A Bayesian approach for inversion of hydraulic tomographic data

Authors


Abstract

[1] Hydraulic tomography is a promising methodology that shows the potential to map subsurface hydraulic properties at an unprecedented level of detail by interpreting a suite of hydraulic tests. In the present work, we apply the hydraulic tomography concept to a Multiple Pulse Multiple Receiver (MPMR) cross-well configuration, which consists of two boreholes subdivided into intervals by packers. A short pressure (or flow) pulse is applied sequentially to all the intervals, and for each perturbation, the transient heads are recorded at the remaining intervals. The resulting tomograms are inverted within the Bayesian framework by using the pilot point approach. In addition to the values of the hydraulic conductivity at the pilot points, we assume that the stochastic parameters of the spatial variability model (the structural parameters) are unknown. Using synthetic two-dimensional and three-dimensional examples, we demonstrate the effectiveness of the MPMR configuration and the inversion procedure for characterizing the spatial variability of the hydraulic conductivity with limited or no prior information. We observed that increasing the number of source points (the locations at which the pulse is alternatively applied) provides more details on the spatial variability and that the parameters of the hydrogeological model of spatial variability are inferred with an acceptable, although variable, level of accuracy. In particular, the theoretical variance of the log conductivity is estimated with large errors, while the estimate of the anisotropic integral scales depends on the distance between the boreholes. Inversion preserves the overall spatial pattern of hydraulic conductivity, although low conductivity values are less connected in the inferred than in the true conductivity fields.

1. Introduction

[2] Predicting the fate of contaminants in heterogeneous aquifers requires careful characterization of hydraulic parameters, such as hydraulic conductivity and specific storage. Traditionally, pumping tests have been used for this purpose. However, pumping test analysis provides effective hydraulic parameters of large volumes, such that much of the spatial variability on a scale of relevance for transport remains unresolved. Unfortunately, existing tests with small support volume, such as slug tests and laboratory analysis on core samplers, cannot fill this gap of knowledge because of their limited spatial coverage, which makes data fusion a difficult, and often elusive, goal [see, e.g., Rubin et al., 1998; Beckie and Harvey, 2002]. If spatial variability is poorly resolved, as actually occurs in most applications, relevant subsurface features, such as interconnected high or low conductivity zones inducing preferential flow paths, may be missed, thus limiting our capability to predict early (and late) arrival times in the successive modeling efforts [e.g., Knudby and Carrera, 2005]. This bears negative consequences on decisions taken by trusting the outcomes of flow and transport models, as for example in risk analysis.

[3] Several techniques have been proposed in order to better characterize the spatial variability of hydraulic parameters at the local scale, including tracer tests, geophysical techniques and a new way of conducting and analyzing hydraulic tests, which is called hydraulic tomography [see, e.g., Gottlieb and Dietrich, 1995; Yeh and Liu, 2000; Bohling et al., 2007, and references therein]. Hydraulic tomography, in particular, has been proposed as a viable alternative to traditional pumping tests because it can be implemented with a spatial resolution that allows resolving the most relevant scales of variability for transport.

[4] The general hydraulic tomography concept, as explained above, can be exploited in several ways, from the interpretation of constant discharge multiwell pumping tests by type-curve analysis [Neuman et al., 2004], or inversion [Yeh and Liu, 2000; Indelman, 2003; Wu et al., 2005; Firmani et al., 2006; Li et al., 2005, 2007], to multiple pulse multiple receiver (MPMR) transient hydraulic tests in a cross-well configuration, as suggested by Butler et al. [1999] and Bohling et al. [2007]. So far, field applications of the MPMR configuration have been hampered by the complexity of the experimental setup, but recent advances in tubing, pulse emission equipments, and monitoring devices opened interesting opportunities for its practical implementation [Butler et al., 1999; Bohling et al., 2007; Liu et al., 2008]. Therefore in the present work we focus on the MPMR configuration because it showed the potential for resolving intermediate to small-scale variability with only a moderate increase in complexity of the experimental setup with respect to traditional pumping tests.

[5] In a typical MPMR test, a localized flow or pressure pulse is applied sequentially at several depths within a borehole while the response of the aquifer to this perturbation is monitored at several locations in one or more additional boreholes (for this purpose the boreholes are subdivided in intervals, for example by using packers). The recorded head data are then utilized in an inversion procedure to infer spatially variable hydraulic parameters.

[6] The idea of inverting head data in order to obtain the hydraulic parameters is not a new one. An extensive literature exists on this subject, mainly referring to head data collected under natural and forced flow conditions [e.g., Yeh, 1986; Kuiper, 1986; Carrera, 1987; Ginn and Cushman, 1990; Sun, 1994], and Bayesian methods have been suggested to alleviate the often overarching difficulties encountered in solving a mathematical problem that is inherently ill posed [e.g., Dagan, 1985; Carrera and Neuman, 1986; McLaughlin and Townley, 1996; Tarantola, 2005]. In this context, synthetic studies showed that inversion is facilitated if redundant measurements are available, as for example when multiple well testing is utilized. Redundancy allows a better separation of noise from signal, although it leads to a larger computational burden [Giudici et al., 1995; McLaughlin and Townley, 1996; Tarantola, 2005]. The MPMR configuration matches perfectly this philosophy because the aquifer is explored with many pulses following different pathways to the receivers.

[7] In an attempt to alleviate the otherwise high computational burden, Bohling et al. [2002, 2007] utilized the steady-shape approach for inversion of transient hydraulic data, a methodology envisioned by Butler [1988] following an early work of Wenzel [1942], whose justification lies in the observation that a steady-shape configuration with nearly constant (in time) hydraulic gradients is established very rapidly in many field settings [Butler, 1988; Kruseman and de Ridder, 1990; Bohling et al., 2002]. In such a situation the spatial gradients are stationary (time independent) and can be approximated by the space derivatives of the heads obtained by solving the steady state flow equation. An important advantage of this methodology, in comparison with the classical analysis of steady state drawdown data, is a much shorter duration of the hydraulic test, which comes at the expense of inability to determine the specific storage. Furthermore, the need to approximate space derivatives with differences, often computed by using data from a few measurement locations, introduces an additional source of uncertainty that, to the best knowledge of the authors, has not been investigated so far. On the same line of thought, i.e., avoiding replicate solutions of the computationally expensive transient flow equation, is the inversion of drawdown temporal moments [Li et al., 2005, 2007; Zhu and Yeh, 2006].

[8] A compelling difficulty in using drawdown data for inversion, or their transformations such as temporal moments, is that the inverse model is underdetermined with many more unknowns (i.e., the hydraulic parameters at the computational cells) than measurements. An effective way to overcome this difficulty is by coupling hydraulic tomography with Bayesian inversion. Firstly, the MPMR configuration reduces the ratio parameters to observations by shifting the volume interrogated at each pulse test, thereby making the problem less underdetermined. Secondly, the Bayesian inversion provides a powerful framework to sort out the space of admissible solutions and retain only those compatible with the observational data [Fienen et al., 2008]. However, a further difficulty emerges because the application of the Bayes' theorem to invert drawdown data leads to a nonlinear relationship between the observational data and the hydraulic parameters, which makes the solution of the inverse problem a difficult task to achieve. In order to facilitate the solution of the problem and reduce the computational burden this relationship is typically linearized by using either the Sequential Successive Linear Estimator (SSLE), developed by Yeh et al. [1996] and Yeh and Liu [2000], and applied in subsequent works [see, e.g., Zhu and Yeh, 2005], or the quasilinear method, developed by Kitanidis [1995] and used in several two and three-dimensional applications [see, e.g., Li et al., 2005, 2007; Fienen et al., 2008].

[9] In this work, we use an alternative approach based on particle swarming, an efficient genetic algorithm, which avoids linearization of the underlying optimization problem, and thus the computation of the sensitivity matrix, which is the most demanding step in the previous approaches. Furthermore, instead of solving directly for the conditional mean, we seek an ensemble of independent realizations conditional to the measurements of the hydraulic properties and the head data, all representing possible realizations of the actual (true) variability of the hydraulic parameters. Inversion is performed by using a Bayesian Maximum a Posteriori (MAP) approach, already shown to be successful with hydrological inversion [see, e.g., Kowalsky et al., 2005] when the geostatistical model of spatial variability is known. Guidelines for obtaining the parameters of the spatial variability model from pumping and tracer test experiments have been provided in a number of studies [see, e.g., Wilson and Rubin, 2002; Bellin and Rubin, 2004; Firmani et al., 2006], typically under ergodic conditions. However, a common assumption in existing works dealing with the inversion of hydraulic tomography data is that they are totally or partially known in advance [Kowalsky et al., 2004, 2005; Nowak and Cirpka, 2004]. In this work we remove this hypothesis and assume that only the type of geostatistical model of spatial variability is known in advance, while structural parameters such as mean, variance, and integral scales of the hydraulic conductivity, are estimated as part of the inversion procedure. A similar approach has been adopted by Li et al. [2007], who inverted head data collected at the Krauthausen (Germany) test site by using the quasilinear method developed by Kitanidis [1995] with the structural parameters that were progressively updated from their prior estimates. However, since their exercise was based on real data, they could not assess the accuracy of their estimates. This is another question we seek to answer with this paper.

[10] Therefore the main objective of the present work is to assess how much spatial variability is granted by hydraulic tomography data collected in a MPMR configuration. Its basic configuration is composed of two boreholes subdivided into intervals, which are used to apply the pulse and to measure the resulting perturbations of the hydraulic head.

[11] In section 2, we discuss the theoretical framework, followed in section 3 by the inversion methodology. In section 4, we discuss a few synthetic examples in a two-dimensional setup to explore the potentiality of the inverse methodology; in section 5, we discuss an application in a three-dimensional setup and finally we present our conclusions in section 6.

2. Inversion of Hydraulic Head Data

2.1. Governing Equation

[12] Groundwater inversion requires a forward model that provides the linkage between aquifer parameters, such as hydraulic conductivity K[LT−1] and specific storage Ss[L−1], and observational data, which in our case are a set of measurements of the transient hydraulic head h[L] at several locations in one or more boreholes. This linkage is provided by the following governing equation:

equation image

with the initial condition of a constant head H0 through the computational domain, which results from neglecting the regional flow, a Dirichlet type boundary condition of constant head at the external vertical planes ΓDir delimiting horizontally the computational domain Ω, and a Neuman type no-flow boundary condition at the horizontal planes, ΓNeu, delimiting the computational domain in vertical direction:

equation image

where n is the unit vector normal to ΓNeu. These conditions should be supplemented by test specific conditions at the boreholes, which will be discussed subsequently.

[13] Furthermore, for simplicity, in the present work we assume that specific storage Ss is constant over the computational domain.

2.2. Model of Spatial Variability

[14] We assume the hydraulic log conductivity Y = ln K distributed as a stationary Random Space Function (RSF) with mean mY and variance σY2, both constant, and the following axisymmetric exponential covariance function:

equation image

where r = (r1, r2, r3) is the two-point separation distance, ri, i = 1, 2, 3 are the Cartesian components of the vector r, and IYh and IYv are the horizontal (i.e., in the (x1, x2) plane) and vertical (x3) integral scales, respectively.

[15] We use this model for illustrative purposes, although recently it has been indicated as representative of a restricted class of weakly heterogeneous depositional environments, such as the low energy depositional formation at Borden [Ritzi and Allen-King, 2007]. However, our inversion procedure is not limited to this model of spatial variability and other models, such as facies models, can be accommodated, provided that they can be identified by a set of parameters.

2.3. Experimental Setup

[16] Let us consider a tomography experiment in a confined aquifer. The experiment is conducted by using two boreholes, each subdivided into Np/2 intervals, for example, by packers. The experiment involves creating a pressure pulse in one of the intervals, while recording the resulting transient head at the remaining Np − 1 locations. This operation is then repeated at the remaining intervals, thereby obtaining Np(Np − 1) transient head signals called tomograms. Note that this configuration is similar to the multiple offset configuration utilized in cross-well ground penetrating radar (GPR), with the only difference that in our case the signal is recorded also at the borehole, where the pulse is applied. The signals recorded at locations along the same borehole emitting the pulse warrant a better investigation of the vertical spatial variability (as shown in the sensitivity maps of Figure 2); on the contrary, the signals recorded at locations along the other borehole explore a larger volume between the two boreholes.

[17] This experimental configuration shares with GPR tomography the advantage of allowing the use of existing wells, or limiting the need for new drillings, but requires the isolation of several intervals along the boreholes, a situation that is amenable but with technical difficulties. We recognize this as the main difficulty to overcome in field applications of this methodology. However, valuable solutions are offered by the recent dramatic advances of the direct push technology (DP) and related monitoring systems [see, e.g., Dietrich and Leven, 2005; Butler et al., 2007; Dietrich et al., 2008; Liu et al., 2008], which promote development of cost-effective and easy to use tools for conducting cross-hole hydraulic tomography tests.

[18] The experiment is thus simulated numerically on a hydraulic conductivity field, which is assumed as the actual (true) conductivity field, obeying the model (equation (3)) of spatial variability with the parameters listed in Table 1. This conductivity field, as well as the other fields utilized in the inversion procedure, is generated by using HYDRO_GEN, a random field generator developed by Bellin and Rubin [1996].

Table 1. Ensemble Average and Standard Deviation of the Parameters of the Spatial Variability Model for Cases 1, 2, and 3a
 mYIYh (m)IYv (m)σY2Ss (m−1)
  • a

    The statistics are computed with 20 Monte Carlo realizations, and the standard deviation is shown within round brackets. The true values of the structural parameters (i.e., those imposed to obtain the reference true field) and the upper and lower bounds of the parameter space explored by PSO are also shown.

True parameters−11.515.00.21.01e−5
Upper bound−8.515.02.02.02e−4
Lower bound−14.51.00.10.15e−7
Case 1 (1) Np = 6−10.9 (1.12)5.31 (1.84)0.51 (0.225)0.59 (0.37)
Case 1 (2) Np = 10−10.58 (0.935)4.62 (1.54)0.52 (0.275)0.93 (0.3)
Case 1 (3) Np = 22−10.01 (0.445)5.96 (1.0)0.59 (0.212)0.3 (0.27)
Case 2 Np = 10−11.04 (0.685)8.07 (0.86)0.25 (0.08)0.28 (0.1)
Case 3 Np = 10−9.8 (0.27)5.01 (0.65)0.7 (0.23)0.37 (0.285)2.1e−5 (5.5e−6)

[19] In order to limit the computational burden we performed a first set of simulations by using a two-dimensional profile (vertical) model, followed by a set of fully three-dimensional simulations with the experimental setups that showed the better performance in the two-dimensional simulations.

[20] For the sake of generality we transform equation (1) in the following dimensionless form:

equation image

where x1 = x1/IYh, x2 = x2/IYh, x3 = x3/IYv, h′ = h/IYh, K′ = K/KG. In addition, e = IYv/IYh is the anisotropy ratio, KG = image is the geometric mean of K and x1, x2 and x3 are the main anisotropy axes. The two-dimensional model is obtained from equation (4) by neglecting the dependence from the horizontal spatial coordinate x2.

[21] Equation (4) is solved by using the Finite Element Galerkin scheme. The computational grid is composed of triangular three-node elements with linear shape functions and eight-node block elements with quadratic shape functions in two-dimensional and three-dimensional configurations, respectively.

[22] We performed the simulations by using the codes Sat2D, developed by M. Putti (personal communication, 2005), and Frac3D [Therrien and Sudicky, 1996] for the two-dimensional and three-dimensional cases, respectively. Solving equation (1) in an anisotropic formation with computational cell dimensions set to a fraction of the respective integral scales would result in elongated computational cells, which lead to nonphysical local extrema in the numerical solution [Putti and Cordes, 1998]. The most efficient way to overcome this problem is by solving the dimensionless counterpart (equation (4)) of equation (1), which allows using a Voronoi mesh with cells of equal size along the coordinate axes, which do not suffer from the above limitations [Putti and Cordes, 1998]. Because the parameters of the spatial variability model are unknown and estimated as part of the inversion procedure, the computational grid is regenerated each time an integral scale is changed according to the procedure described in section 3. This ensures that the constraint that the grid size should be smaller than a fraction of the integral scale is satisfied. For the two-dimensional simulations presented in this study, we used dimensions of the computational cell as small as IYh/25 and IYv/25, in horizontal and vertical directions, respectively. These cells are ten times smaller than the larger cell dimensions granting an accurate reproduction of flow and spatial plume moments as indicated by Bellin et al. [1992], and confirmed later by other numerical investigations [see, e.g., Rubin, 2003, for a review of numerical studies]. The resulting finite element mesh is composed of triangles obtained by “cutting” each square of the dimensionless computational grid along the diagonal from the upper left to the lower right angle. The value of the hydraulic conductivity generated at the center of the square block is then assigned to both triangular elements. In the three-dimensional simulations the vertical cell's dimension is enlarged to IYv/10.

[23] Consistent with equation (2) boundary conditions are of imposed head at the left and right vertical boundaries, while no-flow conditions are assumed for the upper and lower horizontal boundaries.

3. Methodology of Inversion

3.1. Statement of the Problem

[24] Our starting point is the forward operator equation image mapping the vectors of the unknown parameters a and equation image into a set of observational data z [McLaughlin and Townley, 1996; Tarantola, 2005]:

equation image

where equation image stands for the numerical solution of equation (1) or equation (4), a is the vector of the unknown hydraulic conductivities, θ is the vector of the unknown parameters of the geostatistical model of spatial variability, and finally v is the vector of the measurement errors. The vector z contains the head measurements. In principle, point measurements of the hydraulic conductivity should also be included into z (all the measurements of hydraulic conductivity that one can envision are the results of some type of inversion), but in the applications discussed in the present work we use them only as prior information in order to minimize their influence on the inversion.

[25] In the present work, we invert equation (5) by using the Maximum a Posteriori (MAP) approach, which was extensively utilized in geophysical imaging (see, e.g., the comprehensive review of McLaughlin and Townley [1996] and the book by Rubin [2003]), but different from existing approaches we assume unknown both a and equation image.

[26] In principle, the vector a contains the hydraulic conductivity at the computational grid blocks, but in our case this would lead to a severely underdetermined problem with a number of unknown parameters larger than the number of measurements. To overcome this shortcoming (see, e.g., McLaughlin and Townley [1996] for a comprehensive treatment of the problem), and to reduce the ratio parameters to measurements we used the pilot point approach. In essence, this method utilizes a number of independent realizations of the RSF Y(x) = ln K(x), conditional to the values of Y at a given number of locations, called pilot points. In each realization, the values of Y at the pilot points, which are now the only unknowns in the vector a, are perturbed, consequently varying the entire field, such as to improve the match between measured and simulated observational data with respect to the original field. The pilot point concept has been discussed by de Marsily [1978], RamaRao et al. [1995], and Gomez-Hernandez et al. [1997], and extensively applied in subsurface hydrology [see, e.g., Certes and de Marsily, 1991; Gomez-Hernandez et al., 1997; LaVenue and Pickens, 1992; RamaRao et al., 1995]. In order to perform the inversion under conditions as close as possible to applications, we modified the approach suggested by Kowalsky et al. [2004] for joint inversion of hydrological and ground-penetrating radar data in the vadose zone, to handle the case for which the parameters of the spatial variability model are unknown and no reliable prior information of them is available.

[27] Stated in a more formal way, the objective of the inversion is to obtain (multiple) optimal independent estimates of the vector a = {Ypp(xpp,j), j = 1, …, Npp}, where Npp is the number of pilot points, and jointly the vector of the model parameters θ = (mY, σY2, IYh, IYv), that better match the transient hydraulic heads recorded during the hydraulic tomography test. Inversion is performed within the MAP framework by minimizing the likelihood function (A5) specialized to the case in which no prior information is available for θ, because this is a piece of information that is difficult to obtain in applications (see Appendix A for details). Therefore the function to minimize is the following:

equation image

where β is a constant controlling the weight of the regularization term with respect to the error term; in our simulations we assumed β = 1. Equation (6) results from maximizing the a posteriori pdf of observing a and θ given the observational data z, under the hypothesis that the pdfs of a and z in the Bayesian chain (equation (A1)) are Gaussian and that θ is characterized by a noninformative uniform prior pdf.

[28] The first right-hand term of equation (6) is a measure of the mismatch between actual and simulated observations, while the second term is a regularization term, which penalizes solutions with pilot point values, a, that differ largely from their prior estimates (equation image). A consequence of the nonuniqueness of the inverse problem, as stated in Appendix A, is that the space of the observations z maps into a subspace of the solution space, which is called space of admissible solutions [see, e.g., Tikhonov and Arsenin, 1977; Tarantola, 2005], thereby yielding for a and θ an infinite number of admissible solutions, whose likelihood is measured by the a posteriori probability distribution (equation (A1)) rather than a single deterministic solution.

[29] Given our objective of assessing the potential of cross-well hydraulic tomography to characterize spatial variability at the local scale, we consider the idealized situation in which the measurements of K are error free, while the head measurements are affected by an uncorrelated random Gaussian error with zero mean and standard deviation σv,h = 0.01 m. This standard deviation is in line with the accuracy of automatic pressure sensors for head measurements. Therefore the matrix Cv has the diagonal terms equal to σv,h2 = 0.0001 m2 and the off-diagonal terms equal to zero.

[30] The prior information equation image at the Npp pilot points, xpp,j, j = 1, …, Npp, is obtained by kriging the log-transformed hydraulic conductivity measurements Yk(xk,j) = ln Kk(xk,j), where xk,j, j = 1, …, Nk are the locations of the measurements. Consistent with this estimation, in equation (6)Ca is the error covariance matrix with the diagonal terms given by the kriging variances: σok2(xpp,j), j = 1, …, Npp. In the absence of measurements of K, the vector equation image is the null vector and σok2(xpp,j) = σK2 = exp(2mY + σY2)[exp(σY2) − 1]. To simplify the analysis we set to zero the off-diagonal terms of Ca, owing to the sharp decrease of the correlation in the vertical direction [Kowalsky et al., 2004]. Note that both equation image and Ca should be updated each time a parameter of the spatial variability model changes in the iterative process utilized to compute the minimum of the objective function L, equation (6), because these parameters are estimated jointly with a.

[31] As stated before, Bayesian inversion seeks the posterior pdf of the unknown parameter, given prior information and the observational data, rather than a unique optimal value. The posterior pdf is obtained in a Monte Carlo framework through the optimization of a number, say MC, of independent realizations of the log conductivity field conditional to the available Y measurements (i.e., the log-transformed of the K measurements) and the values at the pilot points [Gutjahr et al., 1994; Hanna and Yeh, 1998; Kowalsky et al., 2005].

[32] The minimum of the objective function L, equation (6), is obtained by using the Particle Swarm Optimizer (PSO). PSO is a robust searching algorithm based on a social-physiological metaphor, which has been proven to perform well in problems with highly non linear objective functions with many local minima [Robinson and Rahmat-Samii, 2004]. PSO has been applied successfully in information technology [Donelli and Massa, 2005] and surface hydrology [Gill et al., 2006]. Contrary to gradient methods, such as the Levenberg-Marquardt searching algorithm [Nowak and Cirpka, 2004], and genetic algorithms, PSO is insensitive to both the initial conditions and the shape of the objective function. The space of admissible solutions is explored by a number of particles that move according to simple rules mimicking the behavior of a swarm of bees seeking for flowers. The space that the particles explore has dimensionality n = Npp + Nθ, where Nθ is the number of parameters of the spatial variability model (structural parameters), and is bounded by the extreme values of the parameters, therefore the coordinates of a point in the solution's space are given by the union of the vectors a and θ. A detailed description of the PSO algorithm is provided in the Appendix B, and the bounds of the parameters are provided in the Table 1.

[33] In order to reduce the computational effort both the PSO and Kriging algorithms were paralleled and the Np independent flow equations, resulting from applying the pulse at each one of the intervals within the two boreholes, were solved by using Np nodes of the LINUX cluster (Intel Xeon 3 GHz clock speed) of the CINECA consortium (www.cineca.it). For the two-dimensional simulations discussed in section 4.1, the computational time for each Monte Carlo simulation is 4 hours, while for the three-dimensional example discussed in section 5 the computational time is 9 hours.

3.2. Implementation Steps

[34] Inversion is performed on synthetic observational data obtained by simulating numerically the hydraulic tomography test in a known K field. A reference (true) log conductivity field Y(x) = mY + Y′(x), where Y′ are the fluctuations around the mean mY obeying the spatial model (equation (3)), is thus generated by using HYDRO_GEN and transformed through the following relationship K(x) = exp[Y(x)], to obtain the reference (true) conductivity field. The parameters θ = (mY, σY2, IYh, IYv) used for the generation are assumed as the true parameters of the model of spatial variability to be compared with the parameters obtained by inversion of tomographic data. Measurements of K are extracted by sampling the reference conductivity field at Nk selected position xk,j, j = 1, …, Nk along the two boreholes. These values are then log-transformed: Yk(xk,j) = ln Kk(xk,j), j = 1, …, Nk to be utilized subsequently in the generation of the conditional conductivity fields used as prior information in the inversion procedure. In the present work, we use Nk = 10 in all simulations including measures of hydraulic conductivity.

[35] The next step is the simulation of the hydraulic tomography test. For a given experimental configuration the flow equation (4) is solved as many times as the number, Np, of intervals from which the pressure (or flow) pulse is emitted, and contemporaneously the transient head is recorded at the monitoring positions, which coincide with the remaining Np − 1 intervals.

[36] The resulting observational data are then inverted through the following steps:

[37] 1. A new log conductivity field is generated conditional to the log conductivity measurements Yk(xk,j), j = 1, …, Nk by using the two-step procedure suggested by Kitanidis [1997, p. 76]: first a unconditional field Yu is generated using HYDRO_GEN, then the Nk residuals Yk(xk,j) − Yu(xk,j) at the conditional points are distributed to the grid points by kriging and added to Yu in order to obtain the prior log conductivity field Yc,p, conditional only to the measurements of the hydraulic conductivity. At this time a first set of model parameters θ = (mY, σY2, IYh, IYv) is randomly generated respecting the predefined bounds. Upper and lower bounds for each parameter are shown in Table 1 together with the true parameters (i.e., the parameters utilized to generate the reference K field).

[38] 2. The log conductivity field is further updated such as to honor the values of Y at the pilot point positions (xpp,1, xpp,2, …, image These values are provided by the PSO algorithm in the process of minimizing the objective function L (equation (6)). Similar to the previous step, this is accomplished by distributing the residuals Ypp(xpp,j) − Yc,p(xpp,j), j = 1, … Npp, where Ypp(xpp,j) is the pilot point value at x = xpp,j, and Yc,p(xpp,j) is the value that the log conductivity assumes at the pilot point in the prior generation performed as described at the step 1.

[39] 3. The forward model (equation (4)) is then applied to the resulting log conductivity field conditional to both the K measurements and the pilot point values to compute the transient heads at the observation points to be confronted with the observational data.

[40] 4. The value of the objective function (equation (6)) is computed and new pilot point values as well as new parameters θ are drawn from PSO.

[41] The steps 2 through 4 are repeated until the minimum of the objective function (equation (6)) is obtained. Note that the pilot point values of step 2 are generated randomly at the first iteration of the above procedure. Furthermore, the change of an integral scale at step 4 due to PSO, causes the following iteration to start from step 1 instead of step 2. All computations are performed in dimensionless form, such that a change of one of the integral scales results in a new dimensionless computational domain. Changes of model parameters other than the integral scales do not require restarting from step 1, but only to update both equation image and Ca in the second term of the objective function (equation (6)).

[42] The above procedure is repeated MC times by changing the seed utilized for generating the first unconditional random field (first part of the step 1). In the present study MC = 20 realizations are enough to stabilize the first two spatial moments of the log conductivity. This relatively small number of realizations, if compared with hundred to thousand simulations needed to stabilize moments in forward groundwater modeling, has been observed in other studies [see, e.g., Kowalsky et al., 2004] and is indicative of the effectiveness of observational data in sorting out the space of possible solutions. In other words, one of the benefits of Bayesian inversion is that it filters out realizations that are incompatible with the observational data, thus reducing uncertainty and the need of a large number of MC realizations to stabilize the statistics.

4. Results of Two-Dimensional Simulations

4.1. Sensitivity Analysis

[43] Figure 1 shows the true K field utilized for the simulation of the hydraulic tomography test with two boreholes at a separation distance of 2 m, that with true horizontal integral scale of IYh = 5 m listed in Table 1, corresponds to a dimensionless separation distance of 0.4. Furthermore, inversion is performed by assuming that Nk = 10 measurements of the hydraulic conductivity are available at the boreholes. These measurements may be obtained, for example, by slug or direct push tests or by sample laboratory analysis [see, e.g., Bohling et al., 2002; Butler et al., 2007]. The locations of the measurements and of the Npp = 14 pilot points are shown in Figure 1 by solid black triangles and solid white circles, respectively. In order to explore how the number of observational data influences the inversion we considered the following three cases: (1) Np = 6 borehole intervals (three for each borehole), with a uniform vertical spacing of Δx3 = 0.75 m, (2) Np = 10 and Δx3 = 0.375 m, and finally (3) Np = 22 and Δx3 = 0.15 m, which result in Np(Np − 1) = 30, 90 and 462 transient head signals, respectively. Hereafter these three configurations are referred to as case 1.

Figure 1.

Color map of the true K field. The locations of the K measurements, Nk = 10, and pilot points, Npp = 14, are indicated by solid triangles and circles, respectively.

[44] Preliminary simulations showed that inversion is facilitated if the pilot points are distributed uniformly in the region between the two boreholes at the nodes of a regular grid as suggested by Gomez-Hernandez et al. [1997]. This choice is supported by the rather flat sensitivity function within this region, (see Figure 2b and the ensuing discussion). The main result of these preliminary simulations is that relatively short pulses, say less than 60 s, with the hydraulic head sampled from 3 to 5 times during the pulse emission, suffice to obtain accurate inversions and that no significant improvements are obtained by using longer pulses recorded at a higher frequency and for a longer time. This result is in line with the findings of Vasco et al. [2000], who showed that recording the traveltime of the disturbance to the detection sensor suffices to obtain a satisfying reconstruction of the underlying K field. According to these preliminary results we fixed pulse duration to 30 s, while the head at the monitoring positions was sampled three times at 10 s, 20 s, and 30 s since the 1 m rise of the head at the emission point.

Figure 2.

Modified sensitivity maps of the head at the receivers to changes of K when the pressure pulse is created in the central interval of the left borehole for Np = 10 and t = 30 s and the signal recorded at the locations along the (a) left and (b) right boreholes. The star and the bullets indicate the location where the pulse is emitted and the recording intervals, respectively.

[45] The computational domain used for these simulations is 8 m wide and 6 m high, which implies a computational grid with an average of 30,000 cells. The two boreholes are at a separation distance of 2 m and centered with respect to the computational domain (Figure 1). The distance of the two boreholes from the left and right boundaries is 3 m (note that in Figure 1 the domain is cut around the boreholes in order to obtain a better representation of the central part, which is the most interesting for the inversion).

[46] The impact of the experimental configuration and the pilot point distribution on inversion can be analyzed through sensitivity maps showing the sensitivity of the hydraulic head recorded at the observation points to variations of the hydraulic parameters. According to the adjoint state method, and assuming, as customary, that the hydraulic conductivity is constant within the computational cell, the sensitivity of the hydraulic head to variations of the hydraulic conductivity of the ith cell assumes the following form [Sun, 1994; Sykes et al., 1985]:

equation image

where, the subscript i indicates the computational cell, ψ is the sensitivity function and tf is the total duration of the test. It can be shown that ψ satisfies the following partial differential equation [see, e.g., Sun, 1994]:

equation image

with final and boundary conditions given by:

equation image

where xobs are the coordinates of the monitoring (observation) point, tobs is time at which the head is observed, and K is the true conductivity field. As for the flow equation (1) ΓDir includes the vertical boundaries of the computational domain, while ΓNeu includes the upper and lower horizontal boundaries. Furthermore, the final condition ψ = 0 is applied over the entire computational domain. Equation (8) is solved separately for each monitoring position, with a time step of 1 s. The resulting ψ field is replaced into equation (7) to compute the sensitivity of the hydraulic heads to the variations of K. A sensitivity map is then obtained for each couple of emissions and recording positions at tobs = 30 sec. In order to summarize the results of this analysis, Figures 2a and 2b show a modified sensitivity map for the pulse applied to the center of the left borehole. The map is obtained by assigning to each cell the largest (in absolute value) sensitivity selected among the maps of sensitivity of each observation point. Figures 2a and 2b show the sensitivity map for the monitoring points of the left and right boreholes, respectively. As shown in Figure 2a the head signals recorded at the same borehole from which the pulse is emitted are sensitive to vertical variations of the hydraulic conductivity around the borehole, while the signals recorded at the other borehole show a roughly uniform sensitivity to the hydraulic conductivity of the area between the two boreholes (see Figure 2b). These results suggest the opportunity of using in the inversion the hydraulic head recorded at several locations of the same borehole from which the pulse is emitted in order to obtain information on vertical spatial variability, and that the pilot points can be uniformly distributed within the area of constant sensitivity between the two boreholes.

4.2. Statistics of the Optimized (Inferred) Conductivity Fields

[47] Figure 3a shows one of the 20 optimized fields obtained by inverting the observational data collected with Np = 6, i.e., three vertical intervals for each borehole. Similarly, Figures 3b and 3c show the result of the inversion with Np = 10 and 22 intervals, and the other conditions remaining the same. Visual inspection of Figures 3a, 3b, and 3c evidences that although large-scale variability is adequately captured in all cases, the case with Np = 22 shows a better resemblance to the true field. In particular, Figure 3c shows a better reconstruction of the thin low-conductivity lenses near the top and the bottom of the boreholes and the thin high conductivity strip at the intermediate depth.

Figure 3.

Color maps of the optimized field obtained by inverting the observational data with (a) Np = 6, (b) Np = 10, and (c) Np = 22. (d) The prior conductivity field, conditional to the measurements only, used in all the inversions.

[48] Figure 4a shows the scatterplot of the estimated (shown in Figure 3c) versus the true log conductivities at ng = 1200 grid points of the rectangular area encompassed by the borehole intervals, for Np = 22. This choice is justified by the behavior of the sensitivity function shown in Figures 2a and 2b, which drops quickly outside this area. Furthermore, the spatial moments of an enlarged area obtained by adding a 0.5 m wide strip around the above area showed larger relative differences with the respective true values, probably because the external area is less affected by the pilot point values. Figure 4b repeats Figure 4a but for the ensemble average of 20 optimized fields. Since no appreciable differences are observed between these two scatterplots one can conclude that the overall distribution of the local error is the same in all the realizations. The Spearman's rank correlation (ρ) between estimated and true log conductivities is 0.76 in both cases, and the relative difference between true and inferred conductivities does not exceed ±20%.

Figure 4.

(a) Scatterplot of the true versus the inferred log conductivity in one of the Monte Carlo simulations. (b) Scatterplot of the true versus the ensemble average of the inferred log conductivity obtained with MC = 20 Monte Carlo realizations. In both cases, Np = 22. Furthermore, ρ is the Spearman's rank correlation coefficient and equation image is the spatial average of the relative error between the true and the estimated log conductivity values.

[49] A measure of the local error is provided by the spatial statistics of ɛi = ∣(Y(xi) − YT(xi))/YT(xi)∣; the absolute value of the relative difference between Y(xi), obtained by inversion, and the corresponding actual (true) values YT(xi) at the same ng = 1200 locations utilized in Figures 4a and 4b. The spatial average of the relative error reduces from equation image ɛi/ng = 0.057, for Np = 6, to 0.044, for Np = 22, while the standard deviation σɛ = equation image reduces from 0.047 to 0.034, respectively. Although these statistics show that increasing Np from 6 to 22 has a small impact on the local error, the latter case resembles better the spatial pattern of hydraulic conductivity.

[50] Because we used a relatively large number of hydraulic conductivity measurements in the inversion, one may argue that a simple geostatistical approach would provide similar results. This is explored in Figure 3d, which shows the prior conductivity field (i.e., the field generated in the last repetition of step 1 of the inversion procedure), that through inversion yielded the optimized field shown in Figure 3c. We observe that inversion warrants a much better reproduction of the actual spatial variability (compare Figures 3d and 3c to Figure 1 showing the true K field), and the benefit of inverting hydraulic head data is particularly evident in the zone halfway the two boreholes, where the influence of the measurements is the smallest.

[51] We compare the ensemble average of 20 optimized log conductivity fields to the true values along the vertical section at x1 = −0.5 m for Np = 6, 10 and 22 in Figures 5a, 5b and 5c, respectively. The pilot points are aligned along this vertical section. The same initial set of 20 conductivity fields, conditional to 10 K measurements along the two boreholes, was used in these three cases. The ensemble average of these 20 optimized fields provides the optimal estimate of the large-scale variability that is warranted by the observational data and K measurements, while local uncertainty is fully characterized by the posterior pdf (equation (A1)). Overall, we observe a good reproduction of the spatial variability. As expected, at large Np, observational data are more informative and inversion warrants a better reconstruction of small-scale variability, thereby reducing the interval of confidence.

Figure 5.

Comparison between the ensemble average of the optimized log conductivity fields (solid squares) and the true log conductivity field (red line) along the vertical section at x1 = −0.5 m for (a) Np = 6, (b) Np = 10, and (c) Np = 22. In all cases, the dashed lines indicate the ±2 standard deviation confidence interval.

[52] Table 1 shows the ensemble average and the standard deviation of θ, i.e., the parameters of the model of spatial variability, obtained with 20 Monte Carlo realizations for Np = 6, 10 and 22 (case 1). The smallest standard deviation is obtained with Np = 22, thereby confirming that a finer scanning of the conductivity field leads to better constraints of the structural parameters.

[53] Although the parameters of the geostatistical model of spatial variability reflect the general statistical properties of the log conductivity field, it is not expected that they are good descriptors of the actual spatial variability of the investigated area between the two boreholes. A better perception of how much spatial variability is warranted by inversion of transient head data is provided by the spatial moments. Table 2 compares the ensemble average of the first two spatial moments (the mean, equation image and the variance SY2) of the optimized Y fields to the corresponding moments of the true field for the portion of the domain between the boreholes (i.e., the area swept by the tomography test). We note that the ensemble average of the spatial moments is in good agreement with the moments of the true field, while the small standard deviations suggest the same for each single optimized field. The practical implication of this result is that at least the general features of the spatial variability, as described by the spatial moments of Y that characterize the true field in the zone between the two boreholes, are reproduced in all the optimized fields. Table 2 compares the ensemble average of the spatial second moment 〈SY2〉, computed as average of the moments of 20 realizations, with the second spatial moment of the ensemble mean field SY2. We observe that 〈SY2〉 is closer to the corresponding moment of the actual field than SY2. Moreover, for Np = 22, the uncertainty in the estimation of the second spatial moment is small (see Table 2, case 1). This analysis shows once again that a set of optimal estimated fields is more informative than their ensemble average.

Table 2. Ensemble Average and Standard Deviation of the Spatial Statistics Referred to the Area Encompassed by the Pulse Intervals for Cases 1–5a
 Descriptionequation imageSY2SD(equation image)SD(SY2)Sequation imageYequation image2
  • a

    The spatial statistics are limited to the spatial mean equation image and variance SY2, and the latter is compared with the variance Sequation imageYequation image2 of the field obtained by averaging the 20 realizations of Y resulting from inversion.

Case 1 (L = 2 m)true field−11.630.62
optimized fields Np = 6−11.660.950.230.190.52
optimized fields Np = 10−11.610.640.270.220.4
optimized fields Np = 22−11.530.610.190.0940.52
Case 2 (L = 10 m)true field−11.750.93
optimized fields Np = 10−11.480.70.230.20.33
Case 3 (L = 2 m)true field−11.630.62
optimized fields Np = 10−11.260.470.470.10.58
Case 4 (L = 2 m)true field−11.591.033
optimized fields Np = 10−11.670.770.120.210.6
Case 5 (L = 2 m)true field−11.591.033
optimized fields Np = 10−11.570.90.350.320.69

[54] Additional simulations performed with boreholes at a larger separation distance (i.e., L = 10 m) and Np = 10 showed similar accuracy in the reproduction of the spatial moments in the area between the two boreholes, as shown in Table 2 (case 2). With respect to the case 1 (Np = 10), we observe a slight deterioration of the match between the ensemble average of SY2 and the corresponding true value, accompanied by a better reproduction of vertical integral scale (Table 1, case 2). This result is inherent to the experimental configuration. For boreholes at a short separation distance (i.e., the distance is smaller than the horizontal integral scale) the pulses explore only a small portion of the vertical variability typical of the ensemble, therefore the vertical integral scale obtained from the observational data reflects the local variability and may differ significantly from the true value. However, by increasing the distance between the boreholes, more vertical spatial variability characterizing the ensemble is encapsulated into the observational data and the resulting vertical integral scale approaches the theoretical value. In other words, multiple vertical profiles, if independent, provide a better representation of the vertical variability in a statistical sense as described by the structural parameter. This is obtained when the boreholes are at a distance larger than the horizontal integral scale. A similar argument can explain why the horizontal integral scale deteriorates with large distances between the two boreholes.

4.3. Are Spatial Patterns and Connectivity Preserved by Inversion?

[55] The analyses conducted so far suggest that, although the parameters of the model of spatial variability may differ from the respective theoretical values, the first two spatial moments of the log conductivity are preserved by inversion. Although encouraging, these results are not sufficient to guarantee that the optimized fields reproduce the spatial patterns of high and low conductivities correctly. These spatial variations have been shown to influence transport. For example, recent studies showed that important transport features, such as early arrivals and tailing in a breakthrough curve (BTC), are to a large extent controlled by the spatial organization of high and low conductivity zones [see, e.g., Zinn and Harvey, 2003; Knudby and Carrera, 2005].

[56] The concept of connectivity has been used to identify the relevant hydraulic features that influence transport phenomena in fractured formations (see, e.g., de Marsily et al. [2005] for an extensive review on models reproducing connectivity), and has been recently linked to anomalous (non-Fickian) transport in both fractured [Berkowitz and Scher, 1997] and porous formations [Liu et al., 2004]. A connectivity function τ(r) can be defined as a quantity that measures the probability of observing K higher than a given threshold Kl at two connected locations, separated by the distance r [Western et al., 2001]. According to this definition two cells are connected if one can go from one to the other following a path that crosses only cells with K > Kl.

[57] In order to compute τ(r) the following indicator variable is defined at each cell:

equation image

Similarly, the connectivity of values lower than the threshold can be measured by defining the following complementary indicator variable: IL(x) = 1 − IH(x). The connectivity function τ decreases slower with the separation distance in random fields with connected rather than disconnected patterns of the hydraulic conductivity. Consequently, τ is an indicator suitable to identify organized patterns of hydraulic conductivities [Western et al., 2001; Knudby and Carrera, 2005].

[58] Following the work of Western et al. [2001], the connectivity function was evaluated along the streamlines from the left to the right borehole in the flow fields resulting from imposing a unit head gradient in the x1 direction in the 20 optimized fields with Np = 22 as well as in the true field. Connectivity of both low and high conductivities was obtained by setting the threshold to Kl = K25 (i.e., the numeric subscript indicates the ith percentile of the true conductivity pdf) with the indicator IL and Kl = K75 with the indicator IH, respectively. The results of this analysis are shown in Figures 6a and 6b.

Figure 6.

Comparison between the connectivity function of the true field and the ensemble average of the connectivity functions of the optimized fields with Np = 22 for (a) high hydraulic conductivity (K > K75) and (b) low hydraulic conductivity (K < K25). In all cases, the dashed lines indicate the ±2 standard deviation interval of confidence.

[59] We note that the optimized fields are on average more connected at high values of the hydraulic conductivity than the true field, as shown by the slower decay of the ensemble average of their connectivity functions (Figure 6a). On the other hand, Figure 6b shows that low conductivity values are less connected than in the true field at small separation distances, but becomes slightly more connected at large separation distances.

[60] In order to further analyze to what extent our inversion methodology captures the spatial patterns of hydraulic conductivity with a substantial impact on BTCs at accessible locations we simulated the BTC at the right borehole (see Figure 1) resulting from an instantaneous injection of solute in the left borehole. Injection of solute mass along the borehole is proportional to the local flux, which, as discussed by Demmy et al. [1999], is the injection mode that better reproduces the typical tracer test conducted for aquifer characterization. Flow is induced by applying a constant mean head gradient of 0.1 from left to right in Figure 1, and neither boreholes pumps water in or out of the aquifer. For each optimized field we compute the cumulative BTC, i.e., the cumulative solute mass that enters the right borehole, to be compared with the cumulative BTC obtained by repeating the transport experiment in the true conductivity field. In these experiments we neglect pore-scale dispersion because our interest is in assessing the capability of the inversion methodology to detect spatial patterns with an influence on transport rather than exploring the nonlinear interplay between connectivity and pore-scale dispersion in shaping the BTC. In order to add generality to our analysis we work with the cumulative BTC normalized to the total solute mass which, for a passive solute and in the absence of pore-scale dispersion, coincides with the cumulative frequency distribution (CFD) of the particle traveltime [Dagan, 1989; chap. 5.8]. The number nb,j of particles injected in the jth computational cell along the borehole is given by: nb,j = Kjnb/(Kmnl), where Kj is the hydraulic conductivity of the jth cell, nb is the total number of particles injected within the borehole, and Km = equation imageKj/nl, where nl is the total number of cells (layers) belonging to the borehole. Furthermore, the nb,j particles are distributed uniformly within the cell j. Preliminary simulations showed that nb = 2000 particles suffice to obtain a stable CFD of the particle's traveltime at the right borehole.

[61] In Figure 7a we compare the traveltime CFD resulting from the tracer experiment conducted in the true conductivity field (thick solid line) to the CFDs obtained by repeating the tracer experiment in the pool of 20 optimized fields obtained by inversion with Np = 22. The CFDs are a good match with the true CFD at small to intermediate traveltimes, but slightly overestimate the probability of exceeding large traveltimes.

Figure 7.

Comparison between the cumulative BTC obtained by simulating the transport experiment in the true field (thick solid line) and the BTCs (thin solid line) obtained by repeating the transport experiment in the 20 optimized fields obtained with (a) Np = 22 and (b) Np = 6.

[62] In order to investigate the benefit of inverting tomography data on solute transport modeling we repeated the above experiment by using the 20 optimized conductivity fields obtained by inversion of head data with Np = 6, and the results are shown in Figure 7b. The most important effect of reducing the amount of observational data is that uncertainty increases as an effect of a less detailed reconstruction of the spatial variability. In this case, a large uncertainty means a large difference between the BTCs obtained in different realizations. However, in both cases the ensemble BTC matches very well the true BTC with only a slightly longer tailing. These results show that inversion captures the most relevant features influencing transport, as described by BTC curves at compliance planes. Furthermore, uncertainty reduces with the amount of observational data (i.e., large Np), with the consequent reduction of the differences among BTCs obtained in different Monte Carlo realizations.

4.4. Inference of the Specific Storage

[63] So far, inversion has been performed by assuming that Ss is known and constant over the computational domain. In a further set of inversion experiments, referred as case 3, we relax this assumption assuming Ss constant in space but unknown. Therefore Ss is placed in the vector θ of the unknown parameters. The inversion of the specific storage has been attempted by Zhu and Yeh [2005], Liu et al. [2007]; and Li et al. [2005], among others. The inversion with the configuration adopted in case 1 with Np = 10 (to make the results comparable we used the same true K field and the same initial set of 20 log conductivity fields for the inversion), yields optimized fields that are, on average, less heterogeneous than in case 1. This is shown by the smaller ensemble average of the spatial variance SY2 (0.47 against the value of 0.64 for case 1 with Np = 10; Table 2), and a lower value of the parameter σY2, as shown in Table 1, thereby resulting in a smaller equivalent conductivity [see, e.g., Rubin, 2003, chap. 5] than in case 1. Consequently, we may conclude that the lack of knowledge of Ss exerts a limited impact on the reconstructed hydraulic conductivity, which is still smoother than the companion field obtained under the assumption that Ss is known.

5. Inversion in a Three-Dimensional Setup

[64] The above two-dimensional analysis provides useful suggestions on the experimental setup and a general assessment on how much variability is warranted by tomography data. However, two-dimensional simulations, although valid in relative terms, are not fully representative of data that may be collected in field experiments. In order to assess the performance of our inversion methodology better, we repeat the inversion exercise by considering the same experimental setup as in the case 1, but in a more realistic three-dimensional flow configuration. Simulations, which are referred as case 4, are thus performed in a 3-m-thick three-dimensional computational domain (i.e., 15 IYv with the parameters shown in Table 1) with a horizontal projection that is a square with side of 4 m. The theoretical parameters of the model of spatial variability are the same used in case 1 (Table 1). Furthermore, inversion is performed by using the same number of pilot points as in case 1. The dimensionless computational grid is characterized by cells with spacing Δx1 = Δx2 = 0.04 and Δx3 = 0.1 in horizontal and vertical directions, respectively. The resulting dimensionless domain is discretized on average into 60,000 cells. Pressure pulses are applied sequentially to Np = 10 intervals, by raising the head to 20 m for 30 seconds, while the head perturbation is recorded at the remaining observation points, for a total of 90 transient head curves. The specific storage is constant and equal to Ss = 2 · 10−4 m−1.

[65] Figure 8a shows the true hydraulic conductivity distribution within the vertical plane normal to the x2 direction dividing the domain in two parts of equal dimensions. In addition, Figure 8b shows for the same section the ensemble average of 20 optimized fields, conditional to 10 hydraulic conductivity measurements uniformly distributed along the two boreholes. The ensemble average is a smoother version of the actual field shown in Figure 8a with wiped out small-scale variability, but a correct identification of high and low conductivity zones. The first two spatial moments are preserved by the inversion as shown by the relatively good match between the ensemble average of the spatial moments and the small standard deviation (Table 2, case 4). The influence of conductivity measurements was explored by inverting the same tomography data in the absence of K measurements. The ensemble average of the resulting 20 optimized fields is shown in Figure 8c and the statistics (i.e., ensemble average and standard deviation) of the spatial moments are shown in Table 2 (case 5). Visual inspection of Figures 8a, 8b and 8c reveals that adding hydraulic conductivity measurements does not warrant a better prediction of the actual field as may be expected. In both cases most of the reconstructed spatial variability relies on the information present in the observational data, i.e., in the transient head measurements. This is confirmed by the statistics of the spatial moments (Table 2, case 5) and the spatial variability of the coefficient of variation of the absolute error of Y. In fact, as shown in Figures 9a and 9b, the effect of the hydraulic conductivity measurements is to reduce the coefficient of variation of the absolute error of Y in a small volume surrounding the measurements points, while their impact on the reconstruction of the overall spatial variability is small to negligible.

Figure 8.

Color maps of the spatial distribution of the hydraulic conductivity within the vertical plane at x2 = 0 of the three-dimensional domain used in cases 4 and 5. (a) True K field. (b) The ensemble average of 20 optimized K fields obtained by inversion of tomography data with Np = 10 using 10 K measurements (case 4). (c) Ensemble average of 20 optimized K fields obtained by inverting the same tomography data in the absence of K measurements (case 5).

Figure 9.

Color maps of the coefficient of variation of the absolute error ɛ between inferred and true values of Y for (a) case 4 and (b) case 5.

6. Conclusions

[66] We investigated the effectiveness of hydraulic tomography to assess spatial variability of the hydraulic conductivity in heterogeneous aquifers. Inversion of tomography data collected during a multiple head pulse test conducted in a cross-well configuration was performed by using a Bayesian Maximum a Posteriori approach. Besides applying a MPMR configuration, typical of geophysical imaging to hydraulic tomography, we considered unknown the parameters of the model of spatial variability, which are then inferred jointly to the spatial variability of the hydraulic conductivity. In order to reduce the ratio between parameters and measurements we used the pilot point method, which in essence consists of perturbing the prior conductivity field at a few selected points and propagating the perturbations to the entire field by conditional simulation.

[67] The results of a first set of simulations conducted in a two-dimensional setup confirm the effectiveness of the hydraulic tomography test in mapping the main features of heterogeneous conductivity fields. Moreover, when a larger amount of data is utilized for inversion, for example, by increasing the number of observation points, additional details of the actual (true) conductivity field appear in the ensemble average of the optimized fields, thereby reducing uncertainty.

[68] Inversion is performed with the only prior information concerning the type of log conductivity covariance function, while the parameters of the model of spatial variability, in our case the integral scales, the ensemble average and the variance, are obtained from inversion. While the ensemble average is in a very good match with its theoretical value and the standard deviation is low, the variance is inferred with a larger error. However, this mismatch could be due to the small size of the volume explored by the hydraulic tomography, whose spatial moments differ from the respective ensemble statistics. The same argument can be invoked in order to justify the poor estimation of the vertical integral scale when the distance of the two boreholes is of the order of the horizontal integral scale, or smaller, as in case 1. In fact, in all cases analyzed in this work the spatial moments of the zone explored by the tomography test were nearly the same in the true and optimized fields. Furthermore, we observed slightly better performance for the case with short rather than large distances between boreholes. Since this is accompanied by a similarly good prediction of the connectivity function, one can conclude that a carefully designed hydraulic tomography test provides a good map of the hydraulic properties in the investigated area. Furthermore, horizontal variability is better resolved when the boreholes are placed at small separation distance, while for resolving vertical variability a larger separation distance is preferred. Therefore what our simulations showed is that when the volume investigated is representative of the model of spatial variability (i.e., it is large enough to be ergodic) the structural parameters are estimated with sufficient accuracy. On the other hand, when the investigated domain is small and not representative of the geostatistical model of spatial variability, the estimation of the structural parameters deteriorates. However, this does not impact negatively on the variability reconstructed within the explored volume, which mimics the true spatial variability similarly, or even better, than in the case of a large volume of investigation. This allows us to conclude that inversion is able to extract the information on the spatial variability present in the transient head measurements collected under MPMR configuration.

[69] More realistic three-dimensional simulations confirm what we observed in the two-dimensional simulations; they evidence that the quality of the reconstructed fields depends on the amount of information contained in the head data (i.e., the number and density of the recording intervals in the boreholes) while point measurements of the hydraulic conductivity at both boreholes have a small to negligible effect. Finally, a two-dimensional tracer experiment was conducted in the true field and in the 20 optimized fields obtained through inversion of tomography data. The BTCs obtained by repeating the transport experiment in the estimated fields are in good agreement with the true BTC, with the spreading around the ensemble BTC that decreases as the data utilized for inversion increase. This result suggests that optimized fields obtained by inversion of observational data captures the spatial variability affecting solute transport. The overall quality of the reproduction improves when a finer scanning of the investigation volume is implemented by using more intervals in the two boreholes to create the perturbation and observe the response of the system.

Appendix A:: The MAP Framework

[70] Our inversion procedure is based on the MAP (Maximum a Posteriori) technique, which derives the maximum a posteriori estimate of the parameters by applying the Bayes' theorem under the hypothesis that the vector a = {Ypp(xpp,j), j = 1, …, Npp}, where Npp is the number of pilot points, the vector θ = (mY, σY2, IYh, IYv) of the model parameters, and the vector v of the measurements errors, are independent random vectors [McLaughlin and Townley, 1996]:

equation image

where z is the set of observational head data supplemented by the available K measurements, fa,equation image and fa,equation imagez are the prior and the posterior pdfs of (a, equation image), fza,equation image is the conditional pdf of z given a and equation image, equation image is the forward nonlinear operator and finally fz and fv are the pdfs of the measurements and measurement errors, respectively. Since we assume that the Nk measurements of K are error free, the vector z contains only the head measurements and the measurements of K are used only for computing the prior pdf of a. It is common to assume that the prior fa and fv are multi-Gaussian, with covariance matrices Ca and Cv, respectively:

equation image
equation image

where equation image is the prior mean of a, Cv is the diagonal matrix of the error variances of the head measurements, whereas Ca is also a diagonal matrix with the diagonal terms given by the kriging variances of a, which epitomize the effect of uncertainty associated with the prior information.

[71] After these preparatory steps by substituting the equations (A2) and (A3) into equation (A1) and assuming that equation image is characterized by a noninformative uniform prior pdf we obtain:

equation image

The optimal set of parameters is then obtained by maximizing fa,equation imagez(a, equation imagez), which is equivalent to minimizing the following likelihood function:

equation image

where the first term in the right-hand side is an indicator of the match between the forward model prediction and the measurements. The second term is a regularization term, which penalizes solutions with values of a that differ largely from their prior estimates (equation image).

Appendix B:: The PSO Algorithm

[72] The PSO algorithm is based on the exploration of the space of parameters through N particles (bees). The space dimensionality of the parameters is np. The N particles are introduced randomly into the space of parameters and then tracked along their trajectory. At the iteration step ν the particle number j performs the following jump:

equation image

where w is the inertial weight, which controls the particle memory, c1 and c2 are scaling factors, ɛ1 and ɛ2 are independent random values uniformly distributed between 0 and 1, pbest,j is the personal best of the jth particle (i.e., the location among those visited by the particle with the smallest value of the objective function, equation (6)) and gbest is the global best (i.e., the location among those explored by all the particles with the smallest value of the objective function). The values of pbest,j and gbest are updated at each iteration. The particle is then moved according to the following particle tracking algorithm:

equation image

In principle, the optimal values of the parameters w, c1, c2, the number of iterations NI, and the boundary conditions (i.e., the rules for handling situations in which the particle reaches one of the boundaries of the domain) are problem dependent [Robinson and Rahmat-Samii, 2004]. However, in the present study we utilized the standard setting with c1 = c2 = 2 and w varying linearly with the iterations from w = 0.9 at ν = 1 to w = 0.4 at ν = NI. Furthermore, we utilized the “invisible wall” boundary condition: the particles move without boundaries, but when the particle is outside the search domain the objective function (equation (6)) is not evaluated. The search algorithm given by equations (B1) and (B2) is stopped when more than 50% of the particles collapse around the same position, or when the maximum number NI of iterations is reached.

Acknowledgments

[73] We thank Mario Putti of University of Padua (Italy) for providing the software Sat2D; Dr. Matthew Becker of California State University (Long Beach, California) for the suggestions and interesting discussions on an early version of the manuscript; and the Associated Editor Lee Slater, Geoffrey Bohling, and the other two anonymous reviewers for their thoughtful comments and suggestions. Financial support was provided by the EU project AquaTerra contract 505428 (GOCE) and the project PRIN 2006089309 funded by the Italian Ministry of University and Research.

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