Multiphase Inverse Modeling: Review and iTOUGH2 ... - CiteSeerX

Multiphase Inverse Modeling: Review and iTOUGH2 Applications Stefan Finsterle*

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

ABSTRACT

Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA. Received 26 Sept. 2003. Special Section: Research Advances in Vadose Zone Hydrology through Simulations with the TOUGH Codes. *Corresponding author ([email protected]).

alizing 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.

Published in Vadose Zone Journal 3:747–762 (2004).  Soil Science Society of America 677 S. Segoe Rd., Madison, WI 53711 USA

Abbreviations: IFS, iterated function system; NAPL, nonaqueous phase liquid.

Calibration of a numerical process model against laboratory or field data is often referred to as “inverse modeling.” As the numerical simulation models become more complex, the number of parameters to be estimated generally increases, requiring new testing, modeling, and inversion strategies. The purpose of this survey is to review inverse modeling approaches for unsaturated and multiphase flow models. The discussion focuses on applications rather than theoretical considerations, which have been previously reviewed in the context of saturated flow and transport modeling. We also examine model parameterization issues, specifically the representation of heterogeneity through a limited number of variables that can be subjected to parameter estimation and uncertainty propagation analyses. Different parameterization strategies are illustrated using the multiphase flow simulation–optimization code iTOUGH2. A comprehensive inverse modeling package (such as iTOUGH2, which includes automatic model calibration followed by an extensive residual, error, and uncertainty propagation analysis) is an essential tool to improve test design and data analysis of complex multiphase flow systems.

T

raditionally, the focus of characterization and modeling studies has been on the saturated zone because of its obvious importance for the protection and management of groundwater resources. Soil physicists and agronomists were the first scientists interested in hydraulic and thermal properties of the variably saturated crop root zone. More recently, increased attention has been given to the unsaturated zone from the land surface to the water table. It was recognized that most groundwater contamination problems originate in the vadose zone, either accidentally or by intentional selection of unsaturated sites for disposal of chemical and radioactive wastes. Compressed gas energy storage projects and geologic sequestration of CO2 involve deep injection of gas into geologic formations. Finally, energy recovery from oil, gas, and geothermal reservoirs as well as from coalbed methane and gas hydrate deposits requires dealing with (potentially nonisothermal) multiphase flow systems. The following discussion is concerned with inverse modeling of variably saturated and multiphase flow systems. 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) conceptu-

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INVERSE MODELING FRAMEWORK Inverse modeling is usually defined as the process of estimating model-related parameters by matching a numerical model to measured data representing the system response at discrete points in space and time. Model calibration, history matching, and nonlinear regression are essentially equivalent terms for this estimation procedure, which relates the measurement space to the space of admissible parameters. 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: S⫽

m

兺 ␻(yi;p)

[1]

i⫽1

where ␻ 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: yi ⫽

z*i ⫺ zi(p) ␴i

[2]

where z*i is one of m actual data points (or regularization constraints) at a discrete point i in space and/or time, and zi is the corresponding output variable, which is a function of the input parameter vector p of length n. The weighting coefficient ␴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 quasiNewton) 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, positivedefinite 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).

LITERATURE REVIEW The basic concept of estimating parameters by matching a functional model to observations dates back to Carl Friedrich Gauss, who introduced the method of least squares for the analysis of astronomical and geodetic data during the last decade of the eighteenth cen-


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tury (Gauss, 1821). Gauss made contributions to all aspects of parameter estimation, providing a detailed discussion of measurement errors, a probabilistic justification of the least-squares objective function, advances in computational methods (Gaussian elimination), and an analysis of estimation uncertainty. While the algorithms for identifying the minimum of the objective function have been continually refined, the basic idea and the difficulties associated with solving the inverse problem remain essentially the same. 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. The majority of inverse modeling applications are concerned with laboratory tests (specifically one-step and multistep outflow experiments) for the determination of unsaturated hydraulic properties (most commonly the van Genuchten [1980] parameters). Later studies included more sophisticated water retention functions, such as bimodal (Zurmu¨hl and Durner, 1998) and hysteretic models (Kool and Parker, 1988; Mishra and Parker, 1989; Finsterle et al., 1998b). 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

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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 pres-

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Table 1. Selected studies of unsaturated and multiphase inverse problems. Reference

Experiment

Parameters

Data

Comment


Unsaturated flow, laboratory experiments 1

Zachmann et al. (1981, 1982)

draining column experiment

parameters of characteristic curves

cumulative discharge data

2

Kool et al. (1985a, 1985b); Parker et al. (1985); Toorman et al. (1992) van Dam et al. (1992, 1994)

one-step outflow experiment


cumulative outflow

one-step and multistep outflow experiments


cumulative outflow

4

Eching and Hopmans (1993); Eching et al. (1994a)

multistep outflow experiment, evaporation


cumulative outflow, average water content, pressure

5

Hudson et al. (1996)

imbibition experiment

pressure

6

Garnier et al. (1997)

multistep outflow and evaporation experiments

parameters of characteristic curves parameters of characteristic curves, shrinkage curve

7†

Finsterle et al. (1998b)

radial drainage-imbibition experiment

parameters of hysteretic characteristic curves

cumulative outflow, pressure

8

Zurmu¨hl and Durner (1998) Finsterle and Faybishenko (1999)

multistep outflow experiment radial multistep outflow experiment

parameters of characteristic curves permeability, air entry pressure, pore size distribution index

cumulative outflow, pressure cumulative outflow, pressure

10

Abbaspour et al. (1999)

multistep lysimeter experiments

11

Hwang and Powers (2002) Bitterlich and Knabner (2002)

multistep outflow experiment outflow experiment

permeability, parameters of characteristic curves parameters of characteristic curves parameters of characteristic curves

pressure, water content, cumulative outflow cumulative outflow, pressure cumulative outflow, pressure

13

Bohne and Salzmann (2002)

evaporation experiment

permeability, tortuosity

pressure, water content

14

Dane and Hruska (1983); Zijlstra and Dane (1996) Kool and Parker (1988)

infiltration–redistribution experiment

3

9†

12

cumulative outflow, pressure

synthetic data, examined impact of measurement noise examined nonuniqueness

comparison between onestep and multistep outflow experiments comparison between inversely determined and directly measured parameters upward flow in soil core compared multistep outflow and evaporation methods, swelling soil synthetic data, hysteretic model with air entrapment bimodal hydraulic functions analysis of synthetic and actual data from radial multistep outflow experiments examined different objective functions, global minimization tested different hydraulic functions characteristic curves approximated by spline functions, synthetic example compared nonequilibrium with equilibrium retention data

Unsaturated flow, field experiments

15

infiltration, drainage, and evaporation experiments infiltration experiments

permeability, parameters of characteristic curves parameters of hysteretic characteristic curves

16

Russo (1988); Russo et al. (1991)

17

Mishra and Parker (1989)

infiltration–redistribution and tracer experiment

18

Gribb (1996)

cone penetrometer

19

Sˇimu˚nek and van Genuchten (1996)

disc infiltrometer

20

Yeh and Zhang (1996); Zhang and Yeh (1997); Hughson and Yeh (2000) Pan and Wu (1998)

infiltration experiment

permeability and airentry pressure distribution


Bandurraga and Bodvarsson (1999)

natural state, infiltration, percolation, water redistribution

permeability, parameters of characteristic curves permeability, parameters of characteristic curves

21 22†

Continued next page.

permeability, parameters of characteristic curves permeability, parameters of hysteretic characteristic curves, dispersivity permeability, air entry pressure permeability, air entry pressure, pore size distribution index

water content water content, pressure cumulative infiltration, prior information concentration, water content, pressure

flow rate, pressure cumulative infiltration, water content, pressure head pressure, water content, prior information capillary pressure saturation, water potential, prior information

homogeneous and layered systems, examined uniqueness synthetic data, hysteretic retention function with air entrapment examined different models, well-posedness synthetic data, joint inversion of hydraulic and tracer data synthetic data, examined nonuniqueness axisymmetric model

synthetic data, geostatistical inverse method comparison of optimization algorithms large-scale fracture–matrix system

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Table 1. Continued. Reference


23†

Experiment

Parameters Unsaturated flow, field experiments capillary strength

Data

seepage into tunnel

24

Finsterle (2000); Finsterle et al. (2003); Ghezzehei et al. (2004) Schmied et al. (2000)

25

Ritter et al. (2003)

natural state measurement

26†

Wang et al. (2003)


27†

Kowalsky et al. (2004)


28†

Finsterle and Persoff (1997)

gas pressure pulse decay experiment

29†

Ahlers et al. (1999)

30

Huang et al. (1999)

pneumatic pressures in deep boreholes air-injection tests

31†

Vesselinov et al. (2001a, 2001b) Unger et al. (2004)

cross-hole pneumatic tests

33†

Finsterle and Pruess (1995)

ventilation experiment in tunnel

34

Schultze et al. (1997)

multistep outflow experiment


water potential, gas pressure, cumulative evaporation, prior information cumulative outflow, pressure

35

Vasco and DattaGupta (1997)

production from oil reservoir

permeability

water cut data

36

Chen et al. (1999)

37

Wu et al. (1999)

multistep outflow experiment production from oil reservoir

parameters of characteristic curves permeability

38

Tran et al. (1999)

production from oil reservoir

permeability

39 40

Datta-Gupta and Yoon (2002) Vega et al. (2003)

partitioning interwell tracer test production from oil reservoir

permeability, NAPL saturation permeability

cumulative outflow, pressure water/oil ratio, pressure, prior geostatistical model water cut data, saturation from seismic imaging concentration

41†

Senger et al. (2003)

gas injection tests

42†

Finsterle et al. (1998a)

boiling experiment

43†

Finsterle et al. (2000)

geothermal completion test and production

44†

Engelhardt et al. (2003)

gas injection into heated column

45†

White et al. (2003)

46†

Bjo¨rnsson et al. (2003)

dewatering of open pit mine geothermal production data

32†

observations of natural state in agricultural field

radon in tunnel

† iTOUGH2 (Finsterle, 1999a, 1999b, 1999c) applications.

permeability, parameters of characteristic curves, plant water uptake, N turnover parameters permeability, parameters of characteristic curves permeability, capillary strength, pore size distribution index permeability distribution Gas flow and transport permeability, Klinkenberg parameter, porosity, leakage gas diffusivity

seepage rate

fractured rock, evaporation effects

pressure, NO3 concentrations

estimation of conditioned parameter distribution

water content

used global search algorithm

water content, prior information

evaluated different conceptual models

saturation, GPR, prior information

geostatistics, pilot point method

gas pressure

examined impact of systematic errors

gas pressure

natural pressure fluctuations 3-D pneumatic tomography 3-D pneumatic tomography estimation and optimization problem

permeability

pressure

air permeability, airfilled porosity permeability, porosity, radon concentration, boundary conditions Two-phase flow permeability, parameters of characteristic curves

pressure buildup

permeability, pore compressibility Nonisothermal two-phase flow two-phase flow parameters, thermal properties

Comment

radon concentration in ventilated tunnel

water cut data, travel time, prior geostatistical model pressure, saturation

temperature, pressure, steam saturation, heat flux

field experiment, twophase flow problem

comparison between unsaturated and twophase flow formulations streamline simulator, regularization, simulated annealing air–water, air–oil, and oil–water adjoint-state method sequential self calibration method streamline approach comparison of Baysian vs. deterministic approach calibration and test design

data from below and above boiling conditions, examined impact of heat loss synthetic and actual field data

permeability, porosity, fracture spacing, initial conditions permeability, gas entry pressure, thermal conductivity, heat capacity permeability

pressure, temperature, enthalpy temperature, cumulative outflow drawdown

parallel computing

permeability, porosity, enthalpy, and rate of upflow

generation rate, enthalpy

3-D model, ⬎100 parameters, parallel computing

nonisothermal two-phase flow laboratory experiment


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Fig. 1. Contours of the objective function in the three parameter planes (a) log(kabs)–log(pe), (b) log(kabs)–␭, and (c) ␭–log(pe) (from Finsterle and Faybishenko, 1999). The first row shows the objective function when only cumulative outflow data are used, the second row includes only pressure data, and the third row comprises both the cumulative outflow and pressure data. The planes intersect the parameter space at the global minimum; that is, they contain the best estimate parameter set.

sure 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. Problems of nonuniqueness and parameter identifiability were also analyzed by Russo et al. (1991), using infiltration data, and by Sˇimu˚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)

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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 More´ 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 (Sˇimu˚nek 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 (Fin-

sterle, 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.

PARAMETERIZATION Modeling is concerned with a simplified, abstracted, and parameterized representation of the natural system. While maintaining the salient features of the hydrogeologic system, some of its aspects and processes are lumped together and described by effective and/or averaged parameters. Model conceptualization and parameterization are therefore related, both trying to capture and reduce the complexity of the natural system. For example, complex molecular and pore-scale flow and transport processes are usually formulated based on the continuum approach, which results in effective parameters such as permeability and dispersivity. Moreover, heterogeneity in subsurface properties is approximated by average values assigned to each gridblock of the discretized model. Multiple gridblocks may be grouped together, a process referred to as “zonation.” Heterogeneity may also be described by geostatistical methods in which spatial variability is characterized by a relatively small number of geostatistical parameters (such as variogram parameters, conditioning values at pilot points, or attractor parameters). 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.


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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.

MODELING HETEROGENEITY In the standard zonation approach, constant properties are assigned to groups of elements of the discretized model. This representation of (usually large-scale) heterogeneity requires prior knowledge about the geometry of hydrogeologic units. If stratigraphic information is available, the number of parameters to be estimated can be sufficiently reduced, often yielding a well-posed inverse problem. However, if the subsurface structure is essentially random on the scale of interest, zonation may lead to a calibrated model that does not match the data well, or the estimated parameters are biased because of systematic errors in the model structure. 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

Fig. 2. Fractal permeability field, created from iterated function system (IFS)–generated attractor points (open circles).

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).

Geostatistical Approach and Pilot Point Method Geostatistical methods are widely used to characterize heterogeneity and to generate realizations of spatially variable property fields. A number of interpolation methods (e.g., kriging, Fig. 3a) and simulation techniques (e.g., sequential Gaussian simulation, Fig. 3b and 3c; and sequential indicator simulation) have been incorporated into iTOUGH2 on the basis of the modules provided by the Geostatistical Software Library GSLIB

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(RamaRao et al., 1995) or master points (Go´mez-Hernańdez et al., 1997). In this approach, property values at the pilot points are the parameters of calibration. This approach couples geostatistics and optimization. As shown in Fig. 3b and 3c, changing the permeability at one of the pilot points influences the permeability field in the vicinity of the point within approximately one correlation length. Distributing pilot points across the model domain allows iTOUGH2 to modify the heterogeneous field during an inversion, improving the match of the model output to the measured data, and at the same time acknowledging the geostatistical properties of the field as well as measured permeabilities. An application of the pilot point method using iTOUGH2 was presented by Kowalsky et al. (2004).

iTOUGH2 APPLICATIONS The following examples illustrate some iTOUGH2 applications, in which the model structure (specifically heterogeneity) is examined either through the estimation of effective parameters or through sensitivity and uncertainty propagation analyses.

Water Seepage into Underground Opening

Fig. 3. Spatially correlated permeability field created by (a) kriging, and (b,c) sequential Gaussian simulation. The fields are conditioned on permeability data along two vertical boreholes (black dots). Open circles indicate pilot points. Changing the permeability at the center pilot point affects the field in its immediate vicinity; compare panels b and c.

(Deutsch and Journel, 1992). Internal generation of the heterogeneous property field and its mapping onto the numerical grid make geostatistical parameters accessible to estimation by inverse modeling. This means that geostatistical parameters, such as the correlation length, can be adjusted automatically to directly match pressure and concentration data, rather than to adapt to an empirical variogram. The geostatistically generated property fields can be conditioned on given values at certain locations. While conditioning is usually invoked to honor measured values, it also can be used to adapt so-called pilot points

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). The development of a site-specific discrete fracture network model requires collecting a large amount of geometric and hydrologic data. While part of the geometric information can be obtained from fracture mappings, the description of the network remains incomplete and potentially biased toward fractures of a given orientation and size. Moreover, unsaturated hydrological parameters on the scale of individual fractures are required, along with conceptual models and simplifying assumptions regarding unsaturated flow within fractures


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Fig. 4. (a) Discrete fracture network model for TOUGH2; (b) flow paths inducing seepage into underground opening (after Liu et al., 2002).

and across fracture intersections. Because these databases are often not available and generally difficult or even impossible to obtain for site-specific simulations, such a model must be calibrated against hydrologic data, such as seepage rates collected during a liquid-release test in which water is injected from a point source above the opening. 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

Fig. 5. (a) Discrete-feature model for TOUGH2 and flow through discrete fracture network and seepage into underground opening (after Finsterle, 2000).

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 liquidrelease 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

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Fig. 6. Seepage prediction with calibrated fracture continuum model (solid line) including results from Monte Carlo simulations (dots) and comparison with synthetic seepage data provided by the discrete-feature model (dash-dotted line). Crosses indicate the simulations with the calibrated parameter set and the permeability realization used during the inversion (after Finsterle, 2000).

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 discretefeature model. Synthetically generated seepage data from a liquid-release test simulated with this discretefeature model were used to calibrate a simplified heterogeneous fracture continuum model. The calibrated continuum model was then used to predict seepage rates into a sufficiently large section of an underground opening using distributed, low fluxes representing natural percolation (i.e., conditions significantly different from those encountered during calibration). Monte Carlo simulations were performed to examine prediction uncertainty as a result of uncertainties in the input parameters. Moreover, a new realization of the underlying heterogeneous permeability field was generated for each realization, capturing the impact of variability that can only be characterized stochastically. 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. It should be noted that the two-dimensionality of the models shown in Fig. 4 and 5 amplifies the impact of

discrete flow behavior on seepage. In a two-dimensional discrete fracture network model, all fractures are assumed to be perfectly parallel to the tunnel axis, making it difficult for water to flow around the opening. Moreover, a single asperity contact precludes flow through the respective fracture. In contrast, in a three-dimensional network of randomly oriented fractures, flow diversion around the opening occurs primarily within the fracture planes, and asperity contacts can be bypassed. In-plane flow diversion occurring in multiple fractures of a three-dimensional network can be readily combined into an effective fracture continuum. Using a two-dimensional discrete-feature model to generate synthetic flow and seepage data for the examination of the continuum approach is thus a more rigorous test, and the conclusion that a continuum model is an appropriate representation thus also applies to a three-dimensional system. 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. A pulse of water was uniformly applied for 1 d at the top of the model for |X| ⬍ 3 m. As shown in Fig. 7b, heterogeneity in the fracture continuum leads to an intricate distribution of saturation changes, which is subsequently imprinted on the matrix despite its homogeneity. The saturation changes in the matrix give an indication of the strength of matrix imbibition, which is affected by the residence times of water in the adjacent


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dissipates quickly in the fracture continuum because of its high permeability. 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).

SUMMARY AND CONCLUSIONS

Fig. 7. (a) One realization of the heterogeneous permeability field for the fracture continuum; the matrix is homogeneous. (b) Saturation changes induced by release of a water pulse at the top of the model at |X| ⬍ 3 for 1 d. The model is symmetric about the line X ⫽ 0. While both the fracture and matrix continua occupy the entire model domain, the fracture continuum is shown on the left and the matrix on the right of the symmetry axis.

fractures and the matrix sorptivity properties. Once water enters the matrix, it remains essentially stagnant because the relatively strong capillarity prevents water from flowing back into the fractures after the pulse has passed. The saturation changes disperse gradually (driven by capillary pressure gradients) and slowly migrate downwards in accordance with the low matrix permeability. Consequently, these signals remain visible for a long time. On the other hand, the perturbation

Numerical codes solving combined simulation–optimization problems can be used to support a variety of scientific studies and engineering tasks, including test design, sensitivity analyses, parameter estimation, uncertainty propagation analyses, and testing of alternative conceptual models. A review of the scientific literature indicates that inverse modeling techniques have not been widely applied to unsaturated and multiphase flow problems, except for the analysis of relatively simple outflow experiments designed and optimized to determine unsaturated hydraulic properties. Potential reasons for the small number of formal inversion studies of multiphase flow systems include (i) lack of experimental data tailored for use in multiphase inverse modeling studies; (ii) difficulties in formulating and solving the forward problem; (iii) complexity of the associated, strongly nonlinear inverse problem; (iv) lack of efficient and stable multiphase inverse modeling codes; and (v) lack of computational resources. Many of these problems can be resolved by education and training, and through advances in scientific computing. Conducting suitable multiphase flow tests based on a comprehensive test design supported by modeling is essential to obtain data of sufficient sensitivity to determine the parameters of interest. 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 processrelevant 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 fea-

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tures 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. ACKNOWLEDGMENTS Many people have contributed to the development and testing of iTOUGH2, and new features have been proposed and added in response to specific application needs of the users. I would like to specifically thank Chris Doughty for introducing IFS and Mike Kowalsky for suggesting and testing the pilot point method. Discussions with Chris Doughty and HuiHai Liu and the reviews of Eileen Poeter, Wolfgang Durner, and Jan Hopmans are greatly appreciated. This work was supported by the USDOE under Contract no. DE-AC0376SF00098.

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