-through tests were conducted by fixing the hydraulic gradient across the rock sample and measuring the water outflow rate at the effluent dripping point using a graduated cylinder. The flowrate was measured in both directions (left to right and right to left) by fixing the hydraulic gradient across the fractured block, and for two different hydraulic gradients in each direction under confined conditions. The hydraulic gradient across the rock block was fixed by setting drip points at the end reservoirs at different heights. A peristaltic pump (MasterFlex, model 7550-30, Cole-Parmer, Montreal, Quebec, Canada) was used to supply water continuously into the influent reservoir and the drip point maintained constant head.

3.2. Interpretation of Flow-Through Tests

[21] The flow through tests can be interpreted to obtain the hydraulic aperture of the horizontal fracture through the cubic law [Romm, 1966]:

display math

where, b is hydraulic aperture (L), μ (ML−1T−1) is the dynamic viscosity at room temperature (20°C), Q (L3T−1) is the discharge or flowrate through the fracture, L (L) is the fracture length in the direction of flow, ρ (ML−3) is the fluid density at room temperature, g (LT−2) is the acceleration due to gravity, W (L) is the fracture width perpendicular to the direction of flow, and ΔH (L) is the hydraulic head loss across the fracture plane.

[22] The estimated values of hydraulic aperture of the fracture system from four different flow-through tests varied between 0.047 cm and 0.050 cm and are summarized in Table 1. The hydraulic aperture values are bulk estimates that consider not only the horizontal, but the vertical fractures present in the rock block (Figure 1).

Table 1. Summary of the Flow-Through Tests and the Estimated Hydraulic Parameters
TestFlow DirectionConstant Head at Left Reservoir (cm)Constant Head at Right Reservoir (cm)Hydraulic GradientFlow Rate (mL s−1)Hydraulic Aperture (cm)Keff (cm s−1)
Test 1Left to Right60.055.00.050.280.0501.70 × 10−2
Test 2Left to Right60.148.50.130.500.0471.30 × 10−2
Test 3Right to Left54.660.00.060.270.0481.50 × 10−2
Test 4Right to Left48.460.60.130.510.0471.30 × 10−2

[23] From the flow-through tests, The Keff of the fractured block was estimated using Darcy's Law:

display math

where A (L2) is bulk cross-sectional area and (dh/dl) is the hydraulic gradient across the system. The estimated Keff from four different flow-through tests are also summarized in Table 1, which varied between 1.30 × 10−2 cm s−1 to 1.70 × 10−2 cm s−1.

4. Synthetic Simulations for the Design of Pumping Tests Used for Hydraulic Tomography

4.1. Description of Synthetic Pumping Tests

[24] We next utilized the groundwater flow and solute transport code, HydroGeoSphere (HGS) to simulate the pumping tests on the computer to design the actual pumping tests that will be utilized for the hydraulic tomography survey of the fractured rock block. The domain used for the forward simulation was 91.5 cm in length by 60.5 cm in height and 5 cm thick (one element thick) and was composed of variably sized rectangular elements. The element size varied from 0.05 cm by 0.01 cm to 1.375 cm by 1.375 cm. The finer elements were located along the fracture and the coarser elements were located away from the fractures. Figure S1a in the auxiliary material shows the domain used for simulating the pumping tests and Figure S1b shows the fracture faces. The darker areas in Figure S1a indicate the highly refined areas of the model domain. We assigned constant head boundary conditions (h = 63.5 cm) for left and right boundaries and no flow for the remaining outer boundaries to simulate actual experimental conditions.

[25] The hydraulic aperture obtained from the flow-through tests is assigned to the elements for the horizontal and vertical fractures. HGS then calculates the fracture hydraulic conductivity (Kf) from the hydraulic aperture [Therrien et al., 2009]:

display math

Based on the average hydraulic aperture (0.049 cm) from the flow-through tests, HGS calculated a Kf of 17.46 cm s−1.

[26] Initial estimates of the matrix hydraulic conductivity (Km = 1.00 × 10−7 cm s−1 to 1.00 × 10−4 cm s−1) and specific storage (Ss) values for both fractures (Ssf = 1.00 × 10−9 cm−1 to 1.00 × 10−6 cm−1) and matrix (Ssm = 1.00 × 10−9 cm−1 to 1.00 × 10−6 cm−1) were obtained from the literature [i.e., Schwartz and Zhang, 2003; Singhal and Gupta, 2010]. A range of values were selected to conduct a suite of forward simulations to examine the drawdown behavior in the fractured rock block.

[27] A synthetic simulation of a pumping test was then conducted with pumping taking place at port 5 at a pumping rate of 4 cm3 s−1 in an attempt to gain insight into the drawdown behavior that will propagate through the fractured rock block. Results from the simulation revealed drawdown responses at observation ports located on fractures, while no response was observed at the ports located within the matrix during the duration of the synthetic pumping test.

[28] Following the synthetic simulation of a pumping test, an identical, real pumping test was then carried out at port 5 of the actual fractured rock block and pressure responses at all the monitoring ports were recorded.

4.2. Traditional Interpretation of a Single Pumping Test

[29] The hydraulic parameters of the fractured rock block from the previous section were then adjusted by matching the simulated (dashed) and observed drawdown curves during the real pumping test at port 5 of the fractured rock sample (Figure 2). Examination of Figure 2 reveals that, a good match between the observed and simulated drawdown can be obtained for the bulk of the drawdown responses except for very early time.

Figure 2.

Observed (solid curve) and calibrated (dashed curve) drawdown curves using HGS during a pumping test at port 5.

[30] The manual calibration resulted in higher specific storage values (for both the fractures (Ssf = 8.00 × 10−4 cm−1) and the matrix (Ssm = 3.00 × 10−3 cm−1)) than the literature values. One reason for this may be due to wellbore storage in the ports which is not accounted for explicitly in the numerical model. While these specific storage values are certainly higher, the literature values are representative of pumping tests conducted at a significantly larger scale in the field at greater depths [e.g., Illman et al., 2009; Castagna et al., 2011], and the storage estimates from the fractured rock block nevertheless, yields the best calibrated results.

[31] In order to further verify the high Ss values than what we expect for values obtained in the field, we conducted a Jacob's semilogarithmic analysis [Cooper and Jacob, 1946] of several observation port data from several pumping tests to obtain bulk estimates. Those results yielded a mean K of 2.80 × 10−2 cm s−1 and mean Ss of 1.8 × 10−3 cm−1 which suggests that a higher Ss value can be obtained from this fractured rock block. These parameters (fractures: Kf = 17.46 cm s−1; Ssf = 8.00 × 10−4 cm−1 and the matrix: Km = 1 × 10−7 cm s−1; Ssm = 3.00 × 10−3 cm−1) were then utilized to simulate eight additional pumping tests at ports 3, 7, 12, 13, 15, 16, 18, and 19 at pumping rates ranging from 1 to 4 cm3 s−1 to obtain synthetic drawdown data that were later used for the THT analysis of synthetic data.

5. Description of Sequential Pumping Tests Conducted in the Fractured Rock Block

[32] Upon completion of the flow-through tests and the synthetic simulations of pumping tests, 17 real pumping tests were conducted at each of the fracture ports on the fractured rock block (see Figure 1). A peristaltic pump was used to extract water from the pumping port and a pulse dampener was used between the pump and the pumped port to reduce the oscillations in the pumping rate. Thirty-one pressure transducers recorded pressure responses at different heights of the fractured domain and two of them recorded pressures in the constant head reservoirs during each test. Both reservoirs were fixed at the same level and the constant head was maintained throughout the test by putting the extracted water back to both reservoirs.

[33] The duration of each pumping test ranged from 3 to 10 min. During each test, pressures were recorded at 30 to 31 ports located along the fractures, matrix and constant head reservoirs. Prior to conducting each pumping test, all pressure transducers were calibrated for three different reservoir heads as implemented by Illman et al. [2007, 2008, 2010] and Berg and Illman [2011a]. To develop a static, initial condition, head data were recorded at each pressure transducer for several minutes prior to each pumping test. Then each pumping test was run until steady state conditions which were visually confirmed by observing the stabilization of all head measurements on the data logger connected to a computer. During all of the pumping tests, recorded pressures in ports completed on fractures reached steady state within 2 to 3 s after the water extraction was started and no change in pressure was observed at any of the matrix ports, which was expected due to the very high contrast between the fracture and matrix K. A summary of all pumping tests conducted in the fractured rock block is provided in Table 2.

Table 2. Summary of Pumping Tests Conducted on the Fractured Rock Block
Pumped PortPumping Rate (mL s−1)Duration (min)Maximum Observed Drawdown (cm)Port of Maximum Drawdown
Port 14108Port 3
Port 32.51010Port 15
Port 42.51010.4Port 3
Port 541017.3Port 11
Port 62.51010Port 11
Port 72.557.8Port 19
Port 84109.5Port 7
Port 94105.9Port 8
Port 1141017Port 13
Port 1241019.7Port 13
Port 1341018.7Port 12
Port 142.51016.2Port 15
Port 151.71014.9Port 16
Port 1611011.7Port 15
Port 172.5329.7Port 18
Port 181.71021.3Port 17
Port 1911018.4Port 18

6. Transient Hydraulic Tomography Analyses of Synthetic and Real Pumping Test Data

6.1. Inverse Modeling Approach

[34] We then performed the stochastic inverse modeling of individual pumping tests and THT analysis using the Sequential Successive Linear Estimator (SSLE) developed by Zhu and Yeh [2005]. The forward model included in SSLE is the VSAFT3 code [Yeh et al., 1993] which treats the fractured rock as a porous continuum. SSLE for THT evolved from the SSLE for steady state hydraulic tomography [Yeh and Liu, 2000]. Both use the successive linear estimator (SLE) [Yeh et al., 1995, 1996] concept and successively include data sets, thus reducing the computational burden that would be encountered if all of the data sets were included simultaneously.

[35] SLE starts with (1) cokriging if there are measurements of K and Ss (otherwise, kriging at sample locations) based on the covariance functions for the parameters to obtain conditional mean estimates of the parameter fields and to evaluate the conditional (or residual) covariance functions of the parameters (K and Ss) at every grid block of the inverse modeling domain. (2) Then, the conditional mean parameter fields are used to simulate the responses of the geologic formation (head) based on the governing flow equation. (3) The difference between the observed and simulated heads at head measurement points are subsequently used to improve the estimated parameters at every grid block of the domain. The improvement of the parameter at each block is the weighted sum of all the differences at all head measurement locations, similar to kriging. The weights are calculated based on the cross-covariance (residual) function between the head and the parameters similar to the solution to the kriging system equation. (4) The residual covariances of the estimated are parameters updated, which will be used to calculate the cross-covariance function for the next iteration. (5) If the difference is small than a given tolerance or other criteria are met, the iterative updating is stopped. Otherwise, steps 2 through 5 are repeated. The algorithm of the sequential approach is similar to the SLE except it includes new pumping test data into the SLE.

[36] SLE approach is different from the pilot point approach. SLE distributes information content in the observed head to every block of the aquifer domain according to the spatial cross-covariance (or cross-correlation) function between the head at observation locations and the parameters (K and Ss) at every location. This cross-correlation function is evaluated using the continuously updated parameter residual covariance functions and governing flow equation. They change according to the residual covariance functions of the parameters as well as the flow fields induced by different pumping tests. Estimated parameters at every block in the domain are thus improved if the observed heads have new information. This is the reason that SSLE is capable of mapping the heterogeneity over large area using a small number of observation points during HT. These points are discussed extensively in Huang et al. [2011].

[37] While the simultaneous inversion of pumping tests is also possible with Zhu and Yeh's [2005] THT code extended by [Xiang et al., 2009], the computational requirements are significant for the analyses of our data. Therefore, at this time, we choose to sequentially analyze the pumping tests using the SSLE of Zhu and Yeh [2005].

6.2. Inverse Model Setup

[38] The THT analysis of synthetic data (synthetic THT from now on) was first performed, before performing THT analysis with two sets of laboratory drawdown data (real THT cases 1 and 2 from now on) in order to investigate the capability of SSLE in imaging fractured rock block by computing the K and Ss tomograms with noise-free data. The inversion of the synthetic data is important as it provides us with baseline results that one can expect from the number of pumping and monitoring points utilized in our synthetic and real experiments. These results will later be used to compare against the results from the real THT analysis.

[39] Synthetic THT analysis was conducted by inverting three of the nine synthetic pumping tests conducted by HGS. Upon completion of the synthetic THT, two cases of the real THT were performed by inverting two different sets of pumping tests conducted in the fractured rock block enclosed in a flow cell. Each of the real THT cases 1 and 2 were conducted by inverting three laboratory pumping tests.

[40] All stochastic inversions of pumping tests were performed using 8 to16 processors on a PC-cluster (consisting of 1 master and 12 slaves each with Intel Q6600 Quad Core CPUs running at 2.4 GHz with 16 GB of RAM per slave) at the University of Waterloo. The operating system managing the cluster was CentOS 5.3 based on a 64-bit system.

[41] A 91.5 cm × 60.5 and 5 cm (one element) thick domain was used for the inversion of both synthetic and laboratory data and the domain was composed of variably sized rectangular elements. The domain was discretized into 14,140 nodes and 6,900 elements. The element size varied from 0.5 cm × 0.5 cm to 1.75 cm × 1.75 cm. The finer elements are located along the ports to match the port location and the element center and the coarser elements are located near the boundary. Figure S2 in the auxiliary material shows the computational grid used for both the synthetic and real THT analysis.

[42] The boundary conditions were constant head for left and right boundaries and no flow for the remaining outer boundaries. The hydraulic head for both the left and right boundary of the model domain was set to 63.5 cm for the analysis of both synthetic and real data.

6.3. Input Parameters

[43] Inputs to the SSLE for the synthetic and real THT analyses include mean K and Ss values for the model domain, estimates or guesses of variances and the correlation scales for both parameters, volumetric discharge (Q) from each pumping test and observed pressure head data at various selected times per drawdown curve from each pumping tests. Here, the model was not conditioned with additional data, although point (small-scale) measurements of K and Ss can also be input to the model to condition the estimates.

[44] One can estimate the mean values or initial homogeneous field of K and Ss in a number of ways. For example, literature values of effective hydraulic conductivity (Keff) and specific storage (Sseff) that are considered reasonable for the fractured rock aquifer may be used as initial model input. An alternative could be estimating a geometric mean of the small-scale measurements (i.e., core, slug, and single-hole data), if small-scale data are available. The initial homogeneous K and Ss field can also be selected by obtaining equivalent hydraulic conductivity and specific storage estimates through the analysis of pumping test data by treating the medium to be homogeneous.

[45] Here, the latter option was chosen and the initial homogeneous K and Ss were estimated by coupling PEST [Doherty, 2005] with the forward groundwater model in SSLE [Zhu and Yeh, 2005] and matching the drawdown responses at the pumping port. The homogeneous K and Ss for the fractured rock were estimated by matching the pumping port drawdown for 3 individual laboratory pumping tests conducted at Ports 3, 5, and 7. The model domain and the boundary conditions used for the parameter estimation were the same to those used for the inverse modeling. Estimates of homogeneous K and Ss along with the corresponding 95% confidence intervals are presented in Table 3.

Table 3. Equivalent K and Ss Estimated by PEST From Pumping Tests at Ports 3, 5, and 7
 Port 3Port 5Port 7
Estimated K (cm s−1)2.80 × 10−21.40 × 10−23.00 × 10−2
Max K (cm s−1)3.90 × 10−21.70 × 10−24.00 × 10−2
Min K (cm s−1)2.10 × 10−21.10 × 10−22.00 × 10−2
Estimated Ss (/cm)1.30 × 10−32.00 × 10−24.00 × 10−3
Max Ss (/cm)7.50 × 10−34.00 × 10−22.00 × 10−3
Min Ss (/cm)2.30 × 10−31.00 × 10−29.00 × 10−3

[46] From Table 3, it is evident that the estimates of homogeneous K and Ss from all 3 cases were close to one another. Therefore, the geometric mean of the estimated homogeneous K and Ss (K = 2.3 × 10−2 cm s−1 and Ss = 4.7 × 10−3 cm−1) were incorporated into SSLE as an initial guess for the inversion of the synthetic as well as the laboratory data. The model starts the inversion (for both synthetic and real THT) with these homogeneous estimates of K and Ss and updates them at the end of each iteration and the following iteration starts with these updated values of K and Ss. This process continues until the model goes through the last iteration of the last test.

[47] SSLE also requires the estimates of the hydrogeologic structure (correlation length and the variances of the K and Ss) for inversion. One can assume correlation length and the variances or can conduct a geostatistical analysis of small scale data to estimate the variance and the correlation length. Here, a correlation length of 1 cm and a variance of 5 were assumed for both K and Ss. It is a well-known fact that it is difficult to estimate the variance and the correlation scale accurately and thus the estimation always involves some uncertainty. Here, a unit correlation length (1 cm) was used for the THT analyses, as larger correlation implies a homogeneous field of hydraulic properties. However, a previous numerical study conducted by Yeh and Liu [2000] has shown that the initial guesses of variance and correlation scales have negligible effects on the estimated K field based on hydraulic tomography, because hydraulic tomography utilizes a large number of head measurements, which already hold information of the detailed site-specific heterogeneity [Zhu and Yeh, 2005, 2006; Liu et al., 2007]. Negligible effects of correlation scales on the estimate of fracture patterns in synthetic aquifers were also demonstrated by Hao et al. [2008].

[48] The synthetic THT analysis was performed by inverting hydraulic head data from three synthetic pumping tests sequentially, which included the tests at ports 5, 7, and 3, in that order. The pumping tests utilized in the synthetic THT case were identical to the ones used in real THT case 1. For real THT case 2, we utilized pumping tests at ports 5, 16, and 19, to evaluate whether utilizing different pumping tests could have an impact on the estimated K and Ss tomograms.

[49] The locations for the pumping tests were selected for their ability to stress the entire fractured block. For example, for all the synthetic and real THT cases, the pumping test at port 5 was included first in the inverse model followed by two additional tests with lower pumping rates. The reason for including pumping test at port 5 first was that the pumping test generated drawdown responses at all the ports located on fractures as it is located in the central portion of the fractured block and it had the highest flowrate, thus had the highest signal-to-noise ratio. Illman et al. [2008] showed that including the data with the highest signal-to-noise ratio first into inverse SSLE appeared to improve the results. Results not included here showed that the change in the order of the second (test at port 7) and the third pumping test (test at port 3) did not change the pattern of the resulting K and Ss tomograms significantly.

[50] For the synthetic as well as real THT cases 1 and 2, four data points (at 0.5 s, 1 s, 3 s and 20 s) were extracted from each observation port (both fracture and matrix observation ports) to capture the entire drawdown curve. The total number of observed data points used from each pumping test ranged from 116 to 120. In total, 356 drawdown records from three different pumping tests were utilized to perform synthetic as well as real THT cases 1 and 2. The ports completed within the matrix did not show a response to any of the pumping tests. Therefore, zero drawdown was input to the inverse model for each matrix observation port for both synthetic as well as real THT cases 1 and 2.

7. Results From Transient Hydraulic Tomography

7.1. Inverse Modeling of Synthetic Data

[51] Figures 3a–3c are the K tomograms obtained by inverting the synthetic transient head data one, two, and three pumping tests, respectively, while Figure 3d is the estimated ln K variance ( inline image) map corresponding to the K tomogram of Figure 3c. On Figures 3a–3d, the open black circle represents the pumped port location, while the solid black circles represent the ports which were monitored during the pumping tests. Thin dashed lines on Figure 3c indicate the locations of the horizontal and vertical fractures. Figure 3a shows that with only one pumping test, areas with higher K relative to the background that corresponds to a portion of the horizontal fracture and the vertical fracture in the central portion of the fractured rock block begin to emerge. As more tests are included into the SSLE algorithm, details to the high K zones corresponding to the fracture pattern emerges. In particular, the final K tomogram (Figure 3c) using three pumping tests reveals considerable detail to the fracture pattern and the connectivity of the features away from the two constant head boundaries. Less detail is available near the two constant head boundaries because the drawdowns induced near the boundaries are considerably less than the interior of the fractured rock block. In contrast, details to the fracture pattern are evident near the top and bottom no-flow boundaries where drawdowns tend to be magnified. The inline image map on Figure 3d reveals that the lowest inline image are found along the fracture and the highest within the matrix where the uncertainties are high.

Figure 3.

K tomograms (cm s−1) computed using synthetic data from (a) one pumping test (port 5); (b) two pumping tests (ports 5, 7); (c) three pumping tests (ports 5, 7, 3); while (d) is the estimated ln K variance map associated with Figure 3c. Pumped locations are indicated by the open black circles, while observation intervals are indicated by solid black circles. Thin dashed lines on Figure 3c indicate the locations of the horizontal and vertical fractures. The image in each figure represents the x-z plane through the middle of the domain thickness.

[52] Figures 4a–4c show the corresponding Ss tomograms that were estimated simultaneously. Similar to Figures 3a–3c, details to the fracture pattern where Ss values are lower than the background become more evident as more pumping test data are included into the inverse model. Figure 4d is the estimated ln Ss variance ( inline image) map corresponding to the Ss tomogram of Figure 4c. Similar to Figure 3d, the inline image map on Figure 4d reveals that the lowest inline image are found along the fractures.

Figure 4.

Ss tomograms (cm−1) computed using synthetic data from (a) one pumping test (port 5); (b) two pumping tests (ports 5, 7); (c) three pumping tests (ports 5, 7, 3); while (d) is the estimated ln Ss variance map associated with Figure 4c. Pumped locations are indicated by the open black circles, while observation intervals are indicated by solid black circles. Thin dashed lines on Figure 4c indicate the locations of the horizontal and vertical fractures. The image in each figure represents the x-z plane through the middle of the domain thickness.

[53] One should keep in mind that the synthetic data utilized to generate the synthetic K and Ss tomograms were obtained from forward simulations of pumping tests in the fractured rock block based on Figure S1 in the auxiliary material. In these forward simulations, we assigned uniform values of K and Ss for the fracture and matrix. According to Yeh et al. [2011], if drawdown data are available at every element, one should be able to estimate the K and Ss values perfectly for the fracture and matrix. However, due to lack of drawdown data at all elements, we instead obtain a distribution of K and Ss, with high-K and low-Ss values appearing at fracture locations.

7.2. Inverse Modeling of Real Data

[54] We next examine the results from the inversion of real pumping test data obtained in identical fashion to the synthetic case. As in the synthetic case, the pumping tests took place at ports 5, 7, and 3 (real THT case 1). In reality, we do not know the true K and Ss distributions as in the synthetic case, although the locations of the fractures are visible on Figure 1. Figures 5a–5c and Figures 6a–6c show the sequential improvement of the computed K and Ss tomograms as the number of pumping test data increases from one to three, while Figures 5d and 6d are the corresponding estimated ln K and ln Ss variances, respectively, after the inclusion of three pumping tests in the inverse model. The comparison of the K and Ss tomograms for the synthetic and real cases shows that the correspondence is very good. In particular, as in the synthetic case shown on Figure 3c and 4c, narrow regions of high K (Figure 5c) and low-Ss (Figure 6c) zones that correspond with the actual fracture locations are identified with three pumping tests included in the inverse model. This is despite the fact that the SSLE treats the medium as a heterogeneous porous continuum.

Figure 5.

Case 1 K tomograms (cm s−1) computed using real data from (a) one pumping test (port 5); (b) two pumping tests (ports 5, 7); (c) three pumping tests (ports 5, 7, 3); while (d) is the estimated ln K variance map associated with Figure 5c. Pumped locations are indicated by the open black circles, while observation intervals are indicated by solid black circles. Thin dashed lines on Figure 5c indicate the locations of the horizontal and vertical fractures. The image in each figure represents the x-z plane through the middle of the domain thickness.

Figure 6.

Case 1 Ss tomograms (cm−1) computed using real data from (a) one pumping test (port 5); (b) two pumping tests (ports 5, 7); (c) three pumping tests (ports 5, 7, 3); while (d) is the estimated ln Ss variance map associated with Figure 6c. Pumped locations are indicated by the open black circles, while observation intervals are indicated by solid black circles. Thin dashed lines on Figure 6c indicate the locations of the horizontal and vertical fractures. The image in each figure represents the x-z plane through the middle of the domain thickness.

[55] The THT analysis of pumping tests at ports 5, 16, and 19 (real THT case 2) yielded similar K and Ss tomograms to results from real THT case 1 (Figures 5a–5d; Figures 6a–6d). Those results are included in the auxiliary material (see Figures S3 and S4). In both cases, the first pumping test included into the inverse algorithm was the test at port 5 located near the center of the fractured rock block. In case 1, the two other tests included were at ports 7 and 3, both of which were on the same horizontal fracture as in port 5. In real THT case 2, the latter two tests were located on ports 16 and 19 both of which were situated on the vertical fractures on the upper portion of the domain. This suggests that the pumping location may not be very sensitive to the final results as long as strong drawdown responses can be induced in the surrounding monitoring ports.

8. Discussion

8.1. Comparison of K and Ss Tomograms Obtained From Synthetic Versus Real Pumping Test Data

[56] A visual comparison of the K and Ss tomograms generated by synthetic THT are compared to those generated by the real THT cases 1 and 2. This comparison is made to assess how the estimates from the inversion of noise-free synthetic data match the estimates from laboratory data that contains experimental noise. Our visual assessment reveals that the patterns of estimated K and Ss tomograms generated by the synthetic THT (Figures 3 and 4) and those estimated with real data (Figures 5, 6, S3, and S4) are quite similar suggesting the robustness of the estimates. Figures 7a and 7b represent the scatter plots of K and Ss values, respectively, obtained from the analysis of synthetic data (Figures 3c and 4c) to those obtained from the analysis of laboratory data (Figures 5c and 6c) for real THT case 1. The solid line represents the 1:1 line. Both Figures 7a and 7b reveal that the data points cluster around the 1:1 line with some scatter suggesting generally good correspondence between the two sets of tomograms. Similar results are obtained for the comparison between real and synthetic THT case 2 (Figure S5 in the auxiliary material).

Figure 7.

Scatter plots of (a) K and (b) Ss values from the inversion of real and synthetic data.

[57] Two criteria, the mean absolute error (L1) and mean square error (L2) norms were utilized to quantitatively evaluate the correspondence between the two sets of tomograms. Both L1 and L2 norms were calculated so that the mean absolute and mean square errors can be compared. The L1 and L2 norms are computed as:

display math
display math

where n is the total number of data, i indicates the data number, and inline image and inline image represent the estimates from the two sets of data. To compute the L1 and L2 norms, we utilized the K and Ss values for each element from the tomograms estimated from the synthetic and real data.

[58] The visual assessment of the scatter plots (Figures 7a and 7b) and the L1 and L2 norms suggest a good fit between these two sets of tomograms. This suggests that the THT analyses of the real pumping test data (real THT cases 1 and 2) yield results that are comparable to those from the synthetic data set for this fractured rock block.

8.2. Comparison of K and Ss Tomograms From Two Real Cases

[59] The two sets of estimated K and Ss tomograms obtained from the inversion of pumping test data ports 5, 7, and 3 (real THT case 1) as well as ports 5, 16, and 19 (real THT case 2) are visually compared with the photograph of the fractured dolostone (Figure 1) to evaluate how well the fracture pattern is captured by THT analysis based on the SSLE code. The locations of the horizontal and vertical fractures are also indicated as thin dashed lines on the final K and Ss tomograms. The visual comparison of snapshot of the fractured rock block (Figure 1) and the estimated K and Ss tomograms from the synthetic case (Figures 3c and 4c) as well as the real THT case 1 (Figures 5c and 6c) and real THT case 2 (Figures S3c and S4c) all reveal that the THT analysis of the laboratory pumping tests captured the fracture pattern and their connectivity quite well. In particular, the high K zones in Figures 3c, 5c, and S3c as well as the low-Ss zones in Figures 4c, 6c, and S4c clearly show the fractures and their connectivity, although the high K and low-Ss zone do not continue to the edge of the rock, which may be due to the boundary effect, as pointed out earlier.

[60] A scatterplot of the K values from real THT cases 1 and 2 is shown on Figure 8a. A similar figure (Figure 8b) is shown to compare the Ss values. In both figures, the solid line represents the 1:1 line. Figures 8a reveals that while the scatter is large, the K values from the two cases cluster around the 1:1 line without much bias. On the other hand, examination of Figure 8b shows that the Ss values scatter around the 1:1 line, but there is a noticeable bias suggesting that there are some differences in the Ss values between real THT cases 1 and 2.

Figure 8.

Scatter plots of (a) K and (b) Ss values from case 1 to case 2. The solid line represents the 1:1 line.

[61] These results indicate that the changing the location of the second and third pumping tests did not significantly impact the K estimates from hydraulic tomography in this fractured rock block. However, the Ss estimates may be more sensitive to the location of the pumping tests based on Figure 8b. Here, the pumping test at port 5 was inverted first during both real THT cases 1 and 2. As described earlier, the pumping test at port 5 had the highest flowrate and stressed all the fractures more or less as it is located near the middle of the fractured rock block. Including the pumping test at port 5 in the inverse model followed the recommendation of Illman et al. [2008] that found that including the data with the highest signal-to-noise ratio first into the SSLE code improved the results when there is noise in data. The main reason behind this is that SSLE updates the K and Ss tomograms after each iteration and uses a weighted linear combination of the differences between the simulated and observed pressure heads to improve the estimates. Therefore, the K and Ss distribution estimated in the beginning of inversion process significantly impact the estimated tomograms. A version of the HT algorithm (SimSLE) that includes all data sets simultaneously for interpretation [Xiang et al., 2009], eliminates this ad hoc approach, although this approach can be significantly more computationally expensive.

8.3. Comparison of Estimated Geometric Mean of K From THT to Those From the Flow-Through Tests

[62] To quantitatively compare the K estimates from the real THT cases 1 and 2 with the Keff obtained from the flow-through tests, we calculated the geometric mean of K estimates obtained from case 1 (KG = 1.30 × 10−2 cm s−1) and case 2 (KG = 1.00 × 10−2 cm s−1). The values of Keff obtained through the four separate flow-through tests yielded a range of 1.30 × 10−2 cm s−1 to 1.70 × 10−2 cm s−1. This suggests that the KG from the THT analysis somewhat underestimates the Keff from the flow through tests. One potential reason for this is that the K tomograms from both cases 1 and 2 are smoothed out near the constant head boundary which could have lowered the mean conductivity.

8.4. Prediction of Independent Pumping Tests

[63] The K and Ss tomograms estimated for the real THT cases 1 and 2 were used to predict 5 pumping tests (ports 4, 6, 12, 15 and 18) that were not used in the construction of the tomograms. This is one form of validating the estimated K and Ss tomogram. Forward simulations are again performed using SSLE [Zhu and Yeh, 2005] with the same model domain used for the inversion.

[64] Comparisons of observed (solid) and predicted (dashed) drawdown curves for real THT case 1 are presented in Figure 9 (test at port 12), while similar plots for the other ports are included in the auxiliary material (Figures S6–S9). Plots comparing the observed and predicted drawdown curves for the same five pumping tests for real THT case 2 are included in the auxiliary material (Figure S10–S14).

Figure 9.

Drawdown versus time at the monitoring ports during the pumping test at port 12. The solid curve represents the observed drawdown curve while the dashed curve represents the predicted drawdown curve using the final K and Ss tomograms from case 1.

[65] Examination of Figure 9 and Figures S6–S9 reveals that, during the independent tests, the simulated drawdown matched the observed for most of the monitoring intervals, but there are some exceptions for ports near the pumped port due to excessive noise resulting from pumping. However, the simulated and observed drawdowns seem to match better at later time. At early time, many of the matches are poor and some are of intermediate to fair quality. The drawdown is primarily controlled by K at late time, while early time drawdown is controlled by the diffusivity (α) of the rock (α = K/Ss). As discussed earlier, wellbore storage may have played an important role in delaying the drawdown responses for the observed case which underscores the need to accurately represent borehole geometry. In addition, the estimated tomograms do not exactly replicate the actual values for the fractured rock, which may have led to these deviations between observed and predicted drawdowns especially at early time. However, there was no drawdown (real THT case 1) or a very small drawdown (real THT case 2) at the matrix ports for the simulated cases, which was consistent with the observed drawdowns.

[66] Figure 10 shows the scatter plots of observed versus simulated drawdowns for the 5 independent pumping tests for real THT case 1. Each plot on Figure 10 represents the drawdown values from all the fracture and matrix observation intervals at 0.5, 1, 3, and 20 s since the corresponding pumping test began. The solid line is the 1:1 line indicating a perfect match. The dashed line is a best fit line, and the parameters describing this line, coefficient of determination (R2) as well as L2 norm for the corresponding tests are included on each plot. The R2 values indicate the quality of match between the simulated and observed drawdowns, while the slope and the intercept of the linear model fit are the indication of bias. The L2 norm was calculated using equation (5) with the inline image and inline image now representing the observed and simulated drawdown from each monitoring port. Similar scatter plots for the independent pumping tests for real THT case 2 can be found in the auxiliary material (Figure S15).

Figure 10.

Scatter plots 5 pumping tests using the estimated K and Ss fields from the THT analysis (case 1). The solid line is a 1:1 line indicating a perfect match. The dashed line is a best fit line, and the parameters describing this line as well as L2 norm for the corresponding tests are on each plot.

[67] From Figure 10, it can be seen that for most tests the data points cluster around the 1:1 line with some bias indicated by the slope of the linear model fit. The only exception is the pumping test at port 18, suggesting comparatively poor estimates for the pumping test at that port. In particular, the observed drawdown is somewhat higher than the simulated drawdown in some of the ports which suggests that the K estimates may be too high. The scatter plots for the independent tests for real THT case 2 provided in the auxiliary material (Figure S15) also show similar matches between the observed and simulated drawdowns.

8.5. Uniqueness of Results

[68] The THT analysis conducted with the SSLE code treats the fractured rock as a heterogeneous porous continuum, which makes it challenging to handle the high contrast between the fracture and matrix K and Ss values. Therefore, the estimated K and Ss values for the fractures and the matrix may not exactly correspond to actual values as shown by the results of the synthetic THT case (Figures 3c and 4c). Consequently, this will lead to the deviation of drawdown responses at the monitoring intervals when the tomograms are utilized for predicting pumping tests not used in the construction of the tomograms.

[69] Despite this shortcoming, we found that by and large, the simulated drawdown responses captured the observed drawdown behaviors in the monitoring ports quite well, indicating the robustness of THT analysis of our laboratory data. In addition, the THT analysis of pumping tests revealed the fracture locations and their connectivity quite accurately which is highly encouraging. Based on the work of Illman et al. [2009] and this study, we feel that hydraulic tomography appears to be a promising tool to delineate the K and Ss distributions in fractured rocks, the dominant fracture patterns and their connectivity, both of which has been a challenge for decades. The advantage of hydraulic tomography is that, it does not require measurements of the fracture size, shape, aperture, detailed deterministic or statistical information of the geometry of fractured zone and the spatial distribution of these parameters which are not typically available between boreholes. In particular Neuman [2005] noted that experience shows that the density of fractures counted on surface outcrops is significantly different from those obtained from boreholes implying that information collected along surface outcrops may not be representative of conditions inside the rock mass. In addition, boreholes typically sample a small portion of the rock and the intersection of fractures with a given borehole may not be an indication of the geometry and density of fractures in the intact rock. Rather, hydraulic tomography relies on hydraulic tests conducted across boreholes, which result in the propagation of multiple drawdown signals across a large portion of the fractured rock that are relatively easy to collect. These drawdown signals carry direct information on hydraulic properties of both the fractures and matrix which cannot be obtained from mapping fractures. Thus, hydraulic tomography holds significant potential for characterizing fractured rocks.

[70] Illman et al. [2007] discussed the issue of nonuniqueness of estimated K tomograms, as there could be an infinite number of solutions to the steady state inverse problem for a heterogeneous K field, even when all of the forcing functions are fully specified. Similarly, nonuniqueness of computed heterogeneous K and Ss tomograms could also be an issue for the transient inverse problem. But according to some researchers [Yeh et al., 1996; Yeh and Liu, 2000; Liu et al., 2002; Yeh and Simunek, 2002] when data are available at all estimated locations, the inverse problem becomes well-posed and ultimately lead to a unique and correct solution. More recently, Yeh et al. [2011] suggested that the nonuniqueness issue associated with highly parameterized problems arises from a lack of information required to make the problems well defined. They also suggested necessary conditions for an inverse model of groundwater flow to be well defined, which are: “the full specifications of (1) flux boundaries and source/sink, and (2) heads everywhere in the domain at least at three times (one of which is t = 0) and head change everywhere over the times must be nonzero for transient flow.” Through numerical experiments, Yeh et al. [2011] showed that when the necessary conditions are fulfilled, the inverse problem results in a unique and correct solution.

[71] We acknowledge that there are many possible K and Ss tomograms that can provide equally good matches between the simulated and observed hydrographs for the hydraulic tomography survey in the fractured rock because the conditions listed above for constraining a highly parameterized inverse problem are not fully given. Nevertheless, the SSLE algorithm constrains the estimates with specified mean and covariance function of the K and Ss parameters, and it seeks conditional unbiased mean estimates. That is, the estimates are unique in term of these constrains; they are the statistically unbiased estimates given all these nonredundant information from HT survey. They are, however, not necessarily the true fracture and matrix hydraulic conductivity and specific storage distributions. In spite of the uncertainty (or their deviation from the true), the estimates based on SSLE algorithm have been unequivocally shown to be capable of predicting responses of aquifers due to independent events which were not used in the analysis of hydraulic tomography by previous studies [e.g., Illman et al., 2007, 2008, 2009, 2010; Liu et al., 2007; Xiang et al., 2009; Berg and Illman, 2011a, 2011b; Huang et al., 2011]. Our study here further substantiates its robustness for hydraulic tomography survey of the fractured rock block.

9. Summary and Conclusions

[72] 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 have assessed the performance of transient hydraulic tomography (THT) in a fractured rock block. The block is characterized through flow-through tests and multiple pumping tests. These pumping test data were then analyzed using the Sequential Successive Linear Estimator (SSLE) code of Zhu and Yeh [2005] to conduct THT 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. This study leads to the following major findings and conclusions:

[73] 1. It is possible to delineate permeable fracture zones, their pattern and connectivity through the THT analysis of multiple pumping tests along with the inverse code SSLE developed by Zhu and Yeh [2005]. From the estimated K and Ss tomograms obtained from THT analysis of synthetic and laboratory data, it is evident that THT captured the fracture pattern quite well and they became more distinct with additional pumping tests.

[74] 2. Based on the THT analysis of synthetic data, the estimated K and Ss values for the fractures and the matrix may not exactly replicate the actual values, but the model also provides uncertainty estimates associated with the resulting K and Ss tomograms, which are given in the corresponding variance maps. However, the purpose of the study was to capture the fracture pattern (the pattern of K and Ss tomograms) using THT, which has been achieved here. The high-K and low-Ss zones clearly show the fractures and their connectivity, although the high-K and low-Ss zones do not continue to the edge of the rock, which could be due to the boundary effect.

[75] 3. Two cases of THT analysis were performed using the laboratory pumping tests by changing the location of second and third pumping tests to examine if there was any significant impact of these later pumping test locations on the pattern of resulting K and Ss. Results showed that the patterns of estimated K and Ss tomograms in real THT cases 1 and 2 were similar, although the pumped locations (second and third test) utilized for the inversion were different for two cases. A closer inspection of scatterplots comparing the K values from the two cases revealed that while the scatter is large, the K values cluster around the 1:1 line without much bias. It indicates that, changing the location of second and third pumping tests does not significantly impact the K estimates for this fractured rock block. On the other hand, the Ss values from the two cases scattered around the 1:1 line, but there was a noticeable bias suggesting some differences in the Ss values between the two cases.

[76] 4. Five independent pumping tests not included during the inversion were simulated using estimated K and Ss tomograms generated by each of the two real THT cases to evaluate to what extent they can predict independent pumping tests. For most cases, the predicted drawdown responses from the independent pumping tests captured the observed behavior at later time, while at early time predicted drawdown deviated from the observed for some tests. All the inverse simulations were run for 20 s and the matrix ports had no drawdown (in real THT case 1) or a very small drawdown (in real THT case 2) for all the calibrated and predicted cases, which was consistent with the observed data. SSLE being a porous media code, it is difficult for the model to handle the high contrast between the fracture and matrix K and Ss values. Thus, estimated K and Ss values may not exactly replicate the actual values for the fractured rock, which could have led to some discrepancies between the observed and simulated drawdown values. But overall, simulated drawdown responses captured the observed behaviors both at the fracture and matrix ports quite well, indicating that the K and Ss tomograms estimated using THT successfully captured fracture pattern and their connectivity.

[77] 5. We conclude that the results from this study are encouraging in that the patterns of K and Ss tomograms generated by THT analysis based on the inverse algorithm SSLE, are consistent with the fracture pattern of the rock sample. THT appears to be a promising approach in delineating fractures and their connectivity in the subsurface. This is significant because existing methods of fractured rock characterization typically relies on detailed field mapping of fracture geometry through outcrops and boreholes. However, mapping of such features between boreholes is currently not possible. Geophysical techniques may also provide information on fractures and their connectivity but they do not provide direct information on the hydraulic characteristics, which hydraulic tomography does. However, the development of hydraulic tomography in fractured rocks is still at the early stage given that this study was conducted in the laboratory in a controlled environment. More experiments are needed at the field scale in a variety of fractured rock environments.

Acknowledgments

[78] This research was supported by the Strategic Environmental Research and Development Program (SERDP) project number ER-1610. Additional support for the project was provided to Walter A. Illman by Natural Resources and Engineering Council of Canada (NSERC), Ontario Research Foundation (ORF), Canada Foundation for Innovation (CFI) and Obayashi Corporation. T.-C. Jim Yeh also acknowledges support from NSF grant EAR 1014594. Finally, we acknowledge the constructive reviews by an anonymous reviewer, Junfeng Zhu, Alberto Bellin and Daniel Fernàndez-Garcia (Associate Editor) which improved the manuscript.