High-energy, synchrotron-based x-ray difference micro-tomography (XDMT) was combined with lattice Boltzmann simulations to assess changes in pore-scale flow patterns and bulk permeability resulting from colloid deposition in a granular porous medium. The detailed structural information obtained from XDMT was used to define internal boundary conditions for simulations of pore fluid flow both with and without colloidal deposits. As colloids accumulated in the pore space, the mean tortuosity increased and the tortuosity distribution became multi-modal, indicating the development of macro-scale heterogeneity. These structural changes also produced large reductions in bulk permeability that were not captured by empirical or semi-empirical estimators based on the first-order geometric properties of the porous medium. This work demonstrates that coupling between fluid flow and particle transport produces heterogeneities at the sub-millimeter scale that greatly affect the hydrogeologic properties of natural porous media.
 A variety of empirical and semi-empirical models are used to estimate hydrogeologic properties of natural porous media, but it is not clear how well these estimates capture the changes in permeability resulting from evolution of pore structure. Numerous processes modify pore structure in geologic systems, including compaction and fluidization, chemically induced precipitation and dissolution of the matrix, and the redistribution of fine particles within the porous medium following physical or chemical perturbations. Here we focus on modification of pore structure resulting from deposition of suspended particles. Colloidal particles migrating through a granular porous medium become immobilized by a number of physicochemical mechanisms [McDowell-Boyer et al., 1986]. The deposited particles increase flow resistance and thereby reduce the permeability of the porous medium. This process is very important for a variety of applications including performance of water treatment filters, pathogen and contaminant transport in groundwater, oil extraction, in situ bioremediation of contaminant plumes, and degradation of streambed habitat by siltation [Baveye et al., 1998; Ochi and Vernoux, 1998; Packman and MacKay, 2003; Moghadasi et al., 2004].
 Fine-scale heterogeneity in pore structure and the non-linear coupling between pore fluid flow, colloid deposition, and evolution of the pore geometry make it very difficult to predict the dynamics of the clogging process based on bulk observations [Veerapaneni and Wiesner, 1997; Mays and Hunt, 2005]. A variety of approaches can now be used to directly assess permeability at the local scale. Permeability can be estimated from structure observed by electron microscopy performed on thin sections [Berryman and Blair, 1986, 1987], and three-dimensional x-ray computed micro-tomography has been combined with lattice Boltzmann (LB) simulations of pore fluid flow to calculate macroscopic transport properties of porous media [Auzerais et al., 1996; Zhang et al., 2000; Fredrich et al., 2006]. Recently, we demonstrated the use of x-ray difference micro-tomography (XDMT) to observe the formation of colloidal deposits in granular porous media [Gaillard et al., 2007]. Here we combine XDMT with LB simulations to assess how the evolution of pore structure affects the macroscopic hydrogeologic properties of the porous medium. In particular, we evaluate how the growth of depositional structures produces heterogeneity at the pore scale, and how these effects are propagated to bulk properties assessed at the Darcy scale.
2. Experimental Setup and XDMT Data Acquisition
 We conducted column experiments to deposit colloidal particles in a granular porous medium. Columns were constructed of high-density polyethylene tubes with an inner diameter of 3.2 mm and a length of 30 mm. End-pieces were polyethylene sheets having 90–130 μm pores (Small Parts Int.) The porous medium was composed of naturally packed glass beads that were primarily spherical with diameters between 210 and 300 μm (Potters Industries Inc., P-0120). The columns were initially flushed with a 10 mM calcium chloride (CaCl2) solution at pH = 6.75, and then the influent flow was switched to a suspension containing 200 mg/L zirconia (ZrO2) colloids (Z-Tech LLC, CF-Super-HM) in the same background electrolyte by using a three-way valve. The CaCl2 solution was used to facilitate colloid deposition. Under these experimental conditions, the zeta potential and mean diameter of the colloids were observed to be −7.0 ± 1.5 mV and 1.1 ± 0.2 μm, respectively (Brookhaven Instruments, ZetaPALS). The pore fluid flow was vertically upward at a Darcy velocity of 1 mm/s. Three column experiments were performed with through-flow of 120, 180, and 240 pore volumes of colloidal suspension. These are identified as columns 1, 2, and 3, respectively. Following the colloid deposition phase, the columns were again flushed with the CaCl2 solution in order to remove suspended (non-deposited) colloids from the pore space and then sealed and transferred to the x-ray beamline for analysis.
 Distributions of deposited zirconia particles were imaged by XDMT. The samples were scanned at two monochromatic x-ray energies, 20 eV above and below the Zr K edge (17998 eV). The resulting x-ray absorption maps were then processed to reconstruct the 3D pore structure as well as the distribution of ZrO2 deposits as described by Gaillard et al. . Using these methods, we obtained three-dimensional images having overall dimensions of 7.8 mm cubed at a resolution of 6 μm (pixel length). This resolution was selected to resolve the structure most relevant to Darcy flow in the porous medium and is larger than the size of the individual colloidal particles. Thus, while microscale internal porosity of the colloidal deposits was not resolved, the 6 μm resolution allows calculation of the macroscopic permeability, which is dominated by flow through large pores and is not sensitive to micro-structural features [O'Connor and Fredrich, 1999; Fredrich et al., 2006]. Because the height of each image was considerably less than the column length, each column was scanned in two parts. The first scan covered the first 7.8 mm from the column inlet, and the second scan spanned the range 7.8 mm–15.6 mm. These regions are identified as sample A and sample B for each column, so that samples are identified both by the column number and sample location: 1-A, 1-B, 2-A, etc.
3. Image Analysis and Lattice Boltzmann Simulations
 For many applications, it is necessary to estimate how the hydrogeologic properties of the porous medium change after transport is initiated or perturbed. Most simply, permeability can be correlated to porosity, typically in a power-law relationship. If additional information is available on sample structure, this can be incorporated into the permeability estimate based on generalizations of exact analytical solutions for simple pore geometries. The most commonly used estimator of this form is the Kozeny-Carman (KC) relationship, which relates permeability to the porosity, specific surface area, and tortuosity of the medium. We employed the following form of the Kozeny-Carman equation [Walsh and Brace, 1984]:
where b is a geometric constant; ϕ is the medium porosity defined as the ratio of pore volume to total volume; s is the specific surface area defined as the ratio of the interface area between solid and void to the total volume, and τ is the tortuosity defined as the ratio of the mean fluid flow path length to linear distance traveled in the mean flow direction.
 The porosity was evaluated from each 3D tomographic image by counting the normalized total number of pore voxels in the sample. The specific surface area was first calculated for each 2D horizontal image using the two-point correlation function approach [Berryman and Blair, 1986], and then averaged over the length of the column. The two-point correlation function approach was also employed to estimate mean pore diameters in all samples [Berryman and Blair, 1987], and the Blob3D software package [Ketcham, 2005] was used to obtain complete pore size distributions after deposition.
 We employed a D3Q19 LB model (19 velocity directions in 3D space) to analyze pore fluid flow [Succi, 2001]. The LB method is particularly suitable for this application because it does an excellent job of simulating fluid flow involving complex boundaries [Chen and Doolen, 1998]. To resolve the effects of colloid deposition on pore fluid flow, we directly used the tomographic results to establish internal boundary conditions in the LB model. Thus, use of XDMT allowed us to simulate pore fluid flow both with and without colloidal deposits, and with an LB unit length exactly equal to the XDMT pixel length (6 μm). Permeability was assessed by evaluating the relationship between the imposed head gradient and the total flow through the sample, as described in section 2 of auxiliary material Text S1. The tortuosity was then found by particle tracking based on the LB-simulated flow fields, with at least 1000 virtual particles distributed randomly over the pore space in the influent plane.
 The 3D tomographic reconstructions of sample 1-B before and after colloid deposition are presented in Figures 1a and 1b, respectively. It is obvious that the colloidal deposits are highly heterogeneous at the pore scale (see auxiliary material for additional images and discussion). Considerable ZrO2 accumulation was observed at grain contacts, and the narrow pore throats at these locations became more restricted over time. As porosity decreased due to colloid deposition, the mean pore diameter decreased as a power law with Dp ∝ ϕ0.39, as shown in Figure 2a. The greater mass of colloids delivered into sample 3-A produced lower porosity and smaller pore sizes than in sample 1-A. The mean pore diameters calculated from the size distribution shown in Figure 2b agree with those presented in Figure 2a, indicating that the two-point correlation function approach and the three-dimensional Blob3D analysis approach generate similar results.
 The mean tortuosity of the porous medium also increased as accumulated colloidal deposits changed the geometry of the pore space. Tortuosity increased with decreasing porosity, following a power law with τ ∝ ϕ−0.27, as shown in Figure 3a. However, the mean tortuosity does not fully describe the complexity of the flow paths in the porous medium. The particle tracking simulations reveal that the tortuosity distributions became increasingly multi-modal with increasing accumulation of colloids in the pore space, as shown in Figure 3b. The presence of multiple flow paths having distinctly different travel durations indicates that colloid deposition caused the pore structure to become heterogeneous at the macro-scale.
 The relationship between porosity and permeability obtained from local LB simulations is shown in Figure 4. These permeabilities were evaluated locally within each sample using subdomains having dimensions of 0.96 × 0.96 × 0.96 mm. As found by others [e.g., Costa, 2006], permeability is a strong function of porosity and the relationship can be described by a power law. This is true both for the clean glass beads and for the samples following colloid deposition. However, colloid deposition increased the power-law exponent, with k ∝ ϕ3.2 found for the clean glass beads and k ∝ ϕ3.7 found for the post-deposition samples. It should also be noted that samples closer to the column inlet showed a greater decrease in permeability than those farther from the inlet, demonstrating that increased deposition rates in locations with greater particle flux produced highly clogged regions that dominate the overall sample permeability. For example, after deposition, the inlet region of the column with highest particle influx (sample 3-A) has a permeability nearly two orders of magnitude lower than the downstream region of the same column (sample 3-B).
 The Kozeny-Carman relationship adequately represents the permeability of all clean-bed samples, as well as the samples with relatively low particle accumulation, but physically non-plausible deviations from the KC theory occur in the samples with high particle accumulation (auxiliary material, Figure S4). These samples also show a non-linear increase in head loss with increasing particle accumulation (auxiliary material, Figure S5). Taken together with the observation that these samples have broad and multi-modal tortuosity distributions (Figure 3b), it seems clear that the failure of the Kozeny-Carman relationship comes about because of the increased spatial complexity of the porous medium resulting from colloid accumulation.
5. Discussion and Conclusions
 Direct observations of the evolution of pore structure by means of XDMT readily showed that colloid accumulation was highly heterogeneous at the pore scale. Further, this micro-structural information could be used to quantitatively assess changes in the macroscopic hydrogeologic properties of the porous medium by direct LB simulations. The analyses indicated that colloid accumulation decreased the mean pore size, increased the mean tortuosity, and greatly reduced the permeability of porous medium.
 The Kozeny-Carman relationship relies on measurement of first-order geometric information such as porosity and specific surface area and does not include higher-order geometric and topologic information. These simple structural variables provide only an indirect and incomplete description of pore structure. As a result, KC-based estimates have proven to be very sensitive to the resolution at which the structure is characterized [Berryman and Blair, 1987]. In contrast, LB simulations include a much more detailed description of the pore structure and allow explicit evaluation of the distribution of flow between pores having different morphology and connectivity. Here we showed that colloid accumulation in an initially simple porous medium quickly produced sufficient internal complexity that neither a power-law ϕ-k relationship nor the KC equation could provide good estimates of the permeability even after modest colloid deposition (input of 240 pore volumes of a 200 mg/L suspension under favorable deposition conditions).
 Changes in the tortuosity distribution after deposition indicate that this departure resulted from the development of macro-scale heterogeneity within the porous medium. Colloid deposition effectively isolated small regions of the pore structure, leading to the development of preferential flow paths even at scales less than 1 mm. When evaluated at a larger scale, the observed high rates of colloid deposition produced two-order-of-magnitude variability in permeability over distances of less than 1 cm. This behavior, and the resulting failure of semi-empirical permeability estimating equations, is similar to that found in complex natural porous media such as sandstones when there is a lack of connectivity in the pore space [Fredrich et al., 1993; Lindquist et al., 2000; Fredrich et al., 2006]. Here we have clarified that this behavior readily occurs even through modification of the pore structure of initially simple and homogeneous porous media.
 The XDMT-LB method we outline here can also be used for other applications, such as investigating the internal structure of micro-porous materials, but careful attention must always be paid to matching the resolution to the processes of interest. For example, the scale of roughness relevant to permeability is much larger than the scale relevant to solute sorption. This emphasizes the need for careful consideration of multi-scale process interactions, such as the evolution of larger-scale pore structure and fine-scale mineral structure during passage of a reactive, colloid-laden plume in groundwater. Here we have shown that heterogeneity at the sub-millimeter scale is important to clogging of porous media by colloid deposition, and illustrated how the XDMT-LB approach can be used to assess this type of structural evolution.
 This work is based upon material supported by the National Science Foundation via grant EAR-0310657. 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 the E.I. DuPont de Nemours & Co., the Dow Chemical Company, the National Science Foundation through grant DMR-9304725, 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-31-109-Eng-38. We thank Denis Keane for setting up the equipment, software, and computer cluster used for tomography, Boris Lau and Susa Stonedahl for assisting with the XDMT measurements, Amber Genau and Peter Voorhees for preparing the high-quality 3D images in Figure 1, and Qinjun Kang for testing the LB code.