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Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA
* Corresponding author (SAFinsterle{at}lbl.gov)
Received 26 September 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Abbreviations: IFS, iterated function system • NAPL, nonaqueous phase liquid
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Numerical models support a variety of scientific and engineering tasks, including the study of fundamental physical processes, the design, optimization, and analysis of laboratory and field experiments, and the prediction of the system behavior under natural conditions or in response to management decisions. Regardless of the purpose or application, modeling involves (i) conceptualizing the salient features of the hydrogeologic system of interest, (ii) characterizing these features by means of a finite number of input parameters, and (iii) solving the system of governing equations, which brings the fundamental physical laws together with problem-specific attributes for the estimation of the unknown system states.
Modeling results may be strongly affected by incomplete knowledge about the numerous input parameters, making the second step above (characterizing features by means of a finite number of input parameters) a key element of modeling. For multiphase flow systems, these parameters are usually difficult to determine in the laboratory or the field. Moreover, they may be process and scale dependent; that is, the "measured" parameters are often conceptually and thus numerically different from the effective parameters required by—and most suitable for—the site-specific numerical model. Furthermore, an inappropriate simplification or error in the conceptual model (i.e., failure to succeed in the first step above, conceptualizing the salient features of the hydrogeologic system of interest) is likely to have a significant impact on the simulations and the conclusions drawn from the modeling study. Consequently, the conceptual model must be thoroughly examined and its parameters must be carefully determined to assess the reliability of otherwise accurate numerical predictions.
The two key steps of model development—conceptualization and parameter estimation—are interrelated. Conceptual aspects of a model can be parameterized and thus subjected to parameter estimation and the associated sensitivity and error analyses. The complexity of multiphase flow processes and that of the subsurface itself (mainly its multiscale heterogeneity) combined with the scarcity of characterization data make it necessary to select and examine carefully the parameterization chosen to simplify the multifaceted natural system. Moreover, the recognition that each model is by definition an abstraction of the real hydrogeologic system highlights the need for a thorough error and uncertainty analysis. Inverse modeling is a means to address some issues related to model conceptualization, parameterization, and assessment of various uncertainties.
The next section discusses the inverse modeling framework in very general terms. Examples of unsaturated and multiphase inverse modeling studies found in the scientific literature are summarized next. We then discuss some applications of the iTOUGH2 (Finsterle, 1999a, 1999b, 1999c) multiphase flow simulator, focusing on attempts to identify model structure, specifically heterogeneity.
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INVERSE MODELING FRAMEWORK |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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All parameter estimation methods are "inverse" to a greater or lesser extent, involving a conceptual model and data fitting. For example, even though the determination of the saturated permeability based on flow-rate data is often considered to be a direct measurement, the value is obtained by inverting Darcy's Law, assuming one-dimensional, steady-state flow with a constant pressure gradient (i.e., using a specific model of the flow system). The experiment has to be carefully designed to make sure that the assumptions underlying this simple model are satisfied. Any discrepancies between the experiment and the model lead to an error in the permeability estimate. This simple example illustrates that the design of the experiment has to be adapted to the model used for data analysis, or that the model must be modified if the experimental conditions deviate from the conceptual assumptions in the original model. Flexible parameter estimation techniques, such as inverse modeling, provide opportunities for employing new experimental designs with higher accuracy and efficiency in the determination of unsaturated hydraulic properties.
The following is a short introduction to the "indirect approach" (Neuman, 1973) to inverse modeling, in which an objective function measuring the overall difference between observed data and the corresponding simulation results is minimized by adjusting selected input parameters. The objective function can be expanded to include prior information and regularization terms.
In general, the objective function has the following form:
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is an arbitrary loss function (usually the square or absolute value of the weighted residuals; for additional options see Finsterle and Najita, 1998), and yi is an appropriately weighted residual:
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i is often related to the measurement error, but may include other considerations. The objective function can be derived from maximum likelihood theory. For example, assuming the final residuals follow a Gaussian distribution, maximum likelihood estimates are obtained by minimizing the sum of the weighted squared residuals, that is, by solving a weighted least-squares problem (Carrera and Neuman, 1986a). Exponentially distributed residuals lead to the L1 estimator, which minimizes the sum of the absolute values of the weighted residuals. Additional objective functions derived from maximum likelihood considerations are discussed in Finsterle and Najita (1998).
It is important to realize that the well- or ill-posed nature of the inverse problem is entirely determined by the topology of the objective function in the n-dimensional parameter space. The uniqueness and stability of the inverse problem are directly related to the chosen parameterization and data available for calibration. Adding prior information or regularization terms to the objective function has the sole purpose of making it more convex and smooth, using information not directly contained in the observed data. Nonuniqueness, stability, and robustness of the inverse problem are discussed in detail in Carrera and Neuman (1986b), Yeh (1986), and McLaughlin and Townley (1996).
A number of algorithms are available to minimize the objective function. Most of these methods are local; that is, they cannot guarantee that the global minimum in the admissible parameter space is detected. There is usually a trade-off between generality and efficiency of the minimization algorithm. For example, the solution to a least-squares problem for a linear model can be obtained in a single iteration using the Gauss–Newton method, whereas many function evaluations (i.e., solutions of the forward problem) are needed to find the global minimum of a general objective function of a highly nonlinear model using, for example, the simulated annealing method. Most multiphase flow inverse problems can be categorized as multivariate, nonlinear least-squares problems with simple constraints, which can be solved using a variety of methods. Some minimization algorithms rely solely on function evaluations; examples include the simplex method (Nelder and Mead, 1965), simulated annealing (Metropolis et al., 1953), and genetic algorithms (Holland, 1975). Other methods (e.g., steepest descent, conjugate-gradient, and quasi-Newton) require first derivatives of the objective function with respect to the parameters of interest. Newton's method uses second derivatives. However, curvature information is often derived from a related, positive-definite Hessian matrix that is easier to calculate. Good introductions to these methods from a general, practical perspective are given by Beck and Arnold (1977), Gill et al. (1981), and Press et al. (1992). Derivatives can be computed either numerically by the parameter perturbation method, or by formulating and solving sensitivity or adjoint-state equations. These methods are discussed in detail in Yeh (1986), Kool et al. (1987), and Sun (1994).
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LITERATURE REVIEW |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Numerous research papers discuss the concept of inverse modeling in the context of hydrogeology. They are summarized and reviewed by Neuman (1973), Yeh (1986), Ewing and Lin (1991), Sun (1994), McLaughlin and Townley (1996), and Zimmerman et al. (1998). These articles review inverse modeling concepts and applications to flow and transport problems in the saturated zone. The use of inverse modeling techniques for unsaturated flow problems has been far less extensive, as indicated by the reviews of Kool et al. (1987), Durner et al. (1997), and Hopmans et al. (2002).
Table 1 is a noncomprehensive list of unsaturated and multiphase flow inverse modeling studies reported in the literature. The table includes the experimental setup, the parameters estimated, and the type of data matched during model calibration. The studies are numbered and arranged from simple, one-dimensional unsaturated flow experiments conducted in the laboratory to the analysis of field data from complex, nonisothermal, three-dimensional multiphase flow systems.
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Capillary pressure and relative permeability are among the key characteristics affecting unsaturated and multiphase flow systems. Traditionally, capillary pressure curves have been constructed pointwise by measuring saturation S and capillary pressure pc under equilibrium conditions. For use in numerical models, a functional form is selected and matched to the experimental data. These parametric models can be considered mere fitting functions. However, they contain parameters considered representative of the pore structure. An example is the 
parameter of the Brooks–Corey (1964) model, which can be related to the pore size distribution or fractal dimension of the pore space (Perfect et al., 1996). Consequently, there are attempts to estimate these model parameters from soil texture and other properties that are relatively easy to measure (e.g., Saxton et al., 1986; Schaap et al., 2001). A similar procedure is involved when predicting unsaturated hydraulic conductivity from water-retention curves. Pore-connectivity models relating relative permeability to capillary pressure were introduced by Burdine (1953) and Mualem (1976) and are used in the expressions of Brooks and Corey (1964), van Genuchten (1980), and Russo (1988), among many others.
These conventional methods are time-consuming and expensive. Moreover, they do not permit the determination of both relative permeability and capillary pressure curves during a single experiment, which often leads to inconsistent parameters and thus unreliable model predictions, as pointed out, for example, by Luckner et al. (1989) and Salehzadeh and Demond (1994).
While the approaches discussed above rely on direct observations and make use of geometric models of the pore space, parameter estimation by inverse modeling involves process simulation. That is, it uses the equations governing flow in unsaturated porous media and determines the parameters by minimizing the differences between model predictions and observed data. An advantage of inverse modeling is that any type of data can be used for parameter estimation, provided that the calculated system response is sensitive to the parameters of interest. Furthermore, numerical simulation and inversion techniques impose fewer restrictions on the experimental layout and flow processes considered.
The studies numbered 1 through 13 in Table 1 are concerned with the estimation of unsaturated hydraulic properties from one-step and multistep outflow experiments, which were designed to obtain efficiently the parameters of the capillary pressure and relative permeability functions (referred to as characteristic curves in Table 1). Zachmann et al. (1981) were the first to analyze drainage data using inverse methods. In its simplest form, a one-step outflow experiment consists of a test cell where a stepwise change in the water potential is imposed on one end of the soil column, and the resulting outflow is measured as a function of time.
Several studies (often using synthetically generated data) examined the nonuniqueness of the estimated parameters determined by outflow experiments (e.g., Kool et al., 1985a; Toorman et al., 1992). Modifications to the experimental design were proposed, demonstrating the merit of measuring capillary pressure in addition to cumulative outflow, and performing multiple steps instead of a single step. As an example of such a study, Fig. 1
shows contour plots of the objective function from a radial multistep outflow experiment performed to estimate the logarithm of absolute permeability log(kabs), the logarithm of the air-entry pressure log(pe), and the Brooks–Corey pore size distribution index . The top row of panels shows the objective function obtained when only flow rate measurements are available; the middle row of panels show the objective function obtained when only pressure measurements are available. The bottom row results from combining the two types of observations. The shape, size, orientation, and convexity of the minimum provides information about the uniqueness and stability of the inversion and represents the uncertainty and correlation structure of the estimated parameter set. Furthermore, the potential presence of local minima could be detected readily. The planes shown in Fig. 1 intersect the global minimum. The left column of Fig. 1 reveals that log(kabs) and log(pe) could be estimated jointly from either flow rate or pressure data, whereas the joint estimation of log(kabs) and (central panel of Fig. 1) is likely to be unstable if only pressure measurements are available. The combination of pressure and flow rate data yields a well-defined global minimum for all three parameters. Note that the orientation of the contours around the minima from the flow rate data tend to be orthogonal to those from the pressure data, resulting in a well-developed minimum when combined.
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nek and van Genuchten (1996) for three-dimensional disc permeameter infiltration experiments. The important issue of how unsaturated soil hydraulic functions determined in the laboratory relate to field conditions was discussed by Eching et al. (1994b). Studies 14 through 27 of Table 1 describe applications of inverse modeling techniques to field data using models based on the Richards equation. Most commonly, water content, water potential, and release rates from ponded infiltration and redistribution experiments were used to estimate parameters of soil hydraulic functions. Some studies looked at natural state data. As opposed to the small-scale laboratory experiments discussed above, field-scale studies face the problem of heterogeneity and the associated parameterization issues. The layered structure of most geologic deposits lead some researchers to estimate separate parameter sets for each layer. Stochastic approaches such as those presented in Zimmerman et al. (1998) have not been widely used for unsaturated zone studies. A notable exception is the series of papers by Yeh and coworkers (Study 20; Table 1) and Kowalsky et al. (2004). Model structure identification will be further discussed in Parameterization below.
The relative ease of performing air injection tests provides a means to characterize the permeability distribution of unsaturated fractured rocks. Gas pressure changes in response to atmospheric pressure fluctuations can also be used to infer the gas diffusivity of porous and fractured formations. Finally, laboratory gas injection tests are useful to characterize rocks with very low permeability. Inverse modeling studies of data from single-phase gas flow systems are listed as Studies 28 through 32 of Table 1.
In contrast to the Richards equation, where the gas phase is assumed to be passive at a constant reference pressure, two-phase formulations allow for pressure buildup in both the liquid and gas phases. Moreover, phase transitions of components can be handled, such as dissolution of air in the liquid phase and vaporization of water or a volatile organic compound. Similar governing equations can be developed for two-phase oil–water systems or three-phase systems consisting of a gas, aqueous, and nonaqueous phase liquid (NAPL) phase. Phase transition effects can also be invoked by temperature changes, specifically when above-boiling conditions are reached. The corresponding nonisothermal multiphase flow models are more complex than models based on the Richards equation. They also require additional parameters, such as gas (and NAPL) relative permeabilities and thermal properties. Initial and boundary conditions in such systems are more difficult to prescribe or assess; they are thus more uncertain and may need to be subjected to estimation by inverse modeling.
Only a relatively small number of formal data inversion studies from multiphase flow systems have been reported in the scientific literature (see Studies 33–46 in Table 1). For the characterization of geothermal reservoirs, nonisothermal flow effects have been analyzed using data from small-scale laboratory experiments, well completion tests, and field-scale production. Production data from oil reservoirs are used to infer the structure of the heterogeneous permeability field using fast streamline reservoir simulators in combination with geostatistical approaches.
A large number of computer codes solving inverse and optimization problems have been developed in the mathematical sciences (for an overview, see Moré and Wright, 1993). Specialized codes for applications in hydrogeology mainly deal with automatic well-test analysis and the inversion of pressure head and tracer data. All of these codes are based on flow and transport simulations in the saturated zone. Only a few inverse modeling codes have been developed that are capable of dealing with unsaturated and nonisothermal multiphase flow problems. The program ONE-STEP (Kool et al., 1985c; Eching and Hopmans, 1993) specifically addresses the estimation of soil hydraulic parameters from outflow experiments. The HYDRUS software (imnek et al., 1998, 1999) can be used to estimate parameters of hysteretic retention and hydraulic conductivity functions as well as solute transport parameters from unsaturated flow and concentration data. The iTOUGH2 code (Finsterle, 1999a, 1999b, 1999c) allows estimation of any input parameter to the multiphase flow simulator TOUGH2 (Pruess et al., 1999). Note that any forward model can be linked to general, model-independent, nonlinear parameter estimation packages such as PEST (Doherty, 1994) or UCODE (Poeter and Hill, 1998), or any commercially available optimization package, thus providing inverse modeling capabilities with various degrees of flexibility. For example, PEST was used in combination with the SWAP simulator (van Dam et al., 1997) to estimate effective soil hydraulic properties using evapotranspiration and water content data (Jhorar et al., 2004). An application of UCODE in combination with the STOMP simulator (White and Oostrom, 2000) is documented in Zhang et al. (2002). For the remainder of this paper, we discuss inverse modeling issues using the iTOUGH2 (Finsterle, 1999a, 1999b, 1999c) simulation–optimization code.
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PARAMETERIZATION |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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As long as a feature or process is suitably parameterized, it can be subjected to estimation, optimization, sensitivity, and uncertainty analyses. Parameterization may include not only hydrogeologic properties (such as the heterogeneous permeability field), but also aspects of the conceptual model that are considered uncertain. For example, uncertainty in the initial or boundary conditions can be parameterized and estimated along with hydrogeologic properties. Moreover, a potential systematic error (such as a suspected drift in the data or leakage in a measurement device) can also be parameterized and included in the estimation or uncertainty analysis. An example of this approach is discussed in Finsterle and Persoff (1997).
Sensitivity and uncertainty propagation analyses may involve as many parameters as desired. For inverse modeling purposes, however, it is often not sensible to estimate a large number of strongly correlated parameters on the basis of limited data of insufficient sensitivity. Adding more parameters to vector p always leads to an improvement of the fit. However, if too many parameters are estimated simultaneously, the better reproduction of data comes at the expense of increased estimation uncertainty and thus reduced accuracy of subsequent model predictions. Overparameterization may also mask systematic errors in the conceptual model. To alleviate the difficulties caused by an ill-posed inverse problem, it may be appropriate to include prior information about the parameters or to apply regularization methods.
Because the goodness of fit obtained by model calibration depends on the degree of parameterization and the details of the model structure, it is not obvious which model among a set of alternatives is most appropriate to represent a natural system. Carrera and Neuman (1986a) presented a number of so-called model identification criteria that allow for a more objective ranking of alternative models.
Sun et al. (1998) provided an excellent discussion of the relation between model structure and parameterization. They also proposed a regression methodology to simultaneously determine model structure and model parameters, highlighting the fact that the justifiable complexity of a model is controlled not only by calibration data and prior information, but also by the accuracy requirements of the predictive simulations. In other words, it is the application of the model that determines the acceptable prediction error and thus tolerable estimation uncertainty, which in turn dictates the test design and required measurement accuracy.
Discussing these important issues in detail is beyond the scope of this review article. Here we focus on a few parameterization schemes for representing heterogeneity; they are all available in the iTOUGH2 code, providing the means to identify hydrogeologic structures and to examine their impact on inverse modeling results and/or model predictions.
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MODELING HETEROGENEITY |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Before discussing parameterization and estimation techniques, it should be noted that simulating multiphase flow through highly heterogeneous media is numerically challenging and subject to fundamental difficulties. For example, formation and fluid properties must be appropriately weighted when calculating flow across a domain boundary. The resulting artifacts in the predicted saturation distribution may be accounted for during calibration through the estimation of an effective parameter that partly corrects for systematic modeling errors. Nevertheless, it must be recognized that the estimated parameter distribution is biased and strongly related to the particular numerical model and the applied weighting scheme.
Iterated Function System
Subsurface heterogeneities often exhibit a hierarchical structure described by fractal distributions (e.g., Neuman, 1990; Molz et al., 1997). These fractal hydrogeologic property distributions may be created using a set of affine transformations and an associated set of probabilities, which determine a so-called iterated function system (IFS). Each IFS has a unique attractor, which can be described by a relatively small number of parameters. Following the procedure outlined in Doughty (1995), iTOUGH2 generates fractal sets, which are subsequently mapped to hydrogeologic properties (Fig. 2)
. This method is flexible enough to create fractals with linear, areal, and volumetric structures. Solving the related inverse problem often requires the use of a robust, albeit relatively inefficient search method (e.g., simulated annealing).
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iTOUGH2 APPLICATIONS |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Water Seepage into Underground Opening
Dripping of water into tunnels containing nuclear wastes is a key mechanism affecting the concentration and rate at which dissolved radionuclides migrate away from the repository. In unsaturated formations, water tends to flow around the opening as a result of the capillary-barrier effect (Philip et al., 1989), preventing seepage from occurring or reducing seepage flux below the prevalent percolation flux.
While the capillary-barrier effect is well understood for underground openings in homogeneous formations or layers of uniform porous media, numerical modeling is required to study seepage from fractured rock such as welded tuffs. Fractured rock can be considered a highly heterogeneous medium, where fractures are connected regions of high permeability and low capillarity, interspersed with matrix blocks of low permeability and high capillarity. The location, size, and orientation of fractures, as well as the hydraulic properties within rough-walled fracture planes, tend to be random. The random nature is built into discrete fracture network models, which generate realizations of fracture sets based on their statistical characteristics. An example of this approach is presented by Liu et al. (2002), who generated a two-dimensional TOUGH2 model of discrete fractures (Fig. 4a) and simulated flow and seepage into a circular opening (Fig. 4b).
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A calibrated heterogeneous fracture continuum model is a viable alternative to the discrete fracture network model described above. Because fractures are not perfectly parallel to the tunnel axis, which is an implicit assumption of two-dimensional discrete fracture network models, water can be diverted around the opening also within the fracture plane, leading to a system behavior that is distinctly less discrete and thus appropriately modeled using a continuum approach. Moreover, accepting that parameters used in a numerical model are always (i) related to the conceptual model and its numerical implementation, (ii) specific to the involved physical processes, (iii) dependent on the scale, and (iv) tailored to the prediction variables of interest, the calibration of the model against relevant data yields effective continuum parameters that are not only appropriate, but optimal for the given model and study objectives. This overall approach has been examined for the seepage problem discussed above.
Finsterle (2000) demonstrated that seepage into underground openings excavated from a fractured formation could be simulated using a heterogeneous fracture continuum model, provided that the model is calibrated against seepage-relevant data (such as data from liquid-release tests). To generate synthetic data showing discrete flow and seepage behavior, a two-dimensional high-resolution model was created with multiple sets of elongated features representing heterogeneous fractures, which are embedded in a low-permeability matrix (Fig. 5a) . An excavation-disturbed zone was introduced by increasing permeability around the opening. This model is capable of creating discrete flow and seepage behavior (Fig. 5b) and is thus referred to as a discrete-feature model. Synthetically generated seepage data from a liquid-release test simulated with this discrete-feature model were used to calibrate a simplified heterogeneous fracture continuum model.
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As shown in Fig. 6 , prediction uncertainty is substantial, mainly because of the strong impact of local heterogeneity on seepage, which cannot be described deterministically. Nevertheless, the seepage percentages predicted with the continuum model are consistent with the synthetically generated data from the discrete-feature model. This demonstrates that (i) the calibrated continuum model and discrete-feature model yield consistent estimates of the seepage threshold and average seepage rates and (ii) the continuum approach is appropriate for performing seepage predictions even if extrapolated to percolation fluxes that are significantly lower than those induced by liquid-release tests, which were performed at relatively high injection rates to generate seepage data useable for model calibration.
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The general modeling approach examined by inverting and predicting synthetic data has been successfully applied to the analysis of seepage-rate data from actual liquid-release tests. These three-dimensional iTOUGH2 analyses are discussed in Finsterle et al. (2003) and Ghezzehei et al. (2004).
Water Flow through Unsaturated Fracture–Matrix System
Contaminant transport is strongly affected by the presence of fractures and the degree of fracture–matrix interaction. Measurements of chemical signatures at fractured sites most likely represent matrix concentrations rather than fracture concentrations. It is therefore crucial to understand and accurately model the process of matrix imbibition, which is affected not only by the sorptivity of the matrix, but also by the characteristics of the fractures.
iTOUGH2 simulations of water flow through fractured rock were performed to examine the penetration depth of a large pulse of water entering such a system. The influences of local heterogeneities in the fracture network and variations in hydrogeologic parameters were examined by sensitivity analyses and Monte Carlo simulations. To resolve the pressure and saturation gradients between the fractures and the matrix, the method of multiple interacting continua (Pruess and Narasimhan, 1982, 1985) was employed. A two-dimensional heterogeneous fracture permeability field was generated, exhibiting both local obstacles in the fracture continuum as well as high-permeability channels (Fig. 7a) . These obstacles may represent dead-end fractures, discontinuities in the fracture network, asperity contacts, or heterogeneity in the amount and properties of fracture fillings. The matrix is assumed homogeneous.
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The simulated saturation change in the matrix distribution of Fig. 7b is qualitatively consistent with observed contaminant profiles, which typically show sections of apparently uncontaminated rock interrupted by spikes with high concentration values. Since water that quickly flows through the fracture network may not leave a prominent chemical signal in the matrix, the apparent absence of elevated contaminant concentrations at certain measurement locations along a vertical profile does not necessarily indicate the actual penetration depth of the contamination, which may be much greater.
A detailed sensitivity analysis of contamination signals in a fractured porous medium can be found in Finsterle et al. (2002).
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Despite the physical basis of codes such as iTOUGH2, it should be recognized that modeling always involves a sequence of steps in which the physical system is simplified and reduced to its salient features. As a result of this abstraction process, it is necessary to determine and use effective parameters that are related to the specific conceptual model. This limits the applicability of the model and the generality of its parameters. On the other hand, the estimation of effective parameters by calibrating the model against suitable data of good quality leads to site-specific, model-related, and process-relevant parameters that can be considered optimal for the given modeling task.
Changes in the conceptual model usually have the greatest impact on model predictions and thus on the conclusions of a study. The following iTOUGH2 features support the evaluation of alternative conceptual models:
Automatic model calibration.
The process of matching the model output to observed data by adjusting sensitive parameters helps determine whether or not the conceptual model is a likely representation of the natural system. If the data cannot be matched or the resulting parameter values are unreasonable, the model contains a systematic error that needs to be identified and removed. Moreover, automatic calibration allows for a quick evaluation of multiple alternative models.
Residual and error analyses.
A statistical analysis of the discrepancies between the calculated and observed system response reveals systematic deviations between the behavior of the natural system and its representation in the model, thus identifying aspects of the conceptual model that may need to be refined. The error analysis of the estimated parameters may reveal high estimation uncertainty as a result of strong parameter correlation, indicating that the problem is overparameterized.
Parameterization.
The flexible architecture of iTOUGH2 allows a user to parameterize certain aspects of the conceptual model, and thus be able to submit them to a formalized analysis and optimization. For example, geostatistical tools can be used to describe complex heterogeneity with just a few parameters, which can then be estimated or perturbed in a sensitivity analysis. Similarly, suspected modeling errors or artifacts and trends in the data can also be parameterized (e.g., Finsterle and Persoff, 1997).
Monte Carlo simulations.
Monte Carlo simulations (e.g., Latin Hypercube sampling strategies, which may account for statistical correlations among the parameters) conducted with iTOUGH2 evaluate the prediction uncertainty as a result of uncertainty in the input parameters. Consequently, the impact of parameterized aspects of the conceptual model can also be examined. Moreover, for each Monte Carlo simulation, generating a new realization of the stochastic property field for each simulation examines output variability as a result of unspecified randomness.
Model identification criteria.
iTOUGH2 calculates a number of model identification criteria (Carrera and Neuman, 1986a) for the comparison of alternative models with different numbers of adjustable parameters.
Test design.
iTOUGH2 can be used to perform synthetic inversions in support of experimental design. The information content of potential measurements and the correlation structure of the parameters to be estimated can be determined, indicating whether a proposed design is capable of distinguishing between competing theories or conceptual models.
Solving inverse problems is an inherently difficult task, specifically when dealing with unsaturated and multiphase flow systems. It requires not only proficiency in all modeling aspects, but also a good understanding of the data used for calibration and of the data collection conditions. The results of an inverse modeling analysis must be critically evaluated using statistical tools and examined against our insights into the system behavior. Finally, inverse modeling, and modeling in general, is a tool for the solution of scientific and engineering problems; the appropriateness of its use and the value of the results should thus be judged based on the objectives of each individual study.
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONINVERSE MODELING FRAMEWORKLITERATURE REVIEWPARAMETERIZATIONMODELING HETEROGENEITYiTOUGH2 APPLICATIONSSUMMARY AND CONCLUSIONSREFERENCES |
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imnek, N. Romano, and W. Durner. 2002. Simultaneous determination of water transmission and retention properties. Inverse methods. p. 963–1008. In J.H. Dane and G.C. Topp (ed.) Methods of soil analysis. Part 4. SSSA Book Ser. 5. SSSA, Madison, WI.
