Geophysical Research Letters

Morphology, propagation dynamics and scaling characteristics of drying fronts in porous media

Authors

  • N. Shokri,

    Corresponding author
    1. Department of Earth Sciences, Boston University, Boston, Massachusetts, USA
      Corresponding Author: N. Shokri, Department of Earth Sciences, Boston University, 675 Commonwealth Ave., Boston, MA 02215, USA. (nshokri@bu.edu)
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  • Muhammad Sahimi,

    1. Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California, USA
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  • D. Or

    1. Soil and Terrestrial Environmental Physics, Department of Environmental Sciences, ETH Zurich, Zurich, Switzerland
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Corresponding Author: N. Shokri, Department of Earth Sciences, Boston University, 675 Commonwealth Ave., Boston, MA 02215, USA. (nshokri@bu.edu)

Abstract

[1] Effects of surface wettability on the dynamics and morphology of evaporation-induced drying fronts (DF) receding into hydrophilic (HI) or hydrophobic (HO) sands were studied using highly-resolved neutron radiography. A more pronounced pinning-depinning motion (larger steps and longer waiting times) was observed for the DF in HI sand relative to HO, due to weaker capillary pinning forces in the latter. For length scalesl smaller than a correlation length ξ, quenched disorder dominates DF roughness expressed by a roughness exponent α = 0.77, whereas for l > ξ the fluctuations induced by the DF motion annealed the disorder and result in α= 0.44. Although wettability suppressed evaporation rates in HO relative to HI sand, differences in the evaporation dynamics and pinning behavior did not alter the DF morphology indicators-the fractal dimension and roughness exponent-which remained similar for the HI and HO sands.

1. Introduction

[2] Evaporation from porous media is a ubiquitous phenomenon of fundamental importance for many engineering processes, such as drying of food, wood, and building materials. It is the key for mass and energy exchange between land and atmosphere [Shahraeeni and Or, 2010] shaping hydrological processes, water management, and biological processes in the vadose zone.

[3] Early stages of the evaporation from saturated porous media are marked by high evaporative fluxes supported by capillary induced liquid flow from a receding drying front (DF) (marking the interface between saturated and unsaturated zone) to a vaporization plane at the surface. When the depth of the DF exceeds a characteristic length deduced from the pore size distribution [Yiotis et al., 2004; Lehmann et al., 2008; Chauvet et al., 2010], liquid continuity is disrupted marking the end of stage-1 evaporation and followed by the onset of diffusion-controlled stage-2 evaporation at lower evaporation rates. Quantification of the parameters affecting the DF propagation in porous media is important to elucidate the transport mechanisms that supply different stages of the evaporation rates.

[4] Various factors affect the interactions among the forces that govern the evaporation process including the porous medium texture, its wettability and structure, atmospheric conditions, and the fluid properties. We focus in this study on quantifying the wettability effects on the drying morphology and evaporation dynamics from porous media that differ only by the wettability of their surface. Wettability of natural porous media may be modified by intense forest fires, coatings of surfaces by organic compounds, mineral composition, and roughness of the solid surfaces. Alteration of wettability has significant impact on key hydrological processes such as runoff, water retention, infiltration, flow stability, multiphase flow and evaporation [Doerr et al., 2006]. Wettability alteration affects patterns of liquid-phase distribution and the motion of the DF through impacts on capillarity [Shokri et al., 2009]. Knowledge of the role of wettability on the drying morphology and dynamics is of importance for engineering and natural processes. We report detailed observations and analyses of the effects of wettability on evaporation-induced DF dynamics and spatial evolution of liquid phase distributions in hydrophilic (HI) and hydrophobic (HO) porous media using the neutron radiography technique.

2. Experimental Considerations

[5] Water evaporation from columns packed with HI and HO sand with particle sizes ranging from 0.3 to 0.9 mm was studied. Sand grains were salinized to render them HO using n-octyl triethoxysilane [Shokri et al., 2008a]. The advancing contact angles were measured in the laboratory following the procedure reported in Letey et al. [1962]. The experimentally-determined contact angles were 9 ± 1° and 92 ± 1° for HI and HO sand, respectively. Sand grains were packed in glass Hele-Shaw cells (250 mm height, 75 mm width and 10 mm thickness) with all boundaries closed except the top boundary exposed to air for evaporation. The sand columns were initially saturated with degassed water that was subsequently allowed to evaporate while monitoring the dynamics and morphology of receding DFs into the columns, using neutron radiography technique. Nondestructive neutron radiography measurements were conducted at the NEUTRA radiography facility [Lehmann et al., 2001]. Neutron transmission images were recorded at 5 minutes time intervals and 94 spatial resolution. The top 97 mm of each Hele-Shaw cell was maintained in the field of view for neutron radiography imaging. Evaporation lasted 16.5 and 23.5 hrs for the HI and HO sands, respectively. Variations in imaged gray scale were used to quantify the spatial and temporal distribution of the saturated and partially saturated zones. To delineate the dynamics and morphology of the DF, gray scale images were segmented into binary images to distinguish the interface between saturated and unsaturated zones representing the DF. The segmentation algorithm and the procedure used to compute the tempo-spatial distribution of the liquid phase were similar to those explained byShokri et al. [2008b]; hence they are not repeated here.

3. Results and Discussion

3.1. Dynamics of front Propagations and Liquid Phase Distribution

[6] Due to heterogeneity of pore surfaces, the pore space itself and the wettability properties of sand grain surfaces, regions in a receding DF may become pinned at pore throats requiring higher capillary pressure thresholds PC, whereas other parts of the front readily recede into regions with lower PC, giving rise to a rough interface. The interplay between capillary and gravity forces limits the growth of the DF roughness width, such that in a uniform porous medium evaporation-induced DF would attain a stable roughness pattern [Méhuest et al., 2002; Shaw, 1987]. Moreover, we expect that in HO sands the extent of pinning-depinning of the DF should be milder than in HI due to reduced capillarity.Figure 1 shows traces of the DF sequences in HI and HO sands with regions where the DF was pinned for longer periods marked by thicker lines densities.

Figure 1.

Drying fronts in (a) HI and (b) HO sands represented at equal time intervals of 5 minutes propagating from top to bottom of the column with all boundaries closed except top which was exposed to evaporation. Each white line indicates the position and geometry of DF determined by segmentation of the neutron transmission images. (c) The depth of receding DFs (measured from top of the image) versus time in HI and HO sands.

[7] Numerous studies have shown that macroscopic displacement of the front motion is composed of many pore scale bursts and invasion events that contribute to the apparent gradual and continuous displacement of the macroscopic DF [Haines, 1930; Jacquin and Adler, 1985; Måløy et al., 1992; Furuberg et al., 1996; Shokri and Or, 2011]. Our spatially and temporally resolved observations indicate similar dynamics for evaporation-induced DF propagation. The pinning-depinning motion of the DF was more pronounced in the HI than in HO sand, due to weaker capillary forces in the latter. Differences in the transport processes and liquid configuration in HO sand result in lower evaporation rates relative to HI sand under similar conditions. The effects on the DF velocities are shown inFigure 1c resulting in the DF mean velocities of 4.2 and 2.2 mm/hr in HI and HO sands, respectively.

[8] The DF traces in Figures 1a and 1b exhibit narrower roughness in HO sand expected from the larger Bond number in HO [Méhuest et al., 2002]. Nevertheless, the resulting HO roughness is larger than what is predicted based on the mean wettability properties of HO surfaces using the scaling relations between the Bo number and roughness width as in Tsimpanogiannis and Yortsos [1999]. The discrepancy could be attributed to the effects that result from non-uniform salinization of the irregular sand grains. FollowingTsai et al. [2010] we envision a scenario whereby a receding contact line may become pinned at surface imperfections until the local interfacial curvature overcomes the pinning by mass loss (evaporation from the interface) or by gravitational force exerted by the DF regions receding elsewhere. Recent studies have shown that evaporation induced contact line motion on hydrophobic surfaces may give rise to a range of configurations beyond what would be expected from average contact angle considerations. For example, Gelderblom et al. [2011]showed that diffusion-controlled drying of droplets on hydrophobic surfaces give rise to a range of contact angle hysteresis that span a spectrum of contact line depinning times (depending on initial contact angle at a location or in a pore). The resulting DF dynamics and morphology during diffusion-dominated evaporation from HO sand could be considerably different than behavior during liquid flow-induced drainage fronts.

[9] The narrower DF in HO reflect shorter waiting times between pore-scale invasion of the DF due to lower and more uniform distribution of the pinning forces (PC) involving the DF propagation at smaller length steps [Havlin et al., 1995]. In other words, weaker capillary forces in HO limit the extent of local gravitational gradients within a DF before an invasion event takes place hence resulting in shorter waiting times and limited DF roughness (relative to HI).

[10] In addition to observation of the DF dynamics, neutron transmission images facilitated detailed analysis of the wettability effects on the spatial and temporal liquid-phase distribution during evaporation following the procedure used byShokri et al. [2008b]. Sample results are presented in Figure 2, showing rapid drop in surface saturation in HO sand while HI surface remains wet for a prolonged period. Immediately following the drying of HO surface no continuous capillary liquid pathways remain to connect the receding DF with the surface hence the subsequent evaporation is vapor diffusion-controlled at lower rates (relative to HI) as seen inFigure 1. This result is consistent with data presented by Shokri et al. [2008a] in which a layer of HO sand overlying HI layer suppressed the evaporation rate. Additionally, Figure 2 shows that the unsaturated region above the DF is longer in HI than in HO sand in agreement with the analyses of Tsimpanogiannis and Yortsos [1999] and Yiotis et al. [2004].

Figure 2.

Temporal variations of near surface water saturation at depths ranging from 0 to 10 mm below the surface. The first three curves correspond to depth of 0, 0.5 and 1 mm below the surface. The rest are separated by 1 mm increments.

3.2. Effect of Wettability on the Fractal Dimension

[11] Similarities between drainage (displacement of a wetting fluid by a nonwetting one) and evaporation from porous media made it possible to invoke arguments based on percolation theory [Wilkinson and Willemsen, 1983; Stauffer and Aharony, 1992; Sahimi, 1994] to characterize the DF properties [Shaw, 1987; Tsimpanogiannis and Yortsos, 1999; Xu et al., 2008; Yiotis et al., 2010]. A gray-scale value of each pixel in our experiment corresponds to the average water content across the thickness of the Hele-Shaw cell, hence the imaged DFs may be considered as two-dimensional (2D) fronts. A similar assumption was made byShaw [1987]who pioneered the study of drying processes as immiscible displacement-equivalent processes.

[12] We employed the box-counting method [Feder, 1989] to calculate the fractal dimension Df of DF based on segmented black and white images. Typical results for a DF at a depth of 30 mm below the surface are shown in the inset of Figure 3. The box sizes varied from 8 pixels (8 × 0.094 mm) to the image size (∼75 mm). The dependence of Df on the DF depth for all images is also presented in Figure 3, indicating that, unlike miscible displacement, wettability of sand surfaces in our experiments did not strongly affect Df. The calculated Df in our analysis is smaller than that of Shaw [1987], Df ≅ 1.38 ± 0.02 for a DF receding in a glass column packed with silica spheres and also those reported for drainage fronts in 2D porous media [e.g., Birovljev et al., 1991; Méheust et al., 2002]. Unlike previous studies mentioned here in which Dfwas deduced from 2D media, the neutron transmission images used here are only quasi-2D, as the water content for each pixel represents an average across the 10 mm column by the neutron beams (20 grains in cross section). This may cause some deviations from the morphology of a DF propagating into a 2D column (i.e., packed with a single layer of grains). However, our objective here is not to obtain a precise estimate ofDf for the DF, but to evaluate the effect of wettability on the fractal dimension. Since the same procedure was followed to calculate Df in both HI and HO sand, based on Figure 3, one can conclude that the wettability of the grains in our experiments induced minor effects on Df.

Figure 3.

Evolution of the fractal dimension of DFs as a function of drying front depths. The inset presents the standard box-counting plot, when DF depth was about 30 mm in HI and HO sands.

3.3. Roughness of the Drying Fronts

[13] The irregularity and roughness of an interface between two immiscible fluids is normally characterized by its width defined as [Sahimi, 2011]

display math

where 〈.〉l and 〈.〉x denote, respectively, an average over a horizontal window of size l and over all x, and h(x) is the front height at x. When drying proceeds, the DF becomes rougher, with its width increasing as a power of time, but eventually reaching a steady state characterized by

display math

where α is the roughness exponent of the DF. Since the DF image includes some local overhangs, they are first removed by taking the maximum value of h(xi) at each xi along the DF [Delaplace et al., 1999]. The window size varied from 5 pixels (∼0.5 mm which is close to the average grain size) to the width of the image (∼75 mm). Figure 4 shows the dependence of W on l, indicating that for the initial part of the data, α ≅ 0.77 in both HI and HO sand, whereas α ≅ 0.45 for larger l. The short lresult is different from the prediction of the Kardar-Parisi-Zhang (KPZ) equation [Kardar et al., 1986] which predicts α = 0.5. Experiments on fluid displacement in porous media [e.g., Rubio et al.,1989; Buldyrev et al., 1992] also yielded α > 0.5. The presence of quenched disorder is responsible for the larger values of α [Csahok et al., 1993] at smaller l. Our results for the motion of the DF imply that for length scales smaller than a correlation length ξ- the scale over which the pressure jumps dissipate - quenched disorder dominates the interface roughening, leading toα ≅ 0.77, whereas for l > ξthe fluctuations induced by the DF motion - annealed disorder - determineα, as depicted in Figure 4. The widths shown in Figure 4 were computed by averaging over all the images that may mask the effect of wettability on individual front roughness. Therefore, we calculated the roughness exponent in each image characterizing the DF in smaller length scales (i.e., the range of the window size in the first part of Figure 4 where the average roughness exponent was computed as 0.77); these results are presented in Figure 5. As expected from Figure 1, the roughness exponent was not constant and varied slightly according to the morphology of individual DFs. Nevertheless, Figure 5 shows that for both HO and HI the average roughness exponent was similar with value of α ∼ 0.77 for length scales smaller than a correlation length ξ.

Figure 4.

Scaling of the DF width with l, the horizontal averaging segment length. The width represents an average over 92 DFs in HI and 169 DFs in HO sands.

Figure 5.

The roughness exponent ( characterizing DF for smaller length scales (i.e., the first part of the curves presented in Figure 4) in each image measured during evaporation from HI and HO sand column.

[14] A roughness exponent different from unity implies that the fluctuations in the DF are different in the vertical and horizontal directions. We obtained a roughness exponent of α ∼ 0.77 that is in the range of previously reported values [Rubio et al., 1989; Vicsek et al., 1990]. For example, Rubio et al. [1989] reported α= 0.73 ± 0.03 for water displacing air in a Hele-Shaw cell filled with glass beads. Within their experimental uncertainty, they showed that the exponents were independent of the displacement rates or the size of the glass beads. We note that values for the roughness exponent reported in the literature range from 0.65 to 0.91 [He et al., 1992]. Such a wide range is attributed to effect of quenched disorder generated by the grains [Barabási and Stanley, 1995].

4. Conclusions

[15] Analyses of evaporation from HO and HI sands (identical in all but the wettability properties) demonstrated differences in the velocity of receding DFs, tempo-spatial phase distribution and the extent of capillary liquid flow through unsaturated zone. The disruption of liquid continuity between the receding DF and the surface in HO media results in lower evaporation rates and a slower DF relative to behavior in HI sand.

[16] The DF motion in both media exhibited pinning-depinning motion that is indicative of the interplay between capillary and gravitational forces. The extent of the DF roughness due to detention of parts of the fronts in HO was lower than in HI due to reduced capillarity. Nevertheless, the resulting HO roughness is larger than what is predicted by using the mean wettability properties of HO surfaces, and using the scaling relations between the Bo number and roughness width as inTsimpanogiannis and Yortsos [1999]. The discrepancy may be attributed to effects resulting from non-uniform salinization and coating of the irregular sand grains. FollowingTsai et al. [2010] we envision a scenario whereby a receding contact line may become pinned at surface imperfections until the local interfacial curvature overcomes the pinning by mass loss (evaporation from the interface), or by gravitational force exerted by the DF regions receding elsewhere. This creates opportunities for contact line detention not anticipated from average wettability properties of HO surfaces.

[17] The small scale roughness of the DF for HI and HO sands exhibited fractal characteristics, with relatively minor effect of surface wettability on Df. The roughness exponent α of the DF in both HI and HO sands are very similar (α = 0.77 with a crossover to a smaller α = 0.44 at larger l). Our results indicate that in the presence of quenched disorder (generated by the packing of sand grains), a correlation length ξ quantifies a length scale below which the quench noise dominates the roughening process. For length scales larger than ξ, the fluctuations induced by the DF propagation may affect the resulting roughness [Barabási and Stanley, 1995].

[18] The results presented here shed new lights on the DF displacement patterns and tempo-spatial liquid phase distribution during evaporation from porous media with different wettability distinguishing between the effects of pore space organization (similar for both media) and the specific effects of wettability. Wettability of the grains modified the liquid phase distribution, the extent of the continuous capillary pathways above the DFs, and their propagation velocity, thus considerably affecting the general dynamics of the evaporation process. The primary impact of wettability differences in such displacement processes would be manifested by details of contact lines and evaporation from associated films that would potentially affect deposition processes in HI and HO surfaces as shown byShokri et al. [2008a]. We envision additional effects on, for example, the mobilization of colloidal particles and other interfacial displacement processes, induced by the motion of the DF in drying porous media.

Acknowledgments

[19] We thank Peter Vontobel for his support and expertise at the Spallation Neutron Source of the Paul Scherrer Institute, Villigen, Switzerland. Funding by the Swiss National Foundation projects (2000021-113676/1 and 200021-135077) are gratefully acknowledged.

[20] The Editor thanks the anonymous reviewer for assisting with the evaluation of this paper.