Temporal evolution of pore geometry, fluid flow, and solute transport resulting from colloid deposition

Transcription

1 WATER RESOURCES RESEARCH, VOL. 45,, doi: /2008wr007252, 2009 Temporal evolution of pore geometry, fluid flow, and solute transport resulting from colloid deposition Cheng Chen, 1,2 Boris L. T. Lau, 1,3 Jean-François Gaillard, 1 and Aaron I. Packman 1 Received 1 July 2008; revised 26 January 2009; accepted 16 February 2009; published 12 June [1] Deposition of colloidal particles is one of many processes that lead to the evolution of the structure of natural porous media in groundwater aquifers, oil reservoirs, and sediment beds. Understanding of the mechanisms and effects of this type of structural evolution has been limited by a lack of direct observations of pore structure. Here, synchrotron X-ray difference microtomography (XDMT) was used to resolve the temporal evolution of pore structure and the distribution of colloidal deposits within a granular porous medium. Column filtration experiments were performed to observe the deposition of relatively high concentrations of colloidal zirconia (200 mg/l of particles having diameter 1 mm) in a packed bed of glass beads (diameters mm). Noninvasive XDMT imaging of the pore structure was performed three separate times during each column experiment. The structural information observed at each time was used to define internal boundary conditions for three-dimensional lattice Boltzmann (LB) simulations that show how the evolving pore structure affects pore fluid flow and solute transport. While the total deposit mass increased continuously over time, colloid deposition was observed to be highly heterogeneous and local colloid detachment was observed at some locations in a low ionic strength medium. LB simulations indicated that particle accumulation greatly reduced the permeability of the porous medium while increasing the tortuosity. The colloidal deposits also increased the spatial variability in pore water velocities, leading to higher dispersion coefficients. Anomalous dispersion behavior was investigated by simulation at the scale of the experimental system: weak tailing was found in the clean bed case, and the extent of tailing greatly increased following colloid deposition because of the development of extensive no-flow regions. As a result of this coupling between pore fluid flow, colloid accumulation, and the pore geometry, colloid deposition is expected to strongly influence long-term solute dynamics in cases where solute transport is either accompanied by high colloid influx or where the passage of the solute front mobilizes and then redistributes material from the porous matrix. Citation: Chen, C., B. L. T. Lau, J.-F. Gaillard, and A. I. Packman (2009), Temporal evolution of pore geometry, fluid flow, and solute transport resulting from colloid deposition, Water Resour. Res., 45,, doi: /2008wr Introduction [2] Colloid attachment and detachment is one of the most important processes that modify the internal structure of porous media. Colloidal particles migrating through the porous space become immobilized by a variety of physicochemical mechanisms [McDowell-Boyer et al., 1986]. Accumulated particles increase the microstructural complexity of the porous medium, increase frictional resistance to pore water flow, and consequently reduce the permeability of the porous medium [Gaillard et al., 2007; Chen et al., 2008]. Particle deposition is fundamentally controlled by pore fluid flow [Yao et al., 1971; Rajagopalan and Tien, 1976; Elimelech et al., 1995], leading to complex feedbacks 1 Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois, USA. 2 Now at Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA. 3 Now at Department of Geology, Baylor University, Waco, Texas, USA. Copyright 2009 by the American Geophysical Union /09/2008WR between pore fluid flow and particle deposition dynamics in the porous medium [Mays and Hunt, 2005]. Detachment (reentrainment) of deposited particles also frequently occurs when solution ph increases [Ryan et al., 1999], ionic strength decreases [Roy and Dzombak, 1996], or hydrodynamic shear increases [Ryan et al., 1998]. Even in the absence of these disturbances, particle dislodgement can occur because interstitial velocities increase following restriction of the pore space, resulting in increasing hydrodynamic forces on the deposited particles. Particles detach from collector surfaces when the lift and drag forces become greater than the adhesive forces. Thus, reentrainment tends to occur primarily under conditions of weak particle-particle interactions, and decreases with increasingly favorable attachment conditions [Li et al., 2005]. [3] While a great deal of research has demonstrated that coupling between colloid deposition and fluid flow is a complex process, relatively limited insight is available on the relationship between changing pore geometry, pore fluid flow, and hydrodynamic transport processes in the porous medium [Mays and Hunt, 2005]. Not only are these pro- 1of12

2 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION cesses important for particle migration, but modification of fluid flow and solute transport is also an important issue for a variety of applications including pathogen and contaminant transport, in situ bioremediation of contaminant plumes, management of groundwater resources, and oil extraction [Bear et al., 1993; Lisle and Rose, 1995; Baveye et al., 1998; Ochi and Vernoux, 1998; Kretzschmar et al., 1999; Moghadasi et al., 2004; Zachara et al., 2004; Cortis et al., 2006]. [4] Solute transport in porous media is normally characterized using the advection-diffusion equation (ADE), which predicts symmetric plume spreading with linear growth of the second moment of the solute concentration distribution over time. However, observations of solute migration in natural porous media have consistently shown deviations from the ADE prediction [Silliman and Simpson, 1987; Adams and Gelhar, 1992; Maier et al., 2000; Cortis and Berkowitz, 2004; Zhang and Kang, 2004]. It is now recognized that this anomalous solute dispersion occurs because of internal disorder and heterogeneity within natural porous media, including soils and saturated groundwater aquifers [e.g., Benson et al., 2001; Cortis et al., 2004, 2006; Berkowitz et al., 2006; Zhang and Lv, 2007]. Thus, anomalous dispersion behavior is expected to be strongly influenced by reorganization of the pore space associated with colloid migration, but little information is available on the underlying processes that control the coupling between pore fluid flow, particle migration, and evolution of pore structure. [5] Classically, the key difficulty in evaluating transport in porous media has been the lack of internal observations of pore structure and pore fluid flow. During the past 2 decades, new methods have emerged that enable detailed characterization of both internal structure and the resulting pore fluid flow. A variety of noninvasive experimental methods have been used to characterize deposited colloids within the pore space [e.g., Amitay-Rosen et al., 2005; Li et al., 2006]. Recently, synchrotron-based X-ray difference microtomography (XDMT) has been used to simultaneously resolve the structure of porous media and colloidal deposits at nearmicron-scale resolution [Gaillard et al., 2007]. The observed structural information can then be used to define internal boundary conditions for simulation of the flow field at the pore scale [e.g., Auzerais et al., 1996; Fredrich et al., 2006; Chen et al., 2008]. The lattice Boltzmann (LB) method is particularly suitable for simulating fluid flow with complex boundary conditions such as those found in porous media [Chen and Doolen, 1998; Succi, 2001]. Here, we use these methods to evaluate the temporal evolution of pore geometry, pore fluid flow, and solute transport resulting from colloid deposition in a granular porous medium. We demonstrate how the combined experimental and numerical approach supports direct quantitative analysis of the way in which local-scale interactions between pore structure, pore fluid flow, and particle deposition influence important macroscopic transport properties including permeability, tortuosity, and dispersion coefficients. 2. Observation of Changing Pore Geometry and Colloidal Deposition by XDMT [6] We conducted column experiments to deposit colloidal particles and then observe the resulting pore structure in a granular porous medium at the DuPont-Northwestern-Dow Collaborative Access Team, Advanced Photon Source, Argonne National Laboratory. Columns were composed of high-density polyethylene and had an inner diameter of 3.2 mm and a length of 30 mm. The porous medium was composed of naturally packed glass beads that were primarily spherical with diameters between 210 and 300 mm (Potters Industries, Incorporated, P-0120), and was retained with end-pieces composed of a porous polyethylene sheet having mm pores (Small Parts International). [7] Two experiments were performed to obtain a time series of observations of the evolution of pore structure following ongoing colloid deposition. The matrix structure formed by the packed glass beads remained identical throughout each experiment, and the colloidal deposit structure was observed at three different times with progressively increasing deposit mass. Experiments were performed in a background calcium chloride (CaCl 2 ) solution to facilitate colloid deposition. The columns were initially flushed with the CaCl 2 solution, and then the influent flow was switched to a suspension containing 200 mg/l zirconia (ZrO 2 ) colloids (Z-Tech LLC, CF-Super-HM) in the same background electrolyte by using a three-way valve. The pore fluid flow was vertically upward at a Darcy velocity of 1 mm/s. In the deposition phase, 120 pore volumes of colloidal suspension were pumped through the column, and then the porous medium was again flushed with the CaCl 2 solution in order to remove nondeposited colloids from the pore space. Then the column was carefully moved to the X-ray beamline for XDMT scanning, as described below. Check valves installed at both ends of the column automatically sealed and isolated the porous medium whenever the inflow was stopped. Following completion of the first XDMT data acquisition, an additional 120 pore volumes of the ZrO 2 suspension were pumped through the column, and then the column was again flushed to remove nondeposited colloids and scanned for XDMT. This procedure was then repeated to yield a third observation of the evolving pore structure. Thus, we used the noninvasive XDMT approach to characterize the microstructure of the same sample following cumulative through-flux of 120, 240, and 360 pore volumes of the colloidal suspension. This data set is unique as it includes colocalized, time series observations of the temporal evolution of pore geometry resulting from colloid accumulation in the pore space. This enables explicit modeling of the effects of the evolving pore geometry on pore fluid flow and solute transport, which has not been possible on the basis of previously available data sets where tomography was used to evaluate structure but measurements made at different times were not registered or colocalized [Fredrich et al., 2006; Chen et al., 2008]. The three successive postdeposition pore geometries obtained by X-ray tomography are identified here as the 120 PV, 240 PV, and 360 PV geometries, respectively. [8] The two experiments differed only in the concentration of the background electrolyte: 1 mm and 10 mm CaCl 2 were used, both at ph = Under 1 mm CaCl 2, the zeta potential and mean diameter of the colloids were measured to be 23.0 ± 3.2 mv and 1.1 ± 0.1 mm, respectively (Brookhaven Instruments, ZetaPALS). Under 10 mm CaCl 2, the zeta potential and mean diameter of the colloids were measured to be 7.0 ± 1.5 mv and 1.1 ± 0.2 mm, respectively. 2of12

3 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION [9] The matrix structure (glass beads) and distributions of deposited colloidal particles (ZrO 2 ) were imaged simultaneously by XDMT. The samples were scanned at two monochromatic X-ray energies, 20 ev above and below the Zr K edge (17,998 ev). The resulting X-ray absorption maps were then processed to reconstruct the 3D pore structure as well as the distribution of ZrO 2 deposits. A full description of the XDMT method is given by Gaillard et al. [2007]. Here, X-ray absorption maps were obtained at a resolution of 6 mm, and quantitative analysis was performed on an internal cubic volume of voxels ( mm). Selection of the internal subvolume avoided regions near the column walls where packing artifacts are known to occur, and also regions in the upper and lower portions of the tomograms where decreased intensity of the X-ray beam led to high noise. Note that the resolution used is larger than the diameter of the individual ZrO 2 colloids, and thus does not resolve the microscale pores and surface roughness within the colloidal deposits. As a result, the internal porosity of these deposits was not characterized, the volumetric extent of the deposit cannot be directly converted to deposit mass (as their density varies), and we were unable to simulate water flow and solute transport inside the depositional structures. However, the resolution used here is suitable for simulating the effects of colloid accumulation on fluid flow at the scale of the primary pores, which are on the order of 100 mm [O Connor and Fredrich, 1999; Fredrich et al., 2006]. [10] For the time series measurements, all columns were mounted identically in order to ensure that the pore structure could be exactly compared between scans. A micromanipulation system was used to position each sample with a precision of <10 mm. Samples were manually mounted within the X-ray beam with a precision of 8.3 mm, which is the sample positioning distance associated with the minimum marking on the vernier scale used to set the sample position. X-ray images were used to confirm the sample position, with the images obtained at the same resolution used for XDMT, 6 mm. Sample repositioning during tomographic data acquisition was done by means of an automated micromanipulation system that is reliable at better than the pixel scale. 3. Image Processing and LB Simulations [11] XDMT images were processed using the IDL computational environment (ITT Visual Information Solutions) and Blob3D software package. Blob3D s median smoothing filter and general thresholding filter were used to remove background noise and convert the images to binary (black/ white), respectively, as described by Ketcham [2005]. Geometric properties of the porous medium, porosity, specific surface area, mean pore diameter, and mean grain diameter, were evaluated directly from the processed images using the two-point correlation function approach [Berryman, 1985; Berryman and Blair, 1986, 1987]. The 3D XDMT image data were directly used to define internal boundary conditions in a 3D LB model in order to simulate the porescale fluid flow and solute transport in the porous medium. We employed the D3Q19 (3 dimensions and 19 velocity directions) LB model [Succi, 2001] to analyze pore water flow and solute transport in subvolumes of voxels ( mm) located 500 pixel lengths (3 mm) away from the column inlet. [12] The LB equation for fluid flow can be written as f i ðx þ e i dt; t þ dt Þ ¼ f i ðx; tþ f iðx; tþ f eq i ðr; uþ t ði ¼ 0; 1; Þ ð1þ where f i (x, t) is the particle distribution function, which specifies the number of fluid particles at lattice location x and time t traveling along the ith direction; e i is the lattice velocity vector corresponding to direction i, which is defined as (0, 0, 0)c for i = 0, (±1, 0, 0)c for i = 1, 2, (0, ±1, 0)c for i =3, 4, (0, 0, ±1)c for i = 5, 6, (±1, ±1, 0)c for i =7...10, (0, ±1, ±1)c for i = , and (±1, 0, ±1)c for i = c is defined as dx/dt, where dx is the lattice spacing, here equal to the XDMT resolution (6 mm), and dt is the time step, which is the time that fluid particles take to propagate from one voxel to the adjacent one. In the LB method, dx and dt are used as the spatial and temporal units, respectively. t is the dimensionless relaxation time related to the kinematic viscosity by u =(t 0.5)dx 2 /3dt. f i eq (r, u) is the equilibrium distribution function at location x and time t along the ith direction, which is chosen to recover the macroscopic Navier-Stokes equations and calculated by f eq i " # ðr; uþ ¼ w i r 1 þ 3e i u c 2 þ 9 ð e i uþ 2 2c 4 3u2 2c 2 where w i is the weight coefficient, whose value depends on the specific LB model used. For the D3Q19 model, w i =1/3 for i =0,w i =1/18fori =1...6, and w i =1/36fori = r and u are the macroscopic density and fluid velocity, respectively, and can be calculated by r ¼ X18 u ¼ X18 i¼0 f i i¼0 ð2þ ð3þ f i e i r ð4þ The LB equation (1) recovers the following macroscopic continuity and momentum equations [Qian et al., 1992; Chen et al., þr ð Þ ¼ 0 ðruþ þrðruuþ ¼ rp þr½ruðru þ urþš In addition, the pressure can be calculated by p = c s 2 r, where c s is the speed of sound. For the D3Q19 model, c s 2 = c 2 /3. [13] In order to simulate solute transport, another distribution function g i is used: g i ðx þ e i dt; t þ dt Þ ¼ g i ðx; tþ g iðx; tþ g eq i ðc; uþ ; ði ¼ 0; 1; Þ t s ð7þ 3of12

4 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION where t s is the dimensionless relaxation time related to the solute diffusivity by D =(t s 0.5)dx 2 /3dt. g i eq (C, u) isthe equilibrium distribution function at location x and time t along the ith direction, which is chosen to recover the macroscopic advection-diffusion equation and calculated by g eq i " # ðc; uþ ¼ w i C 1 þ 3e i u c 2 þ 9 ð e i uþ 2 2c 4 3u2 2c 2 where C is the macroscopic solute concentration and can be calculated by C ¼ X18 g i i¼0 The LB equation (7) recovers the macroscopic advectiondiffusion equation [Dawson et al., 1993]: þ ðu r ÞC ¼r ð DrC Þ ð10þ Note that the internal porosity of the colloidal deposits is not captured in the 3D tomographic data used to define boundary conditions in the model, so that the model simulations reflect advection, diffusion, and the resulting dispersion resolved at the scale of the matrix pores only. That is, fluid flow and solute transport is considered to occur around the glass beads and accumulated colloidal deposits, but not within the colloidal deposits. Thus, potential diffusion into and out of the colloidal deposits is not included in these simulations. [14] For fluid flow, we used a periodic boundary condition with a constant pressure difference DP on the inlet (bottom) and outlet (top) of the computational domain [Inamuro et al., 1999], and no-slip boundary conditions on the four lateral sides and interior solid surfaces. Several layers of fluid nodes were added on the inlet and outlet in order to provide smooth inlet and outlet flows. The value of DP was selected to match the Reynolds number of the experiments on the basis of the mean pore water velocity and pore size. Reynolds numbers were always much smaller than one, so that the macroscopic flow was well within the Darcy regime. Also, we confirmed that the small value of DP yielded sufficiently small density variation and Mach number, which are necessary for accurate simulations of incompressible flow and solute transport by the conventional LB method (Q. Kang, personal communication, 2006). As steady flow was reached, the medium permeability k was calculated by Darcy s law: k ¼ runl=dp ð11þ where ru is the mean mass flux through the sample, which is included in order to eliminate the compressibility effect of the conventional LB scheme; n is the kinematic viscosity of fluid; and L is the column length. The tortuosity, t, defined as the ratio of the mean fluid flow path length to linear distance traveled in the mean flow direction, was also found by particle tracking based on the LB-simulated flow fields. The accuracy of our LB code was verified by testing against analytical solutions for several well-defined flow geometries and comparing numerical simulations with experimental observations of permeability obtained using large columns of clean glass beads, as reported in the online supporting information of Chen et al. [2008]. [15] For solute transport, we = 0 at the bottom and top boundaries, where z is the longitudinal flow direction, and no-flux conditions on the four lateral sides and interior solid surfaces. An instantaneous plane source was injected into the porous medium at time zero. The outlet solute concentration was reported over time, yielding a breakthrough curve for conservative solute transport over the length of the simulation domain (0.96 mm). [16] Moment methods were employed to calculate the dispersion tensor of the solute plume D ij ¼ ds 2 ijðþ=2dt t ð12þ where the tensor s 2 ij (t) is the second spatial moment of the solute plume in the i, j directions. The longitudinal dispersion of the solute plume is D L = D zz = ds 2 zz (t)/2dt. In order to avoid errors in estimation of the longitudinal dispersion coefficient based on the limited domain size, we only used the early time data of the concentration field to calculate dispersion coefficients. 4. Results and Discussion [17] The 3D tomographic reconstructions of the pore geometries under both CaCl 2 concentrations at all observation times are presented in Figures 1 and 2. The full tomographic data sets are available via anonymous ftp at It can be seen that the colloidal deposits were highly heterogeneous at the pore scale. There was considerable accumulation of ZrO 2 particles in the narrow throats at grain contacts. The total deposit mass increased over time as a result of ongoing colloid deposition. In addition, the total deposit mass was greater in the 10 mm CaCl 2 background than in the 1 mm CaCl 2 background, as expected because increasing ionic strength favors colloid attachment. While the total deposit mass increased monotonically with colloid influx, more complex behavior was observed locally. In the experiment with lower background ionic strength, local detachment of colloids was observed in some locations within the porous medium. One such region is highlighted in the small circle in Figure 1. In this area, the local deposit mass in the contact of two grains increased between input of 120 and 240 PVs of colloidal suspension, and then decreased from 240 to 360 PVs. This demonstrates that colloid deposition is a highly dynamic process involving both particle attachment and remobilization, and local erosion of colloidal accumulation might be an important factor affecting the evolution of pore geometries. [18] In order to quantitatively assess spatial and temporal variability of colloid deposition and detachment behavior, we evaluated the change of ZrO 2 deposits from the 240 PV geometry to 360 PV geometry for a wide range of averaging volumes, using the method of Zhang et al. [2000] to analyze the statistical representative elementary volume (REV). First, we took a cubic subvolume of 50 3 voxels at the same location inside the 240 PV and 360 PV geometries. In each subvolume, we calculated the ZrO 2 fraction, defined as the ratio of ZrO 2 deposit volume to total volume. The relative 4of12

5 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION Figure 1. Temporal evolution of a 3D pore geometry of voxels ( mm) with 1 mm CaCl 2 background electrolyte: (a) clean bed, (b) following through-flux of 120 PVs of colloidal suspension, (c) following 240 PVs, and (d) following 360 PVs. Glass beads are shown in green and ZrO 2 deposits are red. The evolution in the white circle shows local attachment and detachment of colloids. change of ZrO 2 fraction is defined as (f 2 f 1 )/f 1, where f 1 and f 2 are the ZrO 2 fractions observed in the subvolume for the two different extents of colloid deposition. Then we moved the subvolumes in both geometries by an identical small distance (1 pixel), and obtained the relative change of ZrO 2 fraction for this new pair of locations. This analysis was repeated for every possible subvolume location within the sample. On the basis of these results, the mean and standard deviation of the relative change of ZrO 2 fraction was calculated for this particular subvolume size. This analysis was then repeated for a wide variety of subvolume sizes. The same approach was also used to analyze the relative change of the matrix (glass bead) fraction from the 240 PV geometry to the 360 PV geometry. [19] The mean relative changes of ZrO 2 fraction and glass bead fraction are plotted as functions of subvolume size, with error bars indicating plus and minus one standard deviation (s), in Figure 3. The change in the matrix geometry (glass beads) was evaluated to confirm that the tomograms obtained at different times were adequately registered (colocalized) and that the matrix structure was not perturbed at any time during the experiment. In both columns it was observed that both the relative change in the bead fraction between 240 and 360 PVs and the standard deviation of these observations were very close to zero at all averaging scales. This demonstrates not only that there was very little noise (experimental error) in the XDMT measurement, but also that the structure of the packed column was preserved between the two sets of observations. The mean ZrO 2 fraction increased substantially between 240 and 360 PVs at all scales, as expected because of the ongoing colloid deposition in the porous medium. However, the standard deviation of the ZrO 2 deposit volume was high for small averaging volumes, indicating that there was substantial local variability in particle deposition. At scales of pixel lengths ( mm), negative relative changes of ZrO 2 fraction were observed in many locations under the lower ionic strength background electrolyte, indicating that there was local detachment and redistribution of deposited colloidal particles. [20] These observations strongly suggest that colloid migration through the column was dynamic, with multiple steps of deposition and detachment. It is only when the behavior is averaged over larger volumes that colloid attachment can be seen to dominate, yielding net overall accumulation of colloids in the porous medium. Net accumulation increased with increasing ionic strength, as expected. At the limit of the representative elementary volume, the mean relative change of ZrO 2 fraction was 0.35 under 1 mm CaCl 2 and 0.72 under 10 mm CaCl 2. Moreover, much less spatial variability in ZrO 2 deposits was found in the higher ionic strength background, and negative changes of ZrO 2 fraction were rare even at small length scales. This suggests that more favorable attachment conditions cause colloid deposition to be more homogeneous at the scales considered here, and that deposition is also more 5of12

6 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION Figure 2. Temporal evolution of a 3D pore geometry of voxels ( mm) with 10 mm CaCl 2 background electrolyte: (a) clean bed, (b) following through-flux of 120 PVs of colloidal suspension, (c) following 240 PVs, and (d) following 360 PVs. Much greater colloid accumulation occurred with the higher ionic strength electrolyte (note that each observation is directly comparable between Figures 1 and 2). Glass beads are shown in green and ZrO 2 deposits are red. likely to follow classical (irreversible) filtration behavior under these conditions. [21] LB simulations were performed on two computational domains having dimensions of mm within the 3D reconstructions shown in Figures 1 and 2. Pore water flows were simulated with the same pressure difference in all geometries. Porosities, permeabilities, and pore fluid velocity distributions evaluated by means of image processing and LB simulations are reported in Figure 4. Colloid accumulation in the pore space substantially reduced the bulk porosity of the porous medium. The porosities for the clean bed, 120 PV, 240 PV, and 360 PV geometries with 1 mm CaCl 2 were 0.395, 0.350, 0.316, and 0.288, respectively. For 10 mm CaCl 2, the corresponding values were 0.384, 0.328, 0.263, and 0.171, respectively. Again note that these porosities do not include the microscale internal porosity within the colloidal deposits. Increasing deposits in the pore space also reduced the bulk permeability. The LB-simulated permeability, k, is plotted against the sample porosity, f, in Figure 4a. The permeability decreased with decreasing porosity. It is interesting to note that the results from both experiments can be described with a single power law k / f 3.7. Thus, while increasing ionic strength increased the extent of colloid deposition, and thereby decreased both the bulk porosity and permeability of the porous medium, the consistent porosity-permeability relationship suggests that the deposition had similar effects on the hydrogeologic structure of the porous medium under both electrolyte concentrations. In Figure 4a, permeability results are compared with results from Chen et al. [2008] for conditions that had a similar degree of deposition as found here. That study used the same porous media and colloids in a 10 mm CaCl 2 background solution. The exponent of the power law found from the Chen et al. [2008] data is 3.9, which is similar to that found in the current study. This implies that the power law relationship between porosity and permeability remains valid as long as colloid deposition does not substantially modify the geometric properties of the pore space. [22] Spatial variability in pore fluid flow was assessed in terms of the first moment (mean) and second moment (variance) of the velocity distribution. Results are presented in Figure 4b in terms of the coefficient of variation (CV, the ratio of standard deviation to mean) of the longitudinal velocity, u L. Similar results were found for the magnitude of velocity (u) and the longitudinal component (u L ), but only results for u L are shown here in order to emphasize the longitudinal transport behavior. Increasing deposit mass produced increasing spatial variability in pore water velocities under both ionic strengths again consistently following a power law with porosity, CV / f Thus, accumulation of colloids in the pore space both consistently 6of12

7 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION Figure 4. Variation of (a) LB-simulated permeability, k, and (b) coefficient of variation (CV) for the longitudinal pore flow velocity u L, with sample porosity, f. Open and solid symbols represent observations with 1 and 10 mm CaCl 2 background electrolyte, respectively. Figure 3. Relative fraction changes in colloidal deposits (ZrO 2 ) and matrix structure (glass beads) between 240 and 360 PV geometries versus size of averaging domain. Results are reported as mean values ± 1s of relative fraction changes under (a) 1 mm CaCl 2 and (b) 10 mm CaCl 2. decreased the mean pore fluid velocity (k, u, u L ) and increased its variance. [23] Transport through the porous medium is influenced by the distribution of flow paths through the pore structure as well as the distribution of velocity along those paths. Colloid deposition increased the mean path length for transport through the domain. The mean tortuosity consistently increased with decreasing porosity, following a power law with t / f 0.34, as shown in Figure 5a. Complete tortuosity distributions are shown in Figures 5b and 5c. The increase in the mean tortuosity was accompanied by an increase in spatial variability as well: colloid accumulation caused the variance of tortuosity to increase under all conditions tested, and the distributions gradually became multimodal with increasing deposit mass. This implies that the deposited colloids led to the formation of macroscale heterogeneity in the porous medium, as we found previously [Chen et al., 2008]. Again, these changes were consistent between the experiments with different ionic strength. The only effect of the background electrolyte was that the mean, variance, and multimodality of tortuosity increased faster under the 10 mm CaCl 2 than under the 1 mm CaCl 2 because of the greater rate of accumulation of colloidal deposits. [24] The relationship between the modified structure and transport along flow paths was investigated in detail by examining pore water velocity distributions on a voxel-byvoxel basis, i.e., at every node in the LB simulations. The probability density functions (pdf s) of the longitudinal and transverse velocity components, u L and u T, found with the 1 mm CaCl 2 background are plotted in Figure 6. Because the two transverse velocity components, u x and u y, are very similar, we report u x as u T. The results are presented in terms of velocities normalized using the LB units, u* = u dt/dx. The longitudinal velocity distribution is highly skewed with a remarkable peak close to zero, as shown in Figure 6a. The distribution of the velocity magnitudes (Figure S1) is very similar to the distribution of longitudinal velocity, indicating that the overall velocity depends primarily on the longitudinal velocity component, as expected for the column flow configuration. 1 Colloid accumulation not only reduced the mean longitudinal velocity, but also increased the CVand skewness 1 Auxiliary materials are available in the HTML. doi: / 2008WR of12

8 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION of the velocity distribution. The decrease in the mean velocity and permeability indicates that colloid deposition retarded the fluid flow in the larger pores, but this was not the only effect of deposition. The observation of a large number of stagnant fluid nodes after deposition, combined with the increasing multimodality of the tortuosity distribution, indicates that colloid accumulation also increased the number of isolated and dead-end pores. The transverse velocity distribution has a peak very close to zero (Figure 6b), and its shape is nearly symmetric, as expected because there is no mean flow in the transverse direction. Particle deposition also increased the height of peak near zero. The asymmetric distribution of u L and symmetric distribution of u T found here are consistent with previous findings for fluid flow in porous media [Cenedese and Viotti, 1996; Moroni and Cushman, 2001; Zhang et al., 2005; Zhang and Lv, 2007]. The new results presented here clarify that extensive colloid deposition greatly increases the proportion of stagnant water and that this substantially influences both the longitudinal and transverse velocity distributions. [25] The development of longer pore water flow paths and extensive low-velocity regions within the porous medium make solute transport considerably more complex [Cortis et al., 2004; Zhang and Lv, 2007]. The pore-scale velocity distribution can be related to solute transport through the transition time distribution y(t*), obtained here as the distribution of time required for motion across one pixel length (dx). Following Zhang and Lv [2007], the transition time distribution was evaluated as the distribution of dx/u, where u=(u 2 x +u 2 y +u 2 L ) 1/2. Here the dimensionless time is defined as t* = t/dt. As shown in Figure 6c, y(t*) decayed with time following a very distinct series of power laws as y(t*) / t* b, with b = 2.10, 1.90, 1.80, and 1.76 found for the clean bed and 120 PV, 240 PV, and 360 PV geometries, respectively, in the 1 mm CaCl 2 background electrolyte. The increasingly long power law tails of these transition time distributions are expected to produce similar tailing in solute breakthrough curves [Dentz et al., 2004; Cortis et al., 2004; Zhang and Lv, 2007]. [26] The effects of the increasing structural complexity on transport through the porous medium were directly assessed using pore-scale LB simulations of solute transport. The simulated temporal evolution of the longitudinal dispersion coefficients for solute transport through each pore geometry that developed with 1 mm CaCl 2 is presented in Figure 7a. The same pressure difference was used to simulate solute transport through each of the four pore geometries. Results are presented in dimensionless form. The Peclet number is defined as Pe = Uh/D m, where U is the average longitudinal velocity over the cross section; h is the mean diameter of the grains comprising the porous medium; and D m is the molecular diffusivity of the solute. Different Pe numbers were obtained by adjusting the molecular diffusivity in the LB model (following Zhang and Lv [2007]). The dimensionless time involving solute transport is then defined as t* =td m /h 2. With Pe = , the longitudinal dispersion coefficients were nearly constant over time, implying that 8of12 Figure 5. (a) Variation of ensemble average tortuosity, t, with sample porosity, f. Open and solid symbols represent observations with 1 and 10 mm CaCl 2 background electrolyte, respectively. Cumulative distributions of tortuosity for the observations with (b) 1 mm CaCl 2 and (c) 10 mm CaCl 2, as determined by particle tracking. The curves along the arrow correspond to clean bed, 120 PV, 240 PV, and 360 PV geometries. Tortuosity generally increased and showed increased variance and multimodality with increasing particle deposition.

9 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION they almost reached asymptotic values despite the small spatial scale of the computational domain. However, with higher Pe numbers, the longitudinal dispersion coefficients increased over time, demonstrating the occurrence of anomalous (non-fickian) transport behavior at this scale. This behavior occurred because the solute gradually diffused into isolated pores with nearly stagnant fluid. This caused the spatial distribution of the solute pulse to increase over time, leading to increasing longitudinal dispersion. The magnitude of the longitudinal dispersion coefficient was always greater in the samples with a greater extent of colloidal deposits, and the temporal variability in dispersion was also generally greater with greater colloidal deposits. This is the expected result of the higher spatial variability in pore water flow paths and velocities resulting from colloid accumulation, as presented in Figures 4 6. The longitudinal dispersion coefficient also increased nonlinearly with Pe number, as illustrated in Figure 7b for a fixed early time (t* = 0.006). Figure 6. Probability density functions (pdf s) of (a) longitudinal pore fluid velocity, (b) transverse pore fluid velocity, and (c) the transition time distributions y(t*). The values of b in the power law equation y(t*) / t* b are 2.10, 1.90, 1.80, and 1.76, from clean bed to 360 PV geometry, respectively. The analysis was based on the LB-simulated pore flow fields with 1 mm CaCl 2 background electrolyte. Figure 7. (a) Temporal evolution of the normalized longitudinal dispersion coefficient, D L /D m, and (b) nonlinear dependence of D L /D m on the Pe number at a preasymptotic time. The curves along the arrow correspond to clean bed, 120 PV, 240 PV, and 360 PV geometries. Results are for the 1 mm CaCl 2 background solution. 9of12

10 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION range of temporal and spatial scales. Colloid deposition had a very pronounced effect on solute transport. The increased heterogeneity associated with colloid deposition increased the asymmetry in the breakthrough curves and produced increasingly severe power law tails. The exponents of the power law tails decreased from 7.50 in the clean bed case to 7.00, 5.18, and 4.26 following deposition of 120, 240, and 360 PVs of the colloidal suspension, respectively. Thus, solute transport became more anomalous with increasing extent of colloidal deposits. Figure 8. Breakthrough curves for the four geometries with Pe = The absolute values of slope of the power law tails are 7.50, 7.00, 5.18, and 4.26, from the clean bed geometry to 360 PV geometry, respectively. Results are for the 1 mm CaCl 2 background solution. [27] Dentz et al. [2004] found that the rate of growth of the dispersion coefficient depends on the Pe number as D L /D m / Pe t 2 a, and y(t) / t 1 a = t b. Therefore, on the basis of the distributions of y(t*) shown in Figure 6c, the power law exponents of D L /D m as a function of t* were calculated as 0.90, 1.10, 1.20, and 1.24, from clean bed to 360 PV geometry, respectively. From Figure 7a, when Pe is 5.163, the exponents of D L /D m for the four geometries were 0.301, 0.337, 0.448, and 0.448, respectively. When Pe is 20.65, they grew to 0.671, 0.672, 0.846, and The changes in dispersion coefficients follow the expected trends, demonstrating that the development of stagnant regions as a result of colloid deposition directly result in anomalous solute transport through the porous medium. However, the observed exponents are all smaller than values predicted on the basis of purely advective transport (obtained from the pore water velocity distribution, Figure 6c). This indicates that molecular diffusion is an important mechanism of solute transport even at high Pe. Specifically, molecular diffusion dominates transport in the stagnant regions of the pore space, and thus controls the redistribution of solutes between the stagnant and nonstagnant fluid. [28] The net effect of these processes on solute transport are presented in Figure 8, which shows the breakthrough curves for the four different pore geometries found with the 1 mm CaCl 2 background solution. The outlet concentration, C out, is normalized by the concentration of the instantaneous source, C o. In order to more clearly illustrate the effects of deposited colloids on solute dispersion, Pe = was used to generate all of the breakthrough curves in Figure 8. With this Pe number, the introduced solute does not sample the complete spectrum of pore water velocities before flowing out of the domain. As a result, the averaging necessary for symmetric, Fickian mixing behavior is not achieved, and the breakthrough curves show prominent tailing [Fischer et al., 1979; Benson et al., 2001; Berkowitz et al., 2006]. Thus, asymptotic dispersion coefficients are not found within this 5. Conclusions and Implications [29] The combination of XDMT and LB simulation was used to evaluate both the temporal evolution of the pore geometry of a granular porous medium as a result of the deposition of colloidal particles, and the effects of this structural evolution on pore water flow and solute transport. Although colloid accumulation in the porous medium is dominated by particle attachment to the solid matrix, the time series of XDMT observations revealed that there is substantial local detachment at small spatial scales. Thus, while deposition from suspension leads to a net accumulation of particles in the porous medium, local behavior does not follow the homogeneous, irreversible deposition predicted by colloid filtration theory. Instead, colloidal deposition is clearly a highly dynamic process, and the background chemical conditions strongly affect these dynamics. Conditions that increase particle-particle interaction forces favor stronger attachment of colloids to collector surfaces, and disfavor subsequent release of deposited colloids. We observed substantial erosion of colloidal ZrO 2 deposits at scales up to 700 mm in a 1 mm CaCl 2 background solution, but found that erosion was very rare in a 10 mm CaCl 2 background solution. [30] The evolution of the pore space associated with this colloid deposition strongly influences pore fluid flow and solute transport in the porous medium. LB simulations based on the time series of pore geometry measurements indicated that ongoing accumulation of colloidal deposits not only progressively reduced the bulk permeability of the porous medium, but also increased both the tortuosity of the medium and spatial variability in pore water velocities. Further, colloid deposition also produced substantial regions having nearly stagnant pore fluid. Transition time distributions at the scale of the XDMT resolution (6 mm) followed power laws, and the tailing in the velocity distribution increased (the exponents of the distributions became less negative) with increasing colloid accumulation in the pore space. [31] These changes in the pore water flow field caused solute transport to become more anomalous with increasing colloid accumulation in the pore space, resulting in longer and flatter tails in breakthrough curves for conservative transport at the millimeter scale. Simulations revealed that anomalous dispersion occurred in the clean granular porous medium (prior to colloid deposition) at high Pe numbers. The temporal variability in dispersion coefficients increased with increasing colloid deposition as well as with Pe. The power law distributions found in the local-scale velocities and transition time distributions indicate that the anomalous dispersion occurs because of inhomogeneity in the pore 10 of 12

11 CHEN ET AL.: TEMPORAL EVOLUTION OF PORE GEOMETRY FROM COLLOID DEPOSITION water velocity field. The presence of extensive low-velocity regions means that a long time is required to sample the entire pore fluid velocity distribution, which is required to achieve the averaging implicit in the conventional Fickian advection/diffusion and Taylor advection/dispersion equations. Colloid accumulation increased the spatial variability of the pore flow field, and thereby increased the travel time required for solutes to experience the entire velocity spectrum and achieve the averaging necessary to obtain a constant dispersion coefficient. [32] The combination of XDMT and LB analysis clearly showed how evolution of pore structure influences transport in the porous medium by simultaneously increasing the overall frictional resistance (thereby decreasing the permeability) and increasing the spatial complexity of the pore water velocity field. The effects of extensive colloid deposition on permeability are well known, but predictive tools for assessing this behavior are lacking because of the high degree of complexity involved in pore fluid flow, colloid transport through the porous medium, and colloid accumulation in the pore space (see, e.g., the summary by Mays and Hunt [2005]). In more general terms, an extensive body of theory has recently emerged that predicts that the degree of spatial disorder of porous media plays a critical role in producing anomalous, heavy-tailed solute transport behavior [e.g., Meerschaert et al., 1999; Schumer et al., 2003a, 2003b; Cortis et al., 2004; Berkowitz et al., 2006]. However, little data are available that allow this theory to be evaluated in various types of porous media. The combined experimental and numerical approach implemented here should be extremely useful for advancing theory for transport in porous media, particularly for cases involving evolution of pore structure. The data sets used here have been made permanently and freely available (via anonymous ftp at ) in order to provide test cases for future theory and numerical models. [33] The coevolution of solute transport, particle transport, and pore structure is particularly critical in cases where there is either an extensive reorganization of fine particulate matter or where there is ongoing dissolution and/or precipitation of the porous matrix. Important examples include migration of contaminant plumes, remediation of contaminated aquifers by chemical perturbations intended to either sequester contaminants into solid phases or to mobilize them for subsequent treatment, numerous processes associated with oil extraction, pathogen transmission to drinking water wells, and injection of carbon into the subsurface for geologic sequestration. The methods developed here can be used to evaluate coupling between pore fluid transport and evolution of pore structure in both laboratory and field settings, and should be particularly valuable to relate local-scale evolution of pore structure to overall, site-scale migration of dissolved and particulate materials. [34] Acknowledgments. This work is based upon material supported by the National Science Foundation via grant EAR This work was performed at the Northwestern Synchrotron Research Center located at sector 5 of the Advanced Photon Source. The DuPont-Northwestern-Dow Collaborative Access Team is supported by E. I. DuPont de Nemours and Company, the Dow Chemical Company, the National Science Foundation through grant DMR , and the state of Illinois through grant IBHE HECA NWU 96. Use of the Advanced Photon Source was supported by the Department of Energy under contract W Eng-38. We thank Denis Keane for setting up the equipment, software, and computer cluster used for tomography and Susa Stonedahl for assisting with the XDMT measurements. We thank Wei Zhang for making the high-quality images in Figures 1 and 2. We also thank Rina Schumer and Andrea Cortis for useful discussions regarding relationships between pore structure and anomalous transport behavior and the Editors and three anonymous reviewers for their constructive comments on the manuscript. References Adams, S. E., and L. W. Gelhar (1992), Field-study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis, Water Resour. Res., 28(12), , doi: /92wr Amitay-Rosen, T., A. Cortis, and B. Berkowitz (2005), Magnetic resonance imaging and quantitative analysis of particle deposition in porous media, Environ. Sci. Technol., 39, , doi: /es048788z. Auzerais, F.M., J. Dunsmuir, B.B. Ferreol, N. Martys, J. Olson, T.S. Ramakrishnan, D.H. Rothman, and L.M. 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