Water Resources Research

The configuration of water on rough natural surfaces: Implications for understanding air-water interfacial area, film thickness, and imaging resolution

Authors

  • Tohren C. G. Kibbey

    Corresponding author
    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)

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Abstract

[1] Previous studies of air-water interfacial areas in unsaturated porous media have often distinguished between interfacial area corresponding to water held by capillary forces between grains and area corresponding to water associated with solid surfaces. The focus of this work was on developing a better understanding of the nature of interfacial area associated with solid surfaces following drainage of porous media. Stereoscopic scanning electron microscopy was used to determine surface elevation maps for eight different surfaces of varying roughness. An algorithm was developed to calculate the true configuration of an air-water interface in contact with the solid surface as a function of capillary pressure. The algorithm was used to calculate surface-associated water configurations for capillary pressures ranging from 10 to 100 cm water. The results of the work show that, following drainage, the configuration of surface-associated water is dominated by bridging of macroscopic surface roughness features over the range of capillary pressures studied, and nearly all of the surface-associated water is capillary held. As such, the thicknesses of surface-associated water were found to be orders-of-magnitude greater than might be expected at the same capillary pressures based on calculations of adsorbed film thickness. The fact that capillary forces in air-water interfaces dominate surface-associated water configuration means that interface shapes are largely unaffected by microscopic surface roughness, and interfaces are considerably smoother than the underlying solid. As such, calculations suggest that microscopic surface roughness likely has minimal impact on the accuracy of surface-associated air-water interfacial areas determined by limited-resolution imaging methods such as computed microtomography.

1. Introduction

[2] When an initially water saturated porous medium drains in the presence of air, air-water interfacial area is formed as water is replaced by air in the pores of the medium. Previous researchers have often distinguished between interfacial area corresponding to capillary-held water between porous media grains (sometimes referred to “capillary” or “bulk” area) and water associated with solid grain surfaces (sometimes referred to as “film” area) [e.g., Costanza-Robinson and Brusseau, 2002; Bryant and Johnson, 2004; Porter et al., 2010]. The purpose of this study was to better understand the nature of water associated with solid grain surfaces in media experiencing drainage, with emphasis on understanding how surface roughness impacts both the interfacial area itself and the ability of limited-resolution imaging methods, such as computed microtomography (CMT) to accurately quantify area.

[3] A significant number of studies have been reported where CMT has been used to study fluid-fluid interfaces in porous media as a function of saturation [e.g., Culligan et al., 2004; Brusseau et al., 2007; Costanza-Robinson et al., 2008; Porter et al., 2010]. Some authors have suggested that because the spatial resolution of CMT is limited, CMT measurements of interfacial area may be impacted by the presence of surface roughness on natural surfaces [e.g., Brusseau et al., 2007; Costanza-Robinson et al., 2008]. A hypothesis driving this work was that capillary forces may actually mitigate much of the impact of surface roughness on measured areas in practice, by creating smooth interfaces that bridge much of the higher-frequency microscopic surface roughness. If true, this hypothesis would also have implications for the thickness of surface-associated water.

[4] To study these factors, multiple stereoscopic scanning electron microscope (SEM) images were taken of the surfaces of two types of porous media and used to reconstruct three-dimensional elevation maps of the surfaces. A finite difference algorithm was developed to solve for the shapes of the fluid interfaces that would remain on the solid surfaces following drainage of the medium to a given capillary pressure (Pc, the equilibrium pressure difference between gas and water across a curved interface). Simulated water interface configurations were then characterized to determine area and average water thickness, and results were related to morphologies of the underlying solid surfaces.

2. Materials and Methods

2.1. Materials

[5] Two granular media were used for this work: one natural sand and one type of commercial glass beads. The sand, U.S. Silica (Frederick, MD) F-45, is an unground natural sand with a mean particle size of 0.323 mm [Kibbey and Chen, 2012]. The glass beads, Scientific Industries (Bohemia, NY) SI-BG05, are commercial materials with a mean particle size of 0.629 mm, as determined by image analysis. Both materials were used as received.

2.2. Stereoscopic SEM and Solid Surface Characterization

[6] Stereoscopic SEM involves reconstruction of three-dimensional surface elevation maps from SEM images captured with samples positioned at different angles. The method is a widely used, proven technology and is one of the more robust methods of reconstructing surface elevation maps from SEM data [e.g., Stampfl et al., 1996]. For this work, images were captured with a Zeiss NEON 40 EsB high-resolution scanning electron microscope at the University of Oklahoma Samuel Roberts Noble Electron Microscopy Laboratory. All images used for stereo microscopy were captured at 2000× magnification, to give an approximate final elevation map size of 50 × 40 μm. For each sample, image pairs were taken with sample stages tilted 6° apart (±3° from horizontal). Elevation maps were calculated from image pairs by the 3D module of Scandium (Olympus Soft Imaging Solutions, Münster, Germany) ver. 5.0. The resulting elevation maps are nonoverlapping surfaces with the format z = f(x, y). Elevation maps have a horizontal resolution of 0.558 μm per sample.

[7] Surfaces were characterized by calculating their roughness, their surface areas, and the maximum angle from the horizontal at any point on the surface, an indicator of the abruptness of the roughness elevation changes. Roughness is a measure of the height of surface features above or below a base elevation. For this work, a modified definition of root mean square roughness (referred to as inline image) was used to correct for the fact that most of the surfaces studied exhibited significant deviations from horizontal, simply as a result of the use of natural sand grains. inline image was calculated from equation (1):

display math(1)

where n is the number of (x, y) points on the surface elevation map, and inline image is a corrected surface elevation defined from a base elevation taken at each point from a linear regression of each profile in the y direction (the shorter horizontal dimension for the elevation maps). This method of calculating zi largely corrects for the size dependence of roughness that is sometimes observed for slightly tilted surfaces [e.g., Kim et al., 2012], while still capturing the roughness of natural surfaces.

[8] Solid surface area (As) was calculated from the area of a regular triangular mesh connecting points in the elevation map. The nominal surface area (As-nominal), essentially a smooth-surface approximation of the solid surface area, was calculated from the area of a two-triangle mesh defined by the opposite corner points of the elevation map. Angles from the horizontal (θ) were calculated from the dot product of the surface normal and the vertical (z) axis at points throughout the surface.

2.3. Calculation of Interface Shapes

[9] The algorithm used to solve for interface shapes involves finite difference numerical solution of the Young-Laplace equation (equation (2)):

display math(2)

where R1 and R2 are the principal radii of curvature at each point, γ is water surface tension, and 2H is the mean curvature, given for the air-water interface z = w(x, y) by equation (3):

display math(3)

[10] For this work, two types of boundary conditions are used. Around the perimeter of the solid surface, Neumann boundary conditions are used, with the slope of the calculated air-water interface set equal to the average slope of the solid within a moving window (approximately 23 μm wide × 20 μm deep) around the perimeter of the solid surface elevation map. The moving window approach produces water surface boundaries at the perimeter that are consistent with a solid surface that extends beyond the solid surface elevation map with the same slopes. The approach is necessary because of the significant, spatially varying slopes of the natural solid surfaces examined. Within the boundaries of the solid surface, Dirichlet boundary conditions are used where the water surface is in contact with the solid surface. However, it is important to note that the iterative solution method used (described in the next section) actually produces simulated water surfaces with a zero contact angle adjacent to these internal boundaries.

[11] In the absence of a solid surface, numerical solution of equation (2) is relatively straightforward. However, addition of a solid surface adds considerable complexity because the locations of the boundaries corresponding to contact between the water and solid surfaces vary with the capillary pressure. For this work, the following approach is used. An initial guess of the air-water interface configuration is created at a low capillary pressure (10 cm water for all simulations reported here) by applying the pore morphology algorithm described by Hilpert and Miller [2001]. In short, the method involves passing a sphere of radius 2γ/Pc over the solid surface, “eroding” water away from the locations the sphere is able to access. The water interfaces produced by the method are highly inaccurate for calculating water interface shapes on solid surfaces (the method was designed for a different application), in that the resulting mean curvature varies widely over the interface and is always smaller than the true, target mean curvature defined by capillary pressure (equation (2)). However, the fact that the method underestimates mean curvature makes the resulting water interface a perfect starting point for the iterative solution used here, in that the water interface must be allowed to recede from the initial guess to satisfy equation (2).

[12] Once the initial air-water interface has been created, the true interface is solved for all water interface points not in contact with the solid surface by iterative solution of equation (4):

display math(4)

where ϕ is an iteration parameter (units of length squared). As 2H approaches Pc/γ on the water interface, the left side of equation (4) goes to zero, and the resulting interface is a solution to equation (2), the Young-Laplace equation. During each solution step for equation (4), the water interface elevation is compared with the solid surface elevation over the entire surface. If any points exist where the water surface falls below the solid surface, those points are marked as contact boundaries, and the iteration step is repeated for all points not in contact with the solid surface. If done with a sufficiently small iteration step size (Δϕ), this approach allows the interface to gradually recede to its true configuration, leaving zero contact angles adjacent to the contact (Dirichlet) boundary points. Solution of each step is achieved with successive over-relaxation (SOR). All second derivatives in equation (3) are evaluated at the current iteration step, while first derivative terms are lagged from the previous step. Calculations show that the resulting air-water interfaces have a highly uniform mean curvature, typically corresponding to within 0.1 cm water or better of the target Pc across the water interface. It should be noted that tests conducted starting with a flat water interface in contact with the high point on the solid surface (as opposed to an interface calculated with the Hilpert and Miller model) produced identical final water interfaces, but solution took considerably longer. Once the water interface shape has converged for a given target Pc, the target Pc is increased, and the previous water interface is used as the starting point for solution at the next Pc. For this work, calculations were conducted for Pc values ranging from 10 to 100 cm water in 1 cm intervals. All calculations were conducted using a water surface tension of γ = 72.9 mN/m.

[13] Air-water interfaces calculated with this procedure correspond to the configuration of water that would exist in contact with a solid surface if water were drained from a porous medium under the influence of capillary forces. While it is possible for different water configurations to be reached by other methods (e.g., evaporation, condensation), understanding water configuration on solid surfaces following capillary drainage is important because this condition is likely very common in the subsurface.

[14] It should be noted that others have simulated the configurations of capillary-held water on rough surfaces, although to the author's knowledge this work is the first reported simulation of interface shapes on actual natural surface geometries. For example, Sweeney et al. [1993] used a finite element approach to solve for the shapes of fluid interfaces on simplified surfaces with triangular and sinusoidal roughness. They solved the augmented Young-Laplace equation—a variation on equation (2) that incorporates both capillary forces and disjoining pressure to simulate both capillary-held and adsorbed water. In contrast, the method used here does not explicitly solve for the shapes of adsorbed films but rather considers them to exactly follow the contours of the solid surface in regions not occupied by capillary-held water. It should be emphasized that both the shapes of interfaces and average thicknesses of surface-associated water generated by the two methods should be essentially indistinguishable at the moderate capillary pressures considered here.

[15] Average film thicknesses (t) were calculated from the average distance between the calculated water interface and the solid surface elevation map at all points. Areas of air-water interfaces (Aw) reported in this paper were calculated from the area of a regular triangular mesh connecting points in the elevation map. Areas corresponding to an approximately 12.8 μm sampling resolution (Aw (12.8 μm)) were calculated from a regular triangular mesh of triangles spanning 23 grid units in each direction. This size was selected because 12.8 μm is within the range of typical CMT voxel sizes (typically 5–20 μm for reported area measurements, [e.g., Culligan et al. 2004; Wildenschild et al. 2005; Costanza-Robinson et al. 2008]), and 4 × 3 multiples of the 23 grid unit span (92 × 69) are close to the grid dimensions of the elevation maps (92 − 100 × 74). To allow direct comparison with Aw, the final row and column of triangles used in the calculation of Aw(12.8 μm) were expanded slightly to reach the edge of the computational space.

3. Results and Discussion

3.1. Solid Surface Features

[16] Figure 1 shows low-magnification SEM images of the two media studied for this work. At these magnifications, the significant differences between the surfaces of the two materials are apparent. In the case of the F45 sand (Figure 1a), significant spatial variability exists in the nature of the surface roughness, with angular, crystalline, regions coexisting with more amorphous, pitted surfaces. Within the angular portions, there are not only large flat patches but also many ridges with abrupt elevation changes. In contrast, the surfaces of the BG05 glass beads (Figure 1b) have a much smoother, more homogeneous appearance at this scale, with a relatively small number of surface defects apparent.

Figure 1.

Low-magnification SEM images of the two types of solid media studied in this work: (a) F45 sand and (b) BG05 glass beads.

[17] Figure 2 shows higher-resolution images of the eight solid surface samples for which elevation maps were determined by stereoscopic SEM. The solid surfaces consist of seven images of F45 sand, taken from various locations on different sand grains, and one image of the BG05 glass beads. The images in Figure 2 are arranged in order of decreasing roughness inline image. Calculated properties for the surfaces are given in Table 1 (Red-cyan anaglyphs of the images in Figure 2 are included in the accompanying online supporting information section, along with plots of the spatial distribution of surface angles.)

Figure 2.

High-magnification (×2000) SEM images of the eight surface regions selected for study: (a)–(g) F45 sand; (h) BG05 glass beads.

Table 1. Properties of Solid Surfaces Used for This Work
SurfaceRoughness, inline image (nm)Maximum Angle from Horizontal, θmax (deg.)Solid Surface Area, As (µm2)Nominal Solid Surface Area, As-nom (µm2)As/As-nom
  1. a

    F45-3 with added sinusoidal surface in both x and y directions (wavelength = 2.2 µm; amplitude 0.5 µm each dir.).

  2. b

    F45-3 with added sinusoidal surface in y direction only (wavelength = 2.2 µm; amplitude 1 µm).

F45-11494.680.32757.52318.21.19
F45-21251.567.22610.22347.31.11
F45-31140.478.92702.12234.51.21
F45-4708.167.12437.62279.01.07
F45-5554.255.52277.12152.71.06
F45-6442.264.22496.72297.11.09
F45-7373.844.22363.62271.91.04
BG05266.748.92427.92253.31.08
Added sinusoidal roughness
F45-3+R1a1198.178.13662.02241.61.63
F45-3+R2b1348.379.54344.32238.11.94

[18] Note that the nature of the roughness differs considerably among the solid surfaces in Figure 2, with some exhibiting angular features and abrupt elevation changes (e.g., F45-1 and F45-3), and some exhibiting more gradual ripples and amorphous roughness features (e.g., F45-2 and F45-5). Several also show the presence of attached fines, likely clay particles (e.g., F45-3 and F45-7), although it should be noted that much of the solid surface area and microscopic roughness that would be attributed to the attached fines is below the 0.55825 μm resolution of the solid surface elevation maps derived from stereoscopic SEM. While the nature of the roughness in the eight surface samples selected for the study varies considerably, it should be noted that selected samples do not include some of the most angular regions on sand (e.g., as seen in Figure 1), because many of these features are too large to be encompassed in a 2000× magnification SEM image.

3.2. Water Configuration

[19] Figure 3 illustrates the calculated air-water interface configurations during drainage for the F45-3 sand. At low capillary pressure (10 cm water, Figure 3b), the air-water interface is extremely smooth and pinned to the high points of the solid surface. As capillary pressure increases, the corresponding increase in mean curvature causes the interface to recede toward the solid surface. The number of contact points also increases as the capillary pressure increases. It is important to note, however, that even at Pc = 100 cm water, much of the surface-associated water remains capillary-held, pinned by larger surface roughness features, and far from the solid surface.

Figure 3.

 Calculated air-water interface as a function of capillary pressure (Pc) for F45-3 sand: (a) underlying solid surface and (b)–(f) air-water interfaces.

[20] Figure 4 illustrates the calculated water interface configurations during drainage for the BG05 glass beads. As was observed for the F45-3 sand, the air-water interface is extremely smooth at low capillary pressures but becomes rougher with increasing capillary pressure, as contact with high points on the solid surface increases. It is interesting to note that despite the fact that the BG05 solid surface has considerably lower roughness than the F45-3 solid surface (Table 1) and visibly lacks many of the major macroscopic surface roughness features (Figure 2), the water remains largely capillary-held at Pc = 100 cm water on the BG05 glass beads. In fact, there is less calculated contact between the BG05 surface and the air-water interface at 100 cm water than is observed for the F45-3 sand surface.

Figure 4.

Calculated air-water interface as a function of capillary pressure (Pc) for BG05 glass meads: (a) underlying solid surface and (b)–(f) air-water interface.

[21] This result is highlighted by Figure 5, which shows the contact points between solid surface and air-water interface for both F45-3 and BG05 surfaces as a function of capillary pressure. The percentages listed on the figure correspond to the fraction of total air-water interface that is in contact with the solid surface (i.e., a contact boundary condition), calculated from comparison between triangular mesh surfaces for the solid surface and air-water interface. It is interesting to note just how little contact there is in both cases. The results mean, for example, that at Pc = 100 cm water, the surface-associated water in the case of F45-3 is 95% capillary-held, with its configuration dominated by the presence of macroscopic surface roughness features. Only 5% of the surface-associated water might be considered to be a true adsorbed surface film. And it is important to note that even that value would be reduced significantly if all microscopic surface roughness could be resolved in the elevation maps (That is, the calculated contact points would themselves likely largely correspond to capillary water spanning microscopic roughness features.). As mentioned previously, in the case of the BG05, an even smaller fraction of the surface-associated water is calculated to be in contact with the solid surface. This result is somewhat counterintuitive, in that smoother surfaces might be expected to exhibit a greater fraction of adsorbed surface films compared with rougher surfaces; however, the finding results from the more uniform height and spacing of the roughness features in the BG05 preventing the interface from approaching the solid surface.

Figure 5.

Points where the air-water interface contacts the solid, shown for F45-3 sand and BG05 glass beads. Percentages listed correspond to the fraction of total water surface area in contact with the solid. Note that contact points themselves do not occupy space; they are shown with expanded areas in the figure for visibility.

3.3. Thickness of Surface-Associated Water

[22] Figure 6 shows the calculated average thickness of surface-associated water (t) as a function of capillary pressure for the eight surfaces in Figure 2. In all cases, the thickness decreases with increasing capillary pressure as the air-water interface approaches the solid surface. The functional form is a near-linear relationship between 1/t and Pc, as evident from the inset in Figure 6. Note that t tends to be greatest for rougher surfaces, and t varies more significantly with changing Pc on rougher surfaces.

Figure 6.

Surface-associated water layer thickness versus Pc for the eight solid surfaces in Figure 2.

[23] Of particular note in the results shown in Figure 6 is the fact that the calculated thicknesses are much greater than previously reported film thicknesses [e.g., Tokunaga, 2011; Baveye, 2012]. Note that calculated adsorbed film thicknesses might be expected to be on the order of 10–20 nm for this capillary pressure range, depending on surface potential and solution ionic strength, with lower thicknesses observed at higher capillary pressures [e.g., Gregory, 1975; Tokunaga, 2011]. In contrast, the film thicknesses in Figure 6 are nearly two orders of magnitude larger than that range at lower Pc values. The reason for the difference can be attributed entirely to the large fraction of water that remains capillary held (typ. >95%), even at Pc = 100 cm water, and the fact that the configuration of capillary-held water is entirely controlled by macroscopic surface roughness features.

[24] This fact is amplified by Figure 7, which shows the relationship between average thickness of surface-associated water (t) at Pc = 10 cm and surface roughness inline image for the eight surfaces in Figure 2. The line in Figure 7 is a 1:1 line. Note from the figure that inline image is a very good predictor of t, as might be expected for an air-water interface whose configuration is dominated by major surface roughness features. (Note that if the linearized curves in Figure 6 are extrapolated to Pc = 0, results show that t(Pc = 0) ≈ 1.7· inline image).

Figure 7.

The relationship between surface-associated water layer thickness at Pc = 10 cm water and surface roughness for the eight solid surfaces in Figure 2. A 1:1 line is shown.

[25] It is interesting to note that the result in Figure 7 is quite consistent with experimental measurements of surface-associated water thickness reported by both Kim et al. [2012] on relatively smooth surfaces and Tokunaga et al. [2003] on rough gravels. Both studies used synchrotron X-ray fluorescence to measure the average thickness of surface-associated fluids. Kim et al. [2012] measured the thickness of brines on two SiO2 optical discs, one very smooth (Rrms = 1.6 nm), and the other roughened by sandpaper (Rrms = 330 nm). Brine thicknesses were measured at capillary pressures ranging from approximately 2–37 cm water. Although the surfaces studied by Kim et al. were smoother than the surfaces studied here (their rough surface was similar in roughness to the smoothest surface used here, the BG05 glass beads), like the results shown in Figure 7, their measured water thicknesses are very close to their Rrms values at low Pc in both cases: ∼2 nm for their smooth surface, and ∼250 nm for their rough surface. It is also interesting to note that Kim et al. saw only a relatively modest decrease in film thickness with increasing Pc, a result that is consistent with observations for the smoother surfaces studied here, where air-water interfaces span multiple relatively small, evenly spaced roughness features (e.g., BG05 in Figure 6). Tokunaga et al. [2003] measured water thicknesses on very rough gravels (Rrms ∼1 to 2.5 μm, which is rougher than most of the surfaces studied here). Although they report much thicker surface-associated water at near-zero capillary pressure, as a result of the filling of large isolated surface pits, it is interesting to note that the water thicknesses they report at Pc ∼ 10 cm water are comparable to the range of reported values of Rrms, similar to the results observed here.

[26] The implications of the results in Figures 5-7 are that, for conditions where porous medium saturation is reached by drainage, the thickness of surface-associated water is dominated by macroscopic surface roughness features, and nearly all surface-associated water is capillary-held; true adsorbed water films are extremely rare. Note that this result would also apply to a condition where water is imbibed into a previously drained porous medium, as the surface-associated water would remain from the original drainage. Furthermore, it is probable that capillary forces would even create this condition during imbibition from dry conditions, due to the continuity with a water reservoir, although the current model cannot directly simulate that condition. From an environmental standpoint, this means that true adsorbed water films may only exist well above the water table (i.e., very high Pc), and under conditions where significant evaporation occurs. This result could have significant implications for water-phase mobility of nonvolatile contaminants in the unsaturated zone, as the two-order-of-magnitude thicker water layers would presumably produce diffusion within the surface-associated water layers (i.e., along grain surfaces) that is greater by two orders of magnitude as a result of the larger cross-sectional area for diffusion.

3.4. Interfacial Area and Imaging Resolution

[27] Figure 8 shows the calculated air-water interfacial area and solid surface areas for the F45-3 sand surface as a function of capillary pressure. From Figure 8a, it is apparent that the air-water interfacial area (Aw) is close to the nominal solid surface area (As-nominal, essentially a smooth-surface approximation of the solid surface) at low capillary pressure, but increases with increasing capillary pressure. Like Aw, Aw(12.8 μm) (the interfacial area corresponding to ∼12.8 μm sampling resolution) starts near As-nominal at low capillary pressure and increases with increasing capillary pressure. Unlike Aw, however, Aw(12.8 μm) plateaus as capillary pressure increases, approaching a value somewhere between Aw and As-nominal. These trends are consistent across all of the surfaces studied, although the exact shapes of the Aw and Aw(12.8 μm) curves differ slightly depending on the nature of the surface roughness.

Figure 8.

Calculated air-water interfacial area as a function of capillary pressure (Pc) for F45-3, compared with nominal and true solid surface. Interfacial and solid areas corresponding to F45-3 with superimposed sinusoidal roughness (F45-3+R1, F45-3+R1; Table 1) are shown for comparison.

[28] It is important to note that Figure 8a is shown with an expanded scale to show the shapes of the Aw and Aw(12.8 μm) curves. However, it is important to step back and consider the areas on a full-scale axis to understand the relationships between air-water interfacial areas and the underlying solid. Figure 8b shows the interfacial areas on a full-scale axis, adding the solid surface area (As). From Figure 8b, it is apparent that while both Aw and Aw(12.8 μm) do increase with increasing capillary pressure, both remain much closer to As-nominal than to As. That is, both the true air-water interfacial area (Aw) and the air-water interfacial area corresponding to a lower sampling resolution (Aw(12.8 μm)) are both better approximated by the smooth surface approximation to the solid surface area As-nominal than the true solid surface area As; the higher-frequency surface roughness that causes As to be larger than As-nominal does not contribute appreciably to the air-water interfacial area.

[29] To further examine the role of high-frequency roughness features, two different synthetic surfaces were generated by adding roughness to the F45-3 surface elevation map. One synthetic surface, F45-3+R1 (Table 1), superimposes a bidirectional sinusoid with an amplitude of 0.5 μm in each direction and a wavelength of 2.2 μm, while the second, F45-3+R2 (Table 1), superimposes a single-direction sinusoid with an amplitude of 1.0 μm and a wavelength of 2.2 μm. Because the amplitude of the new roughness is on the same order of magnitude of inline image for F45-3, neither of the synthetic surfaces have final roughness values that are appreciably higher than F45-3 (Table 1). However, both synthetic surfaces exhibit significantly greater solid surface area, As, compared with the F45-3 surface. For example, the surface area of F45-3+R2 (4344 μm2) is approximately 60% greater than the surface area of F45-3 (2702 μm2).

[30] Figure 8 shows Aw curves for the two synthetic surfaces. Note from Figure 8a that these curves are very similar to the Aw curve of the original F45-3 (In fact, they are so similar that they are obscured by the line thickness of the F45-3 Aw curve in Figure 8b.). However, from Figure 8b it is also clear that these curves are even farther from the true solid areas (As-F45-3+R1, As-F45-3+R2) than was the case with F45-3. In fact, the true solid surface area appears to have very little impact on the magnitude of the air-water interfacial area.

[31] This result suggests the common-sense conclusion that high-frequency roughness of any kind is unlikely to have much impact on the air-water interfacial area. Previous CMT studies have reported that measured air-water interfacial areas approach smooth solid surface area approximations at low saturations, rather than the sometimes many-times larger surface area determined by gas adsorption measurements [e.g., Costanza-Robinson et al., 2008]. The results in Figure 8 clarify that one important reason for this finding is that the true air-water interfacial area on a solid surface at high capillary pressure remains close to the smooth surface approximation; the configuration of surface-associated water is simply prevented from conforming to high-frequency roughness by capillary forces.

[32] Figure 9 further explores the effect of sampling resolution on air-water interfacial area determination, showing the ratio of Aw(12.8 μm) to Aw as a function of capillary pressure for all of the solid surfaces in Figure 2. As might be expected, the ratio is less than one (i.e., Aw(12.8 μm) is less than Aw) over the entire capillary pressure range, and the ratio decreases with increasing capillary pressure as the configurations of the true interfaces are increasingly influenced by the underlying solid surface roughness. However, it is important to note that in all cases, Aw(12.8 μm) is within 4% of Aw at Pc = 100 cm water, and within 2% for six of the eight surfaces (The two seeming outliers in Figure 9, F45-1 and F45-3, both have solid surfaces with very steep ridges and abrupt elevation changes. This is apparent from the θmax values in Table 1, as well as the anaglyphs and angle maps in the accompanying online supporting information section.). The result in Figure 9 indicates that true air-water interfaces in unsaturated porous media are likely to be smooth enough that the limited resolution of imaging methods such as CMT should not preclude accurate determination of air-water interfacial areas for water associated with rough surfaces.

Figure 9.

Ratio of interfacial area measured at a 12.8 µm resolution to true interfacial area, shown as a function of Pc for the eight solid surfaces in Figure 2.

4. Conclusions

[33] The results of this work indicate that for surface-associated water resulting from drainage of a porous medium, capillary forces cause the configuration of water to be dominated by bridging of macroscopic roughness features on the solid surface. This result means that surface-associated water on many natural solids under many natural conditions may be orders of magnitude thicker than equations for adsorbed film thickness might suggest. Furthermore, true adsorbed films may be rare in the unsaturated zone, existing only far above the water table and under extremely dry conditions where evaporation is significant. Because diffusion is proportional to the cross-sectional area, this finding has the environmental implication that diffusive mass transfer of low-volatility compounds in surface-associated water may be significantly greater than might be expected in true adsorbed surface films.

[34] Previous studies have distinguished between capillary area (area associated with capillary held water between porous medium grains) and film area (area associated with solid surfaces). The results of this work suggest that, at least for surface-associated water left behind by drainage of a porous medium, nearly all surface-associated water is capillary-held over a wide range of capillary pressures. As such, a more accurate terminology for a wide range of conditions (including those used in most reported CMT studies of fluid areas) might be “intergranular capillary area” and “surface-associated area,” in place of capillary and film area, respectively.

[35] Results of this work further suggest that the actual solid surface area of rough surfaces has little connection to air-water interfacial area of surface-associated water; rather, air-water interfacial area is reasonably well approximated by a smooth-surface approximation of the solid surface area. High-frequency surface roughness features that can increase the surface area of a solid significantly over its apparent smooth-surface area have negligible impact on the configuration of surface-associated water, so also have negligible impact on the corresponding air-water interfacial area. Because capillary forces cause air-water interfaces to be extremely smooth, the results of this work suggest that the limited resolutions of some imaging methods (e.g., CMT) likely do not preclude them from providing reasonable measurements of surface-associated fluid interfacial areas on rough natural surfaces.

Acknowledgment

[36] The author would like to thank Preston Larson of the University of Oklahoma Samuel Roberts Noble Electron Microscopy Laboratory for capturing the stereoscopic SEM images used in this work.

Ancillary