Fully coupled hydrogeophysical inversion of synthetic salt tracer experiments



[1] We present a method for the determination of hydraulic conductivity from monitoring of salt tracer tests by electrical resistivity tomography (ERT). To ensure that the underlying principles of flow, transport, and geoelectrics are obeyed in the inversion, we perform a fully coupled hydrogeophysical analysis using temporal moments of electrical potential perturbations. In the predictive mode, we use moment-generating equations with corresponding adjoint equations for the evaluation of sensitivities. For inversion, we apply the quasi-linear geostatistical inversion approach. The method is tested in a synthetic case study mimicking a laboratory-scale quasi two-dimensional sandbox, in which 48 electrodes and 8 piezometers are used. The hydraulic conductivity field is estimated from the mean arrival times of electrical potential perturbations and hydraulic heads. The estimated hydraulic conductivity field reproduces most features with, however, a loss of variability. Even though only the temporal moments of the electrical signals are used for inversion, the transient behavior is satisfactorily recovered. Also, the spatial patterns of concentration arrival times in the true and estimated cases are in good agreement, so that the propagation of the tracer plume can be followed fairly accurately. We test the effects of large measurement errors and erroneous prior information on the performance of the inversion. While prior statistical parameters are of minor importance in detecting the major pattern of hydraulic conductivity, a large measurement error could have an important impact on the solution. Also, the choice of electrode configurations appears to be important. In particular, strictly surface-based geoelectrical surveys do not seem to be very suitable for identifying spatial patterns of hydraulic conductivity by ERT monitoring of salt tracer tests within aquifers.

1. Introduction

[2] The hydraulic conductivity distribution in an aquifer has a major impact on groundwater flow and solute transport and is therefore of great importance in subsurface hydraulics. It is often estimated by performing hydraulic experiments such as pumping tests, flowmeter tests, and tracer tests. The requirement of monitoring wells causes these approaches to be expensive, leading to a limited number of observation points and therefore often to an insufficient spatial resolution [e.g., Li et al., 2007, 2008]. Minimally intrusive geophysical methods can be applied to identify the subsurface structure, are relatively cheap, and lead to a higher resolution, but they do not yield direct information on hydraulic conductivity. The latter may be derived from the estimated geoelectrical, geomagnetic, electromagnetic, or seismic properties by petrophysical relationships, the validity and generality of which is often questionable. Geophysical surveying techniques can also be used to monitor hydraulic experiments, such as infiltration experiments [e.g., Daily et al., 1992], pumping tests [e.g., Sloan et al., 2007; Rizzo et al., 2004], and tracer tests [e.g., Binley et al., 1996], in which the geophysical signal is directly related to the hydraulic stress applied. One of these approaches consists in monitoring salt tracer experiments by electrical resistivity tomography (ERT), leading to the distribution of changes in the bulk electrical conductivity in the aquifer which may subsequently be converted to the concentration distribution [Binley et al., 1996, 2002; Slater et al., 2000; Kemna et al., 2002; Vanderborght et al., 2005; Singha and Gorelick, 2005]. The problem in most approaches is that the geoelectrical inversion lacks constraints from flow and transport, leading to nonphysical results such as a loss of mass during the salt tracer tests [Singha and Gorelick, 2005]. For this reason an approach is needed which considers flow, solute transport, and geoelectrics as a coupled system. Some work in this direction has been done, by using cross-hole geophysical data (ERT and GPR) [Looms et al., 2008] and by the joint use of GPR and hydrological data [e.g., Kowalsky et al., 2004, 2005]. Also, Binley et al. [2002] combined ERT and GPR data during salt tracer tests in the vadose zone, obtaining an estimate of the mean hydraulic conductivity in the layer of interest. However, to our knowledge, although it has been suggested, the actual determination of the spatial structure of hydraulic conductivity from time-lapse ERT monitoring of salt tracer tests has not yet been performed.

[3] Analyzing the full time series of geoelectrical signals obtained during geoelectrical monitoring of salt tracer tests in order to infer the hydraulic conductivity distribution in the aquifer in a fully coupled mode is theoretically possible, but computationally extremely demanding. Even the two-step approach in which geoelectrical inversion is decoupled from the hydraulic interpretation and the inverted concentration fields are further analyzed to obtain hydraulic conductivity, would be beyond standard computer resources, if we consider that an ERT inversion results in thousands of electrical conductivity values, and a time series could consist of hundreds of time points.

[4] As an alternative to an analysis of full transient time series, Pollock and Cirpka [2008] suggested the use of temporal moments of electrical potential perturbations. They showed how these moments can be computed by moment generating equations and presented an efficient method to calculate the sensitivity of the measurements with respect to the hydraulic conductivity. Geophysical inversion using temporal moments of ERT data has been used before [e.g., Day-Lewis and Singha 2008], but the moments were not directly related to the hydraulic conductivity field.

[5] The method by Pollock and Cirpka [2008] requires a linearization of the Poisson equation. The linearization was shown to be admissible when the tracer concentration was chosen carefully (i.e., yielding a strong enough signal while not affecting the linearization too much) and when the ratio of the first over the zeroth temporal moment of potential perturbation (i.e., the mean arrival time of the perturbation) was considered. They therefore suggested using the latter as measurement for the inversion rather than the zeroth and first moment themselves. They tested their forward model by predicting the measurements of electrical potentials, the temporal moments and the sensitivity patterns from the hydraulic conductivity field in a synthetic test case. However, no inversion was performed yet. In this contribution, we want to test this method further by performing a fully coupled inversion of synthetic salt tracer experiments. The approach of calculating the temporal moments of electrical potential perturbations and their sensitivity with respect to hydraulic conductivity is implemented using the quasi-linear geostatistical inverse method of Kitanidis [1995] with a few modifications. A particular advantage of geostatistical inversion is its rigorous derivation by Bayesian theory facilitating not only a best estimate but also the associated uncertainty of estimation. The method requires prior statistical parameters which may be difficult to obtain. Techniques of estimating these parameters from dependent data exist [Kitanidis and Vomvoris, 1983; Kitanidis, 1995; Li and Cirpka, 2006; Li et al., 2007; Nowak and Cirpka, 2006], but are not implemented in the presented inversion method. The synthetic case study mimics real laboratory experiments with realistic data acquisition setup and measurement errors.

[6] The primary objective of the present study is to analyze how well the hydraulic conductivity field of the aquifer can be reconstructed by applying the proposed method of inverting mean arrival times of geoelectrical signals obtained during salt tracer tests. We also analyze the transient behavior of the tracer to see whether the condensed information used for the inversion is sufficient and to what degree the transport of the tracer through the aquifer can be reproduced. We check under which conditions the approach is reliable, i.e., how exact our measurements should be and how large the influence of erroneous statistical prior knowledge is.

[7] Another important issue in geoelectrical surveying is the choice of electrode configurations used in the monitoring. Various studies have been made to analyze the performance of different arrays, both for surface and for borehole measurements [e.g., Zhou and Greenhalgh, 2000; Dahlin and Zhou, 2004; Stummer et al., 2004]. While our particular small-scale two-dimensional synthetic case study differs from typical field applications, we tried to choose and reproduce measurement setups which relate to typical field surveys involving surface electrodes and a few vertical lines of electrodes, typically installed in boreholes, to see whether they can be used in high-quality monitoring of salt tracer tests.

2. Theory and Methods

2.1. Governing Equations

[8] In this section, we review the governing equations for flow, transport and direct current geoelectrics. A detailed description was given by Pollock and Cirpka [2008].

[9] We assume steady state groundwater flow without internal sources or sinks:

equation image

subject to the boundary conditions

equation image
equation image
equation image

where K is the hydraulic conductivity, assumed isotropic, h is the hydraulic head and n is the unit vector normal to the boundary Γ. The latter is subdivided into a Dirichlet boundary, ΓD, with fixed head h0, a Neumann boundary, ΓN, with fixed normal flux q0 (in many cases zero), and a Cauchy boundary, ΓC, with a normal flux linearly depending on the head; the proportionality coefficient λ is known as leakage coefficient, and href is a known reference head. The specific discharge q follows Darcy's law:

equation image

[10] The concentration of a tracer in an aquifer following an injection through the inflow boundary Γin follows the advection-dispersion equation:

equation image

subject to the initial and boundary conditions:

equation image
equation image
equation image

where θ is the porosity, D is the dispersion tensor, c0(t, x) is a known, time- and space-dependent concentration along the inflow boundary, and the boundary Γ = Γin ∪ Γno flow ∪ Γout is subdivided into an inflow, a no-flow, and an outflow section.

[11] A change in solute concentration in the aquifer leads to a perturbation of the bulk electrical conductivity field σ(t, x). The latter can be expressed as the sum of the background conductivity σ0, constant through time and independent of tracer concentration, and the perturbation of electrical conductivity σ′ resulting from the injection of the tracer:

equation image

[12] The perturbation term depends on the concentration of the tracer and can be expressed as

equation image

where κ is a proportionality constant which can be derived from Archie's law [Archie, 1942] and may vary in space. An additional term accounting for surface conductivity has been neglected. This is only applicable if the aquifer has a low clay content. The latter should be valid in most cases: in the presence of clay, the performance of tracer tests would be much more critical and a different approach would probably be needed in any case.

[13] The electrical potential ϕ in a direct current geoelectrical survey follows the Poisson equation:

equation image

subject to

equation image
equation image

where δ() is the Dirac delta function and I is the current injected at location xi and extracted at location xo. Γn.c. is a no-current flow boundary, Γmix represents a mixed-condition boundary, in which the coefficients β and γ are chosen such that the Poisson equation for uniform coefficients in a semi-infinite domain is met at the boundary.

[14] In geoelectrical surveying we generally consider the electrical potential differences Δϕ measured between two locations image and image instead of the absolute values:

equation image

in which Ω is the entire domain.

[15] In the Poisson equation, equation (12), the relationship between electrical potential ϕ and electrical conductivity σ is nonlinear. By subtracting the Poisson equation prior to the injection of the tracer from the one following the injection and neglecting products of perturbations, we obtain a linearized form of the Poisson equation:

equation image

subject to

equation image
equation image

where ϕ0 is the base potential observed in the absence of the tracer, and ϕ′ the potential perturbation induced by the presence of the tracer.

[16] In our analysis of the geoelectrical surveys, we consider temporal moments of electrical potential perturbations. These can be computed by temporal moment-generating equations, which can be derived from equations (6) and (16) [Harvey and Gorelick, 1995; Pollock and Cirpka, 2008]:

equation image

subject to

equation image
equation image
equation image

subject to

equation image
equation image

where mkc and mkϕ′ denote the kth temporal moments of concentration and potential perturbation, respectively.

[17] It has been shown by Pollock and Cirpka [2008] that the linearization used to obtain the moment-generating equation for electrical potential perturbation is acceptable when considering the mean arrival time of the potential perturbation (i.e., the ratio of the first moment over the zeroth moment) as a primary measurement and choosing the tracer concentration such that the perturbation of electrical conductivity is not too big, because of linearization, while guaranteeing a sufficient signal strength [see Pollock and Cirpka, 2008].

2.2. Geostatistical Inverse Method

[18] For inversion, we apply the quasi-linear geostatistical approach of Kitanidis [1995], which we briefly review in this section. We consider the log hydraulic conductivity Y(x) as a spatial function described by the sum of nβ deterministic base functions Xi(x), each weighted by a trend coefficient βi, and spatially correlated random fluctuations Y′(x) about the deterministic trend:

equation image

[19] We assume the distribution of Y′(x) to be multi-Gaussian, with an expected value of zero and the covariance function RYY(x1, x2). The latter depends on a vector equation image of structural parameters, such as the variance and correlation lengths. In our application, the deterministic contribution is spatially uniform, implying that X(x) = 1 throughout the domain. In general, the corresponding trend coefficients βi are uncertain and follow a multi-Gaussian prior distribution with expected value β* and covariance matrix Rββ.

[20] The primary objective is to estimate the distribution Y(x) from a nd × 1 vector of measurements d of hydraulic heads and mean arrival times of electrical potential perturbation. This is done by maximizing the conditional pdf of the random parameter contributions Y′(x) and the trend coefficients β, given the measurements d. In our application, we consider the structural parameters equation image to be known, but it is also possible to infer equation image jointly with the field Y(x) from the measurements [Kitanidis and Vomvoris, 1983; Kitanidis, 1995; Li and Cirpka, 2006; Li et al., 2007; Nowak and Cirpka, 2006].

[21] The spatial domain is discretized into nY elements, leading to a discrete nY × 1 vector Y of log hydraulic conductivity, a nY × 1 vector of random components Y′, and a nY × nβ matrix of discretized base functions X:

equation image

[22] We assume that the residuals, that is, the difference between the measurements d and the model outcome m(Y) using the estimated log hydraulic conductivity values, follow a multi-Gaussian distribution with covariance matrix Rdd, representing measurement and model error. Then, by applying Bayes' theorem, we obtain that the most likely set of log hydraulic conductivity fluctuations, Y′, and trend coefficients, β, minimizes the following objective function:

equation image

in which the first term (likelihood term) penalizes the deviations between the measurements and the model outcome, the second term penalizes spatial fluctuations, and the third term penalizes deviations in the trend coefficients β from the prior mean. To minimize the objective function, we apply a Gauss-Newton approach by successively linearizing the functional relationship m(Y) between the modeled potential moments and hydraulic heads and the log hydraulic conductivity vector Y about the current estimate equation imagek:

equation image

in which the hat denotes the estimate, k is the iteration index, and Hk is the sensitivity matrix

equation image

[23] The computation of the latter is a very important step in Gauss-Newton like geostatistical inversion. Pollock and Cirpka [2008] presented a suitable method for the evaluation of H based on the continuous adjoint state method of Sun and Yeh [1990]. In case of underdetermined problems, this approach provides a large gain in computational costs compared to the calculation by direct numerical differentiation.

[24] The updated estimate equation imagek+1 of the log hydraulic conductivity vector can be computed by

equation image

in which the coefficients βk+1 and ξk+1 are evaluated by solving the system of quasi-linear cokriging equations [Kitanidis, 1995]:

equation image

[25] The estimation of the log hydraulic conductivity vector Y is repeated with an updated sensitivity matrix until a convergence criterion is met. The geostatistical approach requires the multiplication of large matrices (e.g., HkRYYHkT), which can efficiently be computed by spectral methods [Nowak et al., 2003]. For stabilization of the Gauss-Newton method, we apply a line search in the Gauss-Newton direction.

[26] Finally, the conditional covariance matrix RYYd of the log hydraulic conductivity vector Y given the observations d is approximated by

equation image

in which the iteration index k has been omitted.

2.3. Numerical Implementation

[27] The partial differential equations are discretized by the Finite Element Method (FEM) using bilinear, rectangular elements, each with a constant hydraulic conductivity value. To stabilize the simulation of advection-dominated transport, the streamline upwind Petrov-Galerkin (SUPG) method is applied [Brooks and Hughes, 1982]. To achieve computational speedup, the source terms in the adjoint state equations of temporal moments are defined by analytical expressions [Nowak, 2005]. The resulting systems of linear equations are solved using the factorization of the direct solver package UMFPACK [Davis, 2004].

3. Application to Synthetic Data

[28] In this section we present an application of the geostatistical inversion to a two-dimensional artificial test case, mimicking a laboratory-scale sandbox experiment which is currently in detailed planning.

3.1. Setup

[29] We simulate an intermediate-scale quasi two-dimensional sandbox of dimension Lx × Ly × Lz. The domain is discretized by nx × nz elements of size Δx × Δz. The geometric parameters are listed in Table 1. The domain is assumed to be electrically insulated, so that no-current boundary conditions can be applied at all boundaries. The true hydraulic conductivity distribution in the sandbox is synthetically generated by using the spectral method of Dietrich and Newsam [1993] with an exponential covariance function for the log conductivities. The resulting field is illustrated in Figure 1a. The top and bottom of the sandbox are no-flow boundaries, while we assume fixed hydraulic head conditions along the right and left boundaries. We simulate the injection of a tracer solution with concentration cin through the left boundary during a specified period of time tinj. The simulated sandbox is equipped with electrodes with a horizontal and vertical spacing of 10 cm. In our test case we use 48 electrodes (one row of 28 on the top and four columns with another five electrodes each in the vertical direction). All the electrodes extend through the whole width of the sandbox, so that the electrical field arising from current injection is two dimensional. The electrodes situated near the top of the sandbox approximately represent surface electrodes, while the others simulate boreholes. Our laboratory type of setup is of course a simplification, as the electrical field in common field applications would be three dimensional. However, we try to reproduce the common situation in which only a few boreholes at distances larger than the aquifer thickness are available, whereas the surface is accessible for a comparably large number of electrodes. We simulate injection and extraction of a current using pairs of electrodes, while the electrical potential is measured at other pairs of electrodes. Various electrode arrays were tested, including surface, cross-hole, surface-to-borehole and single-hole measurements. The chosen example includes a total of 448 surface (336 configurations, including a mixture of Wenner and Schlumberger arrays) and surface-to-borehole (dipole-dipole) measurements. We also considered only the surface measurements, to see whether the absence of boreholes would still lead to satisfactory results. Eight head measurements, at the top and bottom of each borehole, were also included in the inversion. In a field campaign this would require packing off various sections of the borehole to prevent hydraulic shortcuts in the boreholes.

Figure 1.

(a) Synthetically generated true log10 hydraulic conductivity field, (b and d) estimated log10 hydraulic conductivity field resulting from the inversion, and (c and e) standard deviation of estimation. Open circles, position of the electrodes used in the inversion; crosses, position of the head measurements.

Table 1. Parameters Applied in the Test Case
Geometric Parameters and Discretization
Lx × Ly × Lzlength × width × height of domain2.8 × 0.05 × 0.6 m
nx × nznumber of cells in x and z directions280 × 60
Δx × Δzgrid spacings in x and z directions0.01 × 0.01 m
Hydraulic and Transport Parameters
equation imagehydraulic gradient0.01
cinconcentration in inflow0.4 g/ℓ
αllongitudinal dispersivity0.01 m
αttransverse dispersivity0.001 m
Dmpore diffusion coefficient10−9 m2/s
Geoelectrical Parameters
Icurrent2 mA
σ0base electrical conductivity0.03 S/m
κlinear dependence of electrical conductivity on concentration0.06 Sm2/kg
Time Parameters
tinjduration of injection2 h
Δttime discretization in transient calculation5 min
Geostatistical Values of Log Hydraulic Conductivity Field
λx × λyCorrelation lengths0.4 × 0.05 m
exp(β*)prior mean10−7 m/s
Rββuncertainty of prior mean10
Measurement Errors
σhhydraulic head10−3 m
σRelectrical resistance1%

[30] Generally, a higher number of independent measurements leads to more accurate inversion results, but in real experiments the number of possible measurements is limited. This particularly holds for time-lapse monitoring considered here. Each time-lapse survey requires a certain amount of acquisition time. During this time, the plume keeps on moving through the domain. One advantage of using temporal moments is that we do not need to take into account the moving of the plume during the acquisition cycle, as each single measurement is registered together with the exact time of acquisition, and each temporal moment is calculated from the time series of a single configuration. However, if an acquisition cycle takes too much time, the plume moves too far between two cycles to be captured in the monitoring. For the evaluation of temporal moments it is of importance that key features of the electrical potential time curves, such as the time and magnitude of maximum perturbation, are resolved, implying that a large time step between the measurements could lead to an insufficient amount of data for accurately determining the temporal moments. Mimicking a laboratory experiment using a multichannel ERT device, it is realistic to assume that a single acquisition cycle including a few hundred measurements takes several minutes. We have accounted for that in our simulations by using an appropriate time step.

[31] In our geostatistical inversion method, we need to compute the sensitivity matrix H, i.e., the partial derivatives of all measurements with respect to all parameters. The sensitivities show the effect of slight changes in a parameter on the measurements. Figure 2 shows two sensitivity patterns for a particular electrode configuration: the sensitivity of electrical resistance with respect to tracer concentration and the sensitivity of the mean arrival time of resistance perturbation with respect to log hydraulic conductivity. The former may be interpreted as the measurement function of concentration and shows a pattern typical in geoelectrics with high sensitivities at the electrodes and changing signs. The sensitivity of the arrival time of the electrical signal on log hydraulic conductivity shows a narrow negative sensitivity stripe which follows the main plume path in upstream direction. We can understand this pattern as the convolution of the first sensitivity with the sensitivity of local concentration arrival time on log conductivity [e.g., Cirpka and Kitanidis, 2000]. By combining various electrode configurations different parts of the hydraulic conductivity field become sensitive. This is a prerequisite to resolve the conductivity field by inversion. The sensitivity patterns in our particular application are influenced by the fact that we consider a sandbox which is bounded on all sides. In a field application with boundaries situated at a large distance from the zone of interest, the sensitivity would be smaller, resulting in a more difficult recovery of the hydraulic conductivity field.

Figure 2.

(a) Sensitivity of electrical resistance with respect to tracer concentration and (b) sensitivity of mean arrival time of electrical resistance perturbation with respect to log hydraulic conductivity. Crosses, position of the current electrodes; circles, position of the potential electrodes.

[32] In order to generate synthetic measurements with a realistic measurement error, we solve the transient forward problem on the basis of the “true” hydraulic parameters. We thus obtain the electrical potential curves for each configuration: Δϕ′(t) = ϕ′(t, image − ϕ′(t, image in which image and image are the two locations of electrical potential measurements. To simulate measurement error, we add white noise to each simulated measurement Δϕ′(t) of potential difference. Lab experiments (in which the same measurements were repeated several times) permitted to estimate the measurement error for the electrical resistances. For all the tested configurations the error reached a maximum of 1%. In our synthetic test case, this error was converted to that of electrical potential and added to the error-free breakthrough curves of potential. From the noisy data, we compute the temporal moments of potential perturbation differences by

equation image

[33] The ratio of the first over the zeroth moment gives us the mean arrival time of the perturbation for each electrode configuration. The moments of the noisy data were taken as our measurements in the inversion procedure. In order to assess the uncertainty of the arrival times of resistance perturbations, we follow a bootstrap procedure, in which additional white noise is added to the Δϕ′(t) time curves. For each breakthrough curve we generated a set of 1000 bootstraps, and calculated the temporal moments for each one of them. The resulting standard deviation of the ratio of the first over the zeroth moment of potential perturbation was taken as the uncertainty of the measurement. To simulate noisy measurements of hydraulic heads, we added a random error with zero mean and standard deviation of 1 mm to the error-free hydraulic heads at the corresponding nodes. The value of 1 mm was also considered as measurement uncertainty. While this accuracy might be optimistic for field applications, it should be achievable in laboratory experiments.

[34] In our synthetic case study, we assumed for simplicity that the proportionality factor κ linking concentration to electrical conductivity is constant and known throughout the domain. In real applications, additional experiments may be required to obtain κ as a spatial field. This spatial variability can be accounted for in the inversion. In the planned laboratory experiments, the variability of κ may be obtained by performing two sets of ERT measurements, one when the tracer concentration is zero throughout the box, and the other when it has reached a maximum value everywhere within the box. Standard ERT inversion would yield the electrical conductivity distributions for two concentrations, which can be converted to the spatial distributions of the base electrical conductivity σ0 and the proportionality factor κ. In field applications, this approach may be difficult to apply, because it is much more difficult to enforce a uniform concentration distribution. Additional geophysical surveying of the aquifer might be necessary to obtain an estimate of the spatial variability of κ (e.g., Linde et al. [2006], using GPR surveys). Alternatively, the proportionality constant could be treated as a free parameter in the inversion.

[35] The hydraulic conductivity estimate obtained by the inversion and its uncertainty would be statistically correct if the deviations between the true and estimated parameters had zero mean and covariance matrix RYYd. With 16800 parameters, unfortunately, testing the full conditional covariance matrix is computationally very demanding. For this reason we restrict the test to the estimation variance by computing the normalized error ɛnorm,i for each parameter, which is the difference between the true and estimated value normalized by the standard deviation of estimation equation imagei of each log hydraulic conductivity value:

equation image

[36] The set of normalized errors ɛnorm,i follows a standard normal distribution, if the computed conditional mean and variance are correct.

[37] To quantify the overall error, we also compute the normalized root-mean-square error (NRMSE) from the normalized residuals for the different test cases:

equation image

[38] For an unbiased estimate, the NRMSE should be close to unity. We also compute the root-mean-square error (RMSE), which does not account for the standard deviation of estimation and expresses the overall discrepancy between the real and estimated log hydraulic conductivity values. A low RMSE indicates that the estimated hydraulic conductivity field agrees very well with the true field, regardless of the estimation uncertainty. To test the performance of the inversion approach, we also compare the true mean values and variances of the log conductivity field with the estimated ones.

3.2. Results and Discussion

[39] After 15–20 iterations, the inversion procedure usually converged or else showed minimal further improvement. The computational time for the inversion and the final transient calculation using the estimated hydraulic conductivity field ranged between 10 and 15 h on an Intel Quadcore Q9300 (2.5 GHz, 8 GB RAM, running in 64 bit MATLAB, on 64 bit Linux). The convergence criterion was set as a maximal absolute difference between the log hydraulic conductivity of two consecutive steps of 0.05, and a maximal change in the likelihood value of the objective function, weighted by the number of measurements, of 0.02. Figures 1b and 1c show the best estimate of the log hydraulic conductivity distribution and its corresponding uncertainty for the described test case. A comparison with the true log hydraulic conductivity field shows that most features are reproduced, but the simulated field is, as expected, smoother than the original one and small-scale variability is lost: the variance of the estimated log hydraulic conductivity field is reduced by about 40% in comparison to the true field. Table 2 shows the root-mean-square errors and the comparison of mean values and variances. Figure 3 shows the distribution of the normalized errors ɛnorm,i for our test case. The assumption that they follow a standard normal distribution must be rejected. The distribution is slightly asymmetric leading to a minimal shift of the mean value toward negative values, and the variance reaches a value of approximately 1.05. This is in agreement with equation (32) giving only a lower limit of the estimation variance. The slight overestimation is caused by the linearizations applied in the inversion procedure.

Figure 3.

Distribution of normalized errors ɛnorm. The solid line indicates a standard normal distribution.

Table 2. Performance for Different Test Casesa
Test CaseRMS(ΔY)NRMSE(ɛ)βYσ2Y
  • a

    Shown are root-mean-square (RMS) error of log conductivity field, normalized root-mean-square (NRMS) error of normalized residuals, mean value of log hydraulic conductivity field, and variance of log hydraulic conductivity field.

True log conductivity field  −71
Test case according to Table 10.721.06−6.950.56
False correlation lengths λx = λy = 0.10.831.02−6.860.60
False prior variance σY2 = 20.981.09−6.930.72
False prior variance and correlation lengths0.930.84−6.860.60
False prior variance, correlation lengths, and mean value βY = −50.900.81−6.880.55
Higher measurement error σR = 2%0.801.15−6.890.46
Higher measurement error σR = 5%1.041.39−7.000.27
Surface measurements0.931.14−6.810.48
Pure hydrological inversion0.970.29−7.090.09

[40] In order to assess the influence of the statistical prior knowledge on the solution, the inversion was performed for the same hydraulic conductivity field but assuming false variance and correlation length, respectively. The numerical results are shown in Table 2. Assuming a too high variance σY2 of the log hydraulic conductivity fluctuations leads to an increase in the standard deviation of estimation, but the estimated log hydraulic conductivity field is very similar. On the contrary, an underestimation of the prior variance leads to a large reduction of the variability of the field because the latter is penalized. Wrong correlation lengths logically lead to differently shaped features in the log hydraulic conductivity field and in its standard deviation, for example, by choosing the same correlation length in both directions, the features are less stretched. However, the overall pattern remains the same, with the zones of high and low conductivity being detected quite accurately. In fact, in some areas, particularly in the top part of the sandbox, the result is better than in the case with true correlation lengths. Combining erroneous values about prior statistics produces a slight deterioration of the estimate (although the main areas of low and high conductivity are still detected), but the results nonetheless show consistency, as the uncertainty of the estimation also increases significantly. Because the prior variance of β is very high, an erroneous mean value does not cause any additional deterioration of the results.

[41] Variations of the measurement error also have a considerable impact on the inversion result. While a doubling of the error does not influence the hydraulic conductivity field too much, except for a slight decrease in variability, a 5% error not only causes an even larger reduction in the variability, but also the pattern is poorly reconstructed. The main reason for this is the difficulty of calculating temporal moments from very noisy data. This issue might therefore become more important in field applications, where the measurement errors could be significantly higher.

[42] The analysis of the RMSE logically shows that the smallest error is obtained by the test case using the true statistical parameters. The test case with an increased error (2%) and the one with false prior correlation lengths are comparatively good too. In particular, the latter shows a NRMSE very close to one, indicating that the estimate is practically unbiased.

[43] The comparison between the true and estimated mean values and variances indicates that, while the mean values show little variations and no significant bias, the variance, indicating how well the variability of the field is reproduced, is always reduced, usually by about 40 to 50%, meaning that small-scale features are lost. Even more variability, in particular in the lower part of the sandbox, is lost when using only surface measurements. The results of the inversion for this case are shown in Figure 1d (showing the estimated field) and Figure 1e (showing the corresponding uncertainty). While the top part of the domain is reproduced quite well, the pattern in the bottom part is only poorly recovered. Also, the uncertainty of the estimation in the lower half of the sandbox is significantly higher than in the upper half. For field applications, this implies that electrodes within the aquifer are most likely required in high-quality monitoring of salt tracer tests by ERT. The quality would further decrease if conductive layers of fines were overlaying the aquifer under investigation.

[44] We also performed inversions using other configurations and a different number of boreholes, the results of which are not shown here. The addition of single-hole and cross-hole measurements did not yield any improvement. The inclusion of more boreholes logically led to a higher resolution and greater variability of the inferred hydraulic conductivity field throughout the sandbox. However, this is obviously associated to higher costs.

[45] In general, the estimation of the hydraulic conductivity in the part of the sandbox nearest to the inflow was more accurate than in the parts near the outflow. The main reason for this can be found in the sensitivity patterns described previously, with the higher-sensitivity areas situated near the electrodes and upstream of them. This implies that the zone at the beginning of the sandbox is much better resolved, while in the zone near the outflow only the measurements including the electrodes situated in that area yield information on the hydraulic conductivity distribution.

[46] In order to show the additional value of geoelectrical data compared to the use of hydraulic data only, we performed a pure hydrological inversion using only the eight hydraulic head values as measurements. The result was, as expected, very bad: the hydraulic conductivity field obtained was practically homogeneous. An increase of the number of measurements led to slightly better results, but only the use of a very large number of boreholes, associated with a reduction of the measurement error, permitted to obtain a hydraulic conductivity field comparable to the one resulting from the coupled inversion.

3.3. Comparison of Transient Behavior

[47] At the heart of our approach is the characterization of the measured electrical signals by their mean arrival time. This is the only information used in the inversion. If these data are informative enough to resolve the underlying hydraulic conductivity field, the full time series of resistances should be recovered when performing transient simulations using the estimated field. We test this by simulating and comparing transient breakthrough curves of electrical resistance perturbations using the true and estimated hydraulic conductivity distribution. Two examples for our main test case are shown in Figure 4.

Figure 4.

Comparison of the breakthrough curve of electrical resistance perturbation for two particular electrode configurations, computed using the true (solid line) and estimated (dashed line) hydraulic conductivity distribution: (a) unimodal resistivity time curve and (b) bimodal resistivity time curve.

[48] While most of the breakthrough curves (even the ones with secondary peaks) are reproduced quite well, like the one illustrated in Figure 4a, some curves show qualitative differences, such as the shift of the secondary peak in Figure 4b. Also, the root-mean-square difference between the true and estimated breakthrough curves is generally higher than the measurement error. We conclude that the full time series of time-lapse ERT data contain more information than covered by the mean arrival time. To retrieve this information in fully coupled hydrogeophysical inversion would be computationally quite demanding. Luckily, major features of the hydraulic conductivity field are already resolved by the inversion of lower-order temporal moments presented here.

[49] We also simulated the propagation of the tracer plume through the sandbox using the true and estimated hydraulic conductivity fields and compared the concentrations obtained and the mean arrival time of the plume throughout the entire sandbox. Figure 5 shows the mean arrival time of the plume in the sandbox, i.e., the ratio of the first over the zeroth moment of concentration. The propagation of the plume is reproduced fairly accurately and there are no significant differences between the true solution and the estimation. Even the loss of smaller-scale heterogeneities, caused by the smoothing of the hydraulic conductivity field in the inversion, does not lead to a great error in the prediction of solute transport. By construction, the mass balance of the tracer is met, which is in contrast to two-step geoelectrical inversion [e.g., Singha and Gorelick, 2005].

Figure 5.

Comparison of the mean arrival time (in h) of the tracer plume for (a) the true and (b) the estimated hydraulic conductivity field.

[50] In most tested cases the results were satisfactory. The main problems were encountered when increasing the measurement error too much, which led to significant differences in the plume arrival time in some parts of the sandbox. Also, when all prior statistical parameters (variance, correlation lengths and mean value) differed significantly from the true values, the results obviously deteriorated. If only one of these values was erroneous, plume propagation was still predicted well. This holds even by choosing a wrong correlation length both in horizontal and vertical direction. While this false assumption was clearly visible in the comparison of the true and estimated log hydraulic conductivity field, in which the features were well reproduced but had a slightly wrong shape, practically no effect could be detected in the prediction of solute transport. In the simulation using only the top row of electrodes representing surface geoelectrical surveying, the low resolution in hydraulic conductivity in the bottom part of the domain also led to an almost complete loss of spatial variability in the pattern of solute arrival time.

4. Conclusions

[51] In this study we have shown that temporal moments of electrical potential perturbations obtained during geoelectrical monitoring of salt tracer tests can be used to infer the hydraulic conductivity distribution using a fully coupled hydrogeophysical inversion strategy. By coupling the hydraulic and the geoelectric problems we avoid nonphysical results, whereas the condensation of information to temporal moments and the use of moment-generating equations (which avoids computing the transient transport and geoelectrics in every iteration) facilitates inverting the data within a reasonable amount of computational time. We have used the quasi-linear geostatistical method of inversion, but the approach of inverting temporal moments of resistivity time curves can also be combined with any other inversion scheme applicable to this type of problem.

[52] The test case with synthetic data showed reasonable agreement between the true hydraulic conductivity field and the best estimate obtained by the inversion. The same applies for the reproduction of transient behavior, which may be important in contaminant studies. Problems could arise from erroneous prior knowledge. A potential remedy would be to estimate the statistical parameters from the measurements too (while of course increasing the computational effort) [Kitanidis, 1995, 1996; Kitanidis and Vomvoris, 1983; Li and Cirpka, 2006; Li et al., 2007; Nowak and Cirpka, 2006]. As in all applications, the quality and quantity of available data plays an important role in determining the resolution that can be obtained in the estimation of the hydraulic conductivity field. For this reason it is important to optimize the data acquisition and find a good tradeoff between amount of data and cost efficiency. Even with a large number of measurements, small-scale hydraulic features remain difficult to detect. The example discussed here was limited to four vertical chains of electrodes, in order to show a setup which, when translated to three dimensions, would not be unrealistic in a field study. The inclusion of further vertical electrode lines would enable to obtain a superior resolution throughout the domain. While installing vertical electrode chains in the subsurface is usually done in boreholes, a cost-efficient alternative might be the installation via direct push methods, allowing for a higher density of vertical electrode lines. Our test simulations using only the top row of electrodes in the geoelectrical survey indicated, not very surprisingly, bad resolution at depth. While the depth penetration can probably be improved slightly in the field by increasing the spacing between the electrodes, which is limited in the sandbox by the boundaries, the results nonetheless cast doubts on whether surface geoelectrical surveying might be suitable for detecting patterns of hydraulic conductivity fields by ERT monitoring of aquifer tracer tests. If the main target quantity is the mean effective velocity of the tracer plume within a shallow aquifer, the resolution obtained by surface surveys might be sufficient, which of course saves costs and efforts in comparison to setting up a test field including numerous boreholes.

[53] We restricted the analysis to the zeroth and first moment of potential perturbation during salt tracer tests. This leads to acceptable results in the prediction of transient behavior. The method could also be extended to include higher-order moments, increasing the amount of information. However, this would come at the cost of increased computational time. Also, as could already be seen when a larger measurement error was assumed, higher-order moments are increasingly sensitive to noise, so that a reliable calculation thereof is not ensured.

[54] Altogether, the successful application of the method to synthetic test cases and the possibility, after some necessary modifications, of extending it to three dimensions for field purposes, leads us to consider this approach as very promising for the estimation of hydraulic conductivity distribution in an aquifer. The next step will be to validate it by applying it to real data obtained from salt tracer experiments in a two-dimensional laboratory sandbox.


[55] We thank Associate Editor Frederick Day-Lewis, Andrew Binley, and two anonymous reviewers for their constructive remarks helping to improve the quality of the paper. This study was funded by the Swiss National Science Foundation under grant 200021-113296. The research is an integral part of the project RECORD by the Competence Center Environment and Sustainability of the ETH domain.