Subsurface contaminant transport in the presence of colloids: Effect of nonlinear and nonequilibrium interactions

Transcription

1 WATER RESOURCES RESEARCH, VOL. 43,, doi: /2006wr005418, 2007 Subsurface contaminant transport in the presence of colloids: Effect of nonlinear and nonequilibrium interactions Hesham M. Bekhit 1,2 and Ahmed E. Hassan 1,2 Received 10 August 2006; revised 26 March 2007; accepted 7 May 2007; published 9 August [1] The effect of kinetic nonlinear sorption of contaminants in the presence of colloids is the focus of this study. Different sorption isotherms are considered where contaminant sorption and colloid deposition are assumed to be linear or nonlinear (Freundlich), and contaminant attachment to mobile and immobile colloids is assumed to follow either linear or Langmuir isotherms. Varying combinations accounting for different possibilities are used to investigate effects of different isotherms on contaminant transport. Twodimensional numerical simulations in homogenous media show that the effect of colloids on nonlinearly sorbing contaminant is altered from facilitation to retardation depending on the Freundlich exponent and concentration value. One finding from the study indicates that incorporating the colloid effect on contaminant transport does not necessarily represent a conservative assumption. The study shows that ignoring the fact that colloids have limited sites and describing contaminant attachment to colloids by linear isotherms may lead to inaccurate results. In addition, it is found that assuming colloids are linearly deposited on the solid matrix is a conservative assumption in the applications that focus on peak concentration arrival. However, when small contaminant concentrations are of concern (i.e., early arrival is the quantity of interest), assuming nonlinear colloidal deposition becomes the critical scenario. Citation: Bekhit, H. M., and A. E. Hassan (2007), Subsurface contaminant transport in the presence of colloids: Effect of nonlinear and nonequilibrium interactions, Water Resour. Res., 43,, doi: /2006wr Introduction [2] Analytical and numerical modeling of reactive contaminant transport in subsurface environments has been an active area of research for decades. The studies on contaminant sorption and desorption have evolved from using a simple retardation factor based on equilibrium assumptions to incorporating kinetic reactions. These studies were conducted by incorporating different sorption sites with varying characteristics and using a variety of expressions characterizing physical and chemical nonequilibrium [Sardin et al., 1991]. Through a large body of analytical and numerical modeling studies, solutions were devised for the reactive contaminant transport problem under linear equilibrium and nonequilibrium (kinetic) conditions. Sardin et al. [1991] provide a detailed review of studies focused on modeling nonequilibrium transport of linearly sorbing solutes in porous media. [3] Modeling reactive chemical transport of contaminants experiencing nonlinear equilibrium or kinetic reactions has received little attention as compared to linear reactions. Studies that consider nonlinear interaction show agreement over the importance of the nonlinear sorption processes. These studies concluded that simplifying the effect of 1 Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Orman, Giza, Egypt. 2 Division of Hydrologic Sciences, Desert Research Institute, Las Vegas, Nevada, USA. Copyright 2007 by the American Geophysical Union /07/2006WR nonlinear sorption behavior to linear sorption leads to significant errors in predicting travel distance and could produce misleading results. Because of nonlinearity in the sorption process, the amount of contaminant sorbed onto the solid matrix highly depends on the concentration value of dissolved contaminant where larger concentrations exhibit different sorption and retardation than do smaller concentrations. On the basis of the nonlinear factor, the portion of the plume with high concentration exhibits more or less retardation than the portion having lower concentration. [4] Several studies revealed the importance of the nonlinear behavior of the contaminant [van Genuchten and Cleary, 1982; Bolt, 1982]. One of the examples is the work presented by Weber et al. [1991] who studied the effects of nonlinearity on plume shape. They concluded that the velocity of the center of mass of a contaminant plume experiencing nonlinear sorption is significantly lower than that of a linearly reacting plume. In addition, Serrano [2003] studied the propagation of nonlinear reactive contaminants in porous media. His results indicated that nonlinear reactions may have a significant effect on the shape and spatial distribution of a contaminant plume at any given time. [5] This study investigates the role of nonlinear sorption on contaminant transport in the presence of colloids. Recognition that colloids might be important in facilitating transport of contaminants arose from studies demonstrating that strongly sorbing contaminants could travel much further and faster than anticipated from traditional solute transport models [McCarthy and McKay, 2004]. For example, at the Nevada Test Site, strongly sorbing radionuclides 1of19

2 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS were observed in groundwater at large distances outside nuclear detonation cavities [Coles and Ramspott, 1982; Buddemeier and Hunt, 1988; Kersting et al., 1999]. Likewise, enhanced transport of radionuclides at Chalk River Nuclear Laboratories in Canada [Walton and Merritt, 1980; Champ et al., 1984], Oak Ridge National Laboratory in Tennessee [McCarthy et al., 1998a, 1998b], and a U deposit in Australia [Short et al., 1988] were attributed to association of the contaminants with a mobile colloidal phase. In addition, at two separate sites at Los Alamos, New Mexico, plutonium and americium were detected at much greater distances from the source than predicted by dual porosity transport models [Nelson and Orlandini, 1986]. [6] These findings compelled researchers to develop transport models that account for colloid-facilitated contaminant transport. These models suggest that depending on system conditions, the presence of colloids in porous media may either enhance or retard contaminant transport [e.g., Grindrod, 1993; Smith and Degueldre, 1993; Abdel-Salam and Chrysikopoulos, 1995a, 1995b; Ibaraki and Sudicky, 1995a, 1995b; Baek and Pitt, 1996; Cvetkovic, 2000; Marseguerra et al., 2001a, 2001b; Ren and Packman, 2004, 2005; Bekhit and Hassan, 2005a; Bekhit et al., 2006]. [7] For colloid-facilitated transport to represent an important process in the subsurface, colloids must be present in sufficient quantities in the porous medium and must remain in suspension for sufficiently long times. Grolimund and Borkovec [2004] describe a scenario of this process. A natural porous medium is saturated with monovalent cations at first. Then, the medium is infiltrated with water of low ionic strength. Under these conditions a considerable amount of colloidal particles is released from natural porous media and remains suspended [Grolimund and Borkovec, 1999; Grolimund et al., 2001]. The sudden change in water chemistry (e.g., ionic strength) was also shown to cause a rapid release of colloidal particles that were attached to the solid matrix [Roy and Dzombak, 1998]. This scenario results in high colloid concentration and may be encountered in real field situations. [8] Grolimund and Borkovec [2004] mention two realistic situations, where colloid release is likely. The first situation concerns an aquifer which is in contact with seawater and is infiltrated with freshwater. The second situation concerns a soil or an aquifer saturated with a dump site leachate (typically dominated by monovalent cations), and is then infiltrated with rainwater or groundwater of low salinity. In both situations, a normality front develops, within which the salinity of the pore water drops substantially. Just after the normality front, colloids will be released from the solid matrix [Grolimund and Borkovec, 2004]. If the aquifer is contaminated with strongly sorbing contaminants, they will be transported by the released colloids. The importance of colloid-facilitated transport in this situation has been demonstrated by means of laboratory column experiments [Grolimund et al., 1996; Roy and Dzombak, 1998; Saiers and Hornberger, 1996, 1999; Faure et al., 1996; Flury et al., 2002] and in the field [Kersting et al., 1999]. [9] When looking at colloids and their effect on subsurface transport, two classes of studies can be distinguished. The first class focuses on transport of colloidal particles and their interaction with the solid matrix in groundwater systems. The second class of studies focuses on colloid-facilitated contaminant transport where interactions between colloids, contaminants, and solid matrix are studied. Numerous studies in each of the two classes have been reported for saturated porous media, fractured media, and also unsaturated media. We review in the following some of the experimental and numerical studies reported in the two categories with emphasis on saturated porous media studies. [10] Examples of laboratory studies in the first class include the work of Mills et al. [1991] who summarized modeling approaches developed for colloid transport and presented a test model for porous media. Bradford et al. [2002] conducted saturated soil column experiments to explore the influence of colloid size and distribution of soil grain size on the transport of colloids in saturated porous media. The results presented by Bradford et al. [2002] showed that effluent colloid concentrations and spatial distribution of colloids attached to porous media highly depend on colloid size and distribution of soil grain size. Peak effluent concentrations decreased and surface mass retained by the porous media increased when the colloid size increased and the median soil grain size decreased. [11] Experimental results of other studies showed that dissolved organic matter and other types of colloids demonstrate a high affinity for the solid matrix [Saiers and Hornberger, 1999; Saiers et al., 1994; Dunnivant et al., 1992; Knabner et al., 1996; Corapcioglu and Choi, 1996]. One of the main findings produced from these experiments showed that colloids are reactive, and their transport could be predicted using the advection-dispersion equation and simulating the deposition process by an equilibrium isotherm, a first-order kinetic process or, more complicated, nonlinear kinetics. [12] Sirivithayapakorn and Keller [2003] conducted experiments to simulate colloid transport in saturated porous media. They focused on studying the effects of excluding colloids from areas having small pore sizes. They found that the size exclusion phenomenon occurred when the ratio of pore throat to colloid diameter was less than 1.5. [13] Different mathematical models were developed to describe colloid transport in the subsurface. Johnson and Elimelech [1995] accounted for the possible occurrence of blocking, where they presented a model containing dynamic blocking functions that describe aspects of particle deposition. Saiers et al. [1994] modeled the transport and deposition of colloid anatase, boehmite, and silica in sand columns by employing both first-order and second-order blocking kinetic approaches. Abdel-Salam and Chrysikopoulos [1994] presented an analytical solution for one-dimensional colloid transport in a single rock fracture with and without colloid penetration into the rock matrix. [14] Johnson et al. [1996] incorporated both patchwise geochemical heterogeneity and random sequential deposition dynamics in a one-dimensional colloid transport model. Their model was used in one of the earliest trials where heterogeneity was considered in colloid transport modeling. They compared the model with the results from transport experiments involving silica colloids traveling through a column filled with geochemically heterogeneous sand grains. Sun et al. [2001] developed a two-dimensional colloid transport model for heterogeneous porous media where they coupled the colloid transport equation with the 2of19

3 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS fluid flow equation and used the finite element method to obtain a numerical solution for these equations. Recently, polydisperse (variable size) colloid transport in fractured systems has been investigated [James et al., 2005; Chrysikopoulos and James, 2003; James and Chrysikopoulos, 1999, 2000]. These studies have shown that larger particles travel faster than smaller particles and colloids with lognormal particle size distribution encounter greater spreading than those with uniform distribution. [15] Many laboratory studies were conducted to understand the interactions among contaminants, colloids, and porous media. Laboratory experiments simulated colloids as nonreactive substances by saturating the porous media with colloids before beginning the experiments. The assumption of nonreactive colloids was invoked by many authors [Magee et al., 1991; Abdul et al., 1990; Dunnivant et al., 1992]. As an example, the column experiments of Abdul et al. [1990] focused on determining the washout efficiency for a number of nonpolar organic contaminants. They found that using a humic acid solution instead of water effectively removes nonpolar organic contaminants. They attributed this result to the effect of humic acid (colloids) on enhancing the migration of contaminants. [16] Several models for contaminant transport in the presence of colloids have been developed [e.g., Corapcioglu and Jiang, 1993; Abdel-Salam and Chrysikopoulos, 1995a, 1995b; Corapcioglu and Choi, 1996; Roy and Dzombak, 1998; James and Chrysikopoulos, 1999, 2003; James et al., 2005; Sun et al., 2001; Sen et al., 2004]. For example, Corapcioglu and Jiang [1993] developed a one-dimensional, homogeneous numerical model to describe colloidfacilitated transport in saturated, nonfractured porous media assuming linear equilibrium between the aqueous and solid phase. They compared their model with the results of column experiments. This model was the first attempt to create a contaminant transport model in porous media in the presence of colloids. Johnson et al. [1995] employed the equilibrium model formulation to describe the results of column experiments on enhanced transport of benzanthracene by colloids. Roy and Dzombak [1998] studied the effects of nonequilibrium sorption/desorption on colloidfacilitated transport of hydrophobic organic compounds in porous media. Magee et al. [1991] accounted for the presence of colloids by developing a modified effective retardation factor that could be used in the traditional advection-dispersion transport equation. [17] In a study of reactive solute transport in the presence of dissolved organic carbon (DOC), Knabner et al. [1996] and Totsche et al. [1996] showed the effect of DOC as a contaminant-retarding agent. They reported anthracene breaks through after 50 pore volumes in the absence of DOC, whereas in the presence of DOC, breakthrough occurred after 140 pore volumes. This experiment was conducted under unsaturated flow conditions, and the DOC comprised 34% mobile hydrophilic and 66% immobile hydrophobic components. [18] Sen et al. [2002] also considered the possibility that colloids retard contaminant movement. Their results showed increased retardation of Ni 2+ under the plugging condition. Bekhit and Hassan [2005a] and Bekhit et al. [2006] examined numerically and experimentally, respectively, the cases where colloids act as retardation agents and explored various conditions where colloids may retard the transport of dissolved contaminants. In the context of contaminant transport in rivers, Ren and Packman [2004, 2005] identified the retardation phenomenon, and they termed it colloid-impeded contaminant transport. [19] As shown from the above discussion, an extensive body of literature exists on colloid transport and colloidfacilitated contaminant transport. However, the effect of using different sorption isotherms (e.g., linear, Freundlich, or Langmuir) in analyzing contaminant transport in the presence of colloids has not been thoroughly addressed. To the authors knowledge, only van de Weerd et al. [1998] studied the effect of using the Langmuir isotherm in onedimensional, coupled transport of colloids and contaminants. They showed the effect of nonlinearity by comparing results from using high initial concentration to results using a lower initial concentration. [20] The lack of these types of studies could be attributed to the complexity of the problem, which makes the analytical solution of such problems extremely difficult. In addition, relying on the standard numerical approximation to solve nonlinearity is not convenient for two reasons [van der Zee, 1990]. The first reason involves constraints on the time step and spatial discretization to overcome the instability problem. Secondly in some situations of nonlinear transport, very sharp concentration gradients may develop that require very fine discretization to obtain accurate solutions. [21] This study focuses on the effects of colloids on transport of contaminants that sorb nonlinearly onto porous media. We adapt the finite cell approach developed by Sun [1999] to study the effect of nonlinear sorption on plume transport. The main advantage of using the finite cell approach for this problem is that it does not solve the governing equations with the nonlinear terms. Instead, the finite cell approach simulates advection and dispersion by using the particle-tracking technique while sorption and desorption (whether linear or nonlinear) are calculated directly as mass exchange processes. Further, we examine characteristics of the contaminant movement and distribution under linear and nonlinear sorption processes pertaining to both contaminant and colloids. [22] One of the important outcomes of this study is providing a solution approach to the coupled partial differential equations describing colloid-associated contaminant transport in porous media, for which a major strength has not been realized in prior studies. Although the tool we use here (the finite cell method) was developed and used in few studies before [Sun, 1999, 2002] we highlight in this study one of its major advantages. The method can handle nonlinear reactions as easily as it can handle linear reactions. We therefore capitalize on this strength and show how this tool can be very useful in dealing with complicated nonlinear, nonequilibrium contaminant reactions in the presence of colloids. [23] Another contribution in this study is the investigation of the impact of colloids on a contaminant undergoing nonlinear sorption. We also address in this study the cases where both colloids and contaminant undergo nonlinear deposition/sorption onto the solid matrix. To the best of our knowledge, no prior study addressed this combined nonlinearity. Again the realization of the full strength of 3of19

4 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS the finite cell approach was the key that allowed such investigation. [24] The study examines the impact on the plume shape and illustrates the way in which sorption to aquifer material and the presence of colloids can influence the overall shape and migration rate of contaminant plumes. The study provides new insights into the combined effects of contaminant concentration value, degree of nonlinearity, presence of colloids, and type of colloid deposition isotherm on the plume migration rates and plume shape. [25] A sensitivity of contaminant plume behavior to the Freundlich isotherm exponent is presented. The results of sensitivity analysis explained the effect of Freundlich isotherm exponent on plume travel distance, plume skewness, and breakthrough curves. The effect of nonlinear isotherm on the dilution of peak concentration is also investigated. 2. Model Description [26] The movement of colloids and their interaction with dissolved contaminants in groundwater systems are represented by six variables describing six constituents. The six variables exist in the system and change in a dynamic manner [Bekhit and Hassan 2005a, 2005b]. The first two variables are the mobile colloid concentration and immobile colloid concentration. The remaining four variables belong to the contaminant and describe the different forms or states that the contaminant can take in the system. These components include (1) the mass concentration of contaminant dissolved in the aqueous phase, (2) the mass concentration of contaminant sorbing onto mobile colloidal surfaces, (3) the mass concentration of contaminant sorbing onto immobile colloidal surfaces, and (4) the mass concentration of contaminant sorbing onto the solid matrix Colloid Transport Equations [27] The mass balance equations for mobile and immobile colloids are given by [Corapcioglu and Jiang, ¼ rj c Q c ¼ Q Sc where C c is the mass concentration of mobile colloids in the aqueous phase (ML 3 ); q is the porosity (dimensionless); J c is the specific mass flux of mobile colloids (ML 2 T 1 ), which is the summation of flux due to advective flow, mechanical dispersion, and Brownian diffusion; Q sc is the net rate of colloid deposition on the solid matrix (ML 3 T 1 ), and S c is the mass of captured (deposited or immobile) colloids per unit total volume of porous media (ML 3 ). The specific mass flux J c is given by ð1þ J c ¼ VqC c ½D B þ D MD ŠrðqC c Þ ð3þ where D B is the Brownian diffusion coefficient (L 2 T 1 ); D MD is the mechanical dispersion tensor of the colloids (L 2 T 1 ); and V is the velocity vector (L T 1 ). The in equation (2) gives the mass exchange between mobile colloids and solid matrix. This exchange can be described by different reaction isotherms. A reaction isotherm is determined in laboratory studies by comparing the sorbed concentration of colloids (or contaminants) to the concentration of the same constituent in solution at different levels for the latter concentration value. Then this experimental data set is fitted with an isotherm that may be linear or nonlinear. That isotherm could either be considered in equilibrium (very fast reactions compared to the transport characteristic timescale) or in nonequilibrium (slow reactions) conditions. The nonequilibrium case is more general and is considered here. Thus the reaction of colloids with the porous medium could be represented in general by one of the following reaction equations:- Linear kinetic isotherm: c t ¼ K 1C c K 2 q S c Nonlinear kinetic Freundlich isotherm: c t ¼ K 1Cc NFrn K 2 q S c Nonlinear kinetic Langmuir isotherm: c t ¼ K S max S c 1C c K 2 S max q S c where K 1 is the colloid deposition rate constant with dimension (T 1 ) for equations (4) and (6) and ((ML 3 ) 1 N Frn T 1 ) for equation (5); K 2 is the colloid release rate constant (T 1 ); N Frn is the exponent of the nonlinear Freundlich isotherm; and S max is the maximum amount of colloids that can be deposited onto the solid matrix per unit volume of aquifer (ML 3 ). Similar isotherms can be written for contaminant interaction with colloids (contaminant sorption onto and desorption from colloids) and contaminant interaction with porous medium (contaminant sorption-desorption processes). [28] The linear kinetic isotherm accounts for a situation where sorption is limited by a first-order kinetic model. The linear isotherm is usually valid for dissolved species that are present at concentrations less than one half of its solubility [Lyman et al., 1992]. This isotherm was used to describe the sorption of pesticides [Leistra and Dekkers, 1977; Hornsby and Davidson, 1973] as well as some organics [Davidson and Chang, 1972]. The Freundlich model is describing the reversible nonlinear kinetic sorption, and it was successfully used to describe the sorption of phosphate [Fiskell et al., 1979] and herbicides [Enfield and Bledsoe, 1975]. The nonlinearity of sorption is frequently correlated to the heat of a sorption reaction [Weber and DiGiano, 1996]. Within limited conditions, such as low solute concentration, the nonlinearity of the Freundlich and Langmuir isotherms is negligible so that the linear isotherm can be used without substantial loss of accuracy [Papelis and Um, 2003] Contaminant Transport Equations [29] The mass balance equations for contaminant attached to mobile and immobile colloids can be expressed as [Corapcioglu and Jiang, 1993] ð4þ ð5þ ðs cm C c Þ ¼ r½½d B þ D MD Š rðqs cm C c t rðvqs cm C c ÞþQ 1 Q 2 ð7þ 4of19

5 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS ðs c S cc Þ ¼ Q 3 þ Q 2 ð8þ where S cm is the mass concentration of contaminant sorbing onto mobile colloids per unit mass of colloids (M/M); S cc is mass concentration of contaminant sorbing onto immobile colloids per unit mass of colloids (M/M); Q 1 describes the mass exchange process between contaminant and mobile colloids (ML 3 T 1 ); Q 2 accounts for the deposition and release of contaminant-carrying colloids onto and from solid matrix (ML 3 T 1 ), and Q 3 describes the interaction between contaminant and immobile colloids (ML 3 T 1 ). Lastly, the mass balance equations for dissolved contaminant and for contaminant sorbing directly onto the solid matrix are given qc f ¼r ½ DB þ D MD Š rq C t r Vq C f Q1 Q 3 Q 4 ¼ Q 4 ð10þ where C f is the mass concentration of contaminant dissolved in the aqueous phase (contaminant mass per unit aqueous volume) (ML 3 ); S s is the mass concentration of contaminant sorbing onto the solid matrix per unit total volume of porous media (ML 3 ); and Q 4 accounts for the sorption process between the contaminant and the solid matrix (ML 3 T 1 ). Note that the terms Q 1, Q 2, Q 3, and Q 4 can be represented using kinetic interactions similar to those given in equations (4), (5), and (6). The details of how these interactions are implemented in the finite cell method are presented in the next section. 3. Numerical Solution 3.1. Finite Cell Method [30] A common numerical approach for solving transport problems including reactive transport is the particletracking method [e.g., Tompson and Gelhar, 1990; Bellin et al., 1992; Chin and Wang, 1992; Hassan et al., 1997, 1998, 2001]. Particles representing the contaminant plume are displaced in space over discrete time steps by the action of some driving mechanisms such as a chemical potential or a velocity field. The finite cell method, developed by Sun [1999], is a variation of the particle tracking concept where particle mass is distributed over a volume (a finite cell) rather than at a point. This method represents the different phases in a certain problem with different sets of cells; each cell is assigned mass, concentration, and volume fractions of the particular constituent. Interactions among these cells and exchanges in mass can then be carried out in a manner consistent with the phenomenological understanding of the involved processes. More details about the finite cell method can be found in work by Sun [1999, 2002] Numerical Implementation [31] The solution of the differential equation governing the movement of mobile cells can be approximated by tracking cell movement in time and space. Tracking in two dimensions is based on the standard random walk equations [Kinzelbach, 1988; Valocchi and Quinodoz, 1989; Tompson and Gelhar, 1990; LaBolle et al., 1996, 2000] x tþdt ¼ x t þ V x ðx t ; y t ; p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2D xx Dt Z 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D xy Dt Z 2 y tþdt ¼ y t þ V y ðx t ; y t ; p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2D yx Dt Z 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D yy Dt Z 2 Dt Dt ð11þ ð12þ where x and y are the coordinates of the particle location (L); V x and V y are the velocity components in the x and y directions (L T 1 ), respectively; D ij is the ij component of the dispersion tensor (summation of Brownian diffusion and mechanical dispersion), Dt is the time step (T); and Z is a normally distributed random number having zero mean and unit variance. The second term multiplied by Dt on the right hand side of equation (11) is an effective velocity that combines local velocity at location (x t, y t ) and time t plus the gradient of the dispersion tensor at location (x t, y t ). The last two terms account for the local-scale dispersion and Brownian diffusion. Since we are dealing with homogenous conditions, the terms involving the gradient of the dispersion coefficients are dropped from equations (11) and (12). Only advective and local-scale dispersion terms are used in this study. [32] Mass exchange occurs between dissolved contaminant and mobile colloids carried in the same cell or when a mobile cell containing contaminant or colloids crosses an immobile cell during transport. This exchange is controlled by the kinetic interaction and sorption isotherms, which are given in equations (4) (6). For example, consider that a contaminated mobile cell interacts with the solid matrix according to the nonlinear Freundlich isotherm given by equation (5). This mass exchange can be expressed as Dm Cf ¼ K f C NFrn f K b S s q V w Dt r ð13þ where Dm Cf is the mass change in the mobile cell (M); V w is the volume of water in the mobile cell (L 3 ); K f is the forward rate constant with dimension [(ML 3 ) 1 N Frn T 1 ]; K b is the backward rate constant [T 1 ]; and Dt r is the time of interaction between the mobile and immobile cells (T), which is a fraction of the time step, Dt. After calculating the change in mass, both the mobile and immobile concentrations are updated as C f ¼ M C f Dm Cf V w S s ¼ M S s þ Dm Cf V w =q ð14þ ð15þ where M Cf is the mass of contaminant in the mobile cell at the beginning of the time step (M) and M Ss is the sorbed 5of19

6 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Table 1. Parameter Values for the Test Case Comparing the Finite Cell Solution to the TVD Finite Difference Solution Description Parameter Value Initial mobile colloid concentration C c0 100 g m 3 Initial dissolved contaminant C f0 100 g m 3 concentration Initial concentrations S c0 = S cm0 = S cc0 = S g m 3 Porosity q 0.3 Velocity in x direction V x 0.5 m d 1 Velocity in y direction V y 0.0 m d 1 Longitudinal dispersivity a L 0.1 m Transversal dispersivity a T 0.05 m Colloid release rate K d 1 Contaminant desorption rate K b 0.5 d 1 Contaminant sorption rate K f 1d 1 Reaction rate coefficients (colloid K 1 = K sm = K sc 1d 1 deposition rate and contaminant desorption rate) Rates of contaminant sorption onto colloids K m im a = K a 3d 1 contaminant mass in the immobile cell at the beginning of the time step (M). Note that if Dm Cf is positive, the dissolved contaminant concentration, C f, decreases at the end of the time step, whereas the sorbed concentration, S s, increases. The opposite is true if Dm Cf is negative. 4. Model Verification and Test Problem [33] In order to test the accuracy of the finite cell solution for contaminant transport in the presence of colloids, we compare the finite cell results to the TVD, finite difference solution presented by Bekhit and Hassan [2005a]. The finite difference model was previously verified using mass balance tests and was compared to analytical solutions and experimental data [Bekhit and Hassan, 2005a, 2005b]. [34] A two-dimensional domain of size m 2 is considered. The domain is divided into a uniform grid of square blocks of length Dx = Dy = 0.2 m, which gives a total of 32,768 blocks (128 rows 256 columns). Each grid block is divided into 16 immobile cells, and the water contained within each grid block is divided into 16 mobile cells having equal volume. This gives a total of 524,288 mobile cells. [35] Two types of boundary conditions are employed throughout the model domain: (1) no-flow boundaries along the base and top of the model domain and (2) constant head boundaries along the left and right edges of the model domain. These conditions result in a unidirectional flow (from left to right with a uniform velocity of 0.5 m d 1 ). Having homogeneous conditions, no vertical velocity components exist in the domain. [36] The following initial and boundary conditions are employed for the transport problem Cx; ð y; 0Þ ¼ C 0 ð16þ Cð0; y; tþ ¼ 0 L x; y; tþ ¼ ð x; 0; t x; W y; t ¼ 0 ð19þ where C represents any of the three mobile concentrations (C c, C f, and S cm ); C 0 is the initial concentration; L x is the domain length in the x direction; and W y is the domain width in the y direction. [37] The two models (finite cell and finite difference) are used to simulate an instantaneous pulse of contaminant released into the simulation domain at time t = 0. A colloid plume is assumed to coincide with the dissolved contaminant plume as an initial condition. All interactions are assumed to follow a linear isotherm. Therefore the terms Q 1, Q 2, Q 3, and Q 4 are given by Q 1 ¼ K m a C f K sm S cm C c Q 2 ¼ K 1 S cm C c K 2 q S c S cc Q 3 ¼ K im a C f K sc q S ccs c Q 4 ¼ K f C f K b q S s ð20þ ð21þ ð22þ ð23þ where K m a and K im a are the sorption rate coefficients of contaminant on mobile and immobile colloids, respectively, [T 1 ]; K sm is the desorption rate coefficient of contaminant from the mobile colloids [T 1 ]; and K sc is the desorption rate coefficient of contaminant from the immobile colloids [T 1 ]. The values of rate coefficients and initial conditions are summarized in Table 1. The values of the parameters for this test case are similar to those used by Corapcioglu and Jiang [1993], Roy and Dzombak [1998], Corapcioglu and Wang [1999], and Kim and Corapcioglu [2002]. [38] Figures 1a and 1b compare the spatial distribution of the mobile and immobile colloid plumes at t = 40 days using the finite difference method (solid line) and finite cell method (dashed line). In Figures 1c 1f, the four contaminant constituents (S cm, S cc, C f, and S s ) are compared using finite difference and finite cell methods. As can be seen from Figure 1, a very good match between the two models is obtained for the six plumes. This agreement indicates that the numerical solution for the finite cell model successfully represents colloid transport and associated contaminant transport. It is important to note that the agreement is very good for all six constituents. Presenting these six plumes guarantees the good performance of the solution better than if one only compares total mobile contaminant concentration (i.e., C f + S cm C c ). There are minor oscillations in the finite cell solution, which result from the discrete nature of the cells and random component of their movement. Such oscillations can easily be smoothed out by increasing the number of mobile cells per grid block. 5. Results and Discussion [39] Discussion of the analyses and results is divided into four main parts or subsections. First, the effect of kinetic 6of19

7 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Figure 1. Comparison between the finite cell solution (solid contours) and the finite difference solution (dashed contours) for colloid and contaminant transport in a two-dimensional domain 40 days after plume release. nonlinear Freundlich isotherm is investigated for contaminant migration in a homogenous porous medium. This effect is studied under two levels of nonlinearity, N Frn = 0.5 and N Frn = 1.3, in the presence and absence of colloids. Because of the nature of nonlinear isotherms, the magnitude of retardation depends on the values of both N Frn and the concentration. To explore these effects and investigate the influence of colloids, the analysis is performed with initial concentration of 1.0 g m 3, and the results are compared to solutions with an initial concentration much larger than 1.0gm 3. Second, the impact of nonlinearity in contaminant sorption on plume dispersion and skewness is investigated in thepresenceandabsenceofcolloids.thesamecasesusedinthe first part are used to study plume shape. [40] Third, the impact of limited colloidal sorption capacity on the contaminant transport is studied. In this analysis, the colloids are assumed to have a limited sorption capacity, and the sorption of contaminant onto colloids is represented by the Langmuir isotherm. A reference case is first established where colloids are assumed to have unlimited sites (contaminant sorption onto colloids is assumed to follow a linear kinetic isotherm). The amount of contaminant that sorbs onto colloids in the reference case is computed and denoted as S ref. Four cases are then presented where the maximum capacity for colloids to hold the contaminant is assumed to be 75%, 50%, 25%, and 10% of S ref. Comparisons among these cases and with the reference case are presented. [41] Fourth, the role of colloids to enhance contaminant transport is evaluated when colloid deposition onto solid matrix is nonlinear. To study this aspect, two scenarios are examined. In the first scenario the Freundlich isotherm is used for colloid deposition, whereas the second scenario assumes colloids to have a linear sorption isotherm. In the 7of19

8 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Figure 2. Cumulative mass arrival curves (left axis, solid line) and maximum concentration (right axis, dashed line) of the dissolved contaminant C f (a) in the absence of colloids and (b) in the presence of colloids for C f0 = 100 g m 3. two scenarios, contaminant sorption onto colloids is assumed to follow the Langmuir isotherm. [42] The same two-dimensional domain and flow conditions described in section 4 are used here. For the transport problem, an instantaneous contaminant is released into the simulation domain at time t = 0 with an initial size of 1 3 m. A colloid plume is assumed to coincide with the dissolved contaminant plume at t = 0. In addition, we assign a control plane (5 m away from the source) normal to the flow direction. The cumulative mass arrival across the control plane and the maximum concentration at the control plane are obtained as a function of time. The model parameters have been selected on the basis of literature values. Unless stated otherwise the following reaction rates are used: K 1 =1d 1, K 2 = 0.5 d 1, K a m =2d 1, K a im =2d 1, K sm = 0.5 d 1, K sc =4d 1, K f =1d 1, and K b = 0.5 d 1.In addition, the physical properties remain constant in all simulations where q = 0.3; D xx = D yy = m 2 d 1 ; and total simulation time is 80 days. The term total contaminant concentration is used to denote the summation of dissolved contaminant concentration and concentration of contaminant sorbed onto mobile colloids (i.e., C Total = C f + S cm C c ) Effect of Nonlinear Freundlich Isotherm on Contaminant Retardation [43] We consider an aquifer contaminated with an instantaneous spill with C f0 = g m 3 and assume that the contaminant sorbs onto solid matrix nonlinearly according to a Freundlich isotherm. Two cases are simulated where N Frn = 0.5 in the first case and N Frn = 1.3 in the second case. In both cases, K f is taken as 0.5 (g m 3 ) N Frn d 1. Figure 2a gives the cumulative mass arrival across the control plane (left axis solid lines) and the maximum concentration at the control plane (right axis dashed line) for the two cases in the absence of colloids. Figure 2a shows that an early arrival of the plume in the case of N Frn =0.5 is obtained with a peak concentration that is about 390% higher than the case of N Frn = 1.3. The plume experiences greater retardation in the case of N Frn = 1.3. These results are in agreement with the 8of19

9 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Figure 3. Cumulative mass arrival curves (left axis, solid line) and maximum concentration (right axis, dashed line) of the dissolved contaminant C f (a) in the absence of colloids and (b) in the presence of colloids for C f0 =1gm 3. analytical traveling wave solution for nonlinear nonequilibrium contaminant transport developed by van der Zee [1990]. [44] Figure 2b is similar to Figure 2a, but the contaminant transport results are obtained in the presence of colloids. A colloid plume having C c0 = 100 g m 3 is released and is assumed to coincide with the contaminant plume. However, to investigate the effect of nonlinear reactions separately, the isotherm governing colloid deposition onto the solid matrix is assumed to be linear. It is important to note that the initial colloid concentration value selected for these simulations is consistent with field findings regarding colloid concentration in soils. Results from field studies have demonstrated that soil colloids may be released to drainage water in high concentrations during rainfall events [Ryan et al., 1998; El-Farhan et al., 2000; Villholth et al., 2000; Petersen et al., 2003]. Also, data collected from the field at the Nevada Test Sites [Kersting et al., 2003] show that...colloids were found in every sample, although there was a large variation in concentration ( mg L 1 ). Concentrations in excess of 1 g L 1 (1000 g m 3 ) have been reported during simulated and natural rainfall events [DeNovio et al., 2004, Table 1]. [45] It is evident from Figure 2b that the presence of colloids facilitates transport of the contaminant for the case of N Frn = 1.3. The 50% mass arrival time for contaminant (Figure 2a) is 26 days in the absence of colloids and 16 days in the presence of colloids. On the other hand, when N Frn = 0.5, the presence of colloids retards the contaminant transport (e.g., the 50% mass arrival time increased from 10 to 12 days because of the presence of colloids). The combined effects of nonlinear isotherm and the presence of colloids are discussed at the end of this section. [46] Bekhit [2004] found that the effective retardation of a contaminant plume experiencing nonlinear reaction is highly dependent on the initial concentration. To examine this phenomenon in the presence of colloids, the previous analysis is repeated under small initial concentration. Figure 3 is thus similar to Figure 2, but is based on using a value for C f0 of 1 g m 3. It is shown in Figure 3a that the 9of19

10 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Figure 4. Schematic diagram showing the combined effect of colloids and nonlinear contaminant sorption on the contaminant transport. plume experiences greater retardation in the case of N Frn = 0.5 than in the case of N Frn = 1.3. Fifty percent of the initial mass crossed the control plan after 14.5 days when N Frn = 1.3, whereas only 40% of the initial mass crossed the control plan after 80 days when N Frn = 0.5. Comparing Figures 2a and 3a indicates that the impact of N Frn is altered on the basis of the value of the dissolved contaminant concentration. These results are in agreement with the sorption of organic component data reported by Appelo and Postma [1993] who state that the distribution coefficients for many organic compounds are smaller for high than for low concentration. These results are attributed to the fact that the sorption capacity decreases when the concentration increases, provided that 0 < N Frn <1. [47] As can be seen from the two cases examined in Figure 3, colloids enhance contaminant transport in both cases. However, colloids effect is more pronounced when N Frn = 0.5, where colloids significantly reduce the 50% mass arrival time from more than 80 days in the absence of colloids to only 37 days in the presence of colloids. Also, the peak concentration at the control plane has a value of about 0.04 g m 3 and occurs at 30 days in the absence of colloids, whereas it increases to 0.06 g m 3 and occurs at 22 days when colloids are present. It should be noticed that colloids did not significantly enhance the 50% mass arrival time of the contaminant for the case of N Frn = 1.3. However, colloids increased the maximum concentration crossing the control plane by about 37.5%. Therefore one could claim that colloids enhance the contaminant transport in both cases presented in Figure 3. [48] In the presence of colloids, the total sorption of contaminant consists of two components. The first is the direct sorption of contaminant onto solid matrix, and the second is the indirect sorption of contaminant (i.e., contaminant sorbed onto colloids that are deposited onto the solid matrix). However, in the absence of colloids, only direct sorption exists. The amount of total sorption controls how fast the contaminant migrates and determines the effective retardation factor. Generally, the amount of contaminant directly sorbing onto the solid matrix in the presence of colloids is less than in the absence of colloids. This is because colloids uphold some of the contaminant mass on its surfaces thereby reducing the amount of contaminant sorbing directly onto the solid matrix. [49] Because of the nature of the nonlinear isotherm and under certain conditions (i.e., contaminant concentration, Freundlich exponent, and colloids and contaminant interaction with the solid matrix), colloids could retard the contaminant. Figure 4 shows a schematic diagram that explains the role of colloids in facilitating or retarding the contaminant transport. The top line in Figure 4 shows cases of nonlinear sorption resulting in highly retarded contaminant plume (left), mildly retarded plume (middle), or lowretardation plume (right). The effective velocity line shows the relative position of the plume in the three cases. These cases depend on the combination of C f and N Frn as shown in Figure 4. The middle line in Figure 4 displays a hypothetical location of a colloid plume. When the contaminant and the colloids are put together (bottom line), either colloid-facilitated or colloid-retarded transport is obtained. [50] Therefore, if a contaminant transport is controlled by nonlinear isotherm with a relatively high retardation factor (e.g., C f 1 and N Frn >1orC f 1 and N Frn < 1), the presence of colloids facilitates the contaminant transport. On the other hand, if the contaminant plume experiences nonlinear isotherm resulting in a small retardation factor, the presence of colloids may facilitate or retard the contaminant transport. For example, under the conditions described in Figure 2b (i.e., C f0 = 100 g m 3, C f 1.0 g m 3 and N frn <1), colloids retard the contaminant transport. [51] The results in Figures 2a and 3a indicate that for a contaminant plume undergoing nonlinear sorption and with a certain initial concentration value, the plume is either strongly retarded or mildly retarded (compared to conservative transport) depending on the value of N Frn. For C f0 = 100 g m 3, nonlinear sorption with N Frn > 1.0 yields stronger retardation than with N Frn values smaller than 1.0 (compare the mass arrival curves in Figure 2a to the 50% arrival time of a conservative contaminant shown in 10 of 19

11 BEKHIT AND HASSAN: NONLINEAR CONTAMINANT SORPTION WITH COLLOIDS Figure 5. Contour map showing the zones where colloids facilitate contaminant transport (solid contours) and the zones where colloids retardate transport (dashed contours). Figure 2a). Similarly, for C f0 < 1.0, nonlinear sorption with N Frn values smaller than 1.0 yields stronger retardation than with N Frn values larger than 1.0. [52] When the contaminant experiences mild to little retardation (Figure 4), a large portion of the dissolved plume (C f ) travels with small retardation, and only the plume tail migrates with a large effective retardation factor. Moreover, the contaminant carried by colloids (S cm ) moves with a uniform effective retardation factor (because of the linear isotherm used for colloid deposition) that is less than the retardation factor of the slowest part of the C f plume but larger than the retardation factor of the fastest part. Thus the presence of colloids facilitates part of the plume and retards another part, and therefore, depending on the ratio between the fastest and slowest portions of the C f plume, the overall effect of colloids could be one of retardation (Figure 2b, N Frn = 0.5) or facilitation (Figure 2b, N Frn = 1.3) of the total contaminant transport. [53] An important aspect here is to quantify the cases under which colloids may facilitate or retard contaminant transport. To address this aspect, a contour map is created in the parameter space which shows the ratio of the plume-first moment (a measure for the travel distance of the center of mass) in the presence and absence of colloids after 80 days under different Freundlich exponent and initial concentration values (Figure 5). In Figure 5, y axis gives the natural logarithm of the initial concentration ranging from [0.1 g m 3 to 1000 g m 3 ], x axis gives the Freundlich exponent, and the contour values give the ratio of the first moment in the presence of colloids to the first moment in the absence of colloids (X 1 j with colloids /X 1 j without colloids ). As shown in Figure 5, under the examined conditions, there are clear zones where colloids may facilitate or retard the transport of a contaminant experiencing nonlinear sorption. The threshold values (dashed thick lines) are located around a Freundlich exponent of value equal to 1, and an initial concentration around 31 g m 3. It is expected that the threshold of N Frn = 1 will remain the same under conditions different than what we used in Figure 5. However, the initial concentration will vary on the basis of the different reaction rates Effect of Nonlinear Freundlich Isotherm on the Plume Shape [54] To examine the effect of nonlinear sorption on the shape of the contaminant plume, a snapshot of the contaminant plume is plotted. Figure 6 shows the distribution of a plume resulting from an instantaneous initial source of C f0 = 100 g m 3 80 days after the plume was released. Plume distributions presented in Figure 6 reveal interesting findings. The contaminant plume exhibits positive skewness at N Frn = 1.3 and negative skewness at N Frn = 0.5, both in the presence and absence of colloids. This could be attributed to the nature of the nonlinear isotherm. As discussed previously, with the initial concentration C f0 = 100 g m 3, the higher-concentration area travels faster than the lower-concentration area when N Frn < 1.0. Therefore most of the mass of the contaminant plume travels faster 11 of 19