Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media

Transcription

1 WATER RESOURCES RESEARCH, VOL. 43,, doi: /2007wr005969, 2007 Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media Ram C. Acharya, 1,2 Albert J. Valocchi, 1 Charles J. Werth, 1 and Thomas W. Willingham 1,3 Received 13 February 2007; revised 15 July 2007; accepted 27 July 2007; published 30 October [1] Several studies have demonstrated that the success of natural and engineered in situ remediation of groundwater pollutants relies on the transverse mixing of reactive chemicals or nutrients along plume margins. Efforts to predict reactions in groundwater generally rely on dispersion coefficients obtained from nonreactive tracer experiments to determine the amount of mixing, but these coefficients may be affected by spreading, which does not contribute to reaction. Mixing is controlled only by molecular diffusion in pore spaces, and the length scale of transverse mixing zones can be small, often on the order of millimeters to centimeters. We use 2D pore-scale simulation to investigate whether classical transverse dispersion coefficients can be applied to model mixingcontrolled reactive transport in three different porous media geometries: periodic, random, and macroscopically trending. The lattice-boltzmann method is used to solve the steady flow field; a finite volume code is used to solve for reactive transport. Nonreactive dispersion coefficients are determined from the transverse spreading of a conservative tracer. Reactive dispersion coefficients are determined by fitting a continuum model which calculates the total product formation as a function of distance to the results from our pore scale simulation. Nonreactive and reactive dispersion coefficients from these simulations are compared. Results indicate that, regardless of the geometrical properties of the media, product formation can be predicted using transverse dispersion coefficients determined from a conservative tracer, provided dispersion coefficients are determined beyond some critical distance downgradient where the plume has spread over a sufficiently large transverse distance compared to the mean grain diameter. This result contrasts with other studies where reactant mixing was controlled by longitudinal hydrodynamic dispersion; in those studies longitudinal dispersion coefficients determined from nonreactive tracer experiments over-estimated the extent of reaction and product formation. Additional work is called for in order to confirm that these findings hold for a wider variety of grain sizes and geometries. Citation: Acharya, R. C., A. J. Valocchi, C. J. Werth, and T. W. Willingham (2007), Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media, Water Resour. Res., 43,, doi: /2007wr Introduction [2] Many risk-based strategies for managing groundwater pollution rely upon intrinsic and stimulated in situ biodegradation of organic contaminant plumes. Field and laboratory studies have shown that the overall rate of biodegradation can be controlled by transverse mixing of electron donors and acceptors (e.g., hydro-carbons and oxygen) that are present at the relatively sharp interface between the ambient groundwater and the contaminant plume [Thompson and Fogler, 1997; Grathwohl et al., 1 Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. 2 Currently at Harden Environmental Services Ltd., Canada. 3 Currently at ExxonMobil Upstream Research Corporation, Houston, Texas, USA. Copyright 2007 by the American Geophysical Union /07/2007WR ; Thornton et al., 2001; Rahman et al., 2005; Cirpka et al., 2006; Tuxen et al., 2006]. Transverse mixing is also a key process determining the rate of dissolution of nonaqueous phase liquids in aquifers [Sale and McWhorter, 2001; Eberhardt and Grathwohl, 2002]. Several recent investigations have focused upon contaminant plume evolution under conditions of mixing-controlled reaction using numerical [Cirpka, 1999; Prommer et al., 2002] and analytical [Ham et al., 2004; Liedl et al., 2005] methods. In particular, Liedl et al. [2005] focused upon steady state plume conditions due to its importance for risk management and demonstrated that longitudinal dispersion had relatively little impact upon the steady state plume length. Although early laboratory-scale studies of dispersion reported the more easily measured longitudinal component, recent work has emphasized transverse dispersion due to its recognized importance as a potentially limiting factor for attenuation of plumes originating from continuous sources [e.g., Grathwohl and Lerner, 2001; Sale and McWhorter, 2001; 1of11

2 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Bockelmann et al., 2003; Huang et al., 2003; Benekos et al., 2006; Cirpka et al., 2006; Olsson and Grathwohl, 2007; Knutson et al., 2007]. [3] Most studies of reactive transport, including those cited above, adopt a continuum-based approach where all solute concentrations are averaged over some representative elementary volume (REV) [Bear, 1972; Dagan, 1989]. Inherent in this approach is the use of classical hydrodynamic dispersion coefficients that quantify solute spreading to also quantify solute mixing, where the former causes stretching and deformation of the original plume shape while the latter causes initially separated adjacent plumes to overlap [Kitanidis, 1994; Jose and Cirpka, 2004; Rahman et al., 2005]. Several recent studies indicate that classical hydrodynamic dispersion coefficients may not adequately describe reactive transport. Gramling et al. [2002] showed that model predictions of reaction product predicted with a longitudinal dispersion coefficient determined from tracer breakthrough data exceeded the amount observed through image-based quantification of a colorimetric reaction in a packed bed. This same conclusion was reached earlier by Raje and Kapoor [2000] who investigated bimolecular reaction at the interface of two moving solutes in a one-dimensional column displacement experiment. They indicated that the over prediction is due to sub- REV concentration variability when fast reaction occurs before the solutes have had the opportunity to mix over the scale of the REV. The field study reported by Semprini and McCarty [1991] also found that use of dispersion coefficients measured from nonreactive tracer breakthrough overestimated the amount of mixing and reaction in a pulsed biostimulation experiment. These results lead to the conclusion that longitudinal dispersion coefficients that describe tracer spreading cannot be applied to model mixing in reactive transport. For upscaling from the pore to the smallest possible Darcy scale, Jose and Cirpka [2004] argue that this is a transient effect and that mixing and spreading are equivalent after a characteristic pore-scale dispersion time. It is important to investigate whether these same conclusions are applicable for reactions that are controlled by transverse mixing, since as already noted, it is transverse, rather than longitudinal dispersion that is most critical for determining overall extent of plumes from persistent contaminant sources. [4] In several recent studies, transverse dispersion and transverse-mixing limited reactions were measured [e.g., Rahman et al., 2005; Benekos et al., 2006; Cirpka et al., 2006]. Ham et al. [2007] reported a careful series of two dimensional nonreactive and reactive experiments within a porous medium of glass beads. The reactive experiments allowed development of a steady state chemical plume that emanated from a constant finite-width source along the inlet boundary; a rapid hydrolysis reaction occurred between the source and background solution, and the plume was delineated using a ph indicator. In contrast to studies where longitudinal mixing limited the extent of reaction, Ham et al. [2007] found that transverse dispersion coefficients measured from a nonreactive steady state tracer experiment were able to accurately predict the extent of the reactive plume. [5] In this paper we use pore-scale simulation to investigate whether classical transverse dispersion coefficients can be applied to model mixing controlled reactive transport. That is, we quantify and compare D T,N (for conservative tracers) and D T,R (for reactive solutes) for various pore-scale heterogeneities and transport regimes. We use pore-scale simulation because transverse mixing (reaction) zones observed both in the laboratory and the field can be very narrow, on the order of millimeters to centimeters [e.g., Nambi et al., 2003; Rahman et al., 2005; Cirpka et al., 2006; Tuxen et al., 2006]. Also, ultimately it is molecular diffusion at the pore scale that causes mixing of reactants. We evaluate different porous media geometries because grain size, geometry, and variance may all affect mixing. Use of high-resolution numerical modeling allows these processes to be studied directly, and we can calculate the precise value for the transverse dispersion coefficient rather than relying upon the classical correlations based upon laboratory studies. This study is restricted to steady state conditions, although it is important to recognize that transients in the flow direction can greatly enhance transverse dispersion and reactive mixing [Goode and Konikow, 1990; Prommer et al., 2002; Cirpka and Attinger, 2003; Cirpka, 2005; Cirpka et al., 2006]. We will also limit our study to the ideal case of mean uniform flow (i.e., fundamentally 1D-flow, from left to right, when averaged over space) in a two-dimensional system, with very fast reaction occurring between two solutes that are input side-by-side. In this case, the overall rate of reaction is controlled by transverse mixing between the two solute plumes. We recognize that some mechanisms (e.g., twisting of streamlines) inherent to more complex 3D domains are not considered. Furthermore, as discussed by Rahman et al. [2005], heterogeneity can cause variable horizontal meandering at different vertical levels along the lateral fringes of a three-dimensional plume; the resulting enhancement of the interfacial area of the plume can lead to further mixing that cannot occur in two dimensional systems. [6] The outline for the remainder of the paper is as follows. In Section 2 we introduce the reactive transport model for binary solutes. In Section 3 we describe our choices of porous media and boundary conditions. In Sections 4 and 5 the methods for calculating dispersion coefficients are described. In Section 6 we present the results, followed by discussions and conclusions. 2. Models [7] A schematic of the physical system modeled in this work for one porous media configuration is shown in Figure 1. The lattice Boltzmann method (LBM) is used to solve the steady state Stokes equation in order to obtain the pressure and velocity field. We use a standard singlerelaxation time BGK implementation [e.g., Hou et al., 1995; Noble et al., 1995]. No-slip along the fluid-solid interface is maintained using bounce-back boundary conditions. Zero flow boundary conditions are imposed along the top and bottom of the 2D domain. Fixed pressure boundary conditions are imposed along the inlet and outlet so that there is flow from the left to the right side of the domain. We use the same LBM code as in our previous work [Knutson et al., 2001, 2005]. [8] We assume the two solutes react and form the product, according to the reaction f A A + f B B! f AB AB, where f {.} is the stoichiometric coefficient. Symbols A, B 2of11

3 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 1. A schematic of 2D porous media and boundary conditions used in the numerical simulations. Shown is a random porous medium composed of differently shaped, sized and oriented cylinders. Shown also are the no flow boundaries ( no flow ), fixed pressures at the inlet (P in ) and outlet (P out ), inlet concentrations of the two reactants which participate in a bi-molecular reaction, species A 0 and B 0, and the flow direction (from left to right). and AB denote, respectively, concentrations of solutes A, B, and the product AB. The model describing steady state transport of these reactants and the product in porous media can be expressed by the advection-diffusion-reaction equations, i.e., r:~va D A :r 2 A ¼ f A R r:~vb D B :r 2 B ¼ f B R r:~vab D AB :r 2 AB ¼ f AB R; where the velocity field ~v is defined by the LBM results [Knutson et al., 2001]. The symbol R denotes the reaction rate. For a bi-molecular reaction, R = k [A][B], and k is a rate constant. The symbol D {.} denotes molecular diffusion coefficients of the participating species (A, B and AB). For simplicity, we assume that all species (A, B and AB) have the same diffusion coefficient, and also the stoichiometric constants are assumed to equal one. Referring to the schematic depicted in Figure 1, zero mass flux boundary conditions are imposed along the top and bottom boundaries of the domain, as well as along each solid-liquid interface. Constant concentration boundary conditions for solutes A and B (denoted by A 0 and B 0 ) are imposed along the inlet, and zero-gradient conditions are imposed along the outlet. The product species (AB) forms along the transverse mixing zone between the two reactant species. A standard finite volume method is used to solve the discretized form of the governing transport-reaction Equation (1). Fine discretization is used to minimize numerical errors [see Knutson et al., 2001, 2005, for more details]. Numerical tests showed that a uniform grid size of 5 and 2.5 microns gave similar results for the porous media considered here (see Section 3). Therefore, a 5 micron grid discretization was used. A finer discretization would be required to investigate media having ð1þ smaller pores. Although the LBM can also be used to solve the advection-diffusion-reaction equation, we have found based on our earlier work [Knutson et al., 2001] that an excessive number of iterations are needed to converge to steady state, and therefore we use a conventional finite volume method. [9] We also assume that the reaction rate is very large; that is, the reaction occurs instantaneously as soon as species A and B mix together. Under this assumption, solution of the coupled system of Equation (1) is greatly simplified. It can be seen that (assuming all the diffusion coefficients are equal) the quantity C* = (A f A fb B) is conserved. Under the assumption of instantaneous reaction, it is not possible for both to coexist at the same spatial location. Therefore instead of solving the coupled nonlinear system Equation (1), only the conservative transport equation for C* needs to be solved, and the values of A and B can be easily recovered from the positive and negative values of C*, respectively. This same strategy has been used by several other investigators, including Ham et al. [2007] and Liedl et al. [2005]. In order to solve for the product, AB, we also note that the quantity f AB f A A + AB is conserved; therefore, we also solve for this conserved quantity and then compute AB since we have already solved for A. 3. Porous Media [10] In this work we modeled reactive transport in both simplified and complex porous media such that inferences and comparisons about the impact of porous medium structure on hydrodynamic dispersion and overall reaction rate can be made. Our 2-D numerical media are aimed at representing three general types of physical systems made of non-deformable porous media: periodic media, random media, and macroscopically trending media. Examples of 3of11

4 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 2. Schematic of 2D packed beds: (a) periodic circular cylinder pack; (b) random pack of differently sized and shaped cylinders; (c) trending sinuous pack of differently sized and shaped cylinders. these media are shown in Figure 2. The cases considered in this work are documented in Table 1 and described below. [11] (a) Periodic media: Two periodic media cases are considered; they are both composed of a staggered array of cylinders, but have different cylinder diameters and slightly different porosities (Table 1). An example is shown in Figure 2a. Repeating pore units represent a unit cell, and also the representative elementary volume (REV). Many phenomenological parameters such as dispersion and permeability tensors are defined in terms of this REV size. It is also possible to use ellipses or squares of any orientation, instead of cylinders, when constructing the unit cells. Because the REV (unit cell) repeats throughout the domain, calculation methods are considerably simplified. This helps optimize computational speed and memory. Periodic media models have been widely used in many theoretical investigations [Whitaker, 1967; Eidsath et al., 1983; Souto and Moyne, 1997; Thompson and Fogler, 1997; Knutson et al., 2001, 2005]. [12] (b) Random media: Two random media cases are considered; they were created by placing elliptical cylinders with randomly generated minor and major axes and random orientation of the major axes with respect to the mean flow direction at random locations in space. An example is shown in Figure 2b. The minimum energy concept was used during packing [Visscher and Bolsterli, 1972]. Two different grain size variances and porosities were considered (Table 1), and minimum pore throat sizes were maintained so that the LBM could handle the flow problem (i.e., a minimum of 10 nodes should span a pore throat [Noble et al., 1995]). [13] (c) Macroscopically trending media: A single macroscopically trending media case was created from a random packing of unequal circular and elliptical cylinders in a syncline and anticline structure as shown in Figure 2c. Two different grain size variances were used to create the trending media, but each had the same porosity (Table 1). The abundance of syncline and anticline structures in geological formations indicates that long-range periodic patterns exist in nature. These larger-scale heterogeneities will lead to increased meandering and tortuosity of the plume centerline, which can enhance transverse dispersion. There is also the possibility of enhanced mixing due to macroscopic flow focusing, as discussed by Werth et al. [2006]. [14] It is important to note that mean grain sizes and porosities for the three types of porous media were set approximately equal to each other whenever possible to facilitate comparison. For example, the mean grain size of the periodic media with smaller grains, is within 1% of the mean grain size of the random media with smaller grains, and is within 1% of the smaller mean grain size in the trending media. [15] Steady state flow patterns from the LBM in selected media are shown in Figure 3. The bright colors indicate high velocity paths (preferential flow paths) which play the dominant role in longitudinal spreading, whereas dark colors indicate low velocity regions having longer localized residence times which allow greater opportunity for ratelimited reactions to progress. Hence the architecture of this Table 1. Characteristics of Porous Media Media Granular Periodic Random Cyl136 Cyl240 RandHigh RandLow Trending Sinuous hd g i [mm] & 270 s dg [mm] & 72 e% of11

5 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 3. Flow field and streamlines simulated by the LBM in different porous media: (a) Velocity field in random porous medium composed of differently shaped, sized and oriented cylinders; (b) Streamlines of flow in a randomly packed bed; (c) Streamlines around a staggered cylinder of a periodically packed bed. flow network determines the magnitude of dispersion in general and transverse dispersion in particular. Figures 3a 3c show streamlines in random and periodic media. In the absence of molecular diffusion, solute particles strictly follow these streamlines and are transported advectively [Baumann and Werth, 2004]. However, molecular diffusion causes particles to hop from streamline to streamline enhancing mixing and spreading. 4. Methods for Determining Transverse Dispersion Coefficient for a Conservative Tracer [16] There have been several studies published on determining dispersion coefficients from pore-scale hydrodynamics. For periodic media, the appropriate REV size is the unit cell and it is possible to use volume averaging methods [Eidsath et al., 1983; Whitaker, 1986; Souto and Moyne, 1997] or related and generalized Taylor-Aris-Brenner methods presented therein. For other media, it is more difficult to objectively select an REV [Zhang et al., 2000] and usually random-walk particle tracking methods are used [Salles et al., 1993; Maier et al., 2000]. In the latter case, the spread of a cloud of particles is computed during transit through the domain. In our work we use a technique similar to those cited above for periodic media, but for the nonperiodic media we use a different method that is more compatible with the steady state problem definition illustrated in Figure Methods for Periodic Media [17] In the methods presented by investigators such as [Eidsath et al., 1983; Edwards et al., 1993; Souto and Moyne, 1997], the sub-rev concentration is related to the macroscopic concentration through solution of an auxiliary steady state problem (the so-called B or f field equation). The dispersion tensor is also defined in terms of this same auxiliary problem. Here, we use a slightly different method, since we are also interested in solving the Figure 4. Method for computing mixing in periodic media (properties according to Cyl240, see Table 1, and Pe = 10): (a) Conventional steady state domain; (b) Streamlines along a strip marked in the steady state domain (c) Scheme of wrap-around option at time t = 0; (d) (f) Simulated transverse spreading of non-reactive solute at times as indicated above. 5of11

6 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 5. Method for computing transverse dispersion: (a) Illustration of point-support method; (b) A typical concentration profile across the domain (? to the principal flow direction). reactive transport problem. A steady state non-reactive tracer profile in one of the periodic media cases is shown in Figure 4. We take a strip of width equal to the unit cell size, such that the velocity field is periodic as illustrated by the streamlines in Figure 4b. We solve the time dependent form of the governing Equations (1) with an initial condition having the lower half of the strip filled with solute A and (for the reactive case) the upper half with solute B. Flow is induced from left to right with periodic boundary conditions as shown in 4c; hence, we call this the wraparound option. For example, the status of solute A is shown in Figures 4d 4f at 0.082, 0.34 and s after the flow is induced. It can be shown that the concentration distribution at various times in the strip directly corresponds to the steady state spatial distribution at successive strips from the inlet of the domain. The transverse concentration distribution at each time step is obtained by averaging along the X direction. The method of moments [e.g., Taylor, 1953; Aris, 1956; De Josselin de Jong, 1958] is then used to compute s T 2, the transverse spatial variance of the dispersed front at each time. To compute D T,N, we use the relation D T;N ¼ 1 2 ds 2 T dt ¼ 1 2 s 2 T t ; ð2þ which is commonly called the secant method. We found that the same value of D T,N is obtained by fitting the well-known error function solution [Crank, 1980] to the transverse concentration profile. We used this method in our previous work [Knutson et al., 2007] where we also found that the value of D T,N obtained in this manner is consistent with that obtained using the unit-cell closure techniques (so-called B-field theory) by previous researchers [Eidsath et al., 1983; Souto and Moyne, 1997]. [18] The results D T,N /D m are interpreted as a function of Péclet (Pe) number, which is defined as Pe ¼ Uhd gi D m ; ð3þ where D m is the molecular diffusion of the conservative solute, hd g i denotes the mean cylinder (grain) diameter, and U is the mean intrinsic velocity in the primary flow direction Method for Non-Periodic Media [19] A steady state nonreactive tracer profile for one of the non-periodic media cases is shown in Figure 5. Unlike for periodic media, there is no simple unit cell or REV over which to average the pore-scale concentration. To avoid introducing any artificial smearing, we use the pointconcentrations computed from the pore-scale model (see Figure 5a). The transverse profile at a particular downstream location X* is plotted and the spatial variance is computed. It is important to note that the point-concentration profile across the transverse direction of the domain suffers data gaps due to the presence of solid grains. We use simple interpolation to fill in these data gaps, resulting in the schematic transverse profile illustrated in Figure 5b. Equation (2) is used to retrieve the dispersion coefficient from the spatial variance of the profile. However, the time in this relation is replaced by the overall residence time from the inlet to X*, that is X*/U. In order for the plume to spread across a statistically representative number of pores, the length and width of the domain should be large. The domain size is determined by the Péclet number and heterogeneity of the system. The computation of the flow field by LBM is very time consuming. To obtain the flow field for large domains, we lengthened (and widened) our domain through mirroring of streamlines. 5. Methods for Determining the Apparent Transverse Dispersion Coefficient for a Reactive Tracer [20] To simplify the analysis, we assume that D A = D B = D AB, the stoichiometry is equi-molar (f A = f B = f AB = 1), and the inlet concentrations of A and B are equal (i.e., A 0 = B 0 ). Under these conditions the reaction zone should align approximately along a straight line as demonstrated in Figure 6. In the periodic media, the plume centerline is straight along the principal flow direction (Figure 6a). In the random medium, some fluctuations of the product plume centerline about the principal flow direction appear 6of11

7 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 6. Simulated product plumes in different porous media for Pe = 150: (a) Periodic medium with a porosity of 0.50 and cylinder size of 240 mm; (b) Random medium with porosity and mean cylinder size 135 mm; (c) Trending sinuous porous medium of randomly shaped and sized cylinders (porosity is 0.49 and mean diameters: 135 mm for fine layer and 270 mm for coarse layer). (Figure 6b). In the trending media (the syncline and anticline like domain), the product plume deforms in a complex manner that is consistent with the deformation of the layered heterogeneities (Figure 6c). Also, the largerscale heterogeneity in the trending media causes deviations of the width of the reaction zone. Although not examined in this work, Nambi et al. [2003] presented experimental results showing how differences in diffusion coefficients, stoichiometry, and inlet boundary concentrations cause vertical migration of the mixing zone along the flow path. [21] For the case of an instantaneous reaction in a uniform flow field there is a continuum scale analytical solution for the steady state product species formed by the transverse mixing of two parallel reactant streams. This solution was presented by Nambi et al. [2003] and is related to that used by Gramling et al. [2002]. The solution of Nambi et al. [2003] is based upon earlier works [Ortoleva and Ross, 1973; Ortoleva, 1994] and is valid for the case when the reactants and product have different diffusion coefficients. We use this analytical solution to compute the total transverse integrated product along the flow direction. This is defined as mx ðþ¼ Z þ1 1 ABðx; yþdy: ð4þ f B = f AB = 1), we find that the total product integrated over the cross section is given by rffiffiffiffiffiffiffiffiffiffiffi xd T;R mx ðþ¼2a o for 0 < x < 1; pu where m(x) is the total mass of product integrated over the transverse direction per unit length [ML 1 ] at a distance x from the inlet, A 0 in this case is the input concentration of solute A given as mass per unit depth [ML 2 ]. This solution is valid at the continuum scale and in this work we argue that it is also applicable to the spatially averaged product from the pore-scale simulations. [22] From the pore-scale simulations we compute m(x) by integrating the product over the transverse direction; we then use least squares to find D T,R by fitting the analytical solution given by Equation (5). This fitting will provide us with the apparent transverse dispersion coefficient (D T,R ). In our work we compare the values of D T,R and D T,N for different media and different regimes designated by dimensionless Péclet number Pe. [23] A related and more simple way of computing D T,R is by integrating the product mass (Equation (5)) up to a given longitudinal position, X*. Then we can solve directly for the apparent diffusion coefficient ð5þ Using the analytical solution for the simplified case where D A = D B = D AB and the stoichiometry is equi-molar (f A = D T;R ¼ 1 X * 3 9pU 16A 2 0 Z X* 0 2 mx ðþdx! ; ð6þ 7of11

8 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 7. Coefficients of transverse dispersion during nonreactive and reactive transport in periodic media composed of circular cylinders. Shown is the relative diffusivity evolution for various Péclet numbers (Pe) asa function of space t = x/dg. Porosity e = 49.7% and grain diameter d g = m. Legends D T,N and D T,R denote respectively, nonreactive and reactive relative diffusivity (DT/D m ). The value of D T,R computed by Equation (6) represents an effective value that, when substituted into the analytical solution (5)), yields the same total product mass created from x =0tox = X* from the pore-scale simulation. 6. Results 6.1. Apparent Transverse Dispersion for a Conservative Tracer, D T,N [24] Values of D T,N were computed as a function of dimensionless distance (t = x/d g ) using Equation (2) for each porous media type; the results (along with those for D T,R ) are shown in Figures 7 and 8. The estimated values of D T,N changes with distance from the inlet but eventually stabilize to an asymptotic value. For the periodic media (Figure 7) at Pe = 10, values of D T,N stabilize within a distance of about 10 grain diameters. At higher Pe, stabilization requires a longer travel distance. It is straightforward to show (by considering an advective timescale equal to x/u and a diffusion timescale equal to d g 2 /D m ) that a dimensionless travel distance of t Pe is required for the solute to diffuse over the scale of the grain diameter. A similar development time was reported by Maier et al. [2000] in their simulations of 3D periodic sphere packs, who noted the consistency of their results with the theory of Koch and Brady [1985] and Koch et al. [1989]. [25] For the random and trending media (Figure 8) at Pe = 10, values of D T,N stabilize within about 30 and 80 grain diameters, respectively; this distance increases with Pe. It is interesting to note that the fluctuations in the random media appear random, whereas in the trending medium the fluctuations are periodic. For our steady state transport scenario, the dispersing solute plume samples a relatively small part of the pore system corresponding to the narrow transverse spreading zone adjacent to the injection line. A transition distance is required for the solute along the mixing interface to sample a statistically representative set of velocities; the transition distance is influenced by both advection and molecular diffusion, and it also depends upon the location of the injection line. Note that the results of Maier et al. [2000] do not exhibit the fluctuations and transition distance shown in Figure 8 since they use a completely different approach to compute dispersion where the entire pore space is sampled at every time step. If we average multiple simulations with slightly different locations of the mixing line, then our results are much less erratic and stabilize to a constant dispersion coefficient at a much smaller dimensionless travel distance. This is equivalent to conducting multiple steady state transport simulations in different realizations of the random porous media. Figure 8. Relative diffusivity evolution during reactive and nonreactive transport for various Péclet numbers (Pe) as a function of space t = x/d g (a) in random porous medium with porosity e = 53.7%, mean grain diameter hd g i = m and standard deviation of m. (b) in trending sinuous and random medium with e = 49.7%, mean grain diameter hd g i = m and standard deviation of m. Legends D T,N and D T,R denote respectively, nonreactive and reactive relative diffusivity (D T /D m ). 8of11

9 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 9. Product formed (m) per unit length as a function of distance at Pe = 50 for (a) random medium (RandLow) and (b) periodic medium (Cyl136). The analytical solution (Equation 5) is also shown using D T,N and D T,R Apparent Transverse Dispersion for a Reactive Tracer, D T,R [26] Values of D T,R were also computed as a function of dimensionless distance using Equation (6); the results are shown in Figures 7 and 8. For all cases considered, values of D T,R /D m steadily increase to an asymptotic value. The times (hence the distances) to reach asymptotic values are dependent on the advective conditions, type of porous medium, particularly the variance of grains and the structure of the porous medium (whether or not it contains a large scale trend). For example, the asymptotic values of dispersion coefficient in periodic media (uniform cylinders) are obtained much earlier than in simple random porous media, whereas the distances for the trending media are much larger than in other simple media (see Figures 7 and 8) Comparing D T,N and D T,R [27] Comparison of D T,N and D T,R indicates that there is minimal difference between asymptotic values of these parameters in a given medium and at fixed Pe. This agreement is further illustrated in Figure 9, where the total mass at each cross-section from the pore-scale model is compared with the predicted values using the analytical solution (Equation (5)) with transverse dispersion given by either D T,N or D T,R. We show results for two media: a random case and a periodic case. For the purpose of comparison with the analytical solution, the product is summed up per unit length per unit fluid volume according to Equation (4) and plotted in Figure 9. To this product profile along the x axis, we fit the analytical Equation (5) to get D T,R. As shown, we observe that for random medium the numerical product is fluctuating for a relatively long distance whereas for periodic medium the numerical and analytical products quickly stabilize and become equal. In all cases, the theoretical prediction using the nonreactive dispersion coefficient accurately models the mixingcontrolled reaction Comparison of Transverse Dispersion Coefficients to Literature Values [28] Asymptotic values of transverse dispersion coefficients (either D T,N or D T,R ) normalized by D m are plotted in Figure 10 as a function of the Péclet number. In the same figure, we include two regression fits of D T,N /D m determined from tracer experiments conducted by Olsson and Grathwohl [2007] in a quasi-2d flow tank packed with spherical glass beads. Their experimental system consisted of two quasi-2d acrylic-glass flow-through tanks, a smaller one of dimensions (L H W) 28.0 cm 14.0 cm 1.05 cm and a larger one of 77.9 cm 14.0 cm 0.8 cm. Glass beads and quartz sand were used as the porous media, and experiments were conducted over a range of Péclet number from approximately one to The porosity of our numerically generated media varied from 45% to 55%. Olsson and Grathwohl [2007] present a method to make adjustments for different porosity values, and we used their relation to obtain the two separate regressions for 45% and 55% porosities. Olsson and Grathwohl [2007] also found that their values of D T,N were similar to many other experimental values reported in the literature [e.g., Grane and Gardner, 1961; Harleman and Rumer, 1963]. [29] We find that the results from the random pack consistently agree with their lab results. Spherical bead diameters used in their experiments are similar (100 microns to 250 microns) to our 2D grains. We also find that transverse dispersion coefficients for different porous media agree at low Pe, but diverge as Pe increases. At high Pe, the dispersion coefficient for periodic media is lower than for randomly packed media, even though the mean grain size is the same. The trending medium that represents the syncline and anticline structure produces even more dispersion than the randomly packed media at high Pe. These macroscopic heterogeneities increase tortuosity of the flow paths and create flow focusing regions of enhanced mixing [Werth et al., 2006]. Although our work is for steady state flow, these trends agree qualitatively with other studies that have investigated the impact of plume meandering on macroscopic transverse dispersion under transient flow conditions [Goode and Konikow, 1990; Cirpka and Attinger, 2003]. We also note that our dispersion coefficients for the periodic media, composed of equal sized cylinders, are similar to those reported by earlier workers [e.g., Edwards et al., 9of11

10 ACHARYA ET AL.: PORE-SCALE SIMULATION OF DISPERSION Figure 10. Relative diffusivity (D T,N /D m ) as a function of Pe. Sinuous denotes (as shown in Figure 2c) a packed bed of randomly shaped sized cylinders with inclusions of synclines and anticlines. RandHigh and RandLow denote randomly distributed packing of circular and elliptical cylinders (as shown in Figure 2b). Cyl136 and Cyl240 denote (as shown in Figure 2a) staggered pack of circular cylinders of size respectively 136 mm and 240 mm. O&G_45 and O&G_55 denote new relations based on lab data [see Olsson and Grathwohl, 2007]. 1993; Souto and Moyne, 1997; Wood et al., 2003], who obtained their results by solving the B-field equation. 7. Conclusions [30] Agreement of D T,N and D T,R indicates that for the cases considered, the continuum-based approach does not over-estimate the reaction product since product formation can be predicted using transverse dispersion coefficients determined from the spread of a conservative tracer. This is in contrast to the results from earlier work [e.g., Raje and Kapoor, 2000; Gramling et al., 2002] that showed nonreactive longitudinal dispersion coefficient values are not able to adequately predict mixing controlled reaction product. This conclusion is reasonable since transverse dispersion relies on molecular diffusion to cause solute movement across streamlines for the case of mean one-dimensional steady state flow, and since hydrodynamic dispersion is much weaker in the transverse than longitudinal direction. Carefully controlled continuum-scale experiments reported recently by Ham et al. [2007] also indicate that mixingcontrolled reaction could be predicted using D T,N. We found good agreement between D T,N and D T,R even for the macroscopically trending case with larger-scale plume meandering. This is possibly due to our assumption of steady state flow conditions, since other continuum-scale studies investigating plume meandering due to temporal variability in the flow direction have indicated that transverse macrodispersion coefficients that describe nonreactive plume spreading tend to overpredict actual solute mixing in the transverse direction [e.g., Cirpka, 2005]. For better insight, we need to simulate transport for a wider variety of grain sizes, larger values of Pe, and other conditions, and further experimental confirmation is needed. Future work should also assess the impact of streamline twisting and other mixing mechanisms which can act in three, but not two, space dimensions. [31] Acknowledgments. This material is based upon work supported by the National Scientific Foundation under grant number BES We thank Associate Editor Olaf Cirpka and three anonymous reviewers for their comments which have improved the paper. References Aris, R. (1956), On the dispersion of a solute in a fluid flowing through a tube, Proc. R. Soc. Lond. A., 235, Baumann, T., and C. J. Werth (2004), Visualization and modeling of polystyrol colloid transport in a silicon micromodel, Vadose Zone J., 3, Bear, J. (1972), Dynamics of fluids in porous media, 764, Dover Publications, Inc., New York. Benekos, I. D., O. A. Cirpka, and P. K. Kitanidis (2006), Experimental determination of transverse dispersivity in a helix and a cochlea, Water Resour. Res., 42, W07406, doi: /2005wr Bockelmann, A., D. Zamfirescu, T. Ptak, P. Grathwohl, and G. Teutsch (2003), Quantification of mass fluxes and natural attenuation rates at an industrial site with a limited monitoring network: A case study, J. Contam. Hydrol., 60(1 2), Cirpka, O. A. (1999), Numerical simulation of biodegradation controlled by transverse mixing, J. Contam. Hydrol., 40(2), Cirpka, O. A. (2005), Effects of sorption on transverse mixing in transient flows, J. Contam. Hydrol., 78(3), Cirpka, O. A., and S. Attinger (2003), Effective dispersion in heterogeneous media under random transient flow conditions, Water Resour. Res., 39(9), 1257, doi: /2002wr Cirpka, O. A., A. Olsson, Q. Ju, M. A. Rahman, and P. Grathwohl (2006), Determination of transverse dispersion coefficients from reactive plume lengths, Ground Water, 44, of 11

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(valocchi@uiuc.edu; werth@uiuc.edu) T. W. Willingham, ExxonMobil Upstream Research Corporation, URC- URC-SW254, P.O. Box 2189, Houston, TX 77252, USA. (thomas.w. willingham@exxonmobil.com) 11 of 11