A nonequilibrium model for reactive contaminant transport through fractured porous media: Model development and semianalytical solution

Authors


Corresponding author: N. Joshi, Department of Civil Engineering, Indian Institute of Technology, Roorkee 247667, Roorkee, India. (nitinj3982@gmail.com)

Abstract

[1] In this study a conceptual model that accounts for the effects of nonequilibrium contaminant transport in a fractured porous media is developed. Present model accounts for both physical and sorption nonequilibrium. Analytical solution was developed using the Laplace transform technique, which was then numerically inverted to obtain solute concentration in the fracture matrix system. The semianalytical solution developed here can incorporate both semi-infinite and finite fracture matrix extent. In addition, the model can account for flexible boundary conditions and nonzero initial condition in the fracture matrix system. The present semianalytical solution was validated against the existing analytical solutions for the fracture matrix system. In order to differentiate between various sorption/transport mechanism different cases of sorption and mass transfer were analyzed by comparing the breakthrough curves and temporal moments. It was found that significant differences in the signature of sorption and mass transfer exists. Applicability of the developed model was evaluated by simulating the published experimental data of Calcium and Strontium transport in a single fracture. The present model simulated the experimental data reasonably well in comparison to the model based on equilibrium sorption assumption in fracture matrix system, and multi rate mass transfer model.

1. Introduction

[2] Fractured porous aquifers are important sources of water for domestic, industrial and agriculture use in both developed and developing countries. About 75% of earth's surface consists of fractured or karstic fractured rock aquifer [Plummer et al., 2002]. Therefore, understanding, characterizing and modeling of physical chemical interaction in fractured aquifer becomes increasingly important in terms of water resources development and groundwater contamination [Dietrich et al., 2005]. The permeability of fractures network is greater than the permeability of porous rock therefore, fractures acts as pathways through which the both reactive and nonreactive contaminants can move rapidly [Tsang and Neretnieks, 1998].

[3] The conceptual model representing the fracture porous media can be categorized as equivalent continuum model (ECM) and discrete fracture model (DFM). In the equivalent continuum models, the transport problem is analyzed at the macroscopic scale and the fracture matrix system is conceptualized into one or more overlapping continua [Schwartz and Smith, 1988; Royer et al., 2002].

[4] Continuum fracture models can be classified as single-continuum models, dual continuum models and triple continuum modelsCommittee on Fracture Characterization and Fluid Flow [1996]. Single-continuum models analyze the fracture matrix system by conceptualizing it as an equivalent porous media [Long et al., 1982; Berkowitz et al., 1988]. A dual continuum models can be subclassified into dual porosity single permeability and dual porosity dual permeability model. A dual porosity single permeability model assumes that no flow takes place between the matrix blocks [Barenblatt et al., 1960; Warren and Root, 1963; Zimmerman et al., 1993; Dershowitz and Miller, 1995; Corapcioglu and Wang, 1999] whereas, dual porosity dual permeability models assume that both fracture and matrix contribute to the flow [Zhang and Sun, 2000; Xu and Hu, 2005]. Triple continuum models conceptualize the fracture matrix system into three overlapping continua, consisting of both large and small-scale fractures along with the porous matrix [Wu et al., 2004]. Continuum models are well suited when the scale of the problem is large, rocks have significant matrix permeability and fractures are well interconnected.

[5] In the discrete fracture models, fractures are explicitly modeled. These models are similar to mobile-immobile model [West et al., 2004]. There are a number of analytical [Grisak and Pickens, 1981; Tang et al., 1981] and numerical [Grisak and Pickens, 1980; Noorishad and Mehran, 1982; Huyakorn et al., 1983] studies that use discrete fracture models for modeling reactive contaminant transport through fractured porous media. The discrete fracture models require detailed information of fracture networks and their characterization; they also require large computer storage and computational efforts [Dershowitz et al., 2004]. The Discrete fracture matrix models are used for the study of radionuclide transport and safety assessment studies of high lever nuclear waste [Norman and Kjellbert, 1990].

[6] There have been a number of analytical solutions given for different source conditions, transport processes and fracture/matrix domains. Bibby [1981]derived an analytical expression for solute in the fracture for the case of diffusion for a two-dimensional solute mass transport in dual porosity medium.Tang et al. [1981]gave an analytical expression for advective-dispersive transport equation including equilibrium sorption and first-order degradation coefficient with constant concentration boundary condition and neglecting transverse dispersion.Sudicky and Frind [1982] and Barker [1982] gave the analytical solution for cases of radioactive contaminant transport in a system of discrete multiple parallel fractures. Moreno and Ramuson [1986] gave an analytical solution for a continuous injection of the constant mass flux in the fracture. There have been a number of analytical solutions that neglect the dispersion in the fracture [Neretnieks, 1980; Grisak and Pickens, 1981]. However, Tang et al. [1981] observed that neglecting longitudinal dispersion in the fracture might lead to significant error in case of low fluid velocity. West et al. [2004] gave an analytical solution for systems of parallel fractures, with a strip source of finite width; they also considered transverse dispersion in the fractures. Roubinet et al. [2012]gave a semianalytical solution considering two-dimension dispersion in fracture and two-dimension diffusion in the matrix.

[7] Analytical solutions are mostly limited to simpler cases and can be used as benchmarks for numerical models. In addition, analytical solutions often consist of multiple integrals of an oscillatory function, which is difficult to evaluate numerically [Bodin et al., 2003]. To overcome such a difficulty, numerical inversion of the Laplace spaced solution is widely employed [Neville et al., 2000; Weatherill et al., 2008; Gao et al., 2010]. However, in case of numerical models, Weatherill et al. [2008]found that, if the advective/dispersive transport in the fracture is higher than the diffusive transport in the matrix then, a very fine grid size of the order of the fracture aperture is required near the interface between fracture and matrix. Such a restriction may lead to higher memory requirement and computational efforts while modeling a large-scale fracture matrix system. However, no such limitation exists in the case of analytical solutions.

[8] There have been a number of studies that incorporate nonequilibrium sorption in a porous media [Brusseau et al., 1989, 1992; Selim et al., 1999; Leje and Bradford, 2009]. Brusseau et al. [1989] developed a multi process nonequilibrium (MPNE) model for porous medium where both transport and sorption related nonequilibrium process contributing to the observed nonequilibrium are considered. The model that incorporates both transport and sorption related nonequilibrium processes yields discrete kinetic terms, each representing individual nonequilibrium processes and can be used for process level investigation of solute transport. Whereas, in case of fractured porous media a very few studies exist that incorporate a nonequilibrium sorption in fracture/matrix. Most of the models available for solute transport through fractured porous media neglect the effects of sorption kinetics in fracture and matrix. Xu and Worman [1999] reported that neglecting the sorption kinetic results in errors in both the peak value of the pulse traveling in the fracture and the variance of the residence time and can be up to several hundred percents. Comans and Hockley [1992] showed that equilibration time for sorption of Cesium (Cs) in aquatic sediments vary over weeks in laboratory tests with illite and montmorillonite.

[9] Based on experimental investigations, Neretnieks et al. [1982] found that both fast and slow reactions contribute to the total retardation in the fracture. The fast reaction is on the immediately accessible fracture surface, whereas, the slow reaction is time dependent as it increases with residence time. Maloszewski and Zuber [1990]developed a model for reactive transport in a fracture matrix system where the adsorption in the matrix is assumed to be governed by a first-order nonequilibrium kinetic sorption. However, the tracer adsorption on fissure walls was approximated by instantaneous equilibrium linear adsorption isotherms.Berkowitz and Zhou [1996]developed an approximate analytical solution for the dispersion of sorbing solute with first-order reaction. They investigated that a reversible surface reaction can be treated as irreversible when adsorption is very strong and desorption is relatively weak. However, at small Damköhler number (ratio of reaction rate to molecular diffusion rate) the effect of surface reaction can be neglected, thus, reactive solute transport can be approximated by nonreactive solute transport.Lee and Teng [1993] gave an analytical solution assuming nonequilibrium sorption in the matrix.

[10] With the above background, the objectives of this study are (1) to propose a generalized model which considers nonequilibrium sorption in both fracture and matrix; (2) to develop a semianalytical solution for the new model; (3) to analyze the effects of various fracture and matrix sorptions and mass transfer parameters on the breakthrough curve (BTC); (4) to test the applicability of the model by simulating different experimental data available in the literature.

2. Conceptual Model

[11] The physical nonequilibrium conditions (PNE) affect the transport of both sorbing and nonsorbing solutes and is evident from large tailing and skewed breakthrough curve [Brusseau et al., 1989]. It has been widely reported that the PNE plays a critical role in the fractured porous media, the mass transfer between the mobile fracture and the immobile porous media is described by PNE. The sorption nonequilibrium (SNE) results from both chemical nonequilibrium and intra sorbent diffusion [Brusseau et al., 1989]. The SNE accounts for intra sorbent diffusion and rate limited interaction between the solute and sorbent.

[12] The present study focuses on developing a model that can account for both PNE and SNE in a fractured porous media. The conceptualization of the model is shown in Figure 1; the conceptual model has a similar approach to that of multi process nonequilibrium model developed by Brusseau et al. [1989]. The PNE is accounted by a diffusive mass transfer between the fracture and the adjacent porous matrix, as done elsewhere [Tang et al., 1981; Sudicky and Frind, 1982; Maloszewski and Zuber, 1990]. For SNE, the model used a two-site conceptualization for both fracture and porous matrix. Here, at the first site (denoted by subscript 1), the sorption is assumed to be governed by an instantaneous equilibrium adsorption isotherm whereas; at the second site (denoted by subscript 2) the sorption is described by a rate-limited process, which is represented as a first-order reaction.

Figure 1.

Schematic diagram of the conceptual model. (Subscripts f and m denote fracture and matrix; subscripts 1 and 2 denote instantaneous sorbed and rate limited sorbed domain; λ denotes decay rate; K and k are sorption and mass transfer coefficients.)

3. Governing Equations

3.1. Equation for Fracture and Matrix

[13] The transport process in the fracture matrix system, can be described by two coupled one-dimensional equation one for the fracture and the other for the matrix. The coupling is accounted by a diffusive mass transfer between the fracture and the adjacent porous matrix. In this study, the following processes have been considered: advection and longitudinal dispersion in the fracture, molecular diffusion from the fracture into the matrix, adsorption and decay in the fracture and matrix. The adsorption in the fracture and matrix is assumed to be governed by both an instantaneous equilibrium adsorption isotherm and a first-order nonequilibrium kinetic sorption. The assumptions made while deriving the governing equations are those ofTang et al. [1981]. The governing transport equation for the fracture can be written as

display math

where math formula and math formula are solute concentrations math formula in the fracture and porous matrix, respectively. x is the spatial coordinate taken in the direction of the flow; y is the spatial coordinate perpendicular to the fracture and t is the time variable. math formula is the mean water velocity math formula in the fracture; Df is the dispersion coefficient math formula in the fracture; math formula is the half fracture aperture math formula; math formula is the matrix porosity; Dm is effective diffusion coefficient math formula; math formula and math formula are the mass of sorbate per unit length math formulaof the fracture for which the sorption is governed by instantaneous equilibrium linear isotherm and the first-order kinetic sorption, respectively. Here, subscripts 1 and 2 are used to represent the instantaneous and rate limited sorption sites, respectively. math formulais the first-order degradation rate constant math formula in the liquid phase for fracture. In this study, different degradation rate coefficients are used for instantaneous and rate limited sorption sites. math formula and math formulaare the first-order degradation rate constants math formula in the sorbed phase for the fracture at site 1 and 2, respectively. Ff is the fraction of the sorbent in the fracture for which sorption is instantaneous, its value ranges between 0 and 1.

[14] The instantaneous equilibrium sorption in the fracture is given by

display math

similarly, the first-order nonequilibrium kinetic sorption in the fracture is given as

display math

where math formula and math formula are the distribution coefficient math formulaand the first-order sorption kinetic coefficient math formula in the fracture, respectively.

[15] Substituting equations (1b) and (1c) in equation (1a), we get

display math

where math formula is the retardation factor for the equilibrium sorption site in the fracture.

[16] Similarly, the equation for the porous matrix can be written as

display math

where math formula and math formula are the mass of solute adsorbed per unit mass of solid math formulain the porous matrix for which the sorption is governed by instantaneous equilibrium linear isotherm and the first-order kinetic sorption, respectively; math formula is the density of the porous matrix math formula; math formulais the first-order degradation rate constant math formula in the liquid phase for matrix; math formula and math formulaare the first-order degradation rate constants math formula in the sorbed phase for the porous matrix; and Fm is the fraction of the sorbent in the matrix for which sorption is instantaneous.

[17] The instantaneous equilibrium sorption in porous matrix is given as

display math

[18] The first-order kinetic sorption in porous matrix is given as

display math

[19] Km is the distribution coefficient for the porous matrix math formula and math formulais the first-order sorption kinetic coefficient math formula in matrix.

[20] Substituting equations (3b) and (3c) In equation (3a) we get

display math

where math formula is the retardation factor for the equilibrium sorption site in porous matrix.

3.2. Initial and Boundary Conditions

[21] A generalized form of initial conditions is considered, which assumes that the fracture and matrix are uniformly contaminated.

display math
display math
display math
display math

[22] However, this assumption is not restrictive in nature and one can proceed with the zero values through the fracture and matrix domain. In the present study, a generalized boundary condition similar to that used by Neville et al. [2000] is considered.

display math

where math formula is the source concentration, math formula is the Heaviside step function and it is used for describing step input from time 0 to math formula. math formula represents a constant concentration or Diritchlet type boundary condition and math formula represents a constant flux or Cauchy type boundary condition.

[23] For the outflow boundary condition for the fracture, both finite and semi-infinite domains are considered here. For finite domain the boundary condition is described by

display math

[24] For semi infinite domain the boundary condition is described by

display math

the inflow boundary condition for matrix describing the coupling of fracture and matrix is given by

display math

[25] Similarly for the matrix, both finite and semi-infinite domain are considered. For finite domain the boundary condition is described by

display math

[26] For semi-infinite domain

display math

4. Analytical Solution in the Laplace Domain

4.1. Analytical Solution for Matrix

[27] Taking the Laplace transform of equation (3c) we get

display math
display math

[28] Taking the Laplace transform of equation (4) we get

display math

[29] Substituting equation (8) into equation (9) and simplifying we get

display math

[30] The solution of equation (10) is obtained by assembling complementary solution and particular solution

display math

[31] In equation (11), on the right hand side the first two terms corresponds to complementary solution whereas, the third term corresponds to particular solution. math formula and math formula are constants depending on the boundary conditions and their expressions are given in section 4.1.1.

display math
display math

4.1.1. Case I-A: Semi-infinite Matrix

[32] Inflow and outflow boundary conditions for the matrix with semi-infinite domain are given byequation (6d) and equation (6f), respectively. In equation (11) when math formula, for the solution to be bounded, the value of math formula. The value of math formula will be obtained by using the inflow boundary condition for matrix (i.e., equation (6d)) therefore, equation (11) becomes

display math

[33] Taking the derivative of equation (14) we get

display math

[34] For the semi-infinite matrix domain

display math

4.1.2. Case I-B: Finite Matrix

[35] For the matrix with finite domain, the inflow and outflow boundary conditions are given by equation (6d) and equation (6e), respectively. Using these boundary conditions the constants math formula and math formula are computed as

display math
display math

[36] Substituting the values of math formula and math formula in equation (11) the equation for matrix in Laplace domain is obtained.

display math

[37] Taking derivative of equation (16c), we have

display math

[38] For the finite matrix domain

display math

[39] It is to be noted that the term math formula has a general form of math formula however the math formulavalues differ for matrix with semi-infinite and finite domains and is given byequations (15b) and (17b), respectively.

4.2. Analytical Solution for the Fracture

[40] Taking Laplace transform of equation (1c)

display math
display math

[41] Taking Laplace transform of equation (2) and substituting the value of math formula from equation (19), we get

display math

[42] Substituting the value of math formula from equations (15) or (17) for a semi infinite or finite matrix domain, respectively, we get

display math
display math
display math

[43] The solution for equation (21a) is given by

display math

where

display math
display math

[44] The constants math formula and math formula can be obtained using inflow and outflow boundary conditions.

4.2.1. Case II-A: Fracture Is of Semi-Infinite Extent

[45] For the fracture with semi-infinite domain, the inflow and outflow boundary conditions are given byequation (6a) and equation (6c), respectively. For the semi-infinite boundary, the solution will remain bounded when math formula and math formula is obtained using inflow boundary condition and is given as.

display math

[46] Substituting the value of math formula and math formula in equation (22a) we get

display math

where math formula is given by

display math

4.2.2. Case II-B: Fracture Is of Finite Extent

[47] For the fracture with finite extent, the inflow and outflow boundary conditions are given by equation (6a) and equation (6b), respectively. Using these boundary conditions the constants math formula and math formula are computed as

display math
display math
display math
display math
display math

[48] Substituting equation (24) in equation (22a), we get

display math

4.3. Numerical Inversion of Solutions in Laplace Domain

[49] In order to obtain the solution in time domain, the analytical solution in the Laplace domain is to be inverted. The inversion of the Laplace transform can be done either analytically or numerically. However, analytical inversion of the Laplace transformed solution often leads to expressions having multiple integrals, which ultimately needs to be evaluated numerically [Tang et al., 1981; Sudicky and Frind, 1982]. However, looking to the complex nature of the solution obtained here a numerical inversion of the Laplace transformed solution is adopted. There are a number of algorithms available for numerical inversion; the present study uses the inversion method developed by de Hoog et al. [1982]. The de Hoog's algorithm has been used widely in the flow and transport problem for both advection and dispersion dominated cases [Neville et al., 2000; Gao et al., 2010]. The de Hoog's algorithm approximates the inverse Laplace transform in the form of a Fourier series.

display math

where T defines the period of the approximating Fourier series and its value is math formula [Neville et al., 2000]. The second parameter a is related to the singularities in the transformed solution and it estimated as

display math

[50] The value of math formula, math formula and math formula is taken as suggested by de Hoog et al. [1982]. For accelerating, the convergence of the Fourier series a Quotient differences algorithm is used [de Hoog et al., 1982].

5. Analytical Solution for Temporal Moments

[51] Temporal moments are often used to characterize the solute transport processes in porous media. The solute breakthrough curve (BTC) is more convenient to measure rather than the concentration profile therefore, the temporal moments can be useful in such cases. The analysis of the temporal moments can be a useful tool for understanding the impact of various processes on the solute transport.

[52] Temporal moments were computed by using the method given by Aris [1956], which has been used extensively [Goltz and Roberts, 1986; Valocchi, 1990; Srivastava et al., 2004]. The temporal moments are given by

display math

where Mn is the nth temporal moment of Cf about the origin (t = 0). Following Kendall and Stuart [1958] the normalized first moment about origin, normalized central, second and third moment can be given as

display math

where math formula is the nth temporal moment about the origin (for n = 1) and about the mean (for n = 2 and 3). The analytical expression for concentration in the Laplace domain contains a multiplication of several terms. When taking the logarithm of the concentration each term can be separately computed and then added together [Srivastava et al., 2004]. The mass of solute recovered at the sampling point is given by the zeroth temporal moment math formula, whereas the normalized first moment by the zeroth moment, normalized central second and third moments give the mean arrival time, variance and skewness of the breakthrough curve, respectively.

[53] For fracture and matrix of semi-infinite extent, the analytical solution in the Laplace domain is given byequation (23b) where terms math formula and math formula are given by equations (21b), (22b), and (22c), respectively, and assuming a zero initial concentrations leads to math formula. The expression for the moments are given as

display math
display math
display math
display math

where the prime represent the differentiation of math formula with respect to s and math formula is the value of math formula at s = 0. The first term on the right hand side of equation (28) represents the contribution of math formula; the second term that of math formula and the third term of math formula.

[54] From the above equations following meaningful parameters can be determined

display math
display math
display math

where math formula, math formula, and Cs are effective velocity, effective dispersion coefficient and the skewness, respectively. math formula, math formula, and math formula are the moments obtained after correcting for the finite pulse which can be obtained by neglecting the first term of the right hand side of the equation (28).

[55] Often it is assumed that bacteria, viruses and other larger sized contaminants/colloids do not penetrate into the rock matrix therefore, for such cases, the mass transfer into the matrix can be neglected. Assuming an equilibrium sorption with first type input boundary condition, the mass arrival is given by equation (28a)whereas the expression for the higher-order temporal moments can be simplified to

display math
display math
display math
display math
display math
display math

6. Comparison With Other Solutions

6.1. Comparison With Analytical Solutions

[56] There are a number of analytical solutions available for the reactive transport in the fractured porous medium. Most of the models assume a linear equilibrium sorption in the fracture matrix system, however, the nonequilibrium model developed in this study can incorporate both linear equilibrium and rate limited sorption in the fracture and porous matrix simultaneously. In order to evaluate the accuracy and applicability of the present semi analytical solution, it is desirable to compare the present semianalytical solution with the existing analytical solutions for the fracture matrix system. Therefore, the present semi analytical solution is compared with the two classical analytical solutions of Tang et al. [1981] and Sudicky and Frind [1982]. Since both the solutions were also derived in Laplace domain, therefore the comparison has been done analytically as given in Appendix A. It can be seen that under special cases, the present analytical solution in the Laplace domain is equivalent to the analytical solutions of Tang et al. [1981] and Sudicky and Frind [1982].

6.2. Comparison With the Multi Rate Mass Transfer (MRMT) Model

[57] Multi rate mass transfer (MRMT) model conceptualizes the fracture matrix system into mobile-immobile zone, where fractures and matrix are considered as mobile and immobile zone, respectively. Advection occurs through mobile zone whereas mass is transferred from the mobile zone into immobile zone via diffusion/first-order mass transfer. MRMT is a particular type of continuous time random walk model [Berkowitz et al., 2008; Kuntz et al., 2011]. MRMT model considers numerous types and rates of mass transfer simultaneously [Haggerty and Gorelick 1995]. MRMT model can describe mass transfer into heterogeneous media and is equivalent to diffusion limited mass transfer into spheres, cylinders or layers. Haggerty and Gorelick [1995] and Salamon et al. [2006]gave the first-order coefficients and capacity coefficient, which transform MRMT model into diffusion model of different geometries.

[58] In this study, a discrete approximation of the mass transfer coefficient is considered. The governing equations and analytical expression for the temporal moments for MRMT model are given in Appendix B. Temporal moment and BTC obtained using MRMT model and fracture matrix model are compared in section 7. Since the fracture matrix model and MRMT model are conceptually different; therefore, in order to match the BTC obtained by equilibrium fracture matrix model, MRMT model is used in the parameter estimation mode. The BTC obtained by equilibrium fracture matrix model was considered as observed data and was fitted by MRMT model using parameter estimation.

7. Model Analysis

[59] For the successful application of the model, it is required to analyze its behavior under a variety of situations. The purpose of this section is to evaluate the present model with different cases of sorption and mass transfer. As it is evident from the literature that the temporal moments have been widely used to study the behavior of solute transport processes therefore, the same is also utilized here to reflect the sensitivity of involved processes for a wide range of model parameter values. One of the uses of the sensitivity analysis is ultimately to provide an insight into the use of the model in simulation.

[60] In order to differentiate the transport/sorption processes, five cases of sorption and mass transfer are considered here. An initially solute free medium and a semi-infinite fracture and matrix domain are assumed. Case 1 neglects the mass transfer from fracture into matrix, i.e., the fracture matrix system is approximated as a single domain model. In case 2, a local equilibrium assumption (LEA) with an instantaneous sorption is assumed in both fracture and matrix. In case 3, a nonequilibrium sorption is assumed in matrix, i.e., math formula and an equilibrium sorption is assumed in fracture, i.e., math formula which is similar to that used by Xu and Worman [1999]; Maloszewski and Zuber [1990]. In case 4, a nonequilibrium sorption is assumed in both fracture and matrix math formula. In case 5, the fracture matrix system is conceptualized as a mobile-immobile zone and a discrete MRTM model is used. MRMT model considers an equilibrium sorption and liquid phase decay in both mobile and immobile region. In cases 1–4, decay is considered only in the liquid phase of the fracture and matrix. The model input parameters for all these cases are listed inTable 1.

Table 1. Parameter Values Taken for BTC and Temporal Moment Analysis for Various Cases of Sorption and Mass Transfer Considereda
Serial NumberParameterUnitCase 1Case 2Case 3Case 4MRMT
  • a

    Subscript f and m denotes fracture and matrix for a fracture matrix model. Subscript a and n denotes mobile and immobile region, respectively, for a MRMT model. A discrete approximation of the mass transfer coefficient is considered for MRMT model.

  • b

    For Temporal moment computation.

  • c

    MRMT model parameters.

1Lm1010101010
2Pulse durationday100/1100/1b100/1b100/1b100/1b
3Dmm2 d−11.00E-061.00E-061.00E-06
4θm0.010.010.010.01
5ρg cm−32.552.552.552.552.55
6vf/vacm d−11.01.01.01.01.0
7bμm50505050 
8αm0.50.50.50.50.5
9Kmml gm−10.0130.0130.0130.013
10km2day−10.0020.002
11Fm110.50.5
12Ff1110.5
13Kfm0.000160.000160.000160.00016
14kf2day−10.02
15λf = λmday−10.0030.0030.0030.003
16λa = λncday−10.003
17ω0cday−10.0015
18βtotc1.432
19Number of multirate seriesc30
20Rac4.861
21Rnc1.349

[61] Péclet number (Pe) representing the ratio of dispersive to advective travel time is an important parameter and is given as

display math

[62] A higher value of Péclet number denotes an advection-dominated transport whereas, a lower value of Péclet number represents dispersion dominated transport.

[63] In order to keep the parallelism with the MRMT model, the mass transfer coefficient math formula in the fracture matrix system is described as in [Carrera et al., 1998]

display math

[64] The Damköhler number representing SNE in fracture and matrix is defined as the ratio of hydraulic residence time to reaction time, thus defines the degree of nonequilibrium [Brusseau et al., 1989] and is given as

display math
display math

where L is the distance of interest and math formula and math formula are the Damköhler numbers representing SNE in fracture and matrix, respectively.

7.1. Comparison of the BTC

[65] Figure 2 shows the BTC obtained for the cases listed in Table 1, pulse duration of 100 days was considered and the BTC was measured at distance of 10 m. The BTC were obtained for case 1–4 using the transport parameters given in Table 1. However, for the case of MRMT model the model parameters such as mass transfer coefficient (ω0), capacity ratio (βtot) and retardation factor of mobile (Ra) and immobile (Rn) region were estimated using optimization algorithm as discussed earlier. It is seen from Figure 2 that the BTC obtained by MRMT model matches very well with the equilibrium fracture matrix model; however, a lower concentration at small transport time and at tail were observed in case of MRMT model BTC.

Figure 2.

Comparison of BTC for different cases of sorption and mass transfer at a distance of 10 m and for pulse duration of 100 days.

[66] When the mass transfer between fracture and matrix is considered, (i.e., Case 2–5), the peak of the BTC reduces and larger tailing as compared to the single domain model (i.e., Case 1) is observed. Considering nonequilibrium sorption in the fracture matrix system leads to an earlier arrival time, lower peak and larger tailing in the BTC as compared to an equilibrium fracture matrix and MRMT model.

7.2. Comparison of the Temporal Moment of the BTC

[67] In this section, the temporal moments of the BTC for the five cases of sorption and mass transfer as discussed earlier are presented. At the inlet of the fracture, a Dirichlet type boundary condition is considered and the pulse length math formula is taken as unity. Effects of various nondimensional parameters (given by equations (31)(33)) on the mean arrival time, variance and skewness of BTC are presented. For various fracture matrix models, the expression of the temporal moment are given by equation (28)(30) and for MRMT model the temporal moments are given in Appendix B.

7.2.1. Effect of Péclet Number

[68] In presence of mass transfer, Péclet number significantly affects the shape of the BTC. In case of fracture matrix model, as the Pe is increased, the solute arrival time increases whereas the peak concentration and the tailing reduces. Figure 3a shows the behavior of mean arrival time with Pe for different cases considered. Identical value of mean arrival time was obtained for cases 2–4; therefore, only case 4 is shown here. It should be noted that the value of the Pe was varied by changing the dispersivity while keeping fracture velocity constant. The mean arrival time increases or in other words the effective velocity decreases with an increase in Pe and at higher values of Pe (i.e., Pe > 50) it becomes asymptotic. This shows that at higher values of Pe, dispersion in fracture is not the main transport mechanism. The increase in the Pe or reduction in the dispersion coefficient would reduce the spreading of the plume therefore, the mean arrival time increases with Pe. The decrease in the effective velocity is due to the coupling between spreading and transformation. The spreading increases the volume occupied by the plume and thus the transformation capacity. The nonequilibrium sorption in fracture/matrix does not have much influence on the mean arrival time of the BTC, as reported by Srivastava and Brusseau [1996].

Figure 3.

(a) Behavior of mean arrival time of the BTC with Péclet number for different cases of sorption and mass transfer considered (identical value of mean arrival time was obtained for cases 2–4; therefore, only case 4 is shown here). (b) Behavior of variance of the BTC with Péclet number for different cases of sorption and mass transfer considered (identical value of mean arrival time was obtained for cases 3–4; therefore, only case 4 is shown here). (c) Behavior of coefficient of skewness of the BTC with Péclet number for different cases of sorption and mass transfer considered. (Identical value of mean arrival time was obtained for cases 3–4; therefore, only case 4 is shown here).

[69] Figure 3b shows the behavior of the second temporal moment or the variance of BTC with Pe and it can be seen that the variance of the BTC shows a nonmonotonic behavior. Identical value of variance was obtained for cases 3–4; therefore, only case 4 is shown here. At lower Pe, an increase in the variance of BTC is due to coupling between spreading and transformation whereas the decrease in the variance at higher Pe is due to reduction of the spreading in the fracture as demonstrated by case 1. In case of single domain model (i.e., Case 1), where the spreading in the fracture is the only dispersion process, the variance of the BTC reduces monotonically with an increase in Pe and as Pe→∞, the variance reduces to zero. In case of fracture matrix models, both spreading in fracture and fracture matrix interaction accounts for the dispersion process and at higher Pe the fracture matrix interaction becomes the main dispersion mechanism. As demonstrated by the asymptotic behavior of the variance for cases 2–5 at higher values of Pe.

[70] Figure 3c shows the effect of Pe on the skewness math formula of the BTC. It can be seen here that when mass transfer between fracture and matrix is neglected (i.e., case1) Cs reduces monotonically with an increase in Pe. Whereas, when mass transfer is considered (i.e., Case 2–5), the behavior of Cs becomes nonuniform and at higher Pe, the Cs becomes asymptotic. This again shows that at higher Pe, the effect of dispersion in the fracture becomes negligible. A larger tailing is observed in case of fracture matrix model as compared to MRMT model thus increasing the spreading and skewness of the BTC. When nonequilibrium sorption is considered in both fracture and matrix i.e., Case 4, a higher tailing and the spreading of the BTC is observed thus resulting in an increased variance and skewness as compared to when nonequilibrium is considered only in matrix.

7.2.2. Effect of Mass Transfer Coefficient

[71] The mass transfer coefficient significantly affects the peak and spreading of the BTC. For a fracture matrix model, as the mass transfer coefficient is increased, the peak of the BTC reduces and the arrival time and spreading increases. Figure 4 a shows the variation of the mean arrival time for different values of the mass transfer coefficient. When mass transfer between fracture and matrix increases, the mean arrival time increases or effective velocity reduces. Cases 2–4 show an identical behavior of mean arrival time with the mass transfer coefficient; therefore, only case 4 is shown here. At very small values of mass transfer coefficient, the fracture matrix model behaves as a single region model. At higher values of mass transfer coefficient, the MRMT model BTC becomes symmetric and any further increase in mass transfer coefficient does not affect the mean arrival time therefore the mean arrival time becomes asymptotic.

Figure 4.

(a) Mean arrival time of the BTC with mass transfer coefficient for different cases of sorption and mass transfer considered (identical value of mean arrival time was obtained for cases 2–4; therefore, only case 4 is shown here). (b) Variance of the BTC with mass transfer coefficient for different cases of sorption and mass transfer considered. (c) Coefficient of skewness of the BTC with mass transfer coefficient for different cases of sorption and mass transfer considered.

[72] Figure 4bshows the effect of mass transfer coefficient on the variance of the BTC. For the case of fracture matrix models, at small values of mass transfer coefficient, the variance of the BTC becomes asymptotic. This again shows that at small values of mass transfer coefficient, the fracture matrix model behaves similar to a single region model. However, a higher asymptotic value of variance is observed for case 4, which is due to presence of rate-limited sorption in fracture. In case of MRMT model, the variance shows a nonmonotonic behavior and at higher values of mass transfer coefficient, the variance becomes asymptotic. At higher values of mass transfer coefficient, MRMT model behaves similar to a single region model with the porosity equal to the total porosity of the medium therefore, the BTC becomes symmetric and thus the variance approaches an asymptotic value. Similarly, at very small value of mass transfer coefficient the MRMT model also behaves as a single region model with the porosity equal to the porosity of the mobile domain.

[73] Figure 4cshows the effect of mass transfer on the coefficient of skewness of the BTC. It is seen that for fracture-matrix and MRMT model the behavior of coefficient of skewness is non-monotonic. At low values of mass transfer coefficient, the fracture matrix model and MRMT model behave as a single region model. At higher values of mass transfer coefficient identical values of skewness is obtained for the case of equilibrium and nonequilibrium fracture matrix models. It is also seen that at lower mass transfer coefficient, nonequilibrium sorption in fracture and at higher mass transfer coefficient nonequilibrium in matrix have a significant influence on the variance and skewness of the BTC.

7.2.3. Effect of Damköhler Number

[74] At higher values of Damköhler number or larger mass transfer rate, the BTC with nonequilibrium sorption model behaves similar to an equilibrium sorption model. Mean arrival time of the solute essentially remains constant with the Damköhler number (not shown here). Figure 5 shows the variance of the BTC with different values of fracture and matrix Damköhler number. At higher values of Damköhler number ( math formula = math formula > 1) variance of the BTC becomes asymptotic and identical to that obtained when equilibrium sorption is assumed in fracture and matrix (case 2). However, equilibrium is achieved at a smaller Damköhler number when nonequilibrium sorption is considered only in matrix as compared to the cases when nonequilibrium sorption is considered in fracture. Larger spreading of the BTC is observed when nonequilibrium sorption is considered in both fracture and matrix as compared to the cases when nonequilibrium sorption is considered only in matrix.

Figure 5.

Variance of the BTC with fracture and matrix Damköhler number for different cases of sorption and mass transfer considered.

[75] Cs shows a similar behavior to that of variance as discussed earlier (therefore not shown here). A higher value of Cs is observed when nonequilibrium sorption is considered only in matrix, as compared to the case when nonequilibrium sorption is considered in the fracture. At higher values of Damköhler number (i.e., math formula = math formula > 1) sorption kinetics in fracture and matrix are not so important therefore, local equilibrium assumption can be applied.

8. Model Evaluation

[76] In order to evaluate the applicability of the present model, the model has been applied to simulate the experimental data. The experimental data were simulated using equilibrium fracture matrix model, nonequilibrium fracture matrix model and MRMT model. For simulation of the BTC using MRMT model, STAMMT-L numerical code [Haggerty and Reeves, 2002] is used. STAMMT-L can incorporate both single rate and multi rate mass transfer and considers both discrete and continuous distribution of mass transfer rate.

8.1. Data Set 1

[77] Grisak et al. [1980] conducted laboratory investigation in which Calcium (Ca) and Chloride (Cl) were passed as tracers through a large, relatively undisturbed 0.76 m long and 0.65 m diameter cylindrical sample of fractured clay loam till.

[78] Assuming a semi-infinite fracture and a finite matrix extent, the nonreactive chloride data were simulated using equilibrium model (not shown here), the flow velocity of 29.7 m d−1; math formula; fracture aperture 2b = 40 μm density of the porous matrix math formula and fracture spacing of 6 cm was considered [Grisak et al., 1980]. The adsorption of Cl in the fracture and matrix is neglected therefore, only effective diffusion coefficient (Dm) and dispersivity (α) values were estimated using a Levenberg-Marquardt nonlinear least squares optimization algorithm asDm= 2.40E-06 m2 d−1 and α = 0.114 m. Grisak et al. [1980] fitted the dispersivity value, α = 0.15 m using their model.

[79] Calcium data were then fitted using equilibrium and nonequilibrium adsorption models. The value of α was kept same as estimated for Cl. The value of Dm for Calcium is taken as math formula m2 d−1 which is obtained by multiplying the ratio of free water diffusion coefficients of Calcium and Chloride (i.e., 38%) to the Dm value fitted for the chloride. Equilibrium adsorption model parameters Km and Kf were estimated as Km = 0.063 ml gm−1and math formula. The estimated value of distribution coefficient corresponds to fracture and matrix retardation factor of 16 and 1.30, respectively. The effective diffusion coefficient selected by Grisak et al. [1980] was Dm = 1.64E − 06 m2 d−1 and assumed that reactive transport takes place in matrix only (i.e., math formula). They also concluded that the best fit appears to be at math formula.

[80] For the nonequilibrium transport model, the value of Dm was kept same as used for the equilibrium adsorption model and assuming an instantaneous sorption in the matrix i.e., math formula and math formula. Using optimization algorithm math formula, math formula, math formula and math formula were estimated. A very low value of math formula indicates that the contribution of SNE to total nonequilibrium becomes insignificant and the apparent retardation factor in the fracture as given by math formula this value is similar to that estimated by equilibrium model. The BTC obtained using both equilibrium and nonequilibrium models superimpose over each other as shown in Figure 6.

Figure 6.

Simulation of the experimental data of Calcium [Grisak et al. 1980] input parameters for this simulation is given in Table 2 (equilibrium and nonequilibrium simulation overlay each other).

[81] Using MRMT model the nonreactive chloride data was simulated (not shown here). The mass transfer coefficient (ω0), capacity ratio (βtot) and dispersivity (α) values were estimated using an optimization algorithm and were found to be 0.35 d−1, 27.26 and 0.114 m, respectively. For a nonreactive solute, the capacity ratio is the ratio of immobile zone to mobile zone porosity, i.e., math formula. The Calcium data was then simulated using MRMT model, the value of α and the ratio of porosity of immobile and mobile region were kept same as estimated earlier for Cl. The mass transfer coefficient (ω0) and retardation factor of mobile (Ra) and immobile (Rn) region were estimated using optimization algorithm and are math formula, math formula and math formula. It can be seen that for the case of MRMT model, a higher value of fracture and matrix retardation factor is obtained as compared to fracture matrix model.

[82] Figure 6 and Table 2 compares the BTC obtained and input parameters used for simulation of equilibrium, nonequilibrium and MRMT models and it can be seen that all the three models fit the observed data very well. However, very small difference is observed between MRMT and fracture matrix models, an earlier arrival time and higher peak concentration is observed with MRMT model.

Table 2. Transport Parameters Taken for Data Simulation of Grisak et al. [1980]a
Serial NumberParameterUnitGrisak et al. [1980]Equilibrium ModelNonequilibrium ModelMRMT
  • a

    Subscript f and m denotes fracture and matrix for a fracture matrix model. Subscript a and n denotes mobile and immobile region, respectively, for MRMT model. A discrete approximation of the mass transfer coefficient is considered for MRMT model.

  • b

    MRMT model parameters.

1LM0.760.760.760.76
2Dmm2 d−11.64E-079.12E-079.12E-07
3θm0.350.350.35
4ρg cm−31.71.71.71.7
5vf/vaam d−129.729.729.729.7
6bμm202020
7αM0.150.1140.1140.114
8Kmml gm−10.010.060.06
9km2day−1
10Fm111
11Ff110.57
12KfM00.00030.0005
13kf2day−15.00E-05
14λf = λmday−1000
15λa = λnaday−10
16ω0aday−10.009
17βtota80.12
18Number of multirate series30
19Ra16.27
20Rn47.81
21RMSE0.020.020.02
22r20.980.980.99

[83] The performance of the models were evaluated by considering root mean square error (RMSE) and determination coefficient math formula which are expressed as

display math
display math

where math formula and math formula are observed and computed concentrations respectively, math formula is the mean value of math formula and N is the number of the observed concentration data at a particular observation point. Identical values of RMSE and math formula, i.e., 0.02 and 0.98, respectively, were obtained for equilibrium, nonequilibrium, and MRMT models as given in Table 2.

8.2. Data Set 2

[84] Neretnieks et al. [1982]conducted an experimental investigation of the radionuclide migration in a granite cylindrical core 30 cm long and 20 cm in diameter. They obtained breakthrough curves for the nonsorbing tracers, titrated water and ligno-sulfonate molecule and for sorbing tracers Cesium (Cs+) and Strontium (Sr2+). Neretnieks et al. [1982] considered porosity of the rock math formula; fracture aperture 2b = 0.18 mm, density of the crystalline rock math formula and dispersivity math formula. Using batch sorption experiments, they found that for Strontium the distribution coefficient was math formula. However, they found that for Strontium (Sr2+), the best fit was observed at math formula. The effective diffusivity for Sr2+ math formula was taken which is same as measured by Skagius et al. [1982] for the granite. Neretnieks et al. [1982]in their experiment S-7 considered a injection pulse of 15 min duration and flow velocity was math formula. In the present study, Sr2+data were simulated using equilibrium, nonequilibrium, and MRMT model considering a semi-infinite fracture and matrix domain. The model parameters were estimated using Levenberg-Marquardt nonlinear least squares optimization algorithm.

[85] First, an equilibrium sorption model was fitted to the experimental data, the effective diffusivity and volume equilibrium constant for Sr2+ was taken same as reported by Neretnieks et al. [1982]. The remaining parameters math formula and Kf were estimated as math formula and math formula. Neretnieks et al. [1982] estimated dispersivity math formula and math formula. It can be seen that a higher value of dispersivity was obtained with equilibrium transport model. Similarly, the experimental data were also simulated using the nonequilibrium model. The value of math formula and math formula were taken which are same as used for equilibrium model. When the effective diffusion coefficient of matrix is low, then an equilibrium sorption model in the matrix could be adequate; therefore, the value of math formula and math formula are considered. Using optimization algorithm math formula, math formula, math formula and math formula were obtained. It can be seen that an identical value of Kf as estimated by Neretnieks et al. [1982] were obtained for nonequilibrium and equilibrium model, the other model parameters are listed in Table 3. A lower value of dispersivity was estimated by nonequilibrium model than the value estimated by Neretnieks et al. [1982]. Neretnieks et al. [1982] reported that both fast and slow reactions contribute to the total retardation in the fracture. The value of the fracture Damköhler number is obtained as math formula. As the computed value of math formula therefore, LEA is not valid and the contribution of SNE to total nonequilibrium is significant. The retardation factor in instantaneous and rate limited sorption domain of fracture as given by math formula and math formula, respectively and are math formula and math formula, the total retardation factor math formulais 14.33. It can be seen that about 65% of the retardation in the fracture occurs in the rate limited sorption domain indicating again that a larger portion of sorption is due to first-order reaction.

Table 3. Transport Parameters Taken for Data Simulation of Neretnieks et al. [1982]a
Serial NumberParameterUnitNeretnieks et al. [1982]Equilibrium ModelNonequilibrium ModelMRMT
  • a

    Subscript f and m denotes fracture and matrix for a fracture matrix model. Subscript a and n denotes mobile and immobile region, respectively, for MRMT model. A discrete approximation of the mass transfer coefficient is considered for MRMT model.

  • b

    MRMT model parameters.

1Lm0.300.300.300.30
2Pulse durationh0.250.250.250.25
3Dmm2 h−13.60E-093.60E-093.60E-09
4θm0.010.010.01
5ρg cm−32.652.652.652.65
6vf/vabm h−10.4140.4140.4140.414
7bmm0.090.090.09
8αm0.0250.1260.01580.01
9Kmml gm−1333
10km2h−1
11Fm111
12Ff110.30
13Kfm0.0010.00090.0012
14kf2h−10.16
15λf = λmh−1000
16λa = λnbh−10
17ω0bh−10. 027
18βtotb9.13
19Number of multirate seriesb30
20Rab2.59
21Rnb1.77
22RMSE0.0050.0020.001
23r20.8260.9630.977

[86] The experimental data were also simulated using MRMT model. First, the simulation was performed with math formula and math formula as estimated experimentally by Neretnieks et al. [1982] and dispersivity (α), mass transfer coefficient (ω0) and capacity ratio (βtot) were estimated using optimization algorithm. It was found that very unrealistic estimates of parameters (α and Ra) along with inferior fit of the BTC were observed (not shown here). Therefore, retardation factor in mobile and immobile region along with dispersivity (α), mass transfer coefficient (ω0), capacity ratio (βtot) were estimated using optimization algorithm as math formula, math formula and math formula and the values of retardation factors math formula and math formula were obtained as shown in Table 3.

[87] Figure 7 shows the observed and simulated BTC's and Table 3 lists the input parameters used for simulation of equilibrium, nonequilibrium and MRMT model. It can be seen that nonequilibrium simulates the observed data better as compared to the equilibrium model. However, the best fit was observed for the case of MRMT model when Kn and math formulawere also varied. Unlike the equilibrium model, the peak and early stage BTC was better-simulated using nonequilibrium and MRMT model. However, at larger transport time (∼20 h), the nonequilibrium model overestimates the tail of BTC. The RMSE and math formula values obtained for equilibrium, nonequilibrium and MRMT models are given Table 3 and it can be seen that a higher math formula and a lower RMSE value was observed with the MRMT model. At an earlier transport time the BTC obtained by MRMT and nonequilibrium model are nearly same. The MRMT model simulates the tail of BTC much better as compared to nonequilibrium model as it considers numerous types and rates of mass transfer from fracture into matrix. However, it should be noted that for the case of MRMT model all the parameters were estimated using optimization algorithm whereas, for nonequilibrium model Dm, math formula, b and Km were independently estimated as given by Neretnieks et al. [1982]. It is also seen for MRMT model when Kn and math formula were independently estimated, very unrealistic estimates of parameters and poor fitting of the BTC were obtained.

Figure 7.

Simulation of the experimental data of Strontium [Neretnieks et al. 1982]. The input parameters for this simulation is given in Table 3.

[88] Neretnieks et al. [1982] found that considerable channeling occurs in the fracture and therefore, the BTC exhibited a steep rise in the concentration due to quick channels, which was then followed by the slower channels. The channeling has a considerable influence on the tailing of the BTC due to presence of slower channels. The channeling effects in the fracture have not been accounted in the present model. Neretnieks et al. [1982] also found that the surface of fissure was different from that of the granite and the rock has a different color up to a depth of few millimeters from the fissure surface, than at larger distances. This altered material property may also lead to heterogeneity in matrix transport parameters, which has not been accounted in the present model and this may be the probable reason for slight overestimation of the tail of BTC.

[89] Experimental run S2 of Neretnieks et al. [1982] was simulated using the parameters estimated for the run S7. Only the value of matrix diffusion coefficient was adjusted in order to account for variation in the fracture velocity. Neretnieks et al. [1982] in their experiment S2 considered a continuous type inflow and flow velocity was observed as math formula. Observed data were simulated using equilibrium and nonequilibrium fracture matrix model and MRMT model. The value of Dm for run S2 is taken as math formula which is obtained by multiplying the ratio of flow velocity of S2 and S7 run (i.e., 1.747) to the Dm value fitted for the S7 run. In case of MRMT model, the mass transfer coefficient ( math formula) was adjusted similarly to account for variation in the velocity. The adjusted value of mass transfer coefficient was obtained as math formula. Figure 8 shows the observed and simulated BTC's for run S2. For equilibrium, nonequilibrium and MRMT model, the RMSE values obtained as 0.068, 0.064 and 0.098, respectively; similarly, the r2 values obtained are 0.98, 0.96 and 0.9, respectively. It can be seen that the present model and MRMT model simulated the BTC very well compared to equilibrium model. It is noted that through these simulations again the better performance of present model and MRMT model over equilibrium fracture matrix model was found. However, the performance of nonequilibrium model and MRMT model were found very much similar.

Figure 8.

Simulation of the experimental run S2 of Neretnieks et al. [1982] using the estimated parameters of S7 run.

9. Conclusions

[90] In the fractured porous media, a very few studies exist that incorporate nonequilibrium sorption in fracture/matrix. In this study, a generalized model, which incorporates nonequilibrium sorption in both fracture and porous matrix, is developed. The present model accounts for both physical nonequilibrium (PNE) and sorption nonequilibrium (SNE) in the fractured porous media. A semianalytical solution and the analytical expressions for the temporal moments for this new model is developed. The semianalytical solution can incorporate flexible initial and boundary conditions; in addition, the solution is capable of incorporating both finite or semi-infinite fracture and matrix domain. The present semianalytical solution was compared with the existing analytical solutions for the fracture matrix system and with the MRMT model. It was found that a very good agreement between the present model with the available analytical solutions has been obtained.

[91] In order to differentiate between various sorption/transport process different cases of sorption and mass transfer were analyzed by comparing the BTC and temporal moments. It can be seen that considering the nonequilibrium sorption in fracture matrix system leads to an earlier arrival time and large tailing thus increasing the variance and skewness of the BTC as compared to equilibrium sorption assumption and MRMT model. Nonequilibrium sorption assumption in both fracture and matrix leads to a larger tailing and spreading of BTC as compared to when nonequilibrium sorption is considered only in matrix. It is also seen that at higher value of Damköhler number (Da > 1) the nonequilibrium model behaves similarly to an equilibrium model.

[92] The applicability of the nonequilibrium model was also tested by simulating the experimental data of Calcium and Strontium. The simulation clearly demonstrated the usefulness of the present model over equilibrium fracture matrix models. As the present model incorporates a more detailed transport/sorption dynamics, it is expected to be more useful for process level investigation of solute transport in fractured porous media.

Appendix A:: Equivalence of the Present Model to Existing Analytical Solutions

A1. Equivalence of the Present Model to Tang et al. [1981] Solution

[93] Tang et al. [1981]derived an analytical solution considering fracture and matrix of semi-infinite extent. In the present solution, on substituting the values of math formula; math formula = math formula = math formula = math formula = 0, math formula; and considering boundary conditions described by equations (6a), (6c), (6d), and (6f) the analytical expression in Laplace domain as given by Tang et al. [1981] can be obtained.

[94] After substituting the above values, the concentration in matrix as given by equation (14) is simplified to

display math

[95] Tang et al. [1981] considered same decay coefficient (λ) in fracture and matrix. The decay rate in fracture and matrix in the present model when divided by the retardation factor in fracture and matrix is equivalent to the decay rate considered by Tang et al. [1981]. Therefore, the equivalence relationship becomes

display math

[96] Simplifying the equation (A1) Further, we get

display math

where

display math
display math
display math

[97] It can be seen that the equation (A3) and (A4) is identical to equation (17) and (18) of Tang et al. [1981].

[98] For the Fracture for a semi-infinite fracture domain and constant concentration inflow boundary condition the solution given byequation (22a) can be modified as

display math

[99] On further simplifying the solution

display math

where

display math
display math
display math

[100] The equation (A6a) is identical to equation (25) of Tang et al. [1981].

A2. Equivalence to the Present Model to Analytical Solution of Sudicky and Frind [1982]

[101] Sudicky and Frind [1982]derived an analytical solution assuming the fracture is of semi-infinite extent. On substituting the values of math formula; considering zero initial condition; neglecting decay in sorbed phase i.e., math formula = math formula = math formula = math formula = 0 and using boundary conditions described by equations (6a), (6c), (6d), and (6e) using the present solution the analytical expression in Laplace domain as given by Sudicky and Frind [1982] can be obtained.

[102] for the matrix of finite domain the equation of matrix in laplace domain is given as

display math

[103] Sudicky and Frind [1982] considered same decay coefficient (λ) in fracture and matrix. The decay rate in fracture and matrix in the present model when divided by the retardation factor in fracture and matrix is equivalent to the decay rate considered by Sudicky and Frind [1982]. Therefore, the equivalence relationship becomes

display math

[104] The using equation (A8) the equation (A7) can be further simplified as

display math

where math formula and math formula is given by (A3b) and (A3c), respectively,

display math
display math

[105] For the Fracture for a semi-infinite fracture domain and constant concentration inflow boundary condition the solution given byequation (22a) can be modified as

display math
display math

where math formula, math formula and math formula are given by (A6b), (A6c), and (A6d), respectively.

[106] The expression given by (A13) is identical to equation (19) of Sudicky and Frind [1982].

Appendix B:: Multi Rate Mass Transfer Model

[107] The governing equation of the mobile region can be written as

display math
display math

[108] For immobile region the equation can be written as

display math

for

display math

where Ca and math formula is solute concentration in mobile zone and jth immobile zone, respectively, math formula and math formula is the retardation factor in mobile and jth immobile region, respectively. math formula and math formula are the distribution coefficient math formula of the mobile and jth immobile zone, respectively; f is the mass fraction of the sorbed phase in the mobile zone. math formula and math formula are the porosity of the mobile and immobile region, respectively. va is the flow velocity math formula in the mobile region and Da is the hydrodynamic dispersion coefficient math formula in the mobile region. math formula and math formula is the decay coefficient math formula in the mobile and immobile zone, respectively; math formulais the first-order mass transfer coefficient math formula for the jth immobile zone and math formula is the capacity ratios for jth immobile zones, and math formula is the total capacity ratio of all immobile zones added together and is given by

display math

[109] In the present study, a generalized boundary condition similar to that used by Neville et al. [2000] is considered.

display math

where math formula is the source concentration, math formula is the Heaviside step function and it is used for describing step input from time 0 to math formula. math formula represents a constant concentration or Diritchlet type boundary condition and math formula represents a constant flux or Cauchy type boundary condition.

[110] For the outflow boundary condition, the mobile region of semi-infinite domain are considered here, the boundary condition is described by

display math

[111] Zero concentration initial condition is considered in mobile and immobile region. The analytical solution in the Laplace domain is given by Joshi [2012] as

display math

where

display math
display math

where s is the Laplace operator and math formula is the pulse duration. In order to obtain the breakthrough curve, the analytical solution in the Laplace domain (i.e., equation (B3)) can be numerically inverted.

[112] Using equations (26) and (27) the expression for the temporal moments can be computed. It should be noted that the analytical expressions of the temporal moments for MRMT model are identical to the fracture matrix model and are given by equation (28). In the expressions of temporal moments (i.e., equation (28)) the term math formula and math formula for MRMT model is given by equation (B3).

Notation
math formula

Fracture aperture math formula.

Ca

Concentration of solute in the mobile region math formula.

Cf

Concentration of solute in the fracture math formula.

Cn

Concentration of solute in the immobile region math formula.

Cm

Concentration of solute in the porous matrix math formula.

Cs

Coefficient of skewness.

math formula

Source concentration math formula.

Da

Hydrodynamic dispersion coefficient in the mobile region math formula.

math formula

Effective dispersion coefficient math formula.

Df

Dispersion coefficient in fracture math formula.

Dm

Effective diffusion coefficient math formula.

f

Mass fraction of the sorbed phase in the mobile zone.

Ff

Fraction of the sorbent in the fracture for which sorption is instantaneous.

Fm

Fraction of the sorbent in the matrix for which sorption is instantaneous.

math formula

Heaviside step function.

math formula

Distribution coefficient of the mobile zone math formula.

math formula

Distribution coefficient of the jth immobile zone math formula.

math formula

Distribution coefficient for the fracture math formula.

math formula

First-order sorption kinetic coefficient in the fracture math formula.

math formula

Damköhler numbers representing SNE in fracture.

Km

Distribution coefficient for the porous matrix math formula.

math formula

First-order sorption kinetic coefficient in matrix math formula.

math formula

Damköhler numbers representing SNE in matrix.

Lf

Length of fracture (L).

Lm

Length of matrix block (L).

math formula

Zero temporal moment of the plume.

Pe

Péclet number.

math formula

Retardation factor for the equilibrium sorption site in the fracture.

math formula

Retardation factor for the equilibrium sorption site in porous matrix.

s

Laplace transform variable.

math formula

Mass of sorbate per unit length of the fracture for which the sorption is governed by instantaneous equilibrium linear isotherm math formula.

math formula

Mass of sorbate per unit length of the fracture for which the sorption is governed by first-order kinetic sorption math formula.

math formula

Mass of solute adsorbed per unit mass of solid in the porous matrix for which the sorption is governed by instantaneous equilibrium linear isotherm math formula.

math formula

Mass of solute adsorbed per unit mass of solid in the porous matrix for which the sorption is governed by first-order kinetic sorption math formula.

math formula

Time, (T).

t0

Pulse duration time.

va

Flow velocity in the mobile region math formula.

math formula

Effective velocity math formula.

vf

Fracture pore water velocity, math formula.

math formula

Capacity ratios for jth immobile zones.

math formula

Total capacity ratio of all immobile zones.

math formula and math formula

Decay coefficient math formula in the mobile and immobile zone, respectively.

math formula

First-order degradation rate constant in the liquid phase for fracture math formula.

math formula and math formula

First-order degradation rate constants in the sorbed phase for the fracture at site 1 and 2, respectively math formula.

math formula

First-order degradation rate constant in the liquid phase for matrix math formula.

math formula and math formula

First-order degradation rate constants in the sorbed phase for the porous matrix at site 1 and 2, respectively math formula.

math formula, math formula and math formula

First, second and third time moments, respectively.

math formula, math formula and math formula

First, second and third time moments for the finite pulse, respectively.

math formula

Non dimensional mass transfer coefficient.

math formula

Bulk density of the porous medium, math formula.

math formula

Matrix porosity.

Ancillary