Identifying the dominant mode of moisture transport during drying of unsaturated soils

Diffusion of capillary water and water vapor during moisture loss in an unsaturated soil is impeded by the chemical and geometrical interactions between water molecules/vapor and the soil structure. A reduction in moisture content contracts the diffuse and adsorbed water layers in the partly saturated soil and disturbs the connected capillary network for flow of liquid water. With further drying, the dry soil layer expands and moisture is predominantly lost as vapor through continuous air-flow channels. The water-filled capillary network and air-filled channels are moisture conduits during different stages of soil drying. It is important to identify zones of dominant moisture transport and to select appropriate tortuosity equations for correct prediction of moisture flux. Laboratory experiments were performed to determine moisture flux from compacted soil specimens at environmental relative humidity of 33, 76 and 97% respectively. Analysis of the resultant τ - θ (tortuosity - volumetric water content) relations, illustrated the existence of a critical water content (θcr), that delineates the dominant zones of capillary liquid flow and vapor diffusion. At critical water content, the pore-size occupied by the capillary water is governed by the generated soil suction. Generalized equations are proposed to predict tortuosity factor in zones of dominant capillary liquid flow and vapor transport over a wide range of relative humidity (33 to 97%).

The experimental moisture flux is used to delineate the dominant regions of capillary water flow and vapor transport in an unsaturated soil during moisture loss. Generalized equations are proposed to predict tortuosity factor in regions of dominant capillary liquid flow and vapor transport over a wide range of relative humidity (33 to 97%). Comparisons are made between the experimental and predicted moisture flux by considering appropriate tortuosity factor in Fick's equation.

Materials and Methods
Soil description. Representative soil sample was obtained from 1-m deep pit at the Indian Institute of Science Campus. The specific gravity, liquid limit and plasticity index of the representative soil are 2.66, 34% and 16% respectively 13,14 . Grain size distribution of the representative soil 15 comprised of 52, 25 and 23% of sand, silt and clay sized fractions respectively. The clay fraction of the soil is composed of non-swelling kaolinite. The maximum dry density and optimum moisture content of the representative soil 16  During drying, the bulk (soil solids + water) weight (W t ) of the soil specimens were periodically monitored up to 56 days as they negligibly changed thereafter. The gravimetric and volumetric water contents of specimens subjected to 56 days of drying are designated as w f and θ f respectively (f-final). Gravimetric moisture loss of 3-6%, 8-13% and 10-14% were observed on drying the soil specimens at RH of 97, 76 and 33% for 56 days. Slight reductions in porosity (3-6%) and void ratio by (4-9%) occurred after 56 days. The compacted soil specimens subjected to 56 days of drying are termed as desiccated specimens.
The experimental moisture flux (q vexpt , g/m 2 /day) at t days of evaporation is calculated as: where, W initial is initial mass of soil specimen (t = 0), W t represents the mass of soil specimen after t days of evaporation and A SA is surface area of cylindrical soil specimen (m 2 ); The (W initial − W t ) term represents the loss in gravimetric water content of a compacted specimen on drying at known relative humidity for t days.
The percent variation in W t value of a specimen from the average (three measurements) at any t, ranged between 0.02 to 0.2%. The average W t value of the specimen (range: 148-169 g), at each t, on drying at the desired humidity is utilized in Eq. 1.
Separate batch of A/B/C specimens were tested to obtain SWCC (soil water characteristic curve) plots. The compacted specimens were equilibrated with saturated K 2 SO 4 (RH = 97%), NaCl (RH = 76%), NaNO 2 (RH = 64%), MgCl 2 .6H 2 O (RH = 33%) and NaOH (RH = 7%) solutions in desiccators; the bulk weights of the compacted specimens were periodically measured till the specimens experienced negligible weight loss. SWCC plots are obtained for each compaction series by plotting total suction (ψ) as function of final degree of saturation (S rfinal ). The total suction (ψ) of the specimen was obtained from Kelvin's equation. The experimental SWCC data is fitted using Fredlund-Xing (FX) equation 17 . The residual water contents (θ r ) were obtained from the FX curves by using the procedure of Fredlund et al. 18 and correspond to 0.0217, 0.0197 and 0.0287 for Series A, B and C respectively.

Mercury intrusion porosimetry (MIP) experiments.
Pore size distribution of series A/B/C specimens that were dried for fifty-six days at RH of 97%, 76% and 33% were determined using Quanta chrome (USA) Poremaster -60 over the pressure range of 0.2 to 60000 psi (pounds per square inch). Sample cells having volumes of 0.5 cm 3 and 2.0 cm 3 were used in the tests. Prior to performing the MIP test, the specimens were freeze-dried using a lyophilizer. The MIP test was performed in two steps: a low-pressure step from 0.2 psi (1.38 kPa) to 30 psi (206.8 kPa) and a high-pressure step from 20 psi (138 kPa) to 60000 psi (413685 kPa).

Results
τ calc -θ relation. Utilizing the experimental moisture flux [q v (t), Eq. 1], the moisture diffusion coefficient [D v (m 2 /day)] is obtained as: In Eq. 2, ∇RH is the relative humidity difference between environment (RH env is the RH of the saturated salt solution in the desiccator) and soil pores (RH soil ), RH′ refers to the average relative humidity of environment and soil pores and ρ vsat is the saturated vapor density (22.99 g/m 3 at 298 K).
From a knowledge of D v , the tortuosity factor (τ calc ) at evaporation time t is calculated as: where, D o (m 2 /s) is the diffusion coefficient in the absence of soil matrix and n a is the air-filled porosity at evaporation period, t. Figure 1 plots τ calc -θ relations of series A specimens that were exposed to environmental RH of 33, 76 and 97% for various evaporation periods (5-56 days). The τ calc -θ relations of series B and C specimens are provided in the Supplementary information section (Figs. A2 and A3). Up to a critical water content (θ cr ), a reduction in volumetric water content increases tortuosity (τ calc ); thereafter, a decrease in volumetric water content (θ) or an increase in volumetric air content (θ a ) reduces tortuosity.
A relation is developed between θ cr and w f ( Knowing w f , the θ cr of a soil experiencing moisture loss is obtained from Eq. 4. The final water contents are attained by the compacted specimens after long periods of drying (10 to 50 days, Fig. 3). In comparison, moist powder soils dry quickly and attain final water contents in 1 to 5 days 19 . The final water contents of the powder and compacted specimens exposed to given RH (33 to 97%) are near similar (Fig. A4) and follow the equation: The w f values of compacted specimens can be quickly determined by testing moist powder specimens and employing it in Eq. 4 to obtain θ cr

Discussion
Critical volumetric water content (θ cr ). Re-engaging with Fig. 1, it is probable that in the θ > θ cr region, liquid water molecules escape through the connected capillaries of the partly saturated soil. The critical water content signifies the minimum water content for the existence of inter-connected water filled pores and agrees with continuum percolation model 7,20,21 and the soil-physics concept of a residual water content 22,23 . In θ > θ cr region, the progressive contraction of the diffuse ion layer and adsorbed water layer thickness in soil capillaries contribute to the loss of water-filled pore connectivity. In θ < θ cr region, moisture loss predominantly occurs as vapor diffusion, causing dependence of τ on air-filled porosity.
The θ versus evaporation period (t) plots (Fig. 3) illustrate that the rate of moisture loss is characterized by falling rate segment (segment 1), which results, as the evaporation demand is higher than flow capacity through the connected liquid network 1,2 . The falling rate segment is tailed by near stationary segment (segment 2), wherein, the rate of moisture loss (slope) is very small. The θ cr values are located in segment 2, implying that vapor Figure 1. τ calc versus θ plots for series (A) specimens exposed to environmental RH of 97%, 76% and 33%. www.nature.com/scientificreports www.nature.com/scientificreports/ transport predominates in this segment. The θ cr values have near similar magnitudes as the θ f values (volumetric water content after 56 days of evaporation), which support the development of the relation between θ cr and w f (Fig. 2).
The θ th -soil surface area relation of Moldrup et al. 5 gave a θ th of 0.13 (surface area of soil in this study is 9.78 m 2 /cm 3 ), which is close to the lower range of θ cr values at RH = 97% (0.16 to 0.22). The relation of Moldrup et al. 5 , however, does not consider the influence of initial soil properties (porosity, water content) or environmental humidity in the determination of θ th .
Moldrup et al. 5 and Ghanbarian et al. 7 have recognized that tortuosity increases with reduction in soil water content as the water films surrounding the soil particles become increasingly discontinuous and viscous. At certain threshold soil water content (θ th ), complete breakage in the continuity of water films causes the liquid phase impedance factor (f) to become zero or the tortuosity factor (τ = 1/√f) to become very large 5 . At θ < θ th values the results of Moldrup et al. 5 show that the tortuosity becomes excessively large and tend towards infinity. This would in principle be true, if capillary flow was the only mode of transport in a drying soil, because then, the actual flow path would become infinitely long at critical water content and beyond 24 . However, in the θ < θ th region where vapor transport dominates, the effective path length should reduce from the participation of connected air-voids in vapor transport. Ghanbarian et al. 7 observed that the sample size is not infinitely large in many cases and the finite size effects dominate the simulated results. Further, when the tortuous path length exceeds the sample length, the finite size scaling is an appropriate approach to generate tortuosity predictions. www.nature.com/scientificreports www.nature.com/scientificreports/ The θ cr values ranged between 0.16 to 0.22 on exposure of series A/B/C specimens to environmental RH of 97%. It ranged between 0.077 to 0.1 when the specimens were exposed to environmental RH of 76% and between 0.04 to 0.05 on exposure to environmental RH of 33%. Ghanbarian et al. 7 have observed that critical water contents are not universal but are dependent on the pore structure of the soil.
The coarse (60 to 6 μm), medium (6 to 0.01 μm), fine (0.01 to 002 μm) and very fine pores (<002 μm) contents 19,25 contribute to the porosity of compacted specimens after 56 days of drying (desiccated specimens) at the three relative humidities (Fig. 4). The unit volume of voids in the desiccated, A, B and C specimens correspond to 0.213, 0.185 and 0.185 cm 3 /g respectively. Calculations show that for specimens exposed to 97% RH, 60% of the medium pores and all coarse pores are occupied by capillary water at θ cr (0.16-0.21). With soil specimens exposed to RH of 76%, 80% of medium pores are occupied by capillary water at the critical volumetric water content (0.071-0.078). Similarly, for specimens exposed to RH of 33%, all fine pores are occupied by capillary water at θ cr (0.029-0.039). At the critical water content, occupancy of narrower pores by capillary water at the lower RH is commensurate with the larger suction developed by these specimens after 56 days of drying 19 . Hunt 6 had observed the minimum water content for continuous network of capillary flow in clay soils can be assumed to be 1/6n (n = porosity), plus the volume of pores that are smaller than 0.3 μm radius. θ > θ cr condition. The θ -t relations (Fig. 3) depict the dependence of θ on relative humidity (series A/B/C), initial porosity (Series A and B specimens) and gravimetric water content (Series B and C specimens). A normalized volumetric water content (∅): would compensate for variations in initial porosity and water content of dis-similarly compacted soil specimens 18 . In Eq. 6, θ s is the saturated water content (θ s = n) and θ r is the residual water content (θ r < θ f ); the residual water contents are obtained from the SWCC plots (Section 2.2). At θ r , the wate r phase is discontinuous and exists as thin water films surrounding the soil particles 26 . The τ -∅ relations (Supplementary information section, Figs. A5-A7) of compacted specimens (series A, B, C) exposed to similar humidity (97/76/33%) follow equations: where a and b are empirical constants at given RH. The trend of a parameter suggests that it is related to the rate of desaturation of voids in segment 1 as they also exhibit progressively steep slopes with reduction in RH (Fig. 3). The variations of the empirical constants (a and b) with RH facilitate determination of τ at any RH of the compacted specimens from the equations: 2 θ < θ cr condition. A normalized volumetric air content for θ < θ cr condition is proposed: The ∅ a term compensates for variability in volumetric air content of compacted specimens in the θ < θ cr region. Variations of τ with ∅ a of the compacted specimens (series A, B, C) exposed to similar humidity (97/76/33%), depicted a reduction in τ with increase in ∅ a ; however, the data set of each compaction series (A/B/C) plot separately. At given RH, the inability of specimens to plot uniquely suggests that the ∅ a term is insufficient to characterize vapor phase tortuosity for variable initial porosity and water content conditions. Ghanbarian and Hunt 21 have used relative air-filled porosity as component of universal scaling law for gas diffusion. In the present study, the universal scaling law could explain the variation of τ of compacted specimens belonging to a single series exposed to given RH. However, like ∅ a, the data set of each compaction series (A/B/C) plot separately (do not plot uniquely) at a given RH.
Besides the availability of connected air-voids (∅ a ), the spontaneity of the water vapor to partition between the dry soil layer and atmosphere may contribute to the ease of vapor transport. The distribution coefficient term, K c , represents the affinity of a chemical compound to partition between two phases 27 . It could account the partitioning tendency of the water vapor between the dry soil layer and atmosphere. In the present context, the distribution coefficient is defined as: s e c where, C s represents the mass (g) of moisture lost per 100 g of soil at given relative humidity and temperature, while, C e is the mass of water remaining in 100 g of soil at evaporation time t. At given t, K c is inversely related to RH (Fig. A8, Supplementary information section). The greater spontaniety of the dry soil layer to lose vapor stems from the more negative change in free energy (∆G°) associated with evaporation at lower RH 19 . Hence the ratio of ∅ a /(K c ) 1/RH is expected to represent the combined influence of normalized air-filled porosity and RH dependent partitioning tendency of water vapor in the dry soil layer. The τ -∅ a /(K c ) 1/RH relations (Supplementary information, Figs. A9-A11) of compacted specimens (series A/B/C) exposed to similar humidity (97/76/33%) follow the general equation: 2

Validation of concept
The predictive ability of Eqs. 7, 11 and 12 (θ > θ cr ) and 15-17 (θ < θ cr ) is verified by comparing the experimental moisture flux of the compacted specimens exposed to RH of 97/76/33% (Eq. 1) with the predicted moisture flux (Eq. 2). The Eqs. (11, 12 or 16, 17) facilitated calculation of τ at given RH (Eqs. 7 and 15). The τ calc value specified D v (Eq. 3) which in turn provided q v(pred) at different t values (Eq. 2). The goodness of fit of q v(pred) with q v(expt) is obtained by calculating mean relative percentage of deviation modulus (P), given as 28 : where q v(expt)i and q v(pred)i are experimental and predicted moisture vapor flux of the soil at given RH and time t, and N is number of q v(expt) values measured at various t. For θ > θ cr condition the compacted specimens mostly exhibit P values of <10% over a wide range (97-33%) of relative humidities (Table 1). For θ < θ cr condition and at RH of 97 and 76%, the P values range between 14-31% with majority of the values varying between 14-19%. Much larger P values are observed at 33% RH (34, 34, 52%). The linear forms of τ -∅ a /(K c ) 1/RH relations (not presented) of the compacted specimens are characterized with slopes of 0.26 and 26 at RH of 97 and 33% respectively. The 100-fold variation in slopes indicate that τ is sensitive to small variations in ratio at low relative humidity. The K c values of the compacted specimens range from 0.22 to 0.57 at RH = 97% and from 3.83 to 8.6 at RH = 33%. Possibly, the more spontaneous nature of moisture loss at low RH (large K c values), renders the vapor phase tortuosity sensitive to slight variations in air-filled porosity, leading to higher P values. Ghanbarian and Hunt 21 have observed that the universal scaling exponent is very sensitive to the measured experimental values at low air-filled porosities. Figure 3 reveals that at 33 and 76% RH, bulk (91-97%) of the initial moisture evaporates in the θ > θ cr region. Hence, for soils characterized by falling rate and stationary rate segments that are exposed to evaporation at RH ≤ 76%, considering moisture loss in the θ > θ cr region may be sufficient, as contribution from the θ < θ cr region to overall moisture loss is small.
The P values of series A, B and C specimens ( Table 2) were calculated using τ based on Penman 29 , Fredlund et al. 18 and Moldrup et al. 30 equations. Besides τ, all other parameters for moisture flux prediction (Eq. 2) remain the same. Use of Fredlund's, Penman's and Moldrup's equations give P values ranging 42 to 328%, 28 to 102% and 97 to 100% respectively ( Table 2). The improved P values from equations proposed in this study (Table 1) underlines the importance of identifying the dominant regions of capillary water flow and vapor transport in unsaturated soils experiencing moisture loss for correct moisture flux prediction.

conclusions
The critical water content θ cr , separates dominant regions of capillary water flow and vapor diffusion during moisture loss in an unsaturated soil. At θ > θ cr condition, the capillary flow of water molecules dominates moisture loss. When θ becomes less than θ cr , vapor diffusion through the air-filled pores of the dry soil layer is important. Both, θ cr and w f occur in the stationary rate segment of the drying curve and have near similar magnitudes. These resemblances encouraged estimation of θ cr from w f values. Specimens exposed to RH of 97, 76 and 33% are constrained to occupy progressively narrower pores at the critical water content owing to larger suction developed in the specimens upon drying. The normalized water content (∅) accounts for the influence of variable initial water content and porosity on capillary water flow tortuosity in the θ > θ cr region. Comparatively, the ∅ a /(K c ) 1/RH ratio represents the influence of variable volumetric air content and vapor partitioning tendency on vapor phase tortuosity in the θ < θ cr region. The more spontaneous nature of moisture loss at low RH, renders the vapor phase tortuosity sensitive to small variations in air-filled porosity, leading to larger deviations between experimental A-97% A-76% A-33% B-97% B-76% B-33% C-97% C-76% C-33%  Table 2. Goodness of fit for moisture flux prediction using τ based on other equations.