Early stages of evaporation from porous media are marked by relatively high evaporation rates supplied by capillary liquid flow from a receding drying front to vaporization surface. At a characteristic drying front depth, hydraulic continuity to the surface is disrupted marking the onset of stage-2 evaporation where a lower evaporative flux is supported by vapor diffusion. Observations suggest that in some cases the transition is accompanied by a jump in the vaporization plane from the surface to a certain depth below. The resulting range of evaporation rates at the onset of stage-2 is relatively narrow (0.5–2.5 mm d−1). The objective is to estimate the depth of the vaporization plane that defines vapor diffusion length at the onset of stage-2. The working hypothesis is that the jump length is determined by a characteristic length of connected clusters at the secondary drying front that obeys a power law with the system's Bond number. We conducted evaporation experiments using sands and glass beads of different particle size distributions and extracted experimental data from the literature for model comparison. Results indicate the jump length at the end of stage-1 was affected primarily by porous media properties and less so by boundary conditions. Results show power law relationships between the length of the vaporization plane jump and Bond number with an exponent of −0.48 in good agreement with the percolation theory theoretical exponent of −0.47. The results explain the origins of a relatively narrow range of evaporation rates at the onset of stage-2, and provide a means for estimating these rates.
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 Drying behavior is often conceptualized by stage-1 evaporation supplied by capillary-induced liquid flow from large pores at the receding drying front supplying fine pores at the evaporating surface. During this period, the evaporation rate is limited primarily by atmospheric conditions. The capacity of capillary transport to supply evaporative demand is the key reason why during stage-1 the drying porous medium behaves as if it was virtually saturated [Scherer, 1990]. Details of drying front dynamics and capillary flows were directly visualized in recent studies by Shokri et al. [2008b, 2009a, 2010a]. They showed that during stage-1, liquid menisci in fine pores at the surface remain coupled with atmosphere (as long as menisci curvature does not exceed pore critical invasion pressure). These hydraulically connected evaporating pores support a capillary gradient and draw water from the porous medium interior to supply evaporative demand at the surface. At a certain depth of the drying front, the downward gravity and viscous forces overcome the upward capillary driving force, disrupting hydraulic continuity with the surface [Lehmann et al., 2008]. The disruption of hydraulic continuity marks the end of stage-1, and subsequently liquid menisci recede from the surface to a level below the surface and form a new vaporization plane referred to as the secondary drying front by Shokri et al. [2009a] who defined it as the interface between the wet and dry zone. This transition marks the onset of stage-2 evaporation; a period with a lower evaporation rate limited by diffusion through the porous media [Shokri et al., 2009a]. These mechanisms for stage-1 and -2 evaporation have been experimentally confirmed in recent studies for homogenous [Lehmann et al., 2008], layered [Shokri et al., 2010b], and partially wettable [Shokri et al., 2009b] porous media.
 Notwithstanding progress in quantifying evaporative fluxes, the nature and dynamics of the transition from stage-1 to stage-2 evaporation has received little attention. Questions such as (1) how the grain and pore size distributions affect the transition? (2) what determines the duration and abruptness of the transition? and (3) what determines the dynamics of liquid menisci detachment from the surface? remain largely open. Better understanding of the transition from stage-1 to stage-2 evaporation and identifying the parameters affecting this transition enables us to accurately describe and predict the entire drying curve. This paper also attempts to make use of certain observations to address why evaporation flux post transition to stage 2 is remarkably similar across a wide range of boundary conditions and porous media types. The primary objective of the present study is to quantify key controls for the transition from stage-1 to -2. We seek to identify the depth in which the secondary drying front forms which is equal to the vapor diffusion length at the onset of stage-2 evaporation. Analysis of this length is instrumental for the prediction of evaporative flux at the onset of stage-2. We propose an analytical estimate for the length of menisci “jump” at the end of stage-1 evaporation that defines the length of vapor diffusion length hence evaporation rate at the onset of stage-2. The length is determined from well-established concepts of percolation theory regarding front width or connected liquid cluster length [Stauffer and Aharony, 1991; Sahimi, 1993, 2011]. The proposed model is experimentally evaluated using measurements of evaporation behavior of various materials differing in particle size distribution. Additionally, we extract experimental data from the literature to evaluate the proposed model.
2. Theoretical Considerations
 Invasion percolation models were previously suggested to describe the process of immiscible displacement of a nonwetting fluid by a wetting fluid from porous media in the absence of gravity effects and at a low velocity where the viscous effects were negligible compared to the capillary forces [Wilkinson and Willemsen, 1983]. This type of displacement gives rise to fractal patterns in the displacing phase [Tsimpanogiannis et al., 1999]. However, in many practical applications such as slow vertical displacements, gravity exerts hydrostatic pressure and stabilizes the front, limiting it to the finite fractal regime [Wilkinson, 1984; Chaouche et al., 1994]. The role of gravity is introduced by considering invasion percolation with a gradient [Hulin et al., 1988; Gouyet et al., 1988; Birovljev et al., 1991]. The front morphology for slow displacement rate reflects interactions between gravity and capillary forces which are commonly quantified by the dimensionless Bond number (Bo) [Wilkinson, 1984]:
where is the correlation length exponent from percolation theory ( for a two-dimensional system). In equation (2), mean front width is measured as where is assumed to be the correlation length of the percolation cluster. During drying, the above scaling law characterizes the front width over which the displacement has fractal properties of an invasion percolation interface [e.g., Shaw, 1987; Tsimpanogiannis et al., 1999; Yiotis et al., 2010]. The front morphology and structure in 3-D porous media strongly differs from the 2-D due to the higher connectivity in 3-D [Wilkinson, 1984; Rosso et al., 1986; Clément et al., 1987]. While in the case of 2-D porous media, the interface between the two immiscible fluids is a compact front with finite width; Rosso et al.  have shown that in 3-D porous media, the front between the two phases may extend over the entire sample. They found that in 3-D porous media the maximum front tail is scaled by equation (2) with , resulting in the exponent of −0.47. This relation was also introduced independently by Wilkinson [1984, 1986]. In 3-D porous media, the above scaling law indicates the maximum attainable correlation length that can be developed at a given value of the Bond number and is the maximum length over which the fractal behavior can be observed [Wilkinson, 1986]. In this work, we use the above theoretical considerations to estimate the width of the secondary drying front (i.e., the interface between the unsaturated and dry zone) [Shokri et al., 2009a] and use this as the diffusion length (or vaporization plane depth) to predict the evaporation rate at the onset of stage-2.
 Consider the situation close to the end of stage-1 evaporation where only a few liquid menisci remain pinned to the evaporating surface (Figure 1). The disruption of capillary flow when the evaporation characteristic length is exceeded, leads to interfacial instability and subsequent detachment from the surface and reestablishment at a level below the surface, i.e., a distance defined by the connected liquid cluster.
 As illustrated in Figure 1, the receding liquid menisci form a new vaporization plane (i.e., secondary drying front) at depth , which is estimated from the width of the anchored drying front (from equation (2)) that links the vaporization plane depth at the onset of stage 2 to the media properties (via the Bond number). This plane defines the vapor diffusion length and evaporation rate from a modified version of Fick's law [Shokri et al., 2009a]:
with the evaporation rate at the onset of stage two e2, the vapor diffusion coefficient in porous media D, the vapor diffusion coefficient in free air Datm, the vapor diffusion length which equals to the length of the jump at the onset of stage 2, the porosity , the saturated water vapor density Csat at the tip of the liquid meniscus after detachment, and the water vapor density above surface C∞. In equation (3), the model proposed by Moldrup et al.  was used to estimate the vapor diffusion coefficient in porous media. In section 3, the details of the experimental studies conducted to evaluate the proposed model are described.
3. Experimental Considerations
 We conducted evaporation experiments in the laboratory to test the theoretical prediction. We used rectangular glass columns (250 mm in height, 75 mm in width, and 10 mm in thickness) uniformly packed with sands and glass beads (manufactured by Carlo Bernasconi AG, Zurich, Switzerland) spanning a range of particle size distributions. Particles were wet packed into the column, ensuring initially saturated conditions. All boundaries of the columns were closed except the top, which was exposed to evaporation. The columns were mounted on digital balances connected to a computer to record digitally the column weight at 100 s intervals. An automatic system that included a camera connected to a computer was set up to remotely acquire images of the columns during the measurements to observe general dynamics of the process. To enhance visibility of vaporization plane, formation, and evolution, we have used a diluted solution of brilliant blue (0.1 g L−1). The experiments were conducted in a room with a relatively constant temperature and relative humidity that were recorded each 100 s, using digital sensors (Hygrowin, Rotronic, Switzerland). Table 1 presents some information about the boundary conditions of the measurements conducted in this study.
Table 1. The Materials, Particle Sizes, Humidity, and Temperature Measured During Each Round of the Experimenta
Average Particle Size (mm)
Average Relative Humidity (%)
Average Temperature (C°)
The average particle size, median of the particle sizes (d50), SD of the particle size distribution (), porosity, average relative humidity above the surface, SD of the relative humidity (), average temperature above the surface, and SD of the temperature measured ().
 The particle size distribution of various materials used in our measurement were determined by a laser diffraction particle size analyzer (LS 13 320, Beckman Coulter, Germany). The results are illustrated in Figure 2.
 The average particle size of each porous medium, d, was calculated using the data measured by the laser diffraction particle size analyzer which was used to estimate the typical pore size of the porous media as a ≈ d/3 [Glover and Walker, 2009].
4. Data Analysis and Discussions
4.1. Determination of the Onset of Stage-2 Evaporation
 The data obtained from the balances were used to determine evaporation rates for each column. Figure 3 illustrates the cumulative mass loss versus elapsed time from the beginning of the measurements.
 The slope of the cumulative mass loss versus time indicates the evaporation rate. From Figure 3 it can be deduced that the slope was nearly constant for the early stages of evaporation (as expected) followed by a lower slope corresponding to the stage-2 evaporation. Results showed that in some cases (e.g., columns packed with an average grain size of 0.224 or 0.417 mm) a distinct and “sharp” transition from stage-1 to stage-2 occurred, while in other cases (e.g., the columns packed with an average grain size of 0.064 or 0.771 mm), there was a more “gradual” transition from stage-1 to stage-2 evaporation. This gradual or sharp transition very likely depends, among other things, on the width of pore size distribution.
 We applied the methodology of Brutsaert and Chen  to determine the magnitude of evaporation rate at the onset of stage-2 evaporation. In this methodology, first the inverse of the square of evaporation rate is plotted against time. Then a linear regression is performed to determine the onset of stage-2 evaporation marked by the intercept between the linear line and the curve. Figure 4a shows an example of the analysis performed on data of the column packed with sand with an average particle size of 0.224 mm to determine the onset of stage 2.
Figure 4b illustrates a relatively constant and high evaporation rate during stage 1 supplied by the capillary flow from the drying front to the evaporation surface. During this period, evaporation occurs at the surface. After ∼19 d, the evaporation rate starts to decrease because of the disconnection of the liquid meniscus from the surface and this decreasing trend continues until all the liquid meniscuses are disconnected. This period was indicated as the transition period in Figure 4b. Detachment of the last liquid meniscus from the surface marks the onset of stage-2 evaporation and changing the vaporization plane to a level below the surface.
4.2. Jump of Liquid Menisci
 Clear evidence of abrupt depinning and subsequent formation of the vaporization plane below the surface is seen by the emergence of a discrete dye deposition layer as depicted in Figure 5. Menisci jump and a distinct new vaporization plane was not visible in all of the columns studied. Figure 5 illustrates snapshots of a region close to the sand surface during evaporation from the column packed with sand with an average particle size of 0.417 mm with the potential evaporation rate of ∼4 mm d−1.
Figure 5a corresponds to the end of stage 1, Figure 5b corresponds to the onset of stage-2 evaporation, and Figure 5c shows the last image taken at the end of the measurement. Disconnection and jump of the liquid meniscus from the surface and pinning to a level below the surface is supported by the preferential dye deposition plane formed below the surface at the onset of stage-2 evaporation and the presence of “dye-empty” regions above that plane (marked by the arrows in the images on the second row). We interpret the zones in which depositing did not take place as regions bridged by menisci that were subsequently depinned and jumped to lower depths. The resulting horizontally preferential patterns of dye deposition mark the locations of vaporization planes with a pinned secondary drying front. Interestingly, the images presented on last row of Figure 5 indicate that the jump of the liquid meniscus may happen several times during stage-2 evaporation.
4.3. Estimated Depinning Lengths and Comparison With Theory
 Experimental evaluation of equation (2) requires an independent estimate of the depinning length at the end of stage 1 as not all dye deposition patterns provided clear results. Shokri et al. [2009a] demonstrated that during stage-2 evaporation, a dry layer is formed near the surface when the secondary drying front recedes into the media. They also established experimentally, that during this period evaporation is supplied by capillary flow from the primary drying front to the secondary drying front and the flux continues as vapor diffusion through the dry layer. By using Fick's law they were able to accurately predict evaporation rates during stage 2 from the length of the dry surface layer (as vapor diffusion length) without any fitting parameters. In the present study, we use an inverse procedure to deduce the length from independently measured evaporation rates at the beginning of stag -2 using Fick's law in equation (3). Thus, the depth of the vaporization plane (liquid menisci jump) was calculated for each column and the results are presented in Figure 6a showing the dimensionless lengths (i.e., the physical length divided by the characteristic pore size of each material a) versus the Bond number.
 In Figure 6, in addition to the experimental data measured in this study, more points were extracted from the literature to evaluate the proposed model. Figure 6a indicates that the value of evaporation rate at the onset of stage 2 is mainly determined by the porous media transport properties and not the atmospheric conditions, though the atmospheric conditions probably affect the dynamic of the transition from stage-1 to stage-2 evaporation. Results show that the length of the liquid menisci jump at the end of stage 1 scales as a power law with Bond number with an exponent of −0.48, which is very close to the theoretical value of −0.47 for the 3-D system. Additionally, the physical length or depth of the new vaporization plane is presented in Figure 6b revealing that the transition to stage 2 is marked approximately by a jump in the vaporization plane to 3–14 mm below the surface in all cases which is consistent with the narrow range of stage-2 evaporation rates found experimentally by Shokri et al. [2009a] and supported by the numerical study of Saravanapavan and Salvucci . This estimation could be useful for constraining models and evaporative water loss estimated at field and regional scales.
 Another interesting question regarding the transition from stage 1 to 2 is related to the “abruptness” and dynamics of the transition which has not been addressed in the literature. Our results are not conclusive enough for quantifying the exact nature of the evaporation curve dynamics during the transition from stage 1 to 2. We expect the pore size distribution and atmospheric boundary conditions play a role in the dynamics where a narrow pore size distribution and higher evaporation rates would promote more abrupt transition, and conversely for wide pore size distribution and low rates. Additional studies are needed to accurately characterize the abruptness and dynamics of this important transition in evaporation regimes.
4.4. Generalization of the Proposed Model
 The proposed model enables estimation of upper and lower limits of evaporation flux at the onset of stage 2 on the basis of textural information (particle size) of common soils (from sand to clay). We use typical air entry pressure for clay from the data of Rawls et al.  that assembled data from more than 1300 sites in the United States, reporting 856 mm and 0.475 as air entry pressure and porosity of clay soil, respectively. This air entry pressure corresponds to a cylindrical pore with ∼0.0087 mm in diameter used to estimate the Bond number of clayey soil. Additionally, we used 0.67 mm as a typical pore size of sandstone with particle size of 2 mm according to a ≈ d/3 [Glover and Walker, 2009]. The porosity of sandstone can be as low as 0.2 [Marshall and Holmes, 1988]. Using these values, we determined the Bond number and estimated the maximum and minimum lengths of liquid menisci jump below the surface at the end of stage-1 evaporation. The additional parameters needed to estimate the upper and lower limits of the diffusive flux are: , D0 = 2.5 × 10−5 m2 s, Csat = 27 × 10−3 kgm−3. We assume a temperature of 25°C and ambient relative humidity of 60% in our calculation. Using the above parameters and equation (3), the upper and lower limits of the evaporation rate at the onset of stage-2 evaporation are estimated as 3.37 mm d−1 (clay) and 0.71 mm d−1 (sand), respectively.
Shokri et al. [2009a] used 20 different experimental values of evaporation rate at the onset of stage-2 evaporation from the literature, considering various porous media. They showed that in most cases, the evaporation rate at the onset of stage 2 was between 1 to 3 mm d−1 irrespective of evaporation rate during stage 1 or boundary conditions. These results are in a good agreement with the above predicted range of the upper and lower limits of the evaporation rate at the beginning of stage 2. Figure 7 illustrates this comparison.
 In Figure 7, in addition to the experimental data presented in this manuscript, data reported by Shokri et al. [2009a] were also used to evaluate the predicted upper and lower limits of the evaporation rate at the onset of stage 2. Figure 7 illustrates that all available data fall within the predicted range, and, equally important, there is little dependency on boundary conditions (stage-1 evaporation rate) and the primary factor affecting stage-2 evaporation is porous media properties. These results provide partial support for the proposed model describing the detachment of the liquid meniscus at the end of stage 1 that was used here to predict evaporation rates at the onset of stage-2 evaporation.
 The present study focused on a noncontinuous (jump) retreat of a capillary-held liquid front at the end of stage 1 to a predictable depth below the surface. The results suggest that the resulting new vaporization plane following front noncontinuous retreat defines a predictable diffusion length and associated diffusive flux. Within this context, evaporation experiments were conducted to study the nature of the transition from stage-1 to stage-2 evaporation. Evidence suggests that at the end of stage 1, the vaporization plane moves abruptly below the surface to a very narrow range of depths (3–14 mm). The length of this jump is a function of the Bond number, i.e., a characteristic of the porous media. We have found a power law relation between the length of this jump and the Bond number with the exponent of −0.48, in very good agreement with the theoretical prediction by the percolation theory relating the maximum front tail observed in an immiscible two-phase flow in 3-D porous media by the Bond number with an exponent of −0.47. These new results offer predictability of the depth of vaporization plane at the onset of stage-2 evaporation, hence prediction of the diffusion controlled evaporation rate. Using the proposed model, we estimated the upper and lower limit of the evaporation rate at the onset of stage-2 evaporation from common, natural porous media which was compared against many experimental data reported in the literature. Interestingly, all the experimental data fall into a range determined by the predicted upper and lower limits of the evaporation rate at the onset of stage 2 (irrespective of boundary conditions and evaporation rate during stage 1). Information regarding vaporization depth at the onset of stage 2 can be useful for the study of salt distribution and deposition during evaporation in arid and salt affected regions. In addition, estimation of the length of the jump of liquid meniscus at the end of stage 1 enables us to predict the flux at the onset of stage-2 evaporation. This will be specifically useful to model and describe the entire drying curve from saturated to dry condition.