WATER RESOURCES RESEARCH, VOL. 48, W10519, doi: /2012wr011862, 2012

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1 WATER RESOURCES RESEARCH, VOL. 48, W10519, doi: /2012wr011862, 2012 A pore network model study of the fluid-fluid interfacial areas measured by dynamic-interface tracer depletion and miscible displacement water phase advective tracer methods Tohren C. G. Kibbey 1 and Lixia Chen 1 Received 11 January 2012; revised 22 August 2012; accepted 30 August 2012; published 9 October [1] Fluid-fluid interfacial areas in porous media are of considerable interest due to the impact they have on a wide range of practical applications involving mass transfer between phases, as well as for their importance in understanding unsaturated and multiphase flow behaviors in porous media. Tracer methods provide a low-cost experimental approach for determining interfacial areas in porous media. Although a number of different tracer methods have been developed, uncertainty remains as to exactly what areas they measure. The work presented here uses pore network model simulations to study the behavior of tracers during simulated tracer measurements for two different specific water-phase tracer methods: the dynamic-interface tracer depletion method and the miscible displacement tracer method. The hypothesis driving this work was that different tracer methods likely measure different areas as a result of the very different ways tracers are used. Experimental data sets for six different porous media were used to validate the model and provide comparison with model-simulated tracer-based area measurements. Results of the work suggest that areas measured using the dynamic-interface tracer depletion method closely match total fluid-fluid interfacial areas, as long as the extent of tracer depletion during the method is relatively small. However, areas measured with the miscible displacement method likely fall somewhere between capillary and total fluid-fluid areas. Calculations conducted with realistic diffusion coefficients and film thicknesses indicate that diffusion and head-driven flow in films are insufficient to allow significant tracer access to film area, suggesting the potential importance of other mechanisms. Calculations conducted as a part of the work suggest that Leverett estimates of maximum area formed during drainage may be closer to total areas than previously reported. Citation: Kibbey, T. C. G., and L. Chen (2012), A pore network model study of the fluid-fluid interfacial areas measured by dynamicinterface tracer depletion and miscible displacement water phase advective tracer methods, Water Resour. Res., 48, W10519, doi: /2012wr Introduction [2] Interfacial areas between fluids in porous media are of considerable interest for a range of practical environmental applications, as well as for their implications for understanding and modeling the motion of fluids in unsaturated or multiphase systems. For example, any mass transfer that occurs between phases either mass transfer of a solute, or evaporation/dissolution of bulk fluids themselves must occur at the interfaces, so rates depend strongly on the area of interface present [Abriola et al., 2004; Grant and Gerhard, 2007; Hoggan et al., 2007]. Furthermore, there is significant 1 School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, Oklahoma, USA. Corresponding author: T. C. G. Kibbey, School of Civil Engineering and Environmental Science, University of Oklahoma, 202 W. Boyd St., Rm. 334, Norman, OK 73019, USA. (kibbey@ou.edu) American Geophysical Union. All Rights Reserved /12/2012WR theoretical and experimental evidence that interfacial area may be an important fundamental property of multiphase porous media, essential to understanding and modeling a wide range of fluid behaviors [Hassanizadeh and Gray, 1993; Pyrak-Nolte et al., 2008;Porter et al., 2010;Joekar-Niasar et al.,2010;niessner et al., 2011]. [3] Tracer methods provide a tool for measuring fluidfluid interfacial areas in porous media containing multiple phases. Tracer methods make use of a chemical (the tracer) that adsorbs preferentially to the interface of interest. The tracer can be introduced through a liquid or (in the case of an unsaturated medium) the gas phase, and some measure of its adsorption (e.g., depletion of solution, retardation of transport, inducement of water motion) can be used to calculate interfacial area. A number of different tracer methods have been developed over the years, based on different methods of making use of tracer adsorption to determine area. [4] A long-standing area of controversy surrounding tracer methods is uncertainty about the types of interfacial area that they measure. When describing fluid-fluid interfacial areas in porous media, researchers often distinguish W of13

2 between area of fluids associated with thin wetting fluid films on solid surfaces (film area), and areas of interfaces associated with bulk water (bulk, or capillary area); the sum of those two areas at any given saturation corresponds to the total fluid-fluid interfacial area [Costanza-Robinson and Brusseau, 2002]. Water films in water-wet porous media have been reported to have thicknesses no greater than the tens of nm [Wan and Tokunaga, 1997; Holmes and Packer, 2002; Tokunaga, 2011]. However, the distinction between film area and capillary area tends to be somewhat operational in practice; for example, in microtomography (CMT) measurements, the distinction between types of area is limited by voxel size (typ mm) [Culligan et al., 2004; Wildenschild et al., 2005; Costanza-Robinson et al., 2008], with water films assumed to be below detection limits, and film area taken to be the areas of water-wet solid not adjacent to measureable water phase. [5] The objective of the work described here was to use network model simulations of tracer adsorption and transport in unsaturated porous media to explore areas measured by two different tracer methods: the dynamic-interface tracer depletion method, and the water-phase miscible displacement tracer method. The hypothesis driving the work was that the significant differences in the ways tracers are used in the different methods likely results in different measured areas Water-Phase Interfacial Tracer Methods [6] Major water-phase interfacial tracer methods can be divided into three categories: water mobilization methods, miscible displacement advective tracer methods, and drainage and redistribution methods. Methods in the three categories differ significantly in their operation, the types of tracers used, and the ways in which tracer interactions with interfaces are used to determine fluid-fluid areas. The three classes of methods are described here briefly to provide context for this work. [7] The water mobilization method [Karkare and Fort, 1996; Silverstein and Fort, 1997] is technically not a water-phase method because it makes use of an insoluble surfactant that exists at the interface between the air and water phases (in contrast, true water-phase methods use a water-soluble adsorbing tracer). Nevertheless, the method is mentioned here briefly because many studies of tracer methods have grouped water mobilization data with true water-phase data. The method uses an incompressible monomolecular surface film to induce water motion; the area per molecule at the onset of water motion can then be used to calculate interfacial area [Karkare et al., 1993]. [8] The water-phase miscible displacement method was originally reported in the late 1990s [Kim et al., 1997; Saripalli et al., 1997], and more than a dozen papers making use of the method or its variations have been published to date. In the laboratory, the method involves adjusting the saturation of the porous medium to the desired value (typically by drainage), and then measuring a breakthrough curve of the aqueous tracer (typically an anionic surfactant, such as sodium dodecylbenzenesulfonate (SDBS)) carried by the flowing water phase through the porous medium. As tracer adsorbs to the fluid-fluid interface, its transport is retarded. Comparison with the breakthrough of a nonadsorbing tracer provides a retardation factor, that can be used in combination with the Gibbs adsorption equation and measured surface or interfacial tension versus concentration to calculate interfacial area [Kim et al., 1997]. The method has been used for determination of both gas-liquid and liquid-liquid interfacial areas, and can be applied in the field as well as the laboratory for determination of liquidliquid interfacial areas. [9] Drainage and redistribution methods differ from the other two categories of methods in that the tracer (typically an anionic surfactant) is initially present in the porous medium at the time the interfacial area is created. Two early methods in this category were described by Schaefer et al. [2000] and Anwar et al. [2000]. In both cases, a tall, segmented column is packed with a porous medium saturated with a tracer solution, and then the solution is drained to produce a natural gravity-induced water content distribution above a water table that is located near the bottom of the column. Schaefer et al. [2000] allow the column to equilibrate for 7 days, while Anwar et al. [2000] recirculates the solution in the column. In both cases columns are disassembled at the end of the experiment and the concentration of tracer in each section is analyzed. Interfacial areas are calculated in each section through the use of the measured surface tension versus concentration curve, the Gibbs adsorption equation, and the assumption that the aqueous concentration is the same throughout the column. [10] A third method in this category is the dynamic-interface tracer depletion method [Chen and Kibbey, 2006]. The method is closely related to the methods of Schaefer et al. [2000] and Anwar et al. [2000], in that the tracer (an anionic surfactant) is present in the system as the interface between fluids is formed. However, the method differs in that it uses a very small, membrane-based porous medium cell (1.27 cm high), and calculates interfacial area as a function of saturation by tracking the concentration of tracer leaving the cell in real time during drainage, rather than disassembling a tall column and using equilibrium water content distribution to provide a saturation distribution. This means the method can be applied to much finer materials (i.e., higher air-entry pressures) than most other methods. The method involves starting with a porous medium fully saturated with an anionic surfactant solution. The nonwetting phase is pressurized by a computer-controlled servo-pressure regulator that increases pressure at a slow, predetermined rate. As surfactant solution (the wetting phase) drains from the porous medium, surfactant adsorbs to dynamically created interface, depleting the bulk solution. A continuous measurement of the concentration in solution leaving the cell coupled with a continuous mass balance on the surfactant in the cell is used with the Gibbs adsorption equation to determine the interfacial area as a function of saturation during drainage Fluid-Fluid Areas Measured by Water-Phase Tracer Methods [11] There are three critical challenges associated with evaluating the types of area measured by interfacial tracer methods based on literature data. First, the number of published data sets for any given method can be quite limited. For example, to the authors knowledge, there are a total of three published water-phase miscible displacement method data sets for gas-liquid systems: Kim et al. [1997], Saripalli et al. [1997] (one data point), and Brusseau et al. [2007]. 2of13

3 (A somewhat larger number of liquid-liquid data sets have been published for that method, although nearly all are limited to data at high water saturations.) Any assessment of published data that does not distinguish between methods has the potential to be skewed by the by inclusion of the methods for which more data sets exist. Second, the various methods differ in the saturation ranges of the areas they can detect. For example, water-phase miscible displacement experiments are very difficult to conduct at low water saturations for both gas-liquid and liquid-liquid systems (a total of three data points have been published for water saturations less than 0.5 (all for gas-liquid systems), and nearly all measurements correspond to water saturations of approx. 0.7 or greater.) In contrast, the other two categories of methods can be used down to residual saturation or lower (in the case of the water mobilization method). This can hide differences between areas detected by the different methods if a distinction is not made between methods, with the low saturation data from water mobilization and drainage-andredistribution methods skewing analyses. Finally, all three classes of methods have more sensitivity to experimental artifacts than is sometimes recognized a fact that complicates direct comparison of different data sets. [12] Of the major categories of method, the area determined by water mobilization methods [e.g., Karkare and Fort, 1996] is probably the most straightforward to assess. Because the method relies on insoluble long-chain alcohols that form spreading monolayer films, water mobilization methods almost certainly provide a measurement that approaches total fluid-fluid areas. The tracers move along the gas-liquid surface and simply cannot distinguish between capillary and film areas. In contrast, the types of areas measured by the other methods is less clear. [13] In their original paper describing the use of the water-phase miscible displacement method to measure airwater interfacial areas, Kim et al. [1997] noticed that when they extrapolated interfacial areas measured by their method to zero saturation, the measured areas were lower than the area that would be predicted from the surface area of the porous medium itself (A nearly dry medium where the porous media surfaces are covered by water films would have a total fluid-fluid area nearly equal to the area of the medium itself). Based on this result, they suggested that the method measures the effective hydrodynamically accessible air-water interfacial area essentially the accessible subset of the total area. Because of heterogeneity in the spatial distribution of water content at low saturations, it is reasonable to assume not all of the water volume in a porous medium is accessible to water flow at low saturations, although others have suggested that inaccessibility of water films to flow could produce the same effect [Costanza-Robinson and Brusseau, 2002]. [14] In a 2002 paper, Costanza-Robinson and Brusseau [2002] hypothesized that aqueous interfacial tracers likely measure primarily air-water interfaces formed by capillary water. Their conclusion was based on an analysis of six different water phase data sets. However, with the exception of the data of Schaefer et al. [2000], which was singled out as a diffusion method, no distinction was made between the experimental techniques used to generate the five other data sets studied, which were measured using methods from all three categories. [15] Dalla et al. [2002] used a pore morphology based simulator to model areas formed during drainage in the sand sample used by Kim et al. [1997]. The resulting simulation found that the experimentally determined areas measured by Kim et al. [1997] fell somewhere between the capillary and total areas. However, they concluded that, if sensitivity of calculated capillary areas to discretization level was considered, the experimental data were more consistent with capillary area than total area. [16] Bryant and Johnson [2004] used theoretically calculated capillary and film areas based on pore network model simulations, and arrived at quantitatively similar theoretical area-saturation relationships to the simulated curves presented by Dalla et al. [2002]. By comparing normalized areas from Kim et al. [1997] and Anwar et al. [2000] to the theoretical curves, Bryant and Johnson [2004] concluded that the tracer measurements clearly show that films are being included in the measurement. Note that Bryant and Johnson s interpretation was probably influenced to some extent by the inclusion of three different data sets from Anwar et al. [2000], all of which, because of the nature of the method, include most of their data points at saturation values below the saturation range of the Kim et al. [1997] data, creating a composite picture of tracer-measured area that may differ from the interpretation that would be made if the methods were considered separately. An additional analysis by Bryant and Johnson [2004] examining a total of 11 different area data sets from different methods (including some measured with gas-phase tracers) found that many exceeded theoretical total area, but followed the trend of total area on a log scale. [17] Brusseau et al. [2007] compared microtomography measured areas with areas measured during a water-phase miscible displacement experiment. Their results showed the tracer method-measured areas more than four times higher than the microtomography-determined total area. They attributed this to the tracer method measuring area due to microscopic surface heterogeneity too small to be resolved by the microtomography measurements. [18] Costanza-Robinson et al. [2008] developed an empirical model for predicting total interfacial area based on microtomography measurements with nine different unconsolidated media. The model was then compared against twelve different water-phase tracer data sets five water mobilizationdatasets,fivedrainageandredistributiondatasets, and two miscible displacement data sets. The authors concluded that the tracers appeared to be retained by the same interfacial domains identified by microtomography measurements of total area, although no distinction was made between the different categories of tracer methods in the analysis. [19] The work presented here takes a different approach to understanding the areas measured by specific tracer methods. The work uses pore network model simulations to model tracer experiments themselves, tracking the adsorption and movement of tracers to air-water interfaces created by drainage of unsaturated porous media. The purpose of the work was to provide a mechanism to conduct different tracer methods under carefully controlled conditions where different areas can be precisely identified, and then explore the extent to which the resulting measurements, as well as the underlying assumptions of the model, are consistent with published experimental data. 3of13

4 2. Numerical and Experimental Methods 2.1. Pore-Network Model [20] Pore network models simulate porous media as a collection of pore bodies connected by pore throats. Pores throats and pore bodies are typically assumed to have very simple geometry, allowing complex networks to be studied with minimal computational cost. Pore network models have been used for a range of applications, including studies of mass transport and adsorption [Raoof et al., 2010], as well as studies of the trends in interfacial area formed with saturation, and the relationship between capillary pressure, saturation and interfacial area [Joekar-Niasar et al., 2008, 2010]. Although pore network simulations have been used in the past in the study the likely contributions of film and capillary area to areas measured by tracer methods [e.g., Bryant and Johnson, 2004], no work has been reported to date simulating the behavior of the tracers themselves. Because the emphasis in this work is on modeling transport and adsorption of tracers, it is critical that the model used be able to generate a physically realistic spatial distribution of interfacial area within the simulated porous medium, and yet be simple enough to allow predictions based on experimental media on the basis of a minimal set of experimentally determined model parameters. To achieve this, we use simple three-dimensional cubic (coordination number ¼ 6) pore network models with volumeless pore bodies, and cylindrical pore throats. The connectivity and arrangement of this model is essentially the same as the tube model used by Joekar- Niasar et al. [2008] to study the arrangement of fluid-fluid interfaces in porous media, although details of the calibration differ, as described in section 2.3. A limitation of a model with volumeless pore bodies is that it is best suited to modeling drainage; more complex pore shapes and handling of fluid interface movement in nodes [e.g., Reeves and Celia, 1996; Joekar-Niasar et al., 2010] are typically required for accurate simulation of imbibition. For this reason, we limit ourselves to simulation of tracer data that correspond to primary drainage. Because pore bodies are volumeless in this model, for the remainder of this paper we refer to pore bodies simply as nodes, and pore throats as pores. Sections describe calculation aspects related to flow, area formation, and adsorption and transport in the model, while sections 2.2 and 2.3 describe the data sources and process used to calibrate the model for simulation of experimental tracer measurements in real porous media Drainage and Fluid Flow [21] For the work described here, all pores in the model are initially saturated. Drainage of the model is induced by increasing the applied pressure difference between the nonwetting and wetting phase boundaries. For a pore to drain, one end of the pore must be continuous with the nonwetting phase boundary (dry continuous), the other must be continuous with the wetting phase boundary (wet continuous), and the pressure difference across the pores must be greater than the threshold capillary pressure of the pore, given by the Young-Laplace equation (P c_thr ¼ 2cos /r [Adamson and Gast, 1997], where is surface tension, is contact angle (assumed zero for all models here), and r is the radius of the pore). From the standpoint of determining continuity with wetting and nonwetting phase boundaries, the model assumes that both phases can be transmitted simultaneously by nodes, with the ability of a phase to be transmitted depending only on the saturation state of the connected pores. If a node is connected to at least one pore containing wetting phase, wetting phase continuity can be traced through the node, and if a node is connected to at least one pore containing nonwetting phase, nonwetting phase continuity can be traced through the node. Continuity for both phases is checked by recursive flood-fill algorithms that start at the nonwetting or wetting phase boundary nodes and mark the continuity state of all connected nodes. [22] During drainage, the applied pressure difference is increased in predetermined steps (typ. 1.0 or 0.1 cm water per step for finer materials, 0.01 cm water per step for coarse glass beads). At each applied pressure difference, pores that are both wet and wet-continuous and that have threshold capillary pressures lower than the applied pressure difference are added to a list. This list is then sorted from low to high threshold capillary pressure (i.e., in order of decreasing pore radius). The list is then cycled through repeatedly, as follows: Each time through the list, the first wet pore is tentatively drained (marked as dry), and wet continuity is checked. This step is necessary because the pore itself could be the only path for wet continuity; with the candidate pore marked as dry it is possible to both confirm that wet and dry continuity exist on opposite sides of the pore (predrainage dry continuity information is used for this assessment), and to select the node to which the pore will drain. If wet and dry continuity do not exist on opposite sides of the pore, the pore is once again marked as wet, previous wet continuity is restored, and the next pore in the list is considered. Once a pore has been successfully drained, dry continuity is checked for the next cycle, and the program returns to the start of the list and repeats the process. The reason the program repeatedly cycles through the list is that the process of draining pores may make pores earlier in the list eligible to drain, by expanding dry continuity. The approach described here ensures that pores drain in order of increasing threshold capillary pressure, as soon as continuity allows them to drain, i.e., the same order they d drain if considered one pore at a time. Furthermore, the approach has the advantage that it allows fluid flow calculations to be done less frequently (once per pressure step), reducing the computational cost of the model. [23] Fluid flow for this work includes both pipe flow through pores, and, in selected cases, flow through films in drained pores. Fluid flow in all simulations of the drainage process is limited to pipe flow through pores, while selected simulations of steady state flow through pores add flow through water films in drained pores to explore differences between observed model-predicted and experimental behaviors. Both pipe flow and film flow are modeled as steady state, laminar, head-driven flow (i.e., Poiseuille flow). Equations for pipe flow are based on standard relationships describing Poiseuille flow in cylindrical pipes [Potter and Wiggert, 2002]. Equations describing flow in films are based on a derived laminar velocity distribution of a cylindrical thin film, with zero shear stress at the film air-water interface (See the auxiliary material (Text S1, section S1) for details). 1 1 Auxiliary materials are available in the HTML. doi: / 2012WR of13

5 For both types of flow, nodes are assumed to not create any resistance to flow. Note that the intent is to model the movement of fluid during slow (pseudo-static) drainage (dynamic-interface tracer depletion method), or during steady state flow (miscible displacement method), so the model itself is not a dynamic pore network model, in that no dynamic flow or capillary effects are included. A system of equations is set up describing the wetting-phase mass balance at each node (each equation sums the flow into the node as calculated from the head difference with surrounding nodes, and flow resulting from Poiseuille flow), and the equations are solved for the head at each node. This approach has been used previously by others for modeling flow in pore network models based on cylindrical pores [e.g., Raoof et al., 2010]. [24] Two different sets of boundary conditions are needed for the two types of tracer methods studied. In the case of the water-phase miscible displacement tracer method, the model is drained to a series of predetermined applied pressure differences, and then breakthrough curves are simulated. Flow for the breakthrough curves is created by placing constant head boundaries at the top and bottom boundary nodes (for these measurements, the top nonwetting-phase boundary is replaced with a second wetting phase boundary to create steady state flow through the wet portion of the network). In the case of the dynamicinterface tracer depletion method, the nonwetting phase boundary is a no-flow boundary for the wetting phase, the wetting phase boundary is a constant (zero) head boundary, and volumetric flow from all pores that drained at each pressure level is added at each of the nodes to which they drained. Solution of the flow equations is achieved using successive over-relaxation (SOR) [Press et al., 1988] Area Model [25] Fluid-fluid area forms in any porous medium during drainage from full wetting phase saturation. Leverett [1941] recognized that the total interfacial area created during saturation change is limited by the amount of pressure volume work put into the system (i.e., the capillary pressure saturation relationship). During drainage, the total increase in interfacial energy created by interface generation cannot exceed the energy put into the system to produce the drainage. In a strongly water wet system, where solid surfaces remain coated with a water film, fluid-fluid area created during primary drainage can be calculated from equation (1) [Leverett, 1941]: A A Leverett ¼ n Z S2 1 P c ds (1) where S is wetting phase saturation, S 2 is a water saturation value, A Leverett is maximum possible total interfacial area per porous medium volume added by draining from S ¼ 1.0 to S ¼ S 2, A is true total area per porous medium volume added by draining, is interfacial tension, n is porosity, and P c is capillary pressure. (Note that Leverett s original equation omitted porosity, giving area per pore volume. Equation (1) uses the form suggested by Grant and Gerhard [2007], providing area in the units common in current experimental studies. Equation (1) also presents the relationship as an inequality, to emphasize that the equation gives a theoretical maximum.) [26] Equation (1) can guide selection and evaluation of an area model for a pore network model, in that total area generated by the model during drainage should not exceed A Leverett. For the model used here, capillary area is given by the area of a spherical cap at any end of a wet pore adjacent to at least one dry pore; the radius of curvature of the spherical cap is the minimum of the pore radius or radius of curvature defined by the Young-Laplace equation [Adamson and Gast, 1997]. Nominal film area is taken to be the outer surface area of pores that have drained (i.e., nominal film area ¼ 2rL for a pore with radius r and length L). In order to account for the loss of Haines jumps due to the use volumeless nodes in the model, it is necessary to reduce the nominal film area by applying a small geometrically based correction at each pore based on capillary area formed in adjacent pores. Details of the correction are provided in the auxiliary material (Text S1, section S2). In short, the correction leads to a model that produces a total area that is very close to A Leverett, but which retains both the physically realistic definitions of pore and capillary areas, and the simplicity (from a calibration and prediction standpoint) of a model with volumeless nodes. Capillary and nominal film area definitions are illustrated in Figure Adsorption and Transport [27] Surfactant adsorption to fluid-fluid interfaces is typically well-described by the Langmuir adsorption isotherm (equation (2)), when surfactant concentration is below its critical micelle concentration (CMC): C ¼ C maxkc 1 þ KC where G is the surface excess of surfactant tracer (i.e., moles adsorbed per area, mol m 2 ), K is the adsorption constant (m 3 mol 1 ), C is surfactant concentration in solution (mol m 3 ), and G max is the maximum surface excess at surface saturation (mol m 2 ). [28] All network model simulations conducted here describe tracer adsorption as Langmuir adsorption. Except Figure 1. Schematic of the area definitions used in simulations. True film areas are reduced slightly from nominal film areas by a calculated geometry-based correction to allow the use of volumeless nodes. (2) 5of13

6 where otherwise noted, all simulations are based on Langmuir parameters (K, G max ) determined previously [Chen and Kibbey, 2006] for the surfactant sodium octylbenzenesulfonate (SOBS), a high purity monodisperse alkylbenzenesulfonate anionic surfactant, in the presence of 0.1 M NaCl. These data were selected because of the high quality of the surface tension data, which correspond to 165 individual surface tension measurements (33 quintuplicate measurements), determined using automated drop shape analysis. Surface tension data were fit using the Szyszkowski equation, a combination of the Langmuir (equation (2)) and Gibbs adsorption equations. The earlier work used adsorption in terms of activity [Chen and Kibbey, 2006]. However, if ionic strength is essentially constant, as it is here (excess salt concentration is 2 orders of magnitude greater than surfactant concentration), the activity coefficient is constant and can be combined with K. As such, for this work we convert the K parameter to allow it to be used in terms of concentration (i.e., in equation (2)), rather than activity. In the units used internally by the model, this gives G max ¼ mol m 2 and K ¼ m 3 mol 1. In addition to providing a realistic model of surfactant adsorption, the use of these experimentally determined parameters for SOBS also allows direct simulation of all aspects of experimental dynamic-interface tracer depletion data measured using the same surfactant system. [29] Area formed during drainage, mass retained thorough adsorption, and solution concentration are all associated with the pores. The approach is similar to the model described by Raoof et al. [2010] for modeling adsorption to solid surfaces, in that both models assume complete mixing within pores (i.e., there is only one concentration associated with each pore), but differs in that their model tracks concentration in nonvolumeless nodes in addition to pores. [30] The transport model is based on mass balances in the pores and nodes, shown in equations (3) (5). Equation (3) describes mixing and adsorption at the nodes: V node dc node ¼ 0 dc node ¼ 0 ¼ X i X C i Q i C node Q j (3) where the sum over i is a sum over all nodes flowing into the node, the sum over j is the sum over all nodes leaving the node, C i and Q i are the concentrations and volumetric flow rates in the pores flowing into the node (including recently drained pores, in the case of modeling transport during drainage), and Q j is the flowrate in each of the pores leaving the node. Complete mixing is assumed to occur in the node, so all pores leaving the node contain the same concentration, C node. [31] Mixing and adsorption in pores is described by equation (4): V pore dc pore ¼ QðC pore in C pore Þ dm where V pore is the volume of the pore, Q is the volumetric flowrate in the pore, C porein is the concentration entering the pore (i.e., C node from equation (3) for the node that flows into this pore), C pore is the fully mixed concentration in the pore (and is also the concentration leaving the pore), j (4) and dm is the change in mass due to adsorption, given by equation (5): A dc ¼ dm ¼ Aðk adsc pore k des CÞ ¼ k des A C (5) maxkc pore m 1 þ KC pore where A is interfacial area at the node, m is mass adsorbed at the node, and k ads and k des are adsorption and desorption rate constants for a linear driving force model; when dm ¼ 0, equation (5) reduces to equation (2) (i.e., G/C pore ¼ k ads /k des ). Note that equation (5) assumes that area is not changing rapidly (i.e., A dc C da ), a condition that is met at all pores that are not in the process of draining (da/ ¼ 0). For pores that are in the process of draining, retention of solute by the area created in the draining pore is handled by assuming solute is in equilibrium between solution remaining in the pore and the newly formed interface, with the concentration leaving the draining pore providing a C i value at the node to which the pore drains (equation (3)). A figure illustrating this process is provided in auxiliary material (Text S1, section S3). [32] For this work, solution of equations (3) (5) is achieved through a decoupled finite difference solution method that first updates concentrations at all nodes based on pore concentrations from the previous time step, and then uses node concentrations to calculate concentrations and adsorbed masses in pores. Note that selected miscible displacement tracer method simulations were conducted with the addition of diffusion in both wet and dry pores by diffusion; rearrangement by diffusion is handled separately at each time step after the update of pore concentrations and adsorbed masses. Details of the solution method and diffusion calculations are provided in the auxiliary material (Text S1, section S4). [33] Mass balance calculations were performed on all drainage and breakthrough simulations, and mass balances were better than 99.6% for all runs. Constant concentration boundary conditions are used for both types of simulations. For drainage simulations (dynamic-interface tracer depletion method), concentrations within the draining pores provide C i values for those pores at the appropriate nodes (equation (3)). For breakthrough curve simulations (miscible displacement method), the network initially contains zero concentrations of both the conservative (nonadsorbing) and interfacial tracers. The nonwetting phase boundary nodes are then given a constant concentration of both the conservative and interfacial tracers (i.e., the tracers are introduced in a step at t ¼ 0), and equations are solved simultaneously for transport of conservative and interfacial tracers Area Calculation From Tracer Results [34] For miscible displacement advective tracer method simulations, areas are calculated from breakthrough curves using the method described by Kim et al. [1997]. Breakthrough curves are used to calculate the retardation factor, R f, from the ratio of integrals in equation (6). R f ¼ Z 1 Z ðc in C out Þ ift dv ðc in C out Þ cst dv (6) 6of13

7 where the superscripts ift and cst correspond to the interfacial and conservative tracers, respectively. Interfacial areas are then calculated from R f using equation (7): A ¼ n S ðr f 1Þ=K 0 D (7) where A is the interfacial area per total volume of the porous medium, KD 0 is the apparent linear adsorption coefficient at the initial tracer concentration used for breakthrough curves, C 0 (i.e., KD 0 ¼ C 0=C 0,whereC 0 is surface excess corresponding to C 0 ), and S is the water saturation. [35] For dynamic-interface tracer depletion method simulations, interfacial area per total pore volume is calculated using the method described by Chen and Kibbey [2006] (equation (8)): C init þ A ¼ n Z S 1 C out ðsþds SC out ðsþ C out ðc out ðsþþ where C init is the concentration of tracer initially in the model prior to drainage, and C out ðc out ðsþþ is the surface excess (equation (2)) calculated at the concentration leaving the cell at the saturation of interest (C out (S)) Experimental Data Sources [36] A total of six different porous media were modeled for the work (Table 1). The media span a wide range of grain sizes, ranging from coarse glass beads to a silt. Data sets used for comparison include CMT measurements for two media, miscible displacement data for two media, and dynamic-interface tracer depletion data for three media. CMT data are used to validate the ability of the model to produce quantitatively realistic area predictions, while tracer data are used for comparison with model predictions of tracer measurements. Data from Brusseau et al. [2006, 2007] and Kim et al. [1997] were taken from published tables, where available, or determined from published figures using the public domain program ImageJ (U.S. National Institutes of Health). Because the air-water P c -S curve in Kim et al. [1997] was plotted on one plane of a three-dimensional plot, inverse perspective correction/rotation was used to extract the data from that figure. Data from Culligan et al. [2004] were provided by D. Wildenschild (personal communication, 2011). The data correspond to a reprocessed data set, all of which except for the total area has been previously published by Porter et al. [2009] and Joekar-Niasar et al. [2010]. The reprocessing involved use of improved segmentation algorithms, as well as reconstructing the data to remove additional ring artifacts, and differs slightly in measured area and residual saturation from the original 2004 data [Porter et al., 2009]. Additional details and quantitative evaluation of the algorithms used are given by Porter and Wildenschild [2010]. [37] Dynamic-interface tracer depletion data were taken from previous work by the authors [Chen, 2006; Chen and Kibbey, 2006]. Raw tracer data were reprocessed to correct for adsorption of the surfactant tracer to the water-wet nylon membranes, an experimental artifact that was discovered after the original publication of the work. The calculation was also modified from the original method to correct for fluid (8) Table 1. Sources of Experimental Data, and Pore Size Distributions From Gaussian Fits to Air-Water Pc-S Data Data Types FIT Air-Water Pc-S Source for Fit Displacement (mm) (if Different) CMT Miscible Tracer Depletion r (mm) r Sp b (mm) Data Source Porous Medium d50 (mm) n Culligan et al. [2004] a Glass Beads X Kim et al. [1997] Sand X Brusseau et al. [2006, 2007] Vinton Soil , 0.41 X X Brusseau et al. [2006] Chen [2006] F45 Sand X J. L. Hoggan and T. C. G. Kibbey c Chen and Kibbey [2006] F110 Sand X Chen et al. [2007] Chen [2006] SCS 250 Silt X Chen et al. [2007] a Reprocessed data provided by D. Wildenschild (personal communication, 2011). b Spacing between adjacent parallel pores in final scaled model (model scaled to match experimental porosity, n). c (unpublished data, 2006). 7of13

8 dead volume between the porous medium and fiber-optic detector. Details of the calculation, including supporting isotherm data and a saturated tracer measurement, are presented in the auxiliary material (Text S1, section S5). Area data from all sources were converted to units of cm 2 area per cm 3 total porous medium volume to allow direct comparison Model Calibration and Simulation of Experimental Data [38] Calibration of pore network models to simulate experimental porous media was conducted using the following approach. For the purpose of this work, pores were given a random spatial distribution, with size distributions defined by a Gaussian distribution. All pores have the same length, dictated by the cubic arrangement of nodes. Calibration is based entirely on determining the equivalent pore distribution that would cause a bundle-of-tubes model to duplicate an experimental air-water P c -S relationship for a given porous medium. The procedure involves fitting a cumulative Gaussian distribution to the P c -S relationship, using equation (9): 2 0 S e ¼ þ erf Br equiv r equiv C7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 (9) 2 2 r equiv 13 where S e is the effective saturation (S e ¼ (S-S wr )/(1-S wr )), and r equiv, r equiv and r equiv are the pore radius, mean pore radius, and pore radius standard deviation, respectively, related to P c by the Young-Laplace equation (r ¼ 2=P c, where r, and P c must be in consistent units). While this approach could be adapted to other pore distributions, note that fits to equation (9) (i.e., Gaussian pore size distribution) are very similar in shape to fits to the van Genuchten equation [van Genuchten, 1980], and provide a very good fit to all experimental data used here. The distribution described by r equiv and r equiv corresponds to a numberdistribution of pores in a bundle-of-tubes model. [39] Using r equiv and r equiv from the fit described above produces a network model-generated P c -S relationship that is slightly sharper than the original P c -S, because unlike a bundle-of-tubes model, phase accessibility in a pore network model prevents some fraction of pores from draining at any given P c, even if P c would otherwise be high enough for them to drain. Figure 2 illustrates the fit process and resulting pore network model-generated P c -S for the glass bead data of Culligan et al. [2004]. The red curve shows the fit of equation (9) to experimental data. The fit results in a pore distribution, shown inset in green. Finally, that distribution is used to generate a network model that is used to simulate drainage, resulting in the model-generated P c -S relationship (blue curve). As expected, the model result has a slightly sharper air-entry, and a flatter curve across much of the saturation range, although it is still quite close to the original data. Furthermore, note that the residual saturation predicted by the model comes out quite close to the measured residual saturation. Fits for all of the experimental data studied are shown in the auxiliary material (Text S1, section S6). [40] It should be noted that other equally valid calibration methods could be used. However, this one has the Figure 2. Gaussian fit to the data of Culligan et al. [2004], illustrating model calibration and resulting P c -S relationship. advantage that it produces unique fits, has only two adjustable parameters, and results in predictions that are not particularly sensitive to artifacts in experimental P c -S relationships, making it a good choice for the purposes of the work to conduct realistic simulations of tracer behavior. [41] The final step in calibration for simulating porous media involves scaling the model so that model porosity matches the target porosity of the medium being simulated. This step does not impact the P c -S relationship (e.g., Figure 2), but does impact the relative magnitudes of total and capillary areas formed in the model during drainage. Models are initially generated with arbitrary length dimensions, and then all pore lengths and model pexternal ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensions are scaled by multiplying by n model init =n expt, where n model init is the initial porosity of the model as generated, and n expt is the porosity of the experimental data being simulated (Table 1). This procedure results in a model with a porosity that exactly matches the porosity of the experimental data. For all of the systems studied here, this process resulted in a spacing between adjacent parallel pores (S p ) that was on the same order of magnitude as the mean grain size, d 50 (Table 1). [42] All simulations conducted for this work make use of models with the node dimensions This corresponds to a total of 1521 nodes, and 3536 pores (pores are not allowed to connect two boundary nodes). Comparisons with network models as large as (7260 nodes, 18,788 pores) found negligible differences in model generated area predictions, likely because of the porosity scaling step used in this work. 3. Results and Discussion 3.1. Model Validation Prediction of Experimental Areas [43] Before the model can be used to simulate the behavior of tracers, it is important to first confirm that the areas predicted by the model are physically realistic. Figure 3 compares model-predicted areas with reported CMT-measured areas for two different porous media [Culligan et al., 2004; Brusseau et al., 2006, 2007]. Note that tracer behavior is not 8of13

9 Figure 3. Comparison between model-predicted areas and computed microtomography (CMT)-measured areas reported by (a) Culligan et al. [2004] and (b) Brusseau et al. [2006, 2007]. simulated in Figure 3 the figure simply compares the total and capillary areas predicted by the model as calibrated from the P c -S relationship for each medium, using the physically based area definitions described in section Examining Figure 3, it is apparent that the agreement between the model and CMT-measured areas is excellent for both media, a fact that gives considerable confidence that areas in the model are sufficiently realistic for tracer simulations to provide insight into tracer-based area measurement. The fact that the two media in Figure 3 differ by almost an order of magnitude in measured area provides further confidence in the ability of the model to be used for predictions across a range of realistic media. [44] Note that quantities of film and capillary areas evolve in the model during drainage just as they do in the experimental systems (i.e., both experimental and model area curves in Figure 3 correspond to areas formed during drainage from full saturation). As such, because of the agreement observed in Figure 3, it is reasonable to expect that flow patterns created by drainage and the areas available to interact with tracers in either tracer method at a given saturation are described realistically by the model. Furthermore, note that the spacing between adjacent parallel pores in the model (S p ) is on the same order of magnitude as the d 50 of the media in all cases (Table 1), a fact that suggests that the length scales corresponding to the spatial arrangement of film and capillary areas in the randomly distributed simulated media are realistic. This result is important for accurate simulation of flow and diffusion in films. [45] An interesting observation in Figure 3 can be made from comparison of the CMT-measured total areas and the calculated Leverett areas (equation (1); the maximum theoretically possible area formed by drainage). In both cases, the total area is in very close agreement with the Leverett area, suggesting that the easily calculated Leverett area (equation (1)) may actually be a very good predictor of total area, even without correction. The agreement is especially close for the Culligan et al. [2004] data (Figure 3a), where the P c -S relationship was measured simultaneously with the CMT areas, on the same packing; the P c -S relationship measured by Brusseau et al. [2006] was measured externally on a different packing of the same material, so would not be expected to provide quite as close an agreement with the CMT data. [46] Dalla et al. [2002] defined an efficiency parameter E D (S) ¼ A tot /A Leverett. Based on a pore morphology-based drainage simulation they calculated that E D varied with saturation, ranging from approx at low S to 0.7 at high S. Examination of the experimental data in Figures 3a and 3b suggests that the E D may actually be closer to 1.0 in practice. This result has implications for Leverett-based models of interfacial area that require E D as a parameter [e.g., Grant and Gerhard, 2007]. [47] Although a complete discussion of the limitations of CMT is necessarily beyond the scope of this paper, the use of CMT for model validation here makes it important to acknowledge that CMT area measurements are themselves not exempt from error. While controlled tests have shown very good performance with modern analysis algorithms [e.g., Porter and Wildenschild, 2010], potential issues with grain size and roughness exist [Costanza-Robinson et al., 2008]. The fact that the CMT-measured total areas in Figures 3a and 3b appear to approach values just below the theoretical areas of the two media at zero saturation (6ð1 nþ=d 50 ; 45 and 161 cm 1, respectively) lends confidence to the values for these two CMT data sets; this result is consistent with earlier published CMT work for porous media covering a wide range of grain sizes [see Costanza- Robinson et al., 2008] Dynamic-Interface Tracer Depletion Method [48] In the dynamic-interface tracer depletion method, tracer leaving a small porous medium cell is measured during drainage. Adsorption to interfaces dynamically formed during drainage depletes pore solution, causing concentration leaving the medium to drop. Figure 4 shows both modelsimulated tracer concentrations and experimentally measured tracer concentrations leaving the media during drainage for three different porous media (three model realizations are shown for each medium). As described previously, simulations were conducted using the same concentration and Langmuir adsorption parameters used for the surfactant in the experiments. The resulting agreement between modelpredicted and experimentally measured tracer concentrations 9of13