A novel particle tracking scheme for modeling contaminant transport in a dual-continua fractured medium


Corresponding author: F. Cadini, Dipartimento di Energia, Politecnico di Milano, Via Ponzio 34/3, IT-20133 Milan, Italy. (francesco.cadini@polimi.it)


[1] The estimation of the extent and timing of solute migration in a fractured medium is a fundamental task for verifying the level of protection against contaminant releases (e.g., toxic chemicals or radionuclides) offered by the engineered and natural barriers of a waste repository. In this paper we present a novel approach for modeling solute transport in a fractured medium, based on an extension of the Kolmogorov-Dmitriev theory of stochastic branching processes. The model equations for the expected values of the solute concentration take a form similar to that of classical dual-continua models. On the other hand, the stochastic nature of the modeling approach lends itself to a new particle tracking scheme of resolution, which allows accounting for realistic features of the transport process. The proposed stochastic modeling framework and simulation solution approach are illustrated with reference to the experimental results from a case study of literature. Some of the model parameters are optimally identified by means of a genetic algorithm search aimed at best fitting the experimental data.

1. Introduction

[2] Fractures are the main responsible of contaminant transport in low-permeability formations because they offer preferential pathways to water flow and associated advective transport in the host rock [Painter and Cvetkovic, 2005]. For this reason, the quantitative analyses of the physical and chemical processes affecting transport through fractured media play a fundamental role in the safety assessments of disposal facilities of hazardous wastes. Typical examples of such facilities are land sites and deep wells for chemical or low-level radioactive wastes, or geological repositories for high-level nuclear wastes [Tang et al., 1981; Berkowitz and Scher, 1997; Benke and Painter, 2003; Delay et al., 2005].

[3] To address these issues, in this paper we propose both a novel particle tracking scheme for solute transport in fractured media and an original two-step procedure for the identification of its parameters from measured data. Our work is motivated by the fact that, regardless the rich literature on the subject, quantitative analyses of solute transport in fractured media are still challenging tasks, due to their inherent complexities. In particular, two issues contribute most to the general difficulty of the problem:

[4] (1) It is difficult, if not impossible, to obtain complete and detailed maps of the subsurface fractures; thus, the analyses must rely on stochastic representations of the fracture network based on observations of exposed fractures [Odling, 1997; Berkowitz and Scher, 1998; Davy et al., 2010]. These representations are complicated by the fact that fracture sizes range from micrometers to kilometers and fracture patterns range from relatively regular polygons to apparently random distributions [Berkowitz and Scher, 1998; Bonnet et al., 2001].

[5] (2) In such heterogeneous domains, the mechanical mixing due to local, irregular advective displacements cannot be represented as a Fickian-like diffusion process, as assumed in most predictive models based on some form of the advection-dispersion equation (ADE), and the dispersivities are observed to grow with the solute residence time or displacement distance, as opposed to the classical ADE assumptions of constant flow fields and dispersion coefficients [Sudicky et al., 1985; Gelhar, 1993; Berkowitz and Scher, 1997; Berkowitz and Scher, 1998; Berkowitz et al., 2001; Berkowitz and Scher, 2001; Berkowitz et al., 2002; Dentz et al., 2004]. This phenomenon, generally known as “scale-dependent dispersion,” “anomalous transport,” or “non-Fickian” behavior, has been shown to be due to heterogeneities in the velocity field at every scale of observation: (i) the microscale of the pores [Berkowitz et al., 2006]; (ii) the local scale of a single fracture [Sudicky et al., 1985]; (iii) the large scale of the entire fracture network [Williams, 1992; Berkowitz and Scher, 1997; Painter et al., 2002]. Moreover, chemical interactions, i.e., adsorption/desorption processes between the solutes and the rock matrix and/or colloidal particles, are also shown to be responsible of non-Fickian contaminant migration.

[6] Some authors addressed the problem of quantifying non-Fickian transport by resorting to stochastic generalizations of ADEs, for example by spatially varying velocity fields [Rubin, 1990; Gelhar, 1993]. The use of these methods is suitable when many site-specific geological observations are available, which is rarely true [Gao et al., 2010].

[7] In many applications, the most practical choice is to resort to transport modeling approaches based on the deterministic ADE. Then, efforts have been devoted to the development of models which (i) explicitly incorporate empirical time- or space-dependent dispersivities into the ADE [Pickens and Grisak, 1981; Gao et al., 2010] and (ii) generalize the ADE by including multicontinua/multirate concepts for both water flow and/or solute transport [Barenblatt et al., 1960; Warren and Root, 1963], like mobile-immobile [van Genuchten and Wierenga, 1976], dual-porosity [Simunek et al., 2003], dual-permeability [Gerke and van Genuchten, 1993a, 1993b], multirate [Haggerty and Gorelick, 1995], etc., where a common mathematical framework [Haggerty and Gorelick, 1995] is adopted to represent mass (water and/or solute) exchanges between different phases or, within the same phase, different velocity regions (see Simunek and van Genuchten [2008] for a comprehensive categorization of all these methods). It is to be noted, however, that all these models are based, explicitly or implicitly, to an underlying Fickian transport behavior at some scale [Berkowitz et al., 2002].

[8] Recently, the continuous time random walk (CTRW) theory has been proposed as an alternative (i.e., not ADE-based) description of non-Fickian transport at different scales, both in fractured [Berkowitz and Scher, 1997; Berkowitz and Scher, 1998] and heterogeneous media in general [Berkowitz et al., 2001; Berkowitz and Scher, 2001; Berkowitz et al., 2002; Dentz et al., 2004]. In the CTRW methods, the transport of a solute particle is modeled as a random walk in space and time, with the mass transfers being described in terms of a joint probability distribution math formula of the displacements math formula and the time math formula. In practical applications, problems may arise in identifying this distribution, which should reflect the medium heterogeneities at the proper scales and account for any additional relevant influence on the transfer process. Another, not ADE-based modeling framework, alternative to the CTRW, was proposed byWilliams [1992], where the actual motion of solute particles was intuitively described in analogy with neutron transport.

[9] In all generality, the differential equations of the model adopted have to be solved to give the contaminant concentrations in time and space. The solution of the model equations represents another difficult task.

[10] For a single, longitudinal fracture, i.e., the simplest type of heterogeneity made of a perfect stratification with continuous layers of different hydraulic conductivity [Sudicky, 1983], analytic approaches have been proposed which are based on some form of the ADE. For example, Neretnieks [1980] proposed an analytic solution for contaminant transport in a single fracture under the assumption that dispersion and diffusion along the fracture are negligible; Tang et al. [1981] proposed a more general analytic solution approach taking into account all dispersive and diffusive processes, but where transversal diffusion occurs in an infinite matrix; Sudicky and Frind [1982] extended such solution to the finite matrix case, including also sorption and radioactive decay; Zhan et al. [2009]also included the non-negligible transverse dispersion in the fracture.

[11] Analytic solutions have the advantage of providing an explicit physical explanation of the underlying driving processes. However, for realistic domains, often characterized by complex boundary conditions, analytic solutions are difficult to obtain, if not impossible; moreover, analytic solutions often rely on a Laplace transformation of the relevant flow and transport equations, so that complex numerical antitransformations may be required. In these cases, the adoption of numerical schemes becomes more attractive.

[12] Numerical schemes can in general be divided in two categories, i.e., Eulerian and Lagrangian methods. Eulerian approaches to solve the transport equations are based on the application of finite difference or finite element methods [Grisak and Pickens, 1980; Salamon et al., 2006b]. A wide spectrum of alternatives were proposed in the literature and several well-established commercial tools are available. Specific numerical tools were developed to solve dual-continua and multirate models, also in variably saturated conditions (e.g., HYDRUS-2-D [Simunek and van Genuchten, 2008]). The major drawbacks of Eulerian methods are that (i) mass conservation is not guaranteed and (ii) they may require very fine discretization of the transport domain to reduce instabilities of the numerical solutions and numerical dispersion, thus leading to large computational expenses, in particular for advection-dominated problems [Delay et al., 2005; Salamon et al., 2006b].

[13] To address these issues, several Langrangian (or mixed Eulerian-Lagrangian) random walk particle tracking (RWPT) methods were developed, based on the simulation of the motion of solute particles within the flow field. In the first particle tracking methods proposed, i.e., the space-based particle tracking methods, the particles are moved by random spatial displacements at specified time steps. The most popular of these approaches is the one based on the Fokker-Planck-Kolmogorov Equation (FPKE) [Kinzelbach, 1987; Uffink, 1990; Delay et al., 2005], whose first application to hydrologic problems dates back to the late 1970s and early 1980s. Further extensions of the FPKE-based particle method were developed in the following years to include multicontinua and/or multirate transport representations. For example, a particle tracking algorithm was applied byKinzelbach [1987]within a mobile-immobile scheme for solute transport in single longitudinal fractures;Michalak and Kitianidis [2000]proposed an analogous RWPT method for handling single-rate, first order mass transfer with no mesh cell communication, whereasSalamon et al. [2006a]extended it to the multiple-rate mass transfer case; recently, with regards to fractured porous media,Liu et al. [2000], Pan and Bodvarsson [2002], Hassan and Mohamed [2003], and Roubinet et al. [2010]improved RWPT modeling of contaminant transport in a dual-continua representation by devising fracture-matrix transfer probabilities which account also for diffusion into the matrix blocks in irregular fracture grids.

[14] Other particle tracking approaches are based on the ordinary Langevin equation: They are less common in transport problems in natural porous media, although they are well suited for applications involving colloid transport. It can be shown that the Langevin equation reduces to the classical FPKE under certain simplifying assumptions [Uffink, 1990; Delay et al., 2005].

[15] In very heterogeneous media, such as fractured rocks, space-based random walk calculations may become very time consuming, mainly due to the large number of jumps that have to be simulated for a particle moving in regions of very low velocities [Delay et al., 2005]. More recently, time domain particle methods were shown to offer a possible solution to this problem, consisting of moving the particles by constant spatial displacements at randomly sampled times: This way the particles cross a constant-velocity segment in a single step, with an increase in the computational efficiency [Wen and Gomez-Hernandez, 1996; Delay et al., 2005; Painter et al., 2008].

[16] In this paper we present an alternative approach to the modeling of solute transport in a fractured medium based on an original extension of the Kolmogorov-Dmitriev (KD) stochastic branching model [Kolmogorov and Dmitriev, 1947] applied to the transport of contaminants in groundwater [Marseguerra and Zio, 1997]. The model aims at describing the stochastic transport of the populations of solute particles in the fractures and host matrix by means of a discrete-states, continuous time Markov process based on properly defined transition probability distribution functions. Partial differential equations, known as the forward Kolmogorov equations (FKEs), are then derived for the expected values of the different particles' populations, which are proportional to the solute concentrations. The Markovian formalism then allows a simple implementation of a particle tracking algorithm for solving the FKEs.

[17] In particular, at the scale of a single, longitudinal fracture, the FKEs take a form similar to that of the coupled differential equations of a classical dual-continua description of the migration process [Simunek and van Genuchten, 2008]. In principle, this analogy can allow the identification of the transition rates of the stochastic model as a function of the dual-continua model parameters, as originally proposed byFerrara et al. [1999]for a standard, single-continuum problem.

[18] The Kolmogorov-Dmitriev particle tracking (KDPT) scheme is here applied to model the solute transport in a single-fracture medium, corresponding to the setting of an experimental case study of literature [Sudicky et al., 1985]. The model transition rates for the solute migration within the fracture and the matrix are then identified by a two-step procedure which first exploits the analogy with the corresponding dual-continua representation and then a genetic algorithm for optimal fitting to the experimental data. The proposed modeling scheme and parameter estimation procedure are shown to effectively reproduce the anomalous transport features in the simple heterogeneous layout experimentally observed at the scale of the single fracture.

[19] The paper is organized as follows. In section 2the modeling of branching stochastic processes is specialized to a one-dimensional, dual-continua transport domain and the operative particle tracking simulation procedure for solute concentration estimation is described. Insection 3the method is applied to the experimental data from the single-fracture case study of literature. Conclusions on the capabilities and limitations of the proposed modeling scheme and parameter estimation procedure are drawn insection 4.

2. Kolmogorov-Dmitriev Particle Tracking for Modeling Contaminant Transport in a Dual-Continua Medium

[20] In this section we present an alternative particle tracking algorithm based on the original extension of the KD model of contaminant transport in a porous medium [Marseguerra and Zio, 1997] to the case of a single, longitudinal fracture. We then propose the first of the two-step procedure for the identification of its parameters, whereas the second step, based on a genetic algorithm-based best fitting of measured data, will be presented insection 3.

[21] The key feature of KD modeling is that different types of particles, characterized by different stochastic behaviors, are introduced to represent the solute in its possible different states (physical or chemical) and regions of space (positions). Partial differential equations, known as the forward Kolmogorov equations (FKEs), are then derived for the expected values of the different particles' populations, which are proportional to the solute concentrations.

[22] Within a Markovian description of the stochastic space-time evolution of a system of different particles, the KD model is naturally suited to a particle tracking-based solution of the associated FKEs: from the probability density functions of the model, a large number of realizations of the migration fates of contaminant particles are simulated.

[23] The KDPT approach is conceptually similar to the FPKE and the CTRW (and also to the approach proposed by Williams [1992]), in that the motion of the solute particles is described as a stochastic process in time and/or space. In addition, the discrete state nature of the method allows a straightforward and simultaneous inclusion of many physical and chemical processes by simply introducing new particle types. For example, Ferrara et al. [1999] modeled both linear and nonlinear equilibrium isotherms for sorption/desorption processes, Marseguerra et al. [2003]accounted for radioactive decay and in-growth andGiacobbo et al. [2005]modeled transport in unsaturated porous media. Moreover, although not yet demonstrated, it is to expect that all first order transformation processes (including multiple-rate mass transfers [Haggerty and Gorelick, 1995]) in a mobile-immobile scheme can be straightforwardly included in the KDPT models by introducing new particle types. For example, as byMarseguerra and Zio [1997], different types of immobile particles can be considered to account for different adsorption/desorption mechanisms on the grains surfaces at the pore scale; different species of mobile particles can be introduced to account for (i) “side pockets” or intraparticle porosity effects, at the pore scale, and/or (ii) low permeability zones at larger scales where diffusion may be the dominant transport process. For these extensions, no additional modeling efforts are required; only an increased number of model parameters need to be estimated, e.g., by analogy with multirate mass transfer models of literature, such as those by Carrera et al. [1998], Haggerty and Gorelick [1995], and Salamon et al. [2006a]. Furthermore, KDPT is continuous in time: Thus adaptive time stepping, which is required by space-based methods, is avoided and a larger computational efficiency in presence of heterogeneous or fractured media is potentially achieved [Painter et al., 2008]. Finally, the KDPT approach was also successfully applied to concentration-dependent problems, such as colloidal transport [Marseguerra et al., 2001] and solubility-limited release [Cadini et al., 2011].

[24] For simplicity, in what follows we shall refer to a one-dimensional domain [Ferrara et al., 1999], which is subdivided in math formula discrete zones, math formula. The objective is that of determining the amount of contaminant present at each time in each zone of the porous matrix and system of fractures. Two categories of particles are introduced: the solutons, which are the particles of contaminant migrating in the porous matrix, and the fracturons, which are the particles of contaminant migrating in the fractures. Thus, the system is made up of math formula different kinds of particles, i.e., the solutons and the fracturons in the zones math formula. Each particle may disappear or give rise to one particle of the math formula remaining kinds according to given transition probability laws. It is assumed that (i) the stochastic process is Markovian, i.e., a particle of the math formula th kind, math formula, gives rise to a branching process independently of its past history; (ii) the process is linear, i.e., the particles do not interact among each other; (iii) within a generic time interval math formula, with math formula sufficiently small, only one particle transition may occur.

[25] Within this framework, in the time interval math formula, a soluton can undergo an exchange transition in zone math formula, thus transforming into a fracturon in zone math formula; the probability of occurrence of this transition, conditioned on the fact that the particle was a soluton at time math formula in zone math formula, is math formula, where math formula is the corresponding transition rate. Alternatively, within the same time interval math formula, the soluton can travel to one of the neighboring zones, math formula or math formula, with transition rates math formula (forward) and math formula(backward), respectively. Because of the continuity of the underlying physical phenomenon, only transitions among neighboring zones (i.e., previous and following one in the one-dimensional domain) are taken into account. Similarly, one fracturon in zone math formula can either enter the porous matrix and become a soluton in zone math formula with transition rate math formula or travel to the nearby zones, math formula or math formula, with transition rates math formula and math formula. Finally, the contaminant might undergo a radioactive decay or a chemical reaction with rate math formula, upon which, for simplicity, we assume the particle disappears (the extension to follow chains of radionuclides and chemical species is conceptually trivial but adds complexity in the presentation of the model which are not meaningful for the scope of the presented work).

[26] To describe the space and time evolution of the system of particles, it is possible to write the following system of math formula coupled differential equations (forward Kolmogorov equations) for the evolution of the expected values of the numbers of solutons ( math formula) and fracturons ( math formula) at time math formula in zone math formula [Marseguerra and Zio, 1997]:

display math
display math

with math formula and where we assume to have one soluton in zone math formula at time math formula ( math formula). These equations appeal to our physical intuition as they represent the balance between the production and the destruction processes for both the solutons and the fracturons populations at math formula in math formula.

[27] For any time- and space-dependent contaminant source, the solution of the above system of equations can be found by convolution, due to the process linearity.

[28] In analogy to the approach proposed by Ferrara et al. [1999], a term-by-term relation is set up with the equations of the solute transport in a dual-continua model to identify the transition rates of the KD model. A dual-continua model represents the flow and transport phenomena by assuming that the conducting medium can be modeled as two overlapping, coupled continua with different properties [Barenblatt et al., 1960; Warren and Root, 1963]. The transport phenomenon is therefore assumed to occur in two domains of different hydrological properties, the porous matrix and the fracture system.

[29] Under the assumptions of constant water and solid densities and no swelling and shrinking effects, in the general case of unsaturated flow, the coupled equations describing solute transport in a dual-continua medium with linear adsorption and first order decay can be written as [Gerke and van Genuchten, 1993a, 1993b]:

display math
display math

where the subscript math formula refers to the matrix subsystem and math formula to the fracture subsystem, math formula is the dispersion tensor, math formula is the decay or chemical reaction rate, math formula is the retardation coefficient, math formula is the relative volume of the pore system ( math formula), math formula is the water content, math formula is the Darcy's velocity, math formula is the solute concentration, and math formula is the solute exchange rate between the porous matrix and the fracture system.

[30] Many approaches of increasing complexity have been proposed in literature for the determination of the solute exchange term math formula, e.g., in Gerke and van Genuchten [1993a, 1993b]and the references therein for geometry-based models which account for diffusion into aggregates of known size and geometry, or, with particular regards to particle methods for a single fracture, byLiu et al. [2000], Pan and Bodvarsson [2002], Hassan and Mohamed [2003], and Roubinet et al. [2010]. Since our objective is to present a new numerical scheme and not to investigate on the particle transfer terms, in this work we resort to the simple expression proposed by Gerke and van Genuchten [1993a, 1993b]. Thus, assuming that only the diffusive contribution to the exchange process is taken into account and that the flow field is at steady state, the first order representation of the exchange term math formula is

display math

[31] The coefficient math formula in equation (3) is [Gerke and van Genuchten, 1993a, 1993b]

display math

where math formula is the effective diffusion coefficient at the interface, math formula is a semiempirical geometric factor, and math formulais the characteristic half-width of the porous matrix. The terms inequations (2) are all referred to the relative volume of the corresponding subsystem, with the exception of math formula, which is defined as the mass of solute per unit volume of bulk soil per unit time [Gerke and van Genuchten, 1993a]. For the general form of equations (2) and (3) the interested reader can refer to Gerke and van Genuchten [1993a, 1993b].

[32] Under the hypotheses of (i) one-dimensional domain, (ii) no retardation ( math formula), and (iii) constant water contents ( math formula and math formula), constant pore velocities math formula math formula (where math formula and math formula are the porosity of the matrix and the fractures system, respectively), constant dispersion coefficients math formula and math formula and constant reaction rates math formula, it is possible to write equations (2) in terms of the concentrations referred to the total bulk volume ( math formula and math formula) and, subsequently, to discretize them in space by a centered Euler finite difference method:

display math
display math

[33] These equations are formally identical to (1a) and (1b)and by term-by-term comparison it is possible to write the following relationships for the forward and backward rates of the solutons and fracturons:

display math
display math

[34] Note that since the backward transition rates math formula and math formula cannot be negative, equations (6b) require an upper limit for the spatial discretization cell:

display math

[35] As a result of the proposed identification, the expressions (6a) and (6b) of the forward and backward transition rates allow the representation of advective and longitudinal, diffusive, and dispersive transport in both the fracture and the matrix systems.

[36] Analogously to the forward and backward rates, the exchange rates can be expressed as

display math
display math

[37] The difference in the exchange rates depends mainly on the relative volume of the matrix and fracture subsystem: The larger the volume of one subsystem is, the less probable the transition to the other subsystem is.

[38] The formal analogy leading to the identification of the FKEs parameters is based on a Eulerian discretization of the dual-continua equations, thus introducing numerical dispersion. With regards to particle migration in a compartment model, methods were proposed in literature to theoretically solve this problem, which are based on sampling the particles' residence time distributions obtained by separate, more detailed models [Kawasaki and Ahn, 2006; Cadini et al., 2010a, b]. However, being beyond the main purpose of this paper, this issue is left to further investigations.

[39] The system of difference equations (5a) and (5b) with parameters identified by (6) and (8) can be solved numerically, e.g., as by Ferrara et al. [1999], where a Runge-Kutta method is applied. However, restrictions apply to the choice of the time step in order to avoid numerical problems. Here we propose to resort to a new particle tracking scheme continuous in time, which is directly derived from the underlying Markovian description of the stochastic evolution of the system of particles, whereby the stochastic migration of a large number math formula of contaminant particles are simulated by repeatedly sampling their births from the release sources and their transitions across the medium compartments.

[40] The random walk of the individual radionuclide particle is simulated either until it exits the domain to the math formula compartment “environment,” that is absorbing because from there the particle cannot come back into the migration domain, or until its lifetime crosses the time horizon T of the analysis. The time horizon T is discretized in Nt equally spaced time instants, with time step math formula. Two counters math formula and math formula are associated to each compartment math formula and each discrete time math formula. During the simulation, a one is accumulated in the counters math formula and math formula if a soluton or a fracturon, respectively, resides, during the random walk, in compartment n at time k. At the end of the Msimulated random walks of the contaminant particles, the values accumulated in the counters allow estimating the time-dependent probabilities of cell occupation by a soluton or a fracturon, math formula and math formula, respectively:

display math
display math

[41] The probability that a generic contaminant particle (either a soluton or a fracturon) occupies cell n at time k is

display math

[42] Similarly, the probability density function of release into the environment (compartment math formula), math formula, can be estimated as

display math

3. Results

[43] The stochastic modeling framework and simulation method presented in section 2 has been applied to an experimental case study of literature [Sudicky et al., 1985] concerning the transport of a nonreactive solute in a dual porosity medium made up of a single horizontal fracture embedded in a porous matrix [Wu and Pruess, 2000, Figure 5]. The experimental setup consisted of a Plexiglas box 1.0 m long, 0.2 m thick, and 0.1 m wide. Inside the box, a 0.03 m thick layer of sand, representing the fracture, was placed between two 0.085 m thick layers of silt, representing the porous matrix. Before the experiment is carried out, the medium was completely saturated in order to reach the steady state conditions. The water flow through the box was kept constant during the entire duration of the experiment. A tracer was then injected as a point source in correspondence of the upstream fracture end (left of Figure 5 in Wu and Pruess [2000]) and the properly normalized solute concentration was observed at the opposite end at predefined time instants; the observations give rise to a pointwise breakthrough curve. Table 1 shows the parameters characterizing the hydrology of the experimental setup [Wu and Pruess, 2000].

Table 1. Hydrologic Parameters of the Experimental Setup From Wu and Pruess [2000]
Fracture Aperture (Sand Layer Thickness) math formula
Fracture Porosity math formula
Matrix Porosity math formula
Fracture Permeability math formula
Matrix Permeability math formula
Molecular Diffusion Coefficient math formula
Fracture Transverse (T) and Longitudinal (L) Dispersivity math formula
Matrix Transverse and Longitudinal Dispersivity math formula
Fracturematrix Dispersivity math formula
Fracture and Matrix Tortuosity math formula
Fracture Pore Velocity math formula
Fracture-Matrix Velocity math formula
Radioactive Decay math formula

[44] In Sudicky et al. [1985], a constant concentration chloride solution was injected into the system for 7 days. Then, the incoming flow was switched to a tracer free solution. The solute concentration math formula observed at the downstream end of the fracture was normalized by the inlet concentration math formula, thus giving rise to the pointwise breakthrough curve shown in Figure 1 (triangles) over 12 days.

Figure 1.

Comparison of the KDPT breakthrough curves with analytic and optimized exchange rates (dotted and dashed lines, respectively) and the experimental data at the outlet of the sand layer (triangles).

[45] The KDPT presented in section 2 was adopted here to describe the tracer transport within the fractured medium of the experiment. Assuming saturation as a special case of variably saturated flow, we here applied the KDPT scheme, although its parameters were formally identified in section 2by analogy to the dual-continua solute equations in unsaturated media ofGerke and van Genuchten [1993a, 1993b]. In principle, different first order exchange models may also be readily implemented within this scheme; on the other hand, more complicated models (e.g., those accounting for the diffusion depth into the matrix [Pan and Bodvarsson, 2002; Roubinet et al., 2010]) would require further processing.

[46] The domain was represented as a one-dimensional array of math formula compartments of equal width math formula, describing both the fracture and the porous matrix systems. The time horizon of the simulation math formula was divided into math formula time intervals of uniform width math formula, for data collection purposes. A total of math formula fracturons were injected in the first compartment at math formula to simulate the source term; the convolution technique was then exploited to account for its uniform distribution over the first 7 days, due to the process linearity. The backward and forward transition rates for both the solutons and fracturons, math formula, math formula and math formula, math formula, respectively, were computed by resorting to equations (6a) and (6b) and the parameters values in Table 1, with

display math

[47] For simplicity, only diffusive motion was taken into account for the solutons; the permeability of the silt is in fact four orders of magnitude lower than the permeability of the sand, and the pore velocity in the matrix can therefore be neglected [Wu and Pruess, 2000].

[48] As anticipated in section 2, the objective of this work is to demonstrate the applicability of the KDPT scheme and its flexibility in including different exchange parameters and not to investigate on their possible expressions. To this aim we first identified the exchange rates by completing the analogy with the dual-continua scheme proposed byGerke and van Genuchten [1993a, 1993b]. Thus, the transformation rates from a soluton to a fracturon, math formula, and from a fracturon to a soluton, math formula, were found by applying equations (8a) and (8b), where the water contents were taken equal to the porosity values ( math formula and math formula) since the medium is fully saturated and the relative volumes of the pore system were math formula and math formula. The first order diffusion coefficient math formula was computed by resorting to equation (3), where the geometric parameter for a rectangular slab is math formula according to Gerke and van Genuchten [1993a, 1993b], the characteristic radius is math formula and the effective diffusion coefficient at the interface is [Wu and Pruess, 2000]

display math

[49] The values of the exchange rates thereby obtained are math formula and math formula; the simulation of particles transport with these values led to an estimated breakthrough curve which captures the general time behavior of the curve, but fails to predict the normalized concentration levels, as shown in Figure 1 (dashed line). This is due in particular to the unreliable values of the parameters in equation (4).

[50] Thus, following the suggestions provided by (i) Gerke and van Genuchten [1993a, 1993b]that although a conceptual basis is available the mass transfer coefficients can be usually estimated by fitting solutions of the dual-continua model to measured solute concentration distributions, or in (ii) [Berkowitz et al., 2006] that a simple form of the distribution math formulacan be used to fit the measurements, we pursued an alternative strategy to the identification of the exchange rates: The problem was formulated as a single-objective optimization problem which is tackled by a genetic algorithm (GA) [Koza, 1996]. The objective of the GA search was the minimization of the mean square error between the breakthrough curve estimated by KDPT and the experimental data of Sudicky et al. [1985]. In this view, the choice of a GA as an optimization tool was further motivated by the fact that no a priori educated guesses can be made on the shape of the objective function, which can be critically nonconvex with multiple local minima. In more complicated settings, the objective function may become even more complicated, thus providing a further justification to the GA application.

[51] A population of 100 chromosomes evolving for 100 generations was employed for the GA search. Each chromosome was made up of two genes, coding the values of the parameters math formula and math formula; the number of bits for their binary representation is 10. By a preliminary, coarse verification of convergence of the GA search, the parameter range considered in the search were set equal to [0; 1] math formula.

[52] Figure 1 compares the estimated breakthrough curve obtained with the optimal values of the exchange rates, math formula and math formula(solid line) with the experimental data (triangles); it is seen that the accuracy is satisfactory throughout (first arrival time and peak time, related to the advective transport component, as well as peak value). The slight differences in the results are mainly due to the one-dimensional domain representation: in fact, the KDPT estimate is indeed in close agreement with the results of the “thin layer” model proposed bySudicky et al. [1985], where the discrepancies from the observed data were attributed to incomplete mixing across the sand layer, which cannot be accounted for in a coupled, one-dimensional approach. Other model differences are (i) the fact that the parameters contributing to the dispersion coefficient math formula in the fracture were not directly measured but estimated on the basis of data from literature [Wu and Pruess, 2000] and (ii) the simplifying assumption of linear approximation of the exchange phenomenon at the interface. It is to be noted however, that these differences are actually due to the dual-continua model which forms the basis for the parameter identification and not to the KDPT scheme itself.

[53] Finally, the exchange rates found by the GA search still satisfy the physical condition math formula (see equations (8)), although they are slightly different from those computed with equations (8a) and (8b), probably because of the dispersivity which was set equal to zero at the interface thus reducing the theoretical value of the effective dispersion coefficient Da.

[54] The KDPT model thus obtained was then applied within the same experimental setup in correspondence of different Darcy's velocities, i.e., math formula and math formula (Figure 2, dashed lines). The comparison of the results with those obtained by highly detailed numerical simulations performed by the standard codes MODFLOW [McDonald and Harbaugh, 1988] and MT3DMS [Zheng, 1999] (Figure 2, solid lines) shows a satisfactory agreement. This confirms the capability of the GA search of capturing the fracture-matrix exchange properties, which depend on the relevant geometric features of the problem. The explanation for the differences between the two curves has already been given above. For completeness,Figure 2 also show the results of the simulations performed by the detailed numerical models and the KDPT for math formula (i.e., the experimental case), and the measurements (triangles).

Figure 2.

Comparison between the detailed numerical codes MODFLOW and MT3DMS (solid line) and the KDPT algorithm (dashed line) for the Darcy's velocities 0.25, 0.5, and 0.75 m d−1 (from left to right).

[55] In order to quantify the influence of the dispersion coefficient in the fracture math formula on the breakthrough curve, a second GA optimization was performed in which math formula is also a parameter to be optimized, together with the exchange rates. The search range for math formula was set equal to [ math formula] math formula; all the remaining optimization settings were left unchanged with respect to the previous search. Figure 3 shows the coefficient of variation, i.e., the standard deviation over the mean, of the mean square error objective function (solid line), and of the three optimization parameters of the GA population as a function of the number of search generations. Considerations of sensitivity analysis similar to those by Giacobbo et al. [2002] lead to the conclusion that the dispersion coefficient in the fracture math formula (dashed line) is not particularly important for the objective function variation, as opposed to the exchange rates math formula and math formula(dash-dotted and dotted lines, respectively) which the GA tends to set to optimal values as soon as possible in the search. This confirms the importance of the solute transfers between the porous matrix and the fracture in determining the non-Fickian dispersion observed in the breakthrough curve [Feehley et al., 2000; Berkowitz et al., 2008].

Figure 3.

Coefficients of variation of the objective function and the three parameters of the GA optimization as a function of the number of generations.

[56] The computational times required by a single KDPT simulation were approximately 22 s on a 6 GB RAM computer with an Intel® Core™ i5 CPU M520 @ 2.40GHz processor, whereas those required by the GA optimization were approximately 510 s. As a term of comparison, a single detailed numerical simulation by MODFLOW and MT3DMS required approximately 135 s on a 2 GB RAM computer with a Intel Core2 Duo CPU @ 3.16 GHz.

4. Conclusions

[57] In this paper, a novel approach to modeling the transport of contaminants in fractured domains was proposed, which amounts to solving by a particle tracking simulation scheme an extended stochastic transport model based on the Kolmogorov-Dmitriev theory of branching stochastic processes. The approach provides an intuitive and flexible way for describing the migration of the solute contaminant particles in the transport domain by means of a probabilistic description of the individual processes which may occur. At the same time, its time-continuity feature allows avoiding the numerical problems typically related to the time step setting.

[58] For the identification of the parameters of the KDPT model, we proposed a two-step procedure which first exploits the analogy with the corresponding dual-continua representation and then a genetic algorithm for optimal fitting to the experimental data. The analogy allowed for the representation of advective and longitudinal diffusive and dispersive transport in both the fracture and the matrix systems; the diffusional exchange processes between the two continua was also included as a first order approximation. More realistic matrix diffusion models might in principle be represented by further splitting the matrix continuum into more continua (in other words, by adding new particle types). At the same time, the analogy introduces the problem of numerical dispersion, as opposed to classical RWPT approaches; methods of literature for reducing this effect can be applied to this problem. Further investigations in both directions are needed.

[59] The KDPT model was applied on an experimental case study of literature of a single, longitudinal fracture. The KDPT model results were shown to be comparable to those obtained by a fine grid numerical solution obtained by the standard MODFLOW + MT3DMS codes.

[60] Furthermore, sensitivity analysis-based considerations on the GA best fitting search also showed that the parameters related to the fracture-matrix mass exchanges are the most relevant in determining the shape of the breakthrough curve, in agreement with the theoretical and experimental works of literature.


[61] This work has been funded by the Ente per le Nuove Tecnologie, L'Energia e l'Ambiente (ENEA) within the Framework Program Agreement with the Italian Ministry of Economic Development.