WATER RESOURCES RESEARCH, VOL. 40, W01506, doi: /2003wr002253, 2004

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1 WATER RESOURCES RESEARCH, VOL. 40, W01506, doi: /2003wr002253, 2004 Stochastic inverse mapping of hydraulic conductivity and sorption partitioning coefficient fields conditioning on nonreactive and reactive tracer test data Hai Huang, 1 Bill X. Hu, 2 Xian-Huan Wen, 3 and Craig Shirley 2 Received 11 April 2003; revised 25 September 2003; accepted 6 November 2003; published 8 January [1] A three-dimensional geostatistically based iterative inverse method is presented for mapping spatial distributions of the hydraulic conductivity and sorption partitioning coefficient fields by sequential conditioning on both nonreactive and reactive tracer breakthrough data. A streamline-based semianalytical simulator is adopted to simulate chemical movement in a physically and chemically heterogeneous field and serves as the forward modeling. In this study, both the hydraulic conductivity and sorption coefficient are assumed to be random spatial variables. Within the framework of the streamline-based simulator an efficient semianalytical method is developed to calculate sensitivity coefficients of the reactive chemical concentration with respect to the changes of conductivity and sorption coefficient. The calculated sensitivities account for spatial correlations between the solute concentration and parameters. The performance of the inverse method is assessed by a synthetic tracer test conducted within an aquifer with distinct spatial features of physical and chemical heterogeneities. The study results indicate that our iterative stochastic inverse method is able to identify and reproduce the large-scale physical and chemical heterogeneity features. The conditional study on the geostatistical distributions of conductivity and sorption coefficient are expected to significantly reduce prediction uncertainties for solute transport in the medium. INDEX TERMS: 1832 Hydrology: Groundwater transport; 1829 Hydrology: Groundwater hydrology; 1831 Hydrology: Groundwater quality; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: conditioning, heterogeneity, inverse mapping, stochastic Citation: Huang, H., B. X. Hu, X.-H. Wen, and C. Shirley (2004), Stochastic inverse mapping of hydraulic conductivity and sorption partitioning coefficient fields conditioning on nonreactive and reactive tracer test data, Water Resour. Res., 40, W01506, doi: /2003wr Introduction [2] Accurate predictions of contaminant transport in groundwater require accurate descriptions of the physical and chemical heterogeneities of an aquifer. However, owing to the current measurement techniques and economical limitation, field measurements of hydraulic and chemical parameters can be conducted only at limited locations and scales. The measurement points are usually sparsely distributed, leaving a large volume of the aquifer untested. Therefore the spatial heterogeneity patterns of the hydraulic conductivity and chemical sorption coefficient in an aquifer are typically inferred from the limited measurements of hydraulic conductivity, head and concentration at a few locations. Our ultimate objective is to build a realistic aquifer model that will be used to predict field groundwater flow and contaminant transport, so it is imperative that the 1 Idaho National Engineering and Environmental Laboratory, Subsurface Science Initiative, Idaho Falls, Idaho, USA. 2 Desert Research Institute, University and Community College System of Nevada, Las Vegas, Nevada, USA. 3 ChevronTexaco Exploration & Production Technology Company, San Ramon, California, USA. Copyright 2004 by the American Geophysical Union /04/2003WR W01506 geostatistical distributions of the model parameters will adequately reproduce all existing flow and transport data because these data are the actual responses of the natural aquifer. In the last three decades, numerous inverse methods have been developed to incorporate different types of information to better characterize the spatial variability of the hydraulic conductivity (or transmissivity for horizontal 2-D problems). Zimmerman et al. [1998] and McLaughlin and Townley [1996] reviewed and compared various inverse approaches. Previously, a few surveys were presented by Kuiper [1986], Yeh [1986], Carrera and Neuman [1986a], Carrera [1988], Ginn and Cushman [1990], Ahmed and de Marsily [1993], and Sun [1994]. [3] Given scattered hydraulic head and conductivity (or transmissivity) measurements, a large number of geostatistically based inverse techniques have been developed to map hydraulic conductivity field conditioned to both the conductivity and hydraulic head measurements [Gutjahr and Wilson, 1989; Robin et al., 1993; Gutjahr et al., 1994; Dagan, 1985; Rubin and Dagan, 1987, 1992; Rubin, 1991a, 1991b; Kitanidis and Vomvoris, 1983; Hoeksema and Kitanidis, 1984; Kitanidis and Lane, 1985; RamRao et al., 1995; La Venue et al., 1995; Carrera and Neuman, 1986a, 1986b; Carrera et al., 1993; Yeh and Zhang, 1996; Zhang and Yeh, 1997; Li and Yeh, 1999; Sahuquillo et al., 1992; Gómez-Hernández et al., 1997; Capilla et al., 1997, 1998; 1of16

2 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 Wen et al., 1996, 1999; Vasco and Karasaki, 2001]. While the nongeostatistically based inverse methods seek a single best estimate of the conductivity field, the geostatistically based inverse methods either yield multiple equiprobable realizations of conductivity field honoring both the conductivity measurements and head data [e.g., RamRao et al., 1995; La Venue et al., 1995; Sahuquillo et al., 1992; Gómez-Hernández et al., 1997; Capilla et al., 1997, 1998; Wen et al., 1996, 1999; Vasco and Karasaki, 2001], or provide the conditional mean conductivity field and its conditional variance and covariance [e.g., Kitanidis and Vomvoris, 1983; Hoeksema and Kitanidis, 1984; Yeh and Zhang, 1996; Zhang and Yeh, 1997; Li and Yeh, 1999]. All the results indicate that conditioning on head (or pressure) data will enhance the estimate of conductivity field and it is able to identify the large-scale heterogeneity patterns. Conditioning on a number of head measurements provides better transport predictions than does conditioning on the same number of measured conductivity data alone. However, the head data quickly becomes redundant when a certain level of enhancement is achieved. [4] In comparison with the large effort devoted to simulate hydraulic conductivity (or transmissivity) field conditioning on head data, much less attention has been paid to stochastic simulations of hydraulic conductivity field conditional to transport data (e.g., arrival times, concentration data or interwell tracer test data) [Harvey and Gorelick, 1995; Datta-Gupta et al., 1994, 1995, 1997; Wen et al., 2002; Li and Yeh, 1999; Cirpka and Kitanidis, 2001]. Harvey and Gorelick [1995] presented a cokriging method, combined numerical flow and transport simulations with a linear estimator, to generate 2-D hydraulic conductivity field that sequentially accounts for the measurements of hydraulic conductivity, head and solute travel time. The conditional auto/cross covariances between conductivity and the dependent variables, head and arrival time, were updated sequentially through the first-order approximations of the governing flow and transport equations. They concluded that arrival time, particularly the median arrival time, is robust indicators of flow paths and flow barriers. Li and Yeh [1999] applied the similar cokriging technique to derive the mean conductivity field sequentially conditioned to head data, concentration data and travel times. In their study, they concluded that steady state head measurements are the most useful secondary information for the estimation of log conductivity field. Similar conclusion was also reported by Cirpka and Kitanidis [2001]. Datta-Gupta et al. [1995] used simulated annealing to simultaneously account for flow and transport data in the generation of hydraulic conductivity field. Wen et al. [2002] applied a sequential self-calibrating method (SSC) to construct 2-D geostatistical aquifer models conditioned to both the head and tracer breakthrough data. All their results clearly demonstrate that conditioning to transport data will significantly enhance the estimate of spatial distribution patterns of the conductivity field, particularly the distribution and connectivity of flow channels and flow barriers, which are critical for performance assessment of planned waste disposal projects. Datta-Gupta et al. [1997], by using a perturbation approach, systematically analyzed the relative worth of pressure data and tracer data for conductivity heterogeneity characterization. They claimed that the tracer data have better resolving capability compared with transient pressure response in constraining large-scale permeability variations. One possible reason for causing such disagreement on the roles of concentration data versus head data might be the different inverse methods used by those researchers. The relationship between concentration and conductivity is extremely nonlinear. The cokriging inverse technique [Harvey and Gorelick, 1995; Li and Yeh, 1999; Cirpka and Kitanidis, 2001] is essentially a linear estimator which requires linearization of the flow equation and transport equation, also requires first-order approximation on the cross covariance between concentration and log conductivity. Therefore, even though iterative cokriging [Zhang and Yeh, 1997; Cirpka and Kitanidis, 2001] could somehow alleviate the limitations of linearization, the information of concentration data may not be fully utilized in those cokriging inverse methods. The Monte Carlo type of inverse approaches [Datta-Gupta et al., 1997; Wen et al., 2002] does not rely on the linearizations required by cokriging technique, and concluded that transport data are important indicators of flow paths and barriers. A thorough discussion on this issue is beyond the scope of this paper. Nevertheless, despite such contradictory conclusions, all those researchers agree that concentration data is important information and should be incorporated into the inversion procedure whenever concentration data is available. Furthermore, it is necessary that the aquifer model we build will adequately reproduce the measured concentration data because those data are the actual responses of the natural aquifer. [5] While the most effort has been devoted to characterize the conductivity heterogeneity of the aquifer, much less effort has been devoted to characterize the heterogeneity of the geochemical properties of the aquifer via geostatistically based inverse approach. The importance of chemical heterogeneity, more specifically, the spatial variability of the sorption partitioning coefficient, has been widely recognized in the literature [Garabedian, 1987; Kabala and Sposito, 1991; Robin et al., 1991; Burr et al., 1994; Miralles-Wilhelm and Gelhar, 1996; Hu et al., 1997; Cvetkovic et al., 1998; Huang and Hu, 2001]. In addition to the conductivity heterogeneity, such chemical heterogeneity also plays important and critical roles in subsurface contaminant transport. However, studies on geostatistically based inverse mapping of sorption coefficient field are very limited. Therefore the main goal of this study is to provide an efficient geostatistcally based inverse procedure to construct 3-D aquifer models for hydraulic conductivity K and sorption partitioning coefficient K d by sequentially conditioning to nonreactive and reactive tracer breakthrough curve data. Among various geostatistically based inverse approaches, we follow the sequential self-calibrating method (SSC), originally developed by Sahuquillo et al. [1992], Gómez-Hernández et al. [1997], and Capilla et al. [1997, 1998] to calculate the 2-D conductivity (or transmissivity) maps by honoring the head data. SSC generally consists of two steps: (1) a set of seed realizations of the hydraulic conductivity field is generated, conditional to conductivity measurements and other geophysical data, if available, then, (2) for each seed realization, an optimal perturbation field will be added to the seed field so that the mismatch between the simulated head and observed head will be minimized. The updated field will reproduce the observed head data, not 2of16

3 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 exactly, but within a tolerance range. The SSC method has been extensively applied to better estimate the aquifer/ reservoir conductivity/permeability field conditional to the measured head (or pressure) data and production data [Capilla et al., 1997, 1998; Wen et al., 1996, 1999; Franssen et al., 1999]. In this study, we extend the SSC method to construct 3-D conductivity maps and sorption coefficient maps by conditioning to nonreactive and reactive tracer test data. [6] This paper is organized as follows. In section two, we describe how the inverse method generates the log conductivity field and log-sorption coefficient field conditional to tracer test data. Then in section 3 we will provide a streamline-based semianalytical method to simulate reactive/nonreactive tracer movement in a heterogeneous flow field and the algorithm for fast and efficient calculation of concentration sensitivity coefficients. Similar streamline simulators were also adopted by Wen et al. [1999, 2002] and Vasco and Datta-Gupta [1997]. In section 4 a synthetic tracer test will be conducted in a hypothetical and perfectly known heterogeneous aquifer with distinct spatial features. The tracer test data will then be used to examine the performance of our extended SSC inverse method and the effects of conditioning on transport predictions. Finally, we will provide some conclusive remarks in section Inverse Mapping of Conductivity and Sorption Coefficient Fields Via SSC [7] A detailed description of the SSC method is given by Gómez-Hernández et al. [1997], wherein the SSC method was originally developed to estimate the transmissivity field conditioned to measured head data. In this study, we extend the SSC method to generate multiple realizations of the conductivity field and sorption coefficient field that will reproduce both the nonreactive and reactive tracer test data, yet still honoring all static data, if available. The main steps of the SSC inverse method conditioning on tracer test data are summarized as follows. [8] 1. Construct multiple initial log hydraulic conductivity fields that honor all conductivity measurements. The same number of initial log sorption coefficient fields is also constructed. A number of modern geostatistical tools can serve this purpose [see Deutsch and Journel, 1998]. In this study, a sequential Gaussian simulation technique (not limited to) (see SGSIM program in GSLIB [Deutsch and Journel, 1998]) is used to generate the initial log conductivity fields and log sorption coefficient fields presented later in this paper. Of course, each initial realization does not necessarily reproduce the tracer test data under the same initial and boundary conditions. Each initial realization is then processed one-at-a-time using the following steps (outer iteration). [9] 2. Given a combination of an initial log conductivity field and a log sorption coefficient field, first, we use the initial log conductivity field as input to solve the flow equation subject to initial and boundary conditions, along with external stresses (i.e., pumping and injection), to obtain the groundwater head distribution within the aquifer. Next, the initial log sorption coefficient field is converted into a heterogeneous retardation factor field. Then, a streamlinebased technique will be used to simulate the tracer movement within the heterogeneous flow field and calculate the tracer breakthrough curves at the pumping wells. The calculated breakthrough curves are compared with the observed breakthrough curves. An objective function that measures the mismatch between the simulated and observed breakthrough curves is defined as O ¼ Xnwell nw¼1 Xntime t¼1 Wðnw; t Þ C cal ðnw; tþ C obs 2 ðnw; tþ ð1þ where C cal and C obs are simulated and measured concentrations in sampling well nw at a particular time t. W (nw, t) is the weight assigned to the sampling well nw at sampling time t according to the sampling accuracy. In practice, the magnitude of the weight is proportional to the inverse of measurement error. We say that the conductivity field and the sorption coefficient field are conditioned to the tracer test data when the above objective function is smaller than a predefined tolerance value. Otherwise, an optimal log conductivity perturbation field ln K(x) and sorption coefficient perturbation field ln K d (x) are added to the initial log conductivity field and log sorption coefficient field, respectively. The optimal perturbation fields ln K(x) and ln K d (x) are obtained by minimizing the linear approximation of equation (1) through Taylor s expansion, ^O ¼ O C þ cal Xntime fdg t;k f ln Kg þ Xntime t¼1 þ Xntime t¼1 t t¼1 f ln Kg T ½MŠ t;k f ln K gþ Xntime t¼1 fdg t;kd f ln K d g f ln K d g T ½MŠ t;kd f ln K d g ð2þ Equation (2) is written in matrix notation form, where fdg t;k ¼ 2 fdg t;kd ¼ 2 C cal t C cal ½MŠ t;k ¼ S Cobs Cobs t ½ Š T t;k ½W ½MŠ t;kd ¼ ½SŠ T t;k d ½W t t T ½W Št ½SŠ t;k T ½W Št ½SŠ t;kd Š t ½SŠ t;k Š t ½SŠ t;kd ð3aþ ð3bþ ð3cþ ð3dþ 1;t ;t ; ; ; ; ln K ln K ln K ln K ln K ln K dn 2;t ; ln K ln K ln K ; ; ; ln K ln K ln K dn ½SŠ t;k ¼ ; ½SŠ. t;kd ¼ nw;t nw;t ; 5 4 nw;t 5 nw;t ; ln K ln K ln K ln K ln K ln K dn ð3eþ Here N is the total number of blocks within computation domain. The matrices [S] t,k and [S] t,kd are often called 3of16

4 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 sensitivity coefficients. They are the derivatives of the tracer concentrations in sampling wells with respect to the log hydraulic conductivity and log sorption coefficient. It should be pointed out that the sensitivity coefficient matrix is time dependent. For large 3-D problems, these two sensitivity matrices will become extremely large. In practice, by using the master point concept [Gómez- Hernández et al., 1997], the number of actual blocks with log conductivity and log sorption coefficient been perturbed via the inverse method is reduced to between 1/100 and 1/10 of the number of blocks of entire domain. The locations of the master points can be determined at the beginning of the simulation in a random way. The optimal changes of log conductivities and log sorption coefficients are determined at these master points first, and then smoothly interpolated by kriging to all grid blocks. Adopting the concept of master points significantly reduces the sizes of the sensitivity coefficient matrices and also reduces the dimensionality of the inverse problem. The sensitivity coefficients are calculated as part of the solutions of the flow and transport equations by bookmarking each streamline s trajectory and travel times, which will be described in the next section. The concept of master point is different from the pilot point described by RamRao et al. [1995]. In their approach, pilot points are added sequentially. At each step, their locations are selected optimally, by using adjoint sensitivity analysis. The pilot point is located at the most sensitive location, such that small changes of the lnk value at those points will yield reasonable reductions on the objective function. Then adding other pilot points and repeat the calibration process. The previous process is iteratively repeated until further addition of several pilot points does not decrease the objective function significantly. In master point concept, all the master points are generated at the same time and their locations are arbitrary, often on a pseduoregular grid. Then we seek for a perturbation field by optimizing all master point perturbations at once. Readers should also notice that within this step (inner iteration) we implicitly assume that optimal perturbations on the log conductivity and log sorption coefficient fields will not change the geometries of streamlines. The changes of streamlines geometries are detected by the outer iteration where we recalculate the geometries of streamlines. Therefore the assumption of unchanging streamline geometry is relaxed in the outer iteration. [10] 3. Determine the optimal perturbations of the log conductivity field and log sorption coefficient field at all master locations by solving the inverse problem in the form of equation (2), parameterized by the log conductivity perturbations {lnk} T = (ln k 1,..., lnk np ) and log sorption coefficient perturbations { ln K d } T =( ln k d,1,..., lnk d,np ) at the master points, subject to constraints. A gradient projection-based method [Gómez-Hernández et al., 1997] is applied to solve this inverse problem. The minimization procedure is very similar to the general Gauss-Newton method. The detailed mathematical description is given by Gómez-Hernández et al. [1997]. Note that optimizing equation (2), a linear approximation of the original objective function equation (1), instead of optimizing (1) itself, enables the initial log conductivity and log sorption coefficient fields to be rapidly updated without repeatedly solving the flow and transport equations. [11] 4. The optimal perturbations lnk(x) and lnk d (x) at the master locations are smoothly propagated to all grid cells by kriging to update the initial log conductivity and log sorption coefficient fields. [12] 5. Loop back to step (2) and solve the flow and transport equations with the updated log conductivity and log sorption coefficient fields, and repeat the steps (2) to (4) until the objective function (equation (1)) is sufficiently close to zero, or less than a tolerance prespecified by the user. [13] In practice, the inverse mapping of the hydraulic conductivity fields and sorption coefficient fields is carried out sequentially: (1) First, the nonreactive tracer data is used as conditioning data to update the initial input hydraulic conductivity field; (2) Then the updated hydraulic conductivity field is used as input field for reactive chemical transport simulation, and reactive tracer data is used as conditioning data to update the input sorption coefficient field. Such sequential conditioning not only reduces the parameter space of the inverse problem, but also accelerates the convergence of the inversion. The convergence issue of the SSC inverse method had been extensively studied by many researchers [Gómez-Hernández et al., 1997; Capilla et al., 1997, 1998; Wen et al., 1996, 1999]. Less than 20 iterations are normally required for 2-D problems for calculations conditioned to head (or pressure) data. As will be shown later in this paper, for 3-D calculations conditioned to tracer test data, the inverse method still converges rapidly. Furthermore, the streamline-based tracer transport simulator enables us to quickly simulate the tracer breakthrough curves in sampling wells. The transport simulator also provides efficient and fast calculation of sensitivity coefficient matrices within one single simulation run, which renders the inversion feasible and practical for 3-D problems. In the next section, the streamline-based simulator and the method for fast calculation of sensitivity coefficients are explained. 3. Streamline Simulator and Sensitivity Calculation [14] Two critical issues associated with the inverse problem integrating the dynamic concentration data are: (1) efficient simulation of the tracer movements in a physically and chemically heterogeneous medium, and (2) accurate calculation of the sensitivities of tracer concentrations with respect to the log conductivity and log sorption partitioning coefficient. In this section, a streamline transport simulator developed by Datta-Gupta and King [1995] will be extended to simulate the tracer movement and calculate the sensitivities of tracer concentration. For each outer iteration described in section 2, only one forward simulation run is required to obtain the sensitivities, instead of multiple forward simulation runs required in Datta-Gupta et al. s [1997] perturbation approach Tracer Response in Sampling Wells [15] In this study, we consider nonreactive and reactive tracers undergoing instantaneous sorption. We further assume that the local-scale dispersion is ignored. Thus the tracer movement within the flow field is subjected only to the advection and retardation caused by the sorption. In our 4of16

5 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 approach, the velocity field is numerically obtained by using finite difference method to solve the flow equation. Once the velocity field is obtained, the movement of tracers injected into the aquifer from the injection wells could be approximated by following the streamline approach, tracking the movements of solute particles along various streamlines initiated from the injection wells, with each particle carrying a certain amount of tracer mass. This approximation allows us to calculate the arrival times of tracers to a particular sampling well along different streamlines. Finally, the tracer concentration within a particular sampling well is obtained by summing up the contributions of solute coming along all individual streamlines as follows: C cal ðnw; tþ ¼ X nsnw s¼1 q s f s ðt; t s Þ Q nw where ns nw is the total number of streamlines that arrive at the sampling well nw with an extraction rate [L 3 T 1 ]of Q nw ; q s is the tracer mass flux [MT 1 ] carried by the streamline s; and t s is the travel time [T] for solute transported along streamline s to reach the sampling well nw, and the function f s (t, t s ) is dimensionless and determines the contribution of the streamline s to the tracer concentration at time t. The function f s (t, t s ) takes the following form for the continuous injection scenario, f s ðt; t s Þ ¼ 8 < 1; if t s t : 0; if t s > t and, for the pulse injection with duration t, f s ðt; t s Þ ¼ 8 < 1; if t s t t s þ t : 0; otherwise ð4þ ð5aþ ð5bþ The travel time t s, in the numerical model, is the sum of the travel times for solute traveling along the streamline s through all individual numerical cells from the injection well to the sampling well, i.e., t s ¼ Xns;c t s;c c¼1 where n s,c is the total number of cells crossed by streamline s from the injection well to the sample well, and t s,c is the travel time for the streamline s to cross a particular cell c. t s,c is written in the follow integral form t s;c ¼ Z ls;c 0 R c f c dr k c J ¼ Z ls;c 0 1 þ r bk d;c f c dr where l s,c is the total length of the streamline s within cell c; f c is the porosity of cell c; J is the head gradient along the streamline within the cell c; k c and R c are the conductivity and retardation values of the cell; K d,c and r b are the sorption coefficient value in cell c and bulk density of the medium. f c k c J ð6þ ð7þ 5of16 [16] Equations (4) (7) define the streamline-based tracer movement simulator. The calculation of tracer concentrations within sampling wells reduces to the calculation of solute travel times along all individual streamlines. More details about the streamline approach are given by Datta- Gupta and King [1995]. Vasco and Datta-Gupta [1997] applied the streamline simulator in their stochastic reservoir characterization. Wen et al. [2002] adopted a similar method as the forward model in their 2-D inversion. Compared with the grid-based transport simulators, the streamline simulator is fast and free of numerical dispersion. The advantage of the streamline simulator over the grid-based simulator of the advection-dispersion equation is more obvious for large 3-D inversion calculations. One major disadvantage of the streamline simulator here is that it neglects local physical dispersion. When local dispersion is not negligible, the spreading of the breakthrough curve could be much wider than those the streamline approach simulated. In such case, we suggest using the median arrival time in the objective function. The observed median arrival time can be derived from the breakthrough curve at sample well. Thus we can still use the streamline simulator to calculate the travel times, and consequently the median arrival time. The median arrival time somehow makes the knowledge of local dispersivity values unimportant. Such treatment was also recognized by Harvey and Gorelick [1995] and Cirpka and Kitanidis [2001] Sensitivity Calculation [17] Efficient calculation of the sensitivity coefficients plays a key role for any gradient-based inverse approach. Traditionally, the perturbation method is the simplest way to compute sensitivities [e.g., Datta-Gupta et al., 1997]. For the perturbation method, after finishing the first step of the simulation, an arbitrary perturbation lnk i is added to the conductivity at the master point i (i =1,2,..., number of master points), and the flow and transport simulations are run again to obtain the perturbed tracer concentration. Then the ln k i, are calculated through numerical differencing. The drawbacks of the perturbation method include: (1) it requires a large number of simulation runs (up to 1 + total number of master points), which prohibits the application of this method to a complicated 3-D problem; (2) the calculated sensitivities largely depend on the magnitude of the perturbations lnk i ; and (3) the perturbations and the resulting sensitivity coefficients for all master points are not spatially correlated. Sensitivity coefficient matrices [S] t,k and [S] t,kd can be quickly obtained within the framework of streamline simulator. The method is also free of the previously mentioned drawbacks associated with the perturbation method. [18] From equation (4), it can be seen that the calculation of tracer concentration sensitivities simply becomes the calculation of the sensitivities of travel time t s with respect to the changes of the log hydraulic conductivity and log sorption coefficient within cells. By taking the derivative of (7) with respect to ln k i, the log conductivity within cell i (one of the master points), and applying the chain rule, we s;c ln k Z ls;c c R c f ¼ c ln k ln k ln k ln k i 0 k c ln k i ¼ t s;c v i;c ð8þ

6 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 where v i,c is the kriging weight of master point i to the cell c, which accounts for the spatial correlation of the conductivities between the two cells. Given the covariance function of lnk, v i,c can be evaluated through standard kriging tools [Deutsch and Journal, 1998]. Equation (8) has distinct physical meanings, for instance, the travel time through a cell is inversely proportional to the conductivity of that cell, and if the kriging weight v i,c is too s,c /@ ln k i goes to zero, indicating weak correlation between the travel time t s,c within the cell c and the log conductivity changes at master point i. Therefore we have the sensitivity of travel time t s with respect to log conductivity change given ln k i ¼ Xns;c ln k i ¼ Xns;c c¼1 t s;c v i;c Similarly, by taking the derivative of (7) with respect to ln K d,i, the log sorption coefficient within cell i (one of the master points), we s;c ð9þ ¼ r bk d;c =f i t s;c v i;c ln K d;i 1 þ r b K d;c =f c Thus the sensitivity of travel time t s with respect to the log sorption coefficient ln K d;i ¼ Xns;c c¼1 r b K d;c =f i t s;c 1 þ r b K d;c =f c v i;c ð11þ The right-hand side of equation (11) is always positive, implying that larger K d values along the streamline will result in larger travel time. Equation (11) is consistent with the fact that a larger K d value has a stronger retardation effect on tracer movement. Like the sensitivities to the log conductivity, the sensitivities of travel time to log sorption coefficient ln K d also explicitly account for the spatial correlation of geochemical properties of the aquifer via kriging weight w i,c. Though we use the same notation, this kriging weight in (11) could be different from that in (9), which allows the physical properties (conductivity) and geochemical properties (sorption coefficient) of the aquifer to have different spatial correlation structures. Such flexibility makes the method more applicable for solute transport in natural fields, where cross correlation of conductivity and sorption coefficient is often unclear [Allen-King et al., 1998]. [19] Equations (9) and (11) provide a complete set for calculation of tracer travel time sensitivities to the log conductivity and log sorption coefficient. The solute concentration sensitivities within sampling wells then could be obtained through the travel time sensitivities. The complete set of sensitivity matrices [S] t,k and [S] t,kd can be obtained simultaneously within one single forward simulation run by bookmarking the trajectories and travel times of all the individual streamlines that reach sampling wells. The spatial correlations of log conductivity and log sorption coefficient perturbations among all master points are also accounted for within sensitivity matrices through the kriging weight v i,c. Nevertheless, equations (9) and (11) provide an efficient way to calculate the concentration sensitivity coefficients within one single simulation run, which renders the inverse method feasible to integrate tracer test data, especially for large 3-D problems. 4. Simulation Results and Discussions [20] Assessment of any inverse method under field conditions requires a large number of conductivity, sorption coefficient and concentration measurements. Such data sets rarely exist. Even if these data sets are available, it is difficult to determine the measurement errors in the data sets. Therefore validation of an inverse model is often inconclusive under field conditions. More importantly, verification and validation of an inverse method against a perfectly known synthetic aquifer is always the first step toward field applications. Therefore the performance of our SSC inverse method integrating dynamic tracer test data is demonstrated by using a hypothetical aquifer with perfectly known conductivity and sorption coefficient distributions. In this section, a hypothetical aquifer model with distinct physical and chemical heterogeneities is constructed, and then a synthetic tracer test is performed within the aquifer to obtain the tracer breakthrough curves in sampling wells. Both nonreactive and reactive tracers are injected into the aquifer and sampled within pumping wells. The tracer breakthrough data obtained from the synthetic test are referred to as reference breakthrough curves and used to test the performance of our inverse method and evaluate the effects of conditioning on tracer test data on transport predictions Synthetic Tracer Test [21] Figure 1 shows the aquifer models used for the synthetic tracer test. Figure 1a is the reference log conductivity field ( grid with cell size of 1 m 1m 1 m). The ln K field is heterogeneous and was generated by using SGSIM in GSLIB [Deutsch and Journel, 1998] with a large number of conditioning data points with high ln K values (light color) in the middle of the domain to create a set of well-connected high-conductivity channels. The connected channels form a highly conductive zone inside the aquifer to mimic the fractured zone. Figure 1b shows the corresponding reference ln K d (log-sorption coefficient) field. Unlike the reference ln K field, in the reference ln K d field, a zone with much smaller ln K d values was constructed in the middle of the domain (dark area) to mimic the low retardation effect of the fractured zone, while the rest of the aquifer has larger ln K d values. Generally speaking, the aquifer model has a highly conductive zone with small retardation appearing in the middle of the domain, and the rest of the aquifer is much less conductive and much more highly retarding to reactive chemical transport. One injection well, well 1, and three pumping wells, wells 2, 3 and 4, as shown in Figure 1a, are inserted into the aquifer to conduct the tracer test. The injection well 1 and the sampling well 3 are placed in the middle zone of the domain, and the other two sampling wells 2 and 4 are located in the regions with relatively low conductivity and high sorption coefficient values. Nonreactive and reactive tracers are continuously injected into the aquifer through well 1, and monitored within pumping wells 2, 3 and 4. The monitored tracer breakthrough curves are obtained by running forward flow and transport simulations with the 6of16

7 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 Figure 1. The hypothetical aquifer models used for the tracer test. (a) Reference log conductivity field and configuration of wells and (b) reference log sorption partitioning coefficient field. reference conductivity and sorption coefficient fields. The monitored breakthrough curves of the nonreactive and reactive tracers are shown in Figure 2. For the nonreactive tracer, well 3 samples the tracer at the earliest time and its breakthrough curve has the least spreading (steep slope), reflecting the rapid movement of the tracer from the injection well 1 to well 3 through the highly conductive flow zone. The nonreactive tracer in the other two wells, 2 and 4, needs more time to reach and has much wider spreading. The nonreactive tracer responses in three wells are consistent with the reference conductivity field. The breakthrough curves for the reactive tracer are shown in Figure 2b. The timescale in Figure 2b is much larger than that in Figure 2a. Because well 3 is connected with the injection well 1 through a low ln K d zone, the reactive tracer behaves very similar to the nonreactive tracer. However, the reactive tracer takes much longer time to reach wells 2 and 4, and the breakthrough curves spread much wider than their nonreactive counterparts, because of the strong retardation effect of the aquifer. All the reference curves are consistent with the large-scale physical and chemical heterogeneity features of the aquifer, and clearly demonstrate that solute breakthrough curves carry important information about the transport properties of the aquifer. While the breakthrough curves for nonreactive tracers are determined only by the conductivity distribution, the breakthrough curves for reactive tracers show the combined effects of conductivity distribution and sorption coefficient distribution. These reference curves will be used as conditioning data to test the performance of our inverse method. For simplicity, a constant weight 1 is assigned to all the concentration data Simulation Results [22] A set of 100 initial realizations of the log hydraulic conductivity and log sorption coefficient field were generated by using SGSIM from GSLIB [Deutsch and Journel, 1998] with the same spatial statistics as the reference ln K and ln K d fields, but all the conditioning data points used for generating the reference field were disregarded. Three of the realizations are shown in Figure 3. In comparison with the reference log conductivity and log sorption coefficient Figure 2. Tracer breakthrough curves (normalized by the injected concentration C 0 ) in sampling wells. (a) Nonreactive tracer and (b) reactive tracer. 7of16

8 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 Figure 3. Initial log conductivity and sorption coefficient fields. (a c) Log conductivity field and (d f ) log sorption partitioning coefficient. fields, these individual initial realizations do not have high ln K and low ln K d zone in the middle of the domain, which is considered to be the most important feature of physical and chemical heterogeneity patterns of the aquifer. The initial input fields are statistically homogeneous, without distinct spatial patterns. This is the worst scenario with no specific knowledge about the large-scale features of the aquifer. The SSC inverse method described in sections 2 and 3 is then applied to update the initial fields by conditioning on the tracer test data as shown in Figure 2. 8of16

9 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 We expect that, ideally, all individual updated log conductivity and log sorption coefficient fields would show spatial features similar to those in the reference conductivity and sorption coefficient fields, and yet still reproduce all the observed breakthrough curves. [23] Figure 4 shows the updated log conductivity fields (left column) and the updated log sorption coefficient fields (right column) corresponding to those initial fields shown in Figure 3. As shown in Figures 4a, 4b, and 4c, the updated log conductivity fields indeed exhibit distinct and connected channels with relatively high lnk values (the regions with light color) in the middle of the domain, while the rest of the aquifer in these updated log conductivity fields has relatively low lnk values and remains similar to the initial fields. Though not shown here, other SSC updated log conductivity fields also show similar features after conditioning to the tracer test data. The visual comparisons of the log conductivity fields before and after updating indicates that our SSC inverse method is able to identify the flow channels and barriers by conditioning to the tracer test data. In this particular example, all the updated log conductivity fields successfully reproduce the large-scale features of the reference conductivity field. The updated sorption coefficient fields obtained by conditioning to the reactive tracer transport data are shown in Figures 4d, 4e, and 4f. All three updated sorption coefficient fields exhibit a distinct zone with low ln K d values (the darker region) appearing in the middle of the domain, and the rest of the aquifer seems to have much higher ln K d values. The large-scale features in the reference log sorption coefficient field are well reproduced in these updated log sorption coefficient fields. Similar to the updated log conductivity field, though not shown here, all the updated log sorption coefficient fields exhibit similar features to those shown here. Beside the low ln K d zone, another interesting point about the updated sorption coefficient fields is that the ln K d values of the updated fields are systematically lower than the initial values and they are much closer to the reference sorption coefficient field. Comparison of the log sorption coefficient fields before and after updating also reveals the robustness of our method for efficient calculation of sensitivity coefficients. By conditioning to the reactive/nonreactive tracer test data, our results shown in Figure 4 demonstrate the ability of the SSC inverse method to identify the large-scale physical and chemical heterogeneity patterns of the aquifer. [24] One unique feature of the SSC inverse method is that it can generate multiple equiprobable realizations of the log conductivity field conditioned to the tracer test data, which means that solute transport in every SSC generated conductivity field will reproduce the measured solute breakthrough curves in the pumping wells. Therefore it is worthwhile to compare the ensemble statistics of the SSC updated fields to those of the initial input fields, to determine how well the updated mean fields can reproduce the spatial features of the reference fields. Figure 5 shows a comparison of the mean log conductivity and log sorption coefficient fields before and after updating (both averaged over 100 realizations). While the initial mean log conductivity field (Figure 5a) exhibits low spatial variability, as it should, the updated mean log conductivity field (Figure 5b) exhibits a distinct zone with relatively high lnk values inside the domain, similar to the reference conductivity field. The comparison of the mean log conductivity field before and after updating also suggests that conditioning to tracer test data is able to identify large-scale flow channels and barriers. The comparison between the mean log sorption coefficient fields before (Figure 5c) and after (Figure 5d) updating also shows similar results. The updated mean log sorption coefficient field is very much closer to the true sorption coefficient field than is the initial mean sorption coefficient field, in terms of both spatial patterns and magnitudes. The comparison indicates that characterizing the large-scale chemical heterogeneity of a natural aquifer is possible and feasible by conditioning to reactive tracer test data via our SSC inverse method. [25] In order to more quantitatively assess the performance of our SSC method conditioning to tracer test data, scatterplots (i.e., SSC updated conductivity and sorption coefficient fields versus true fields) are shown in Figures 6a 6d, along with two statistical properties L 1 norm (mean absolute error) and L 2 norm (mean square error) defined as L 1 ¼ 1 N L 2 ¼ 1 N X N i¼1 X N i¼1 ln K i;estimated ln K i;true ð12þ 2 ln K i;estimated ln K i;true where N is the number of grid cells (a total of grid cells for the synthetic aquifer here). Figures 6a and 6b show the scatterplots of the mean conductivity field versus true conductivity field before (Figure 6a) and after updating (Figure 6b). Our initial input conductivity fields are statistical homogeneous, so the initial mean field exhibits small variations and systematically deviates from the true conductivity field (i.e., deviates from the 45 line). However, SSC updated mean conductivity field (Figure 6b) is much closer to the true conductivity field, and the two statistical properties L 1 norm and L 2 norm are also significantly reduced. Similar results are also observed for the scatterplots of the mean sorption coefficient field before (Figure 6c) and after updating (Figure 6d). Given the large number of grid cells and the fact that the initial fields are systematically deviating from the true field, the SSC updated conductivity field and sorption coefficient field still approximate the true fields reasonably well, as shown in Figures 6b and 6d. [26] The mismatch between breakthrough curves before and after updating was also examined. The mean breakthrough curves (both nonreactive and reactive) (averaged over 100 realizations) before and after updating are compared along with the reference ( measured ) breakthrough curves. Figure 7 shows the comparison of nonreactive tracer breakthrough curves before and after updating. Generally speaking, the breakthrough curves after updating (solid lines) match the reference curves very well. The arrival times, magnitudes of the concentration and spreading widths of the curves are reproduced very well by the updated fields. The initial fields actually failed to match the measured breakthrough curves. Large differences are observed between the measured and predicted curves when using the initial input fields, particularly for well 3. In wells 2 and 4, the initial fields predict earlier arrivals and faster movements than the measured ones, but in well 2, the 9of16

10 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 Figure 4. Updated log conductivity and sorption coefficient fields after conditioning to tracer test data. (a c) Updated log conductivity fields and (d f ) updated log sorption coefficient field. initial fields predict much slower tracer movement and wider spreading than the measured ones. Despite such contrary behavior, the updated fields still provide good matches between measured and predicted curves, implying that our method of calculating concentration sensitivities is quite efficient. [27] Figure 8 shows the comparisons between the measured reactive tracer breakthrough curves and those pre- 10 of 16

11 W01506 HUANG ET AL.: INVERSE MAPPING CONDITIONING ON TRACER DATA W01506 Figure 5. The mean log conductivity and sorption coefficient fields before and after conditioning to tracer test data. (a) Initial mean log conductivity field, (b) updated mean conductivity field after conditioning, (c) initial mean sorption coefficient field, and (d) updated mean log sorption coefficient field after conditioning. dicted from: (1) initial conductivity fields and initial sorption coefficient fields (dashed line); (2) updated conductivity fields and initial sorption coefficient fields (dashed line with ); and (3) updated conductivity field and updated sorption coefficient fields (solid line). Surprisingly, in well 2, the breakthrough curves predicted by the initial fields (dashed line) and the updated fields (solid line) are very close and both agree very well with the measured one, despite the fact that the initial conductivity and sorption coefficient fields have no distinct spatial features and are completely different from the reference fields. Furthermore, the breakthrough curve predicted by using the updated conductivity fields and initial sorption coefficient fields (dashed line with ) deviates from the measured curve much more than the curve predicted by using the updated conductivity and updated sorption coefficient fields (solid line), implying that conditioning on nonreactive transport data alone cannot well predict the reactive transport behavior. Similar results are also observed in well 4 (Figure 8c). In well 3, as shown in Figure 8b, the initial fields (dashed line) yield the largest deviations from the measured curve. Conditioning to the nonreactive transport data (dashed line with ) reduces such deviations to a certain degree, however, significant differences in arrival times and spreading are still obvious. Further conditioning to reactive transport data (solid line) yields an excellent match between the predicted and measured curves. Such improvements once again demonstrate the robustness of the concentration sensitivities with respect to the chemical heterogeneity. In addition to the conductivity heterogeneity, the reactive tracer transport is also influenced by chemical heterogeneity. Like the results shown in Figure 7 (nonreactive case), the mean breakthrough curves predicted by using the initial 100 realizations of the conductivity and sorption coefficient fields significantly deviate from the reference curves (except for well 2), while those predicted by using the SSC updated conductivity and sorption coefficient fields are fairly consistent with the reference curves, in terms of both arrival times and spreading widths. [28] Next, we examine the effects of conditioning on the accuracy of transport predictions. The variances of the breakthrough curves before and after updating, corresponding to the mean breakthrough curves shown in Figures 7 and 8, are compared in Figures 9 and 10. Figure 9 shows the comparison between the variances of the breakthrough curves of the nonreactive tracer before and after updating. The SSC 11 of 16