Introduction

Capillary water flow and vapor transport occur during moisture loss from unsaturated soils. Capillary flow transfers water molecules from the soil interiors to the vaporization plane. The vaporization plane defines the boundary between the partly saturated soil and the dry soil layer. Water molecules vaporize at this boundary and escape to the atmosphere through the connected air-voids of the dry soil layer1,2. Increase in moisture loss facilitates the growth of dry soil layer leading to dominance of vapor flow. Impediments to moisture movement in a soil arise from chemical and geometrical interactions between water molecules/vapor and soil structure that is captured by the tortuosity (τ) factor3,4. Moldrup et al.5 observed that the tortuosity to liquid flow is strongly related to the surface area of the soils, while tortuosity to gas transport depends on the connectivity of the air-filled pores. Existence of internal and external surface areas in clay soils would cause considerable fraction of the soil water to not participate in capillary flow and lead to differences in critical volume fractions for percolation of air and water6. Ghanbarian et al.7 expressed tortuosity as a power law function of water content, the critical water content and system size. The results of the authors also indicated that pore connectivity and tortuosity must be treated as two separate properties.

An increase in soil surface area leads to retention of a larger fraction of soil-water in the diffuse ion layer. In a saturated soil, the growth of diffuse ion layer constricts the pore voids and makes the flow-path tortuous3,8. Besides, diffuse ion layer formation, water molecules are adsorbed on the soil particles by hydrogen bonds and van der Waals attraction3,8,9. In an unsaturated soil, reduction in moisture content during drying contracts the thickness of diffuse and adsorbed water layers that disturbs the network of connected water-filled capillaries3,10,11. At very low degree of saturation, moisture movement occurs in the vapor phase; the permeability to vapor transport is high, when the air-phase is continuous2,12. The connected capillary water network and air voids network are channels of moisture transport during different stages of soil drying. It is hence important to identify the dominant mode of moisture loss and to select appropriate tortuosity equations for correct prediction of Fickian moisture flux (Dv)10 in a drying soil.

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% respectively13,14. Grain size distribution of the representative soil15 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 soil16 corresponds to 1.78 Mg/m3 and 16% respectively.

Experimental moisture flux

Saturated K2SO4, NaCl and MgCl2.6H2O solutions maintained constant relative humidity (RH) of 97%, 76% and 33% in desiccators. The environmental humidity imposed by the salt solutions induced moisture loss from the compacted soil specimens (cylindrical specimens of 38 mm diameter and 76 mm length). The desiccators containing the saturated salt solution and compacted soil specimens (placed in the upper half) were maintained at constant temperature of 25 °C (±1 °C) in a temperature controller chamber. Moisture flux under known RH gradient were traced for soil dry density - compaction water content combinations of 1.69 Mg/m3 - 11% (series A); 1.78 Mg/m3 - 11% (series B); and 1.78 Mg/m3 - 16% (series C) respectively.

Triplicate specimens of given series (A/B/C) were horizontally placed on porous polypropylene plate in the upper half of a desiccator (Supplementary information section, Fig. A1) containing the desired saturated salt solution (K2SO4, NaCl or MgCl2.6H2O). Nine specimens (Triplicates × 3 humidity) of a series were tested at the three humidities.

During drying, the bulk (soil solids + water) weight (Wt) 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 wf and θf 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 (qvexpt, g/m2/day) at t days of evaporation is calculated as:

$${{\rm{q}}}_{{\rm{v}}({\rm{expt}})}=\frac{{{\rm{W}}}_{{\rm{initial}}}-{{\rm{W}}}_{{\rm{t}}}}{{{\rm{A}}}_{{\rm{SA}}}{\rm{t}}}$$
(1)

where, Winitial is initial mass of soil specimen (t = 0), Wt represents the mass of soil specimen after t days of evaporation and ASA is surface area of cylindrical soil specimen (m2); The (Winitial − Wt) 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 Wt value of a specimen from the average (three measurements) at any t, ranged between 0.02 to 0.2%. The average Wt 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 K2SO4 (RH = 97%), NaCl (RH = 76%), NaNO2 (RH = 64%), MgCl2.6H2O (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 (Srfinal). The total suction (ψ) of the specimen was obtained from Kelvin’s equation. The experimental SWCC data is fitted using Fredlund-Xing (FX) equation17. The residual water contents (θr) 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 cm3 and 2.0 cm3 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 [qv(t), Eq. 1], the moisture diffusion coefficient [Dv (m2/day)] is obtained as:

$${{\rm{q}}}_{{\rm{v}}({\rm{t}})}=-\,{{\rm{D}}}_{{\rm{v}}}{{\rm{\rho }}}_{{\rm{vsat}}}\frac{\nabla {\rm{RH}}}{{\rm{R}}{\rm{H}}{\prime} }$$
(2)

In Eq. 2, ∇RH is the relative humidity difference between environment (RHenv is the RH of the saturated salt solution in the desiccator) and soil pores (RHsoil), RH′ refers to the average relative humidity of environment and soil pores and ρvsat is the saturated vapor density (22.99 g/m3 at 298 K).

From a knowledge of Dv , the tortuosity factor (τcalc) at evaporation time t is calculated as:

$${{\rm{D}}}_{{\rm{v}}}={{\rm{n}}}_{{\rm{a}}}{{\rm{D}}}_{{\rm{o}}}/{\tau }_{{\rm{calc}}}$$
(3)

where, Do (m2/s) is the diffusion coefficient in the absence of soil matrix and na is the air-filled porosity at evaporation period, t.

Figure 1 plots τcalc - 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 τcalc - 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 (θcr), a reduction in volumetric water content increases tortuosity (τcalc); thereafter, a decrease in volumetric water content (θ) or an increase in volumetric air content (θa) reduces tortuosity.

Figure 1
figure 1

τcalc versus θ plots for series (A) specimens exposed to environmental RH of 97%, 76% and 33%.

Full size image

A relation is developed between θcr and wf (Fig. 2):

$${{\rm{\theta }}}_{{\rm{cr}}}=0.02{{\rm{w}}}_{{\rm{f}}}+0.01({{\rm{R}}}^{2}=0.98)$$
(4)
Figure 2
figure 2

Variation of θcr with wf for series (A–C) specimens.

Full size image

Knowing wf, the θcr 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 days19. 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:

$${{\rm{w}}}_{{\rm{f}}({\rm{compacted}})}=1.02{{\rm{w}}}_{{\rm{f}}({\rm{powder}})}-0.024({{\rm{R}}}^{2}=0.98)$$
(5)
Figure 3
figure 3

(A–C) Variation of volumetric water content with time: Blue, red and green circles refer to initial, critical and final volumetric water contents.

Full size image

The wf values of compacted specimens can be quickly determined by testing moist powder specimens and employing it in Eq. 4 to obtain θcr

Discussion

Critical volumetric water content (θ cr)

Re-engaging with Fig. 1, it is probable that in the θ > θcr 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 model7,20,21 and the soil-physics concept of a residual water content22,23. In θ > θcr 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 < θcr 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 network1,2. The falling rate segment is tailed by near stationary segment (segment 2), wherein, the rate of moisture loss (slope) is very small. The θcr values are located in segment 2, implying that vapor transport predominates in this segment. The θcr values have near similar magnitudes as the θf values (volumetric water content after 56 days of evaporation), which support the development of the relation between θcr and wf (Fig. 2).

The θth - soil surface area relation of Moldrup et al.5 gave a θth of 0.13 (surface area of soil in this study is 9.78 m2/cm3), which is close to the lower range of θcr 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 θth.

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 (θth), 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 large5. At θ < θth 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 beyond24. However, in the θ < θth 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.

The θcr 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) contents19,25 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 cm3/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 θcr (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 θcr (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 drying19. Hunt6 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.

Figure 4
figure 4

(A–C) Frequency distribution plots of desiccated specimens.

Full size image

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 (∅):

$$\varnothing =\frac{{\rm{\theta }}-{{\rm{\theta }}}_{{\rm{r}}}}{{{\rm{\theta }}}_{{\rm{s}}}-{{\rm{\theta }}}_{{\rm{r}}}}$$
(6)

would compensate for variations in initial porosity and water content of dis-similarly compacted soil specimens18. In Eq. 6, θs is the saturated water content (θs = n) and θr is the residual water content (θr < θf); the residual water contents are obtained from the SWCC plots (Section 2.2). At θr, the water phase is discontinuous and exists as thin water films surrounding the soil particles26. 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:

$${\rm{\tau }}={\rm{a}}{\varnothing }^{-{\rm{b}}}({\rm{General}}\,{\rm{equation}})$$
(7)
$${\rm{\tau }}=0.15{\varnothing }^{-2.71}({{\rm{R}}}^{2}=0.98)({\rm{RH}}=97 \% )$$
(8)
$${\rm{\tau }}=0.81{\varnothing }^{-1.22}\,({{\rm{R}}}^{2}=0.75)({\rm{RH}}=76 \% )$$
(9)
$${\rm{\tau }}=7.11\,{\varnothing }^{-0.24}({{\rm{R}}}^{2}=0.74)({\rm{RH}}=33 \% )$$
(10)

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:

$${\rm{a}}=17.96{({\rm{RH}})}^{2}-34.27({\rm{RH}})+16.46[{\rm{RH}}\,{\rm{in}}\,{\rm{decimals}}]$$
(11)
$${\rm{b}}=-\,7.51{({\rm{RH}})}^{2}+5.90({\rm{RH}})-1.37[{\rm{RH}}\,{\rm{in}}\,{\rm{decimals}}]$$
(12)

θ < θ cr condition

A normalized volumetric air content for θ < θcr condition is proposed:

$${\varnothing }_{{\rm{a}}}=\frac{{{\rm{\theta }}}_{{\rm{a}}}-{{\rm{\theta }}}_{{\rm{r}}}}{{{\rm{\theta }}}_{{\rm{cr}}}-{{\rm{\theta }}}_{{\rm{r}}}}$$
(13)

The ∅a term compensates for variability in volumetric air content of compacted specimens in the θ < θcr 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 Hunt21 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, Kc, represents the affinity of a chemical compound to partition between two phases27. 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:

$${{\rm{K}}}_{{\rm{c}}}=\frac{{C}_{s}}{{C}_{e}}$$
(14)

where, Cs represents the mass (g) of moisture lost per 100 g of soil at given relative humidity and temperature, while, Ce is the mass of water remaining in 100 g of soil at evaporation time t. At given t, Kc 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 RH19. Hence the ratio of ∅a/(Kc)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/(Kc)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:

$${\rm{\tau }}={\rm{x}}{(\frac{{\varnothing }_{{\rm{a}}}}{{{\rm{K}}}_{{\rm{c}}}^{1/RH}})}^{{\rm{y}}}$$
(15)

where x and y are empirical constants. Variations of x and y with RH follow the polynomial equations:

$${\rm{x}}=173.75\,{({\rm{RH}})}^{2}+305.58\,({\rm{RH}})+133.14\,[{\rm{RH}}\,{\rm{in}}\,{\rm{decimals}}]$$
(16)
$${\rm{y}}=-10.26\,{({\rm{RH}})}^{2}+12.56\,({\rm{RH}})-1.43\,[{\rm{RH}}\,{\rm{in}}\,{\rm{decimals}}]$$
(17)

Validation of Concept

The predictive ability of Eqs. 7, 11 and 12 (θ > θcr) and 15–17 (θ < θcr) 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 τcalc value specified Dv (Eq. 3) which in turn provided qv(pred) at different t values (Eq. 2). The goodness of fit of qv(pred) with qv(expt) is obtained by calculating mean relative percentage of deviation modulus (P), given as28:

$${\rm{P}}( \% )=\frac{100}{{\rm{N}}}\mathop{\sum }\limits_{{\rm{i}}=1}^{{\rm{N}}}\frac{|{{\rm{q}}}_{{\rm{v}}({\rm{expt}}){\rm{i}}}-{{\rm{q}}}_{{\rm{v}}({\rm{pred}}){\rm{i}}}|}{{{\rm{q}}}_{{\rm{v}}({\rm{expt}}){\rm{i}}}}$$
(18)

where qv(expt)i and qv(pred)i are experimental and predicted moisture vapor flux of the soil at given RH and time t, and N is number of qv(expt) values measured at various t.

For θ > θcr condition the compacted specimens mostly exhibit P values of <10% over a wide range (97–33%) of relative humidities (Table 1). For θ < θcr 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/(Kc)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 Kc 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 Kc values), renders the vapor phase tortuosity sensitive to slight variations in air-filled porosity, leading to higher P values. Ghanbarian and Hunt21 have observed that the universal scaling exponent is very sensitive to the measured experimental values at low air-filled porosities.

Table 1 Goodness of fit for moisture flux prediction based on proposed equations.
Full size table

Figure 3 reveals that at 33 and 76% RH, bulk (91–97%) of the initial moisture evaporates in the θ > θcr 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 > θcr region may be sufficient, as contribution from the θ < θcr 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 Penman29, 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.

Table 2 Goodness of fit for moisture flux prediction using τ based on other equations.
Full size table

Conclusions

The critical water content θcr, separates dominant regions of capillary water flow and vapor diffusion during moisture loss in an unsaturated soil. At θ > θcr condition, the capillary flow of water molecules dominates moisture loss. When θ becomes less than θcr, vapor diffusion through the air-filled pores of the dry soil layer is important. Both, θcr and wf occur in the stationary rate segment of the drying curve and have near similar magnitudes. These resemblances encouraged estimation of θcr from wf 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 > θcr region. Comparatively, the ∅a/(Kc)1/RH ratio represents the influence of variable volumetric air content and vapor partitioning tendency on vapor phase tortuosity in the θ < θcr 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 and predicted moisture flux at RH = 33%. For soils characterized by falling rate and stationary rate segments on exposure to RH ≤ 76%, considering moisture loss only in the θ > θcr region may be sufficient; this is so, as the contribution from vapor transport in the θ < θcr region to overall moisture loss is small (3–9% of initial volumetric water content).