Comparative analysis of formulations for conservative transport in porous media through sensitivity-based parameter calibration


  • Valentina Ciriello,

    Corresponding author
    1. Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Università di Bologna, Bologna, Italy
    • Corresponding author: V. Ciriello, Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Università di Bologna, Bologna 40136, Italy. (

    Search for more papers by this author
  • Alberto Guadagnini,

    1. Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano, Milano, Italy
    2. Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona, USA
    Search for more papers by this author
  • Vittorio Di Federico,

    1. Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Università di Bologna, Bologna, Italy
    Search for more papers by this author
  • Yaniv Edery,

    1. Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot, Israel
    Search for more papers by this author
  • Brian Berkowitz

    1. Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot, Israel
    Search for more papers by this author


[1] We apply a general strategy based on Global Sensitivity Analysis (GSA) and model discrimination criteria to (a) calibrate the parameters embedded in competing models employed to interpret laboratory-scale tracer experiments, (b) rank these models, and (c) estimate the relative degree of likelihood of each model through a posterior probability weight. We consider a conservative transport experiment in a uniform porous medium. We apply GSA to three transport models, based on: the classical advection-dispersion equation (ADE), a dual-porosity (DP) formulation with mass transfer between mobile and immobile regions, and the Continuous Time Random Walk (CTRW) approach. GSA is performed through Polynomial Chaos Expansion of the governing equations, treating key model parameters as independent random variables. We show how this approach allows identification of (a) the relative importance of model-dependent parameters, and (b) the space-time locations, where the models are most sensitive to these parameters. GSA is then employed to assist parameter estimates within a Maximum Likelihood framework. Finally, formal model identification criteria are employed to (a) rank the alternative models, and (b) associate each model with a posterior probability weight for the specific case study. The GSA-based calibration of each model returns an acceptable approximation (remarkably accurate in the case of the CTRW model) of all available concentration data, with calibration being performed using minimum sets of observations corresponding to the most sensitive (space-time) locations.

1. Introduction

[2] Selection of an appropriate quantitative model and associated parameter calibration are key issues in the interpretation of transport experiments in natural and reconstructed porous media. The assessment of model sensitivity to parameter uncertainty, and comparison among different models on the basis of model selection criteria, are at the core of an appropriate methodology to address this problem [e.g., Barth and Hill, 2005a, 2005b]. Key sources of uncertainty associated with modeling of processes governing conservative transport in porous media at different scales of observations include hydraulic and transport parameters (e.g., hydraulic conductivity, porosity, and dispersivity) and boundary conditions (e.g., the concentration at the source location or the fluid flow rate). Sensitivity of model response to these parameters typically varies in space and time. An important step in a parameter estimation procedure is to identify locations in the system where the model is most sensitive to its parameters [e.g., Tiedeman et al., 2003, 2004]. This, in turn, constitutes the basis for model-based experiment design and interpretation [e.g., Fajraoui et al., 2011, and references therein].

[3] A useful approach for the design, analysis, and interpretation of conservative transport experiments in porous media is based on Global Sensitivity Analysis (GSA). GSA provides a convenient and powerful way to (a) analyze the influence of the uncertainty associated with model parameter values on the variability of model response and (b) identify space-time locations where a model is most sensitive to its unknown parameters. GSA is applied in several fields of engineering [Saltelli et al., 2000; Sudret, 2008] and has recently been employed for risk analysis of environmental contamination scenarios [Volkova et al., 2008; Oladyshkin et al., 2012; Ciriello et al., 2013]. It has also been used to illustrate how to improve the design of laboratory-scale experiments and parameter calibration based on the classical Advection-Dispersion Equation (ADE) model [Fajraoui et al., 2011]. Such analyses are typically based on the Sobol indices [Sobol, 1993; Archer et al., 1997]; these are variance-based sensitivity measures and do not require assumptions on linearity or monotonic behavior of the specific model. Their computation is generally performed within a Monte Carlo (MC) framework [Sobol, 2001], thus potentially leading to heavy computational cost for complex models and large parameter number [Sudret, 2008].

[4] Among different alternatives, the Sobol indices can be computed through the Polynomial Chaos Expansion (PCE) technique [Wiener, 1938]. This methodology was introduced in the engineering context through the stochastic spectral finite element analysis by Ghanem and Spanos [1991]. It is a useful tool for high-performance and accurate GSA analysis, relying on the definition of a surrogate model to calculate the sensitivity indices analytically via a simple postprocessing operation and requiring negligible computational time [Sudret, 2008; Ciriello and Di Federico, 2013; Ciriello et al., 2013 and references therein]. A PCE-based GSA has been presented in the literature for the ADE model in diverse settings [e.g., Volkova et al., 2008; Zhang et al., 2010; Shi et al., 2010; Fajraoui et al., 2011, 2012; Ciriello et al., 2013 and references therein].

[5] To the best of our knowledge, a detailed study of parameter sensitivity for the design and interpretation of laboratory scale, conservative transport experiments, based on a set of alternative process-based models, has not been undertaken. Most notably, quantification and comparative analysis of the sensitivity of widely used transport formulations such as (a) a dual-porosity (DP) model with mass transfer between mobile and immobile regions [Huang et al., 2003; Liu et al., 2010], and (b) the Continuous Time Random Walk (CTRW) [Berkowitz et al., 2006 and references therein] to their parameters has not been reported in the literature.

[6] Here we focus on the conservative, one-dimensional transport experiment presented by Gramling et al. [2002], performed within a laboratory chamber filled with a uniform, reconstructed porous medium. As candidate one-dimensional interpretive models, we select (a) the classical ADE, (b) a dual-porosity (DP) scheme with mass transfer between mobile and immobile regions, and (c) the Continuous Time Random Walk (CTRW) formulation. The space-time distribution of solute concentration along the chamber is considered as the system state against which we (a) perform a PCE-based GSA, (b) analyze the sensitivity of each model to its parameters, (c) explore the feasibility of estimating key model parameters based on a limited set of data, measured at locations determined by the results of the GSA, (d) apply model discrimination criteria to quantify (in a relative sense) the ability of these alternative models to interpret experimental observations, and estimate the relative degree of likelihood of the models through a posterior probability measure for the specific case study [Ye et al., 2008; Bianchi Janetti et al., 2012, and references therein], and (e) assess the predictive ability of the selected models.

[7] Our methodology allows quantification of (a) the relative importance of the parameters associated with each model, and (b) the space-time locations where the system state is most sensitive to model parameters. This information is relevant for model-based experiment design and robust parameter calibration at affordable computational cost. In our application, parameter calibration is performed within a Maximum Likelihood (ML) context [e.g., Carrera and Neuman, 1986].

2. Experimental Case Study and Transport Formulations

[8] Gramling et al. [2002] illustrate the results of a conservative transport experiment performed in a laboratory-scale, glass (rectangular) flow chamber, of length L, filled with millimeter-sized grains of cryolite. A solution of CuEDTA2− (at 0.01 M concentration) was injected continuously into the chamber, displacing a solution of Na2EDTA2– with initial concentration c0 = 0.02 M. The authors report the relative concentration profiles of CuEDTA2− at four different times (τ1 = 532 s, τ2 = 1023 s, τ3 = 1523 s, and τ4 = 2023 s), from which about 380 measurements of solute concentration can be derived. The main characteristics of the experiment are summarized in Table 1.

Table 1. Experimental Conditions of the Conservative Transport Experiment of Gramling et al. [2002]
  1. a

    Values calibrated by Gramling et al. [2002] on the basis of the measured concentration profiles.

Chamber length (cm)36
Chamber cross section (cm2)5.5 × 1.8 = 9.9
Average grain size (cm)0.13
Flow rate (mL/min)2.67
Velocity (cm/s)1.21 × 10−2
Hydrodynamic dispersion coefficienta (cm2/s)1.75 × 10−3
Chamber dispersivitya (cm)0.145
Grain Peclet number2.24 × 103
Reynolds number0.157
Observation times (s)532/1023/1523/2023

[9] In the following, we review the key equations of the three models we consider to quantify migration of a conservative solute in a uniform porous medium. Additional details regarding the selected transport models are reported in Appendix A (supplementary material). A one-dimensional transport scenario is considered, following the usual practice adopted in interpretation of flow-through laboratory chamber experiments such as that reported in Gramling et al. [2002].

[10] The classical ADE in Laplace space is

display math(1)
display math(2)

where inline image is solute concentration at location 0 ≤ x ≤ L and time t; v is average flow velocity, and D is hydrodynamic dispersion ( inline image, αL representing longitudinal dispersivity). Here and below, u represents the Laplace parameter, indicates Laplace transform, and primed quantities are derivatives with respect to spatial coordinate x.

[11] The DP scheme with mass transfer between mobile and immobile phases [Huang et al., 2003; Liu et al., 2010] is formulated as

display math(3)
display math(4)

where c* and c are, respectively, solute concentration in the immobile and mobile regions; inline image, f is the fraction of mobile pore space in the porous medium, K is the rate of mass transfer between mobile and immobile fluid flow regions, and D′ = αL q, q being Darcy flux.

[12] The CTRW formulation [Berkowitz et al., 2006] reads

display math(5)

where vψ and Dψ are, respectively, the transport velocity and generalized dispersion coefficient, and inline image represents the memory function which accounts for the unknown heterogeneities below the level of measurement resolution; t1 is a characteristic transition time, while ψ(t) represents the transition time distribution. A truncated power law (TPL) formulation of ψ(t) has been shown to describe transport for diverse sets of physical scenarios [Dentz et al., 2004; Berkowitz et al., 2006]. The TPL can be written as

display math(6)

[13] Here t2 >> t1 is the cut-off time of the TPL, n is a normalization factor, β is a parameter characterizing the nature of the dispersive transport, and Γ(a, s) is the incomplete gamma function [Cortis and Berkowitz, 2005; Berkowitz and Scher, 2009].

[14] The key flow and transport parameters of the different models (the parameters to be calibrated) are represented as independent random variables, to analyze how the uncertainty associated with their values propagates to solute concentrations. Mutual independence of parameters is assumed to exemplify our approach, given that no information regarding possible correlations is available. Table 2 reports the input random parameters and the corresponding probability distributions adopted in our study. For the ADE, effective velocity, v, and dispersivity, αL, are considered as the two parameters affected by uncertainty. Darcy velocity, q, the mobile porosity of the medium, f, the mass transfer rate, K, and the longitudinal dispersivity, αL, are identified as the four sources of uncertainty in the DP model. Five uncertain parameters are considered for the CTRW formulation, i.e., the transport velocity, vψ, the generalized dispersion coefficient, Dψ, the exponent, β, and the times, t1 and t2. We generally assume that uncertain parameter values are distributed normally, with the exception of parameters whose range of variability may entail negative values that have no physical meaning. A lognormal distribution is adopted for these latter parameters. Mean parameter values were selected on the basis of (a) calibration results obtained by Gramling et al. [2002] (with reference to dispersivity, effective velocities, and medium porosity), or (b) preliminary calibration against the complete data set. Values of parameter standard deviation were selected to ensure that relatively wide intervals in the parameter space were explored, while minimizing the possibility of sampling negative values in the case of normal distributions.

Table 2. Model Parameters and Adopted Sampling Distributions
ParameterModelDistributionMeanStandard Deviation
Effective velocity (v)ADENormal1.21 × 10−4 m/s1.00 × 10−6 m/s
Longitudinal dispersivity (αL)ADE/DPLognormal1.45 × 10−3 m4.50 × 10−4 m
Flux (q)DPNormal4.356 × 10−5 m/s5.11 × 10−7 m/s
Mobile porosity (f)DPNormal0.363.00 × 10−3
Mass transfer (K)DPNormal1.00 × 10−5 1/s3.00 × 10−6 1/s
Transport velocity (vψ)CTRWNormal1.21 × 10−4 m/s1.00 × 10−6 m/s
Generalized dispersion coefficient (Dψ)CTRWNormal1.75 × 10−7 m2/s5.44 × 10−8 m2/s
Exponent of TPL distribution (β)CTRWNormal1.979.85 × 10−2
Characteristic transition time (t1)CTRWLognormal6.6 s3.3 s
Cut-off time (t2)CTRWLognormal100 s50 s

[15] Parameter distributions were chosen on the basis of preliminary analyses, following the above criteria. This choice influences the PCE representation of the models, in terms of selection of the polynomial basis, as described below. We note that the GSA results are affected mainly by the variance associated with each parameter; the type of distribution has little effect (data not shown).

3. Methodology

[16] Here we present (a) the variance-based sensitivity indices and their analytical definition through the PCE technique (for more details, see Ciriello et al. [2013, and references therein]), (b) the main concepts of ML-based parameter calibration and model identification criteria, and (c) the overall modeling and interpretation strategy.

3.1. Variance-Based Global Sensitivity Analysis (GSA)

[17] Let inline image represent a stochastic model response associated with the vector X of M input parameters. The latter are modeled as mutually independent random variables. The model inline image is assumed to be a square integrable function. The space-time variability of the model response is omitted below, for brevity. Variances and sensitivity measures defined in the following are referred implicitly to a specific space-time location (x, t).

[18] The total variance, V(Y), associated with the model response can be resolved into the contributions of the M random variables through ANOVA decomposition [Archer et al., 1997; Saltelli et al., 2000]:

display math(7)

[19] Here, inline image is the partial variance characterizing the contribution of parameter Xi to the uncertainty of Y, inline image being the expectation of the model response when the value of Xi is set to inline image (the variance is calculated referring to all possible values inline image); the term inline image is the partial variance associated with the influence of the interaction between the uncertain parameters Xi and Xj on V(Y); similarly, Vijk represents the contribution of the interaction between parameters Xi, Xj, and Xk, to the total variance; the number of terms in (13) is 2M−1, the last one, corresponding to V12…M, represents the contribution of the interaction amongst all parameters to V(Y).

[20] The generic s-order Sobol index [Sobol, 1993] is defined as

display math(8)

[21] Here, inline image identifies the set inline image of random model parameters, with inline image and inline image. The sum of the indices defined in (8) is unity. The principal sensitivity index Si (s = 1) is a measure of the influence of the uncertainty associated with Xi, when considered individually (i.e., it is the reduction in the variance of Y if Xi is not uncertain).

[22] The total sensitivity index, inline image, is a measure of the overall contribution of the variability of Xi, including its interactions with the other model parameters, to the total system variance (i.e., it is related to the expected residual variance of Y if only Xi is uncertain) and is defined as

display math(9)

where X−i is the vector collecting all the parameters inline image [Sobol, 1993; Saltelli et al., 2000]. By definition, the sum of total sensitivity indices is larger than unity. The larger this sum, the more appreciable the variance due to interactions among parameters.

[23] The Sobol indices must be computed for each location (x, t) of interest; this is traditionally achieved through Monte Carlo simulation [Sobol, 2001]. This involves multiple integrations to compute the variances in (7). For this reason, the computational cost of a complete GSA could represent a practical limitation, especially in the presence of a complex model and/or a large number of parameters [Sudret, 2008].

3.2. Polynomial Chaos Expansion (PCE)

[24] The PCE technique is based on approximating the system response Y through a finite series of orthogonal polynomials, which are chosen on the basis of the distributions associated with the model parameters collected in X [Wiener, 1938]. While the Hermite basis is suitable for Gaussian processes, and also for certain types of non-Gaussian inputs (e.g., lognormal distributions), different types of orthogonal polynomials are required for optimum convergence rate in the case of other distributions [Xiu and Karniadakis, 2002].

[25] The PCE-surrogate model inline image converges to Y in the L2-sense, according to Cameron and Martin [1947], if Y belongs to the Hilbert space of second-order random variables. In general, inline image does not depend directly on X but on the vector inline image. The latter includes M independent random variables whose distribution is linked to the selected polynomial basis by [Ghanem and Spanos, 1991; Xiu and Karniadakis, 2002; and Soize and Ghanem, 2004]:

display math(10)

[26] Here Ψj denotes a multivariate orthogonal polynomial, aj represents a coefficient of the expansion, and inline image, p being the maximum degree of the expansion. The random variables collected in Ω are related to the corresponding random variables collected in X (i.e., the original model parameters), through an isoprobabilistic transform of the kind inline image [Sudret, 2008].

[27] Calculation of the coefficients aj is performed by a nonintrusive, regression-based approach. The selection of the optimum set of regression points is based on the methodology embedded in the Gauss quadrature technique, i.e., combinations of the roots of the next higher order polynomial (p + 1) are used as the points at which the approximation is solved [Webster et al., 1996; Sudret, 2008]. Specifically, the regression is performed upon minimization of the variance of a residual, defined as the difference between the solution given by the original model and the surrogate model response, i.e., inline image, with respect to the vector of the unknown coefficients a: inline image. This condition can be rewritten in matrix format as: inline image, inline image, inline image, where N is the number of regression points, Y′ is the vector denoting the model response at these points, while the product inline image defines the so-called information matrix [Sudret, 2008 and references therein].

[28] Correlation among random input model parameters can be accommodated in the methodology by applying the Nataf transform [Nataf, 1962]. In this case, knowledge of the marginal probability density functions of the parameters and the associated correlation matrix is required.

[29] Once the Polynomial Chaos representation, inline image, of the model Y is available, the Sobol indices can then be determined analytically. The mean of the approximated response inline image corresponds to the first coefficient of the expansion, a0, while the associated total variance is:

display math(11)

[30] The general expression for a given Sobol index calculated through the PCE is [Sudret, 2008]:

display math(12)

[31] ϕ denoting a generic term that depends only on the variables specified by the subscript inline image. Calculation of inline image can be performed following, e.g., Abramowitz and Stegun [1970].

3.3. Maximum Likelihood (ML) Parameter Estimation and Model Quality Criteria

[32] Let N be the number of available observations of the model response Y collected in the vector inline image. The covariance matrix of measurement errors, BY, is here considered to be diagonal with nonzero terms equal to the observation error variance inline image [Carrera and Neuman, 1986]. Denoting by inline image, the vector of model predictions at locations where measurements are available, the ML estimate inline image of the vector of the M uncertain model parameters can be obtained by minimizing with respect to X the negative log likelihood criterion:

display math(13)

where inline image. The criterion (13) includes the weighted least square criterion [Carrera and Neuman, 1986; Bianchi Janetti et al., 2012, and references therein]. Here minimization of (13) is achieved using the iterative Levenberg-Marquardt algorithm as embedded in the code PEST (model-independent Parameter ESTimation and uncertainty analysis) [Doherty, 2002].

[33] Alternative (competing) models which can be used to interpret available system states can be ranked by various criteria [e.g., Neuman, 2003; Ye et al., 2004, 2008; Neuman et al., 2012; Bianchi Janetti et al., 2012, and references therein], including:

display math(14)
display math(15)
display math(16)

where the Akaike information criterion, AIC, is due to Akaike [1974], AICc to Hurvich and Tsai [1989] and KIC to Kashyap [1982]. Note that the lowest value of a given model identification criterion indicates the most favored model (according to the criterion itself) at the expense of the remaining models. Our definition of KIC (16) is consistent with (14) of Ye et al. [2008] when no prior information is considered to be available and Q represents the Cramer-Rao lower-bound approximation for the covariance matrix of the parameter estimates, i.e., the inverse expected Fisher information matrix [see Ye et al., 2008 for details]. Such a covariance matrix provides a measure of the quality of parameter estimates and of the information content carried by data about model parameters. Note that KIC favors models with relatively small expected information content per observation, when comparing models having equal numbers of parameters, equal values of the minimum of NLL, and prior probability of parameters evaluated at the optimum [Hernandez et al., 2006; Ye et al., 2008; Riva et al., 2011].

[34] These model discrimination criteria can also be employed to assign posterior probability weights to the various tested models, thus quantifying prediction uncertainty. The posterior probability related to model Mk (k = 1,…, NM, with NM the number of available process models) is calculated as [Ye et al., 2008]:

display math(17)

[35] Here ΔICk = ICk − ICmin, with ICk being either AIC (14), AICc (15), or KIC (16) and ICmin = min{ICk} its minimum value over the competing models considered; p(Mk) is the prior probability associated with each model. If no prior information is available, then p(Mk) = 1/NM, so that all models are associated with equal prior probability.

[36] The adoption of model identification criteria and posterior model probabilities allows ranking of the candidate models analyzed on the basis of their associated posterior probabilities and discrimination among them in a relative sense. To the best of our knowledge, such a study has not yet been conducted with reference to the interpretation of laboratory-based transport experiments in conjunction with a sensitivity-driven calibration of model parameters.

3.4. Sensitivity-Based Modeling Strategy

[37] Our modeling and interpretation strategy is designed to be adopted in different contexts. To this end, the key operational steps below provide a general, flexible guideline.

[38] 1. Identify the competitive models with respect to the physical processes involved. Here, we consider three conservative transport models, i.e., the ADE, DP, and CTRW formulations (see section 'Experimental Case Study and Transport Formulations').

[39] 2. Identify and model sources of parametric uncertainty and of the probability distributions for M model parameters, whose values are uncertain and need to be calibrated against measurements of state variables. Here, we consider M = 2, 4, and 5, respectively, for the ADE, DP, and CTRW models. Table 2 reports the uncertain parameters considered for the three models, together with the adopted probability distributions.

[40] 3. Extract the space-time locations (x, t) at which experimental observations are available. With reference to the experiment of Gramling et al. [2002], one-dimensional concentration profiles are available at four observation times, for a total of 380 data points.

[41] The next steps are followed for each model.

[42] 4. Compute the PCE approximation through (10), at each location (x, t) identified at step 3. In the transport application, we employ a Hermite Chaos expansion. For each model, we perform the analysis by employing a PCE at different orders (p = 2, 3, 4).

[43] 5. Derive (analytically), via PCE, the Sobol indices for the selected model at each (x, t) through (12). In our application, we adopt the total sensitivity index defined in (9) as a reference sensitivity measure for the definition of the influence of model parameters. As explained in section 'Variance-Based Global Sensitivity Analysis (GSA)', when the sum of the total sensitivity indices of the model parameters is significantly larger than unity, the effect of interactions among parameters might not be negligible. If this occurrence is noted throughout the system, then the effect of such interactions needs to be computed through (12) with s > 1.

[44] 6. Analyze the results provided by GSA. Two kinds of information are identified for each model parameter: (a) its overall influence on the model response, calculated through the arithmetic average of the values of the associated total sensitivity index at the locations of interest (step 3); and (b) the (x, t) locations at which local maxima of such total sensitivity index occur. Observations available at these locations are included in the sensitivity-based subsets employed for parameter calibration.

[45] 7. Identify the sensitivity-based calibration data sets. These data sets include observations at the locations identified at step 6. A collection of both unimportant and important observations should be excluded from each data set and remain usable for subsequent model validation. In our application, this leads to identifying subsets of the concentration measurements of Gramling et al. [2002, their Figure 4], corresponding to the (x, t) locations, at which the three transport formulations are most sensitive to the uncertainty in their parameters. Note that the sensitivity-based data sets generally include different observations for each model.

[46] 8. Compute ML estimates (13) of model parameters. Here, we consider the effect of calibrating the model on (a) sensitivity-based data sets identified at step 7, or (b) fixed subsets which include the same collection of observations for all models. More specifically, these latter subsets are formed by all concentrations corresponding to the first, second, and third observation times presented in Gramling et al. [2002, their Figure 4].

[47] Steps 4 to 8 are repeated for each of the models. The steps following model calibration are illustrated below.

[48] 9. Compute model quality criteria (14)-(16) and posterior probability weights (17) to rank the interpretive capability of the models for each calibration set. This step provides information about the ability of each model to approximate the observations associated with the highest information content, for purposes of calibration. We compare the interpretative capability of the models when only important (sensitivity-based) observation, or a combination of both important and seemingly unimportant observations, are considered in the calibration. As we will show in section 'Results and Discussion', observations identified as unimportant for calibration purposes in the context of GSA may be relevant to model selection criteria.

[49] 10. Assess the predictive capability of each model, and for each calibration subset, by comparison of model results against either observations collected at space-time locations which are not employed in the parameter calibration procedure, or the entire available data set. This step is typically termed “model validation” and is key to the characterization of model interpretive capabilities. While model quality criteria provide information about the best model with respect to the observations employed for parameter calibration, this step enables testing of the predictive performance of each model with respect to the entire available data set. If steps 9 and 10 provide contrasting results, when model calibration is based on the sensitivity-based subsets identified at step 7, this might imply that while a model has the best capability to approximate its most sensitive observations, such observations are less important for the interpretation of the overall physical process on the basis of the model structure, i.e., the intrinsic interpretive capability of the model is not adequate against the specific case study.

4. Results and Discussion

[50] Here we present the results of application of our sensitivity-based strategy to the specific case study described in section 'Experimental Case Study and Transport Formulations'. We also report implications of our findings to model-based experimental design.

4.1. Global Sensitivity Analysis (GSA) of the Selected Transport Models

[51] Table 2 reports the parameters that we consider as uncertain for the three selected models, together with the adopted probability distributions. For each model, the analysis was performed by employing a PCE at different orders (p = 2, 3, 4). For illustration purposes, we report here the results obtained for a PCE of order 2. These do not differ significantly from those obtained with higher order PCE (data not shown). Note that a minimum number of model runs equal to 6, 15, and 21, respectively, for the ADE, DP, and CTRW formulations, is required to identify the PCE at each (x, t) through the regression-based approach described in section 'Polynomial Chaos Expansion (PCE)'. The quantitative results illustrated in this section are tied to the specific experimental case study. While different experimental settings (in terms of, e.g., flow domain and configuration and transport scenario) might lead to different results, application of the GSA-based methodology we present is general and allows discrimination of the relative effects of the different model parameters.

[52] Figure 1 juxtaposes the concentration profiles, c(x), of the conservative experiment reported by Gramling et al. [2002] and the related total sensitivity indices of the parameters associated with the ADE, for given observation times. Curves in Figure 1 represent the spatial distribution of the total sensitivity indices (9) associated with the ADE parameters and calculated for each observation time on the basis of the PCE technique described in section 'Polynomial Chaos Expansion (PCE)'. These curves allow identification of the locations in the chamber where the ADE is highly or poorly sensitive to its parameters, depending on the local values of the total sensitivity indices. Corresponding depictions for the DP and CTRW formulations are presented in Figures 2 and 3, respectively.

Figure 1.

Space-time concentration profiles from Gramling et al. [2002, Figure 4] and total sensitivity indices (ST(Ω), Ω = v, αL) associated with the parameters of the ADE model.

Figure 2.

Space-time concentration profiles from Gramling et al. [2002, Figure 4] and total sensitivity indices (ST(Ω), Ω = q, αL, f, K) associated with the parameters of the DP model.

Figure 3.

Space-time concentration profiles from Gramling et al. [2002, Figure 4] and total sensitivity indices (ST(Ω), Ω = vψ, Dψ, β, t1, t2) associated with the parameters of the CTRW model.

Figure 4.

Concentration profile at time τ1 from Gramling et al. [2002] and sensitivity-based observation subset 4 (Table 4) selected for the ADE, DP, and CTRW models.

[53] Figure 1 shows that the sensitivity indices of the two ADE parameters (i.e., v and αL) are in general anticorrelated. Locations where the effects of a parameter are dominant are clearly identifiable by a sharp local peak or by persistently high values of the total sensitivity index. In general, the uncertainty associated with the velocity dominates the transport process in the proximity of the inflection point of the concentration profiles. This location corresponds to solute center of mass and is associated with virtually vanishing sensitivity to dispersivity. In contrast, dispersivity appears to play a dominant role close to the tails of the concentration profile. This result is consistent with the format of the ADE, where dispersivity is linked to the spreading of the concentration distribution around the center of mass, while the displacement of the center of mass is typically governed by advective processes.

[54] For the DP formulation, Figure 2 reveals that the role of the mass transfer coefficient, K, appears to be less significant at the advancing solute front than at locations in the upstream part of the concentration profiles. This observation is consistent with the main effect of this parameter, which is associated with delayed diffusion of solute from immobile to mobile regions. The total sensitivities to Darcy flux and porosity are very similar for the four observation times, the system state appearing to be slightly more sensitive to the former than to the latter. Dispersivity plays a relevant role in the DP formulation, being the most important parameter for earlier times. The effect of dispersivity on the variance of the system response tends to decrease with time, consistent with the increased impact of advective processes with solute residence time.

[55] Figure 3 clearly shows that the parameter β appearing in the CTRW formulation always plays the most prominent role, the importance of Dψ being significant only for the observations available at earliest times. The transport velocity vψ and the two characteristic times t1 and t2 display a similar behavior, appearing to be only marginally relevant in this case study and for the selected variability interval. This behavior is likely due to the relative uniformity of the reconstructed porous medium, wherein the entire spectrum of transition times can be sampled by solute particles migrating through the system. These findings imply that β and Dψ are the only relevant parameters for model calibration in the experimental setting considered here, because they encapsulate the key information on variability of the system response.

[56] Comparison of Figures 1 and 2 indicates that the main features of the spatial distribution of the parameter sensitivities (i.e., dispersivity and velocity/flux) that appear within both the ADE and DP formulations are qualitatively similar. These formulations are sensitive to all parameters, albeit with various degrees and at different locations. On the other hand, it is noted that the CTRW model is essentially sensitive to only two parameters.

[57] To complete the analysis, Table 3 reports the mean values of the total sensitivity indices associated with the uncertain parameters (see step 6 of section 'Sensitivity-Based Modeling Strategy'), as calculated on the basis of the complete available data set (i.e., considering the four concentration profiles presented in Gramling et al. [2002]). This allows a ranking of the global importance of each model parameter and provides valuable information for the parameter calibration step.

Table 3. Mean Values of the Total Sensitivity Indices Calculated on the Complete Set of Available Concentration Data
ModelParameterMean Values of ST
ADELongitudinal dispersivity (αL)0.581
ADEEffective velocity (v)0.435
DPLongitudinal dispersivity (αL)0.356
DPFlux (q)0.346
DPMobile porosity (f)0.178
DPMass transfer (K)0.147
CTRWExponent of TPL distribution (β)0.905
CTRWGeneralized dispersion coefficient (Dψ)0.174
CTRWCharacteristic transition time (t1)0.012
CTRWCut-off time (t2)0.012
CTRWTransport velocity (vψ)0.010

[58] One can observe that, for each model, the sum of the mean values of the total sensitivity indices (9) associated with the parameters is generally larger than unity. This is due to contributions of parameter interactions to the variance of the model output (sections 'Variance-Based Global Sensitivity Analysis (GSA)' and 'Sensitivity-Based Modeling Strategy'). Table 3 suggests that this contribution is globally negligible for the three models, as the sum of the mean values of the total sensitivity indices associated with the parameters of a given model is close to unity. Also, Figures 1 and 2 indicate that the principal and total sensitivity indices, calculated respectively by (8) and (9), virtually coincide at almost all space-time locations for the ADE and DP models. On the other hand, Figure 3 shows the occurrence of a nonnegligible mutual influence between β and Dψ. The effect of interaction between these two parameters is confined within a small region close to the domain boundaries, and at early observation times where terms of order higher than one appearing in (8) have some influence on the variability of the model output. Because this effect is modest and restricted to very limited areas, second-order Sobol indices are not computed.

[59] As illustrated in section 'Polynomial Chaos Expansion (PCE)', the PCE technique provides a surrogate model, formulated in terms of the model parameters. The quality of the approach and results presented here were assessed by comparing concentration profiles obtained by a given model and the corresponding PCE approximation, for several sets of parameter values sampled randomly within the ranges of variability indicated in Table 2. It was found that the concentration profiles calculated with the complete model and its PCE approximation were essentially identical in all cases (data not shown).

4.2. Parameter Calibration and Model Identification Criteria

[60] Calibration of the parameters of the three models against available concentration data is performed on the basis of the results and observations presented in section 'Global Sensitivity Analysis (GSA) of the Selected Transport Models', which are employed to identify the sensitivity-based calibration subsets. No prior information is available due to the lack of replicates of experimental results, from which one could provide an estimate of the observation error variance [see, e.g., Bianchi Janetti et al., 2012]. Consequently, calibration is performed by assigning equal weights to the observations.

[61] We consider different subsets of the available database upon which model calibration is performed, to assess the influence of measurement (space-time) location on the quality of parameter calibration results and application of model discrimination criteria analysis. Table 4 lists the different data subsets, including the number of data points associated with each. As an example, Figure 4 depicts the location of the measurement points selected for subset 4 in Table 4. We consider separately the first three available concentration profiles in their entirety (sets 1, 2, and 3 in Table 4) to investigate time dependence of the parameters. We then apply the GSA methodology described in section 'Methodology' and identify sets of observation points that are most sensitive to the parameters. We do so by selecting such sets within different concentration profiles (sets 4 and 5 in Table 4) and considering different sample sizes (sets 5 and 6 in Table 4). This procedure enables (a) investigation of the possibility of optimizing the use of information content associated with observations for calibration purposes, and (b) validation of model predictions (step 10 in section 'Sensitivity-Based Modeling Strategy') using both sensitive observations and those associated with lack of sensitivity [see, e.g., Foglia et al., 2013]. Here, model validation is performed considering the entire data set of Gramling et al. [2002]. Another possibility would be to include in the validation step only the observations omitted from the regression.

Table 4. Calibration Sets for the Three Selected Models
SubsetDescriptionNumber of Observations
1All observations from concentration profile at τ1110
2All observations from concentration profile at τ273
3All observations from concentration profile at τ392
4Most sensitive observations from concentration profile at τ120
5Most sensitive observations from concentration profiles at τ2 and τ320
6Most sensitive observations from concentration profiles at τ2 and τ340

[62] Table 5 reports the values of the model parameters obtained upon performing calibration on the basis of the different data subsets presented in Table 4. For each estimated parameter, Table 5 also reports the ratio, R, of the difference between the upper, U, and lower, L, limit identifying the 95% estimate confidence limits and the estimated value, C, i.e., inline image. Confidence intervals are calculated on the basis of the same linearity assumption adopted to derive the equations for parameter optimization implemented in PEST [Doherty, 2002]. As expected, this ratio is smallest for the model with the smallest number of parameters. In particular, it is noted that the quality of the estimate of parameters t1 and t2 of the CTRW model appears to be relatively poor, consistent with the observation that the model is not sensitive to these two parameters as revealed by the GSA, i.e., these parameters do not affect significantly the performance of the formulation against the observation data set (Figure 3).

Table 5. Calibrated Values, C, of Model Parameters and Ratio, R, of the Difference Between the Lower and Upper Limits Identifying the 95% Estimate Confidence Limits and C
ModelParameterSubset 1Subset 2Subset 3Subset 4Subset 5Subset 6

[63] From examination of Table 5, we observe that the (calibrated) value of the velocity, v, in the ADE model does not depend on the particular choice of subset. A similar observation can be made with reference to the flux, q, in the DP model, which exhibits variations of only a few percent among different subsets. On the contrary, calibration of dispersivity, αL, for the ADE and DP models appears to be impacted by the choice of the data subset (subsets 1, 2, and 3). Calibrated porosity, f, in the DP model virtually coincides with the average value of its distribution in all calibration subsets. The mass transfer coefficient, K, exhibits a calibrated value associated with subsets relative to early observation times (subsets 1 and 4), which is significantly higher than that resulting from calibrating the model against data taken at later times.

[64] In the CTRW model, the cut-off time, t2, is remarkably stable for all subsets considered, regardless of the lack of model sensitivity to this parameter. A similar observation can be made for the characteristic transition time, t1, except for the scenario corresponding to adoption of the first concentration profile as a calibration subset (subset 1). The generalized dispersion coefficient, Dψ, is associated with calibrated values which virtually coincide with the average of the selected distribution employed for GSA, with the exception only of the early observation times (subsets 1 and 4). The calibration values of the exponent of the TPL distribution, β, and transport velocity, vψ, show opposing trends over time. We note that large calibrated values of vψ are found for early observation times, while the reverse is true for β (subsets 1, 2, and 3).

[65] From Table 5, we can observe that the confidence intervals related to the estimates of v for the ADE model tend to overlap for all data subsets with the exception of the case associated with early observation times (subsets 1 and 4) where the calibrated velocity is associated with relatively large confidence intervals. Along the same lines, dispersivity values calibrated upon considering the sensitivity-based subsets (subsets 4, 5, and 6) and subsets 2 and 3 are statistically indistinguishable. This indicates that selection of a smaller set of data points does not affect notably the values of the parameters estimated for the ADE. Confidence intervals associated with the DP model appear to be relatively small for the first two data subsets, where a large amount of data is adopted and tailing behavior associated with delayed diffusion is visible in the experimental concentration curves. All subsets selected render statistically equivalent results for the calibration of q, f, and K. Dispersivity calibration results observed for the ADE also hold for the DP model.

[66] For the CTRW model, it is noted that all confidence intervals associated with the estimated parameters tend to overlap significantly. The parameter β, which is also the most influential to the system behavior, is best estimated for this experimental setting. As the cut-off times t1 and t2 are not influential in this case study, we also performed model calibration by setting t1 and t2 as the mean values of their selected distributions and estimating the remaining three parameters. In this case, the estimated values of vψ, Dψ, and β virtually coincide with those listed in Table 5. As expected, the width of the resulting confidence intervals decreases significantly, with values of R which are generally one order of magnitude lower than those presented in Table 5 (data not shown). Note that parameters which appear as unimportant for calibration purposes may be relevant in the context of assessing prediction uncertainty [Tiedeman and Hill, 2007].

[67] Comparison among the competing models for each calibration set is then possible on the basis of the model identification criteria (14)-(16) and posterior probabilities (17). Table 6 presents the value of NLL (13) together with model identification criteria results (i.e., AIC (14), AICc (15), KIC (16)) for the selected transport models and each calibration subset. The posterior probability calculated on the basis of the AIC (14) criterion is also included for completeness. Evaluating the posterior probability according to the other discrimination criteria does not produce significantly different results. This is observed because these criteria are dominated by the NLL values in our example, and no prior information is available which could influence the KIC values.

Table 6. Results From Model Calibration and Identification Criteriaa
SubsetNLL (13)AIC (14)AICc (15)KIC (16)Posterior probability (17)
  1. a

    For each model and data subset, the smallest values of NLL (13), AIC (14), AICc (15), KIC (16) and the largest of posterior probability (17) are emphasized.


[68] We first note that the posterior model weights indicate that one model always has a markedly high degree of likelihood at the expense of the remaining two, depending on the set of observations considered. For example, the second and the third concentration profiles (subsets 2 and 3), respectively, clearly render the DP and CTRW as the best interpretive models. In contrast, extracting only the most sensitive observations from these two profiles (subsets 5 and 6) results in the ADE being clearly preferable. The DP emerges as the best modeling choice for the early-time observations (subsets 1 and 4).

[69] It is interesting to observe that the identification criteria AIC (14) and AICc (15) render almost identical values, and very close to NLL (13), for all of the scenarios tested. This implies that the contribution of NLL dominates over the influence of the number of parameters associated with the selected models (M = 2, 4, and 5 for ADE, DP, and CTRW, respectively) in the calculation of these model selection criteria. The identification criterion KIC (16) is generally in line with the results of the remaining criteria, with the exception of subsets 2 and 4, for which ADE and CTRW are, respectively, favored over DP. Note that KIC values differ from NLL (13), as KIC also contains the expected information content through the parameter covariance matrix.

[70] The predictive capability of the selected models is then explored by comparison of calibrated model outputs against concentrations values and profiles which were not employed during the calibration step. For the purpose of illustration, the calibration values of parameters resulting from subsets 2 and 5 are considered below.

[71] Figure 5 (first row) compares the four measured concentration profiles and the ADE modeling results when the parameters are calibrated on the basis of the most sensitive observations taken at the second and third concentration profiles (i.e., subset 5). The insert in each figure is a scatterplot of the model results versus measurements. Figure 5 (second row) presents corresponding results when parameter calibration is performed on the basis of the complete set of observations available for the second observation time (subset 2). Figures 6 and 7 illustrate corresponding results for the DP and CTRW models, respectively. The picture is complemented by Table 7, which reports the mean square error (MSE) between data and model predictions calculated for each observation time and model, for calibration subsets 2 and 5.

Figure 5.

Comparison among the four concentration profiles of Gramling et al. [2002] and modeling results obtained through the ADE model with parameters calibrated on the basis of (top) the most sensitive observations taken at τ2 and τ3, i.e., subset 5 in Table 4, or (bottom) the complete set of observations available for τ2, i.e., subset 2 in Table 4.

Figure 6.

Comparison among the four concentration profiles of Gramling et al. [2002] and modeling results obtained through the DP model with parameters calibrated on the basis of (top) the most sensitive observations taken at τ2 and τ3, i.e., subset 5 in Table 4, or (bottom) the complete set of observations available for τ2, i.e., subset 2 in Table 4.

Figure 7.

Comparison among the four concentration profiles of Gramling et al. [2002] and modeling results obtained through the CTRW model with parameters calibrated on the basis of (top) the most sensitive observations taken at τ2 and τ3, i.e., subset 5 in Table 4, or (bottom) the complete set of observations available for τ2, i.e., subset 2 in Table 4.

Table 7. Results of Model Validation in Terms of Mean Square Error (MSE) for Each of the Four Concentration Profiles (Corresponding to Observation Times τi, i = 1, 2, 3, 4) and Observation Subsets 2 and 5 (Table 4)a
  1. a

    The smallest values of MSE for each model are emphasized.


[72] Figures 5-7 (first rows) reveal that the sensitivity-based calibration of each model returns an acceptable approximation (and particularly accurate in the case of the CTRW model) of all four profiles, even though only 20 observations are used out of a total of 380 data points available. It is remarkable to note that the best predictive power, assessed through Figures 5-7 and Table 7, is associated with the CTRW model, even though the posterior model probability weight associated with the ADE is clearly dominant in this case (Table 6).

[73] The analysis performed on the basis of calibration subset 2 (second rows in Figures 5-7) shows that the DP model is the best alternative for fitting the observations (Table 6), but is not equally able to predict the remaining concentration profiles, especially at late time. This appears to be linked to the observed tendency of the mass transfer coefficient to be associated with larger values when its estimate is based on early time observations. The CTRW model, which includes the ADE and the DP model as particular cases, appears to return the best prediction capability also in this case. This is supported by inspection of Table 7, which indicates that the CTRW formulation has the lowest MSE for all observation times and for both subsets 2 and 5. On the other hand, MSE values for the ADE and DP are virtually coincident. The difference between the CTRW-based results and those based on ADE or DP formulations tends to increase when longer observation times are considered. MSE is lower for subset 2 than for subset 5 for all models and observation times.

4.3. Implications for Experiment Design

[74] The sensitivity-based methodology presented here has direct implications for analysis of the interpretive capability of models, for a given case study. GSA allows identification of (a) the parameters that may play an important role in model interpretation, thus answering the question of which parameters can be estimated; (b) convenient space-time locations where measurements should be collected, for use during the model calibration step, thus indicating where one should concentrate measuring efforts, and how much data should be collected; and (c) reduced sets of observations with relevant information content for parameter calibration (observations that are deemed as unimportant during the calibration phase may be considered in the context of the application of model selection criteria).

[75] The first key point above has been shown to be relevant, in this case study, for, e.g., the CTRW model, because GSA reveals a markedly different degree of influence of the (uncertain) parameters on the model output. Note that the parameters which are less relevant for this model are associated with the poorest calibration results, in terms of relative width of confidence intervals (see Table 5); this suggests the possibility of excluding these parameters from the analysis of the model interpretive capability. As we note in section 'Parameter Calibration and Model Identification Criteria', excluding t1 and t2 from model calibration results in significant narrowing of the confidence intervals associated with the remaining parameters. This supports the relevant role of the GSA-based approach in the parameter identification process.

[76] The second and the third key points are particularly relevant in light of the need to optimize the number of measurements. This becomes particularly relevant when the analysis is performed in a multimodel context, as done here. When a set of multiple models is employed, it becomes relevant to explore the possibility of optimizing the set of measurement points to properly calibrate the parameters associated with each model, given that each model can display large sensitivity to parameters within different space-time intervals. This kind of analysis is exemplified in Figure 4, where one can observe that regions with high sensitivity to parameters overlap for the different models, thus guiding optimization of the experimental effort, in terms of the number and space-time location of measurements to be collected.

[77] A key point for sensitivity-based experiment design is to assess the ability to measure the state variable of interest, at locations suggested by the GSA. In this sense, the analysis of observability represents the first step in experimental design. Assessment of the limitations of an experimental setup can be performed jointly with the GSA, identifying the possibility of collecting reliable observations under the condition suggested by sensitivity criteria.

5. Conclusions

[78] Application of a complete methodology for sensitivity-based parameter calibration applied to transport models in porous media has been illustrated. The proposed sensitivity-based modeling strategy may be employed in different contexts involving parameter calibration and model assessment, and the methodology we describe includes general guidelines for application of our approach. We demonstrated the potential of the methodology for model-driven experimental design through application to a conservative transport experiment. The methodology is articulated according to the following steps: (a) selection of one or more competing interpretive models for the transport problem considered; (b) identification of space-time locations which are most influenced by the uncertainty in model input parameters, via a complete GSA performed through the PCE method; (c) calibration of model parameters within a Maximum Likelihood context, considering subsets of measurements associated with the space-time locations which are most sensitive to model parameters; (d) ranking of selected models by means of model quality criteria and estimating the relative degree of likelihood of each model by means of a weight, or posterior probability; and (e) model validation against the available observations which are not employed in the calibration step or against the entire data set.

[79] We have shown how this GSA-based approach allows identification of (a) the relative importance of model-dependent parameters, and (b) the observations carrying the largest information content for parameter calibration and model identification purposes. Our investigation on the interpretive capability of three conservative transport models (i.e., ADE, DP, and CTRW) through the methodological framework and specific case study, leads to the following key results and conclusions:

[80] 1. Results from the ADE model are most sensitive to velocity at locations close to the solute center of mass, while sensitivity to dispersivity is largest close to the tails of the concentration distribution. The role of the mass transfer coefficient in the DP model is less significant at the advancing solute front than at the upstream tail of the concentration profiles. Dispersivity is the most important parameter in the DP model for earlier times, its effect decreasing with time. While the ADE and DP models are sensitive to all parameters, albeit with various degrees and at different locations, the CTRW model is sensitive chiefly to β, characterizing the nature of the dispersive transport; the role of Dψ is of some importance only for observations at earliest times.

[81] 2. Posterior model weights indicate that one model always has a markedly high degree of likelihood, at the expense of the other two models, depending on the set of observations. Model ranking is highly dependent on the selected subset of observations. The DP model renders the best approximation for the early-time observation subsets, while the ADE is preferable when the GSA-based observation sets are considered. The CTRW model is not excessively penalized in the ranking based on the adopted identification criteria despite its larger number of parameters.

[82] 3. The best predictive power, assessed through the validation results presented in Figures 4-6 and Table 7, is always associated with the CTRW model, even in cases where the posterior probability weight associated with either the ADE or the DP model is clearly dominant. The GSA-based calibration of each model returns an acceptable approximation (remarkably accurate in the case of the CTRW model) of all available concentration profiles even as calibration is performed using minimum sets of observations corresponding to the most sensitive (space-time) locations.

[83] 4. For the selected dataset, it was shown that observations identified as unimportant in the phase of parameter calibration might be relevant when evaluating model selection criteria.


[84] The authors thank Ming Ye and Mary Hill for insightful and constructive comments on previous versions of the manuscript. V.C. acknowledges partial support from Marco Polo Program 2011 of the University of Bologna. B.B. gratefully acknowledges support from the Israel Science Foundation (grant 221/11).