Figure  presents vertical cross-sections of the drainage and imbibition simulations at three stages of water saturation for the five selected snow samples described in Table . The pore-scale distribution of liquid water in the microstructures can be observed. For imbibition, the small pores are first filled, and the larger pores are filled last. For drainage, it is the other way around, with water escaping the large pore first. Residual liquid water at the end of the drainage simulation is shown in pink. Air bubbles entrapped in the ice skeleton are shown in yellow (for instance, in the bottom right of the 0A (RG) sample). This figure also highlights the fact that, for a given saturation, the liquid water distribution is different depending on the snow types, as well as between the imbibition and drainage processes.

Figure a and b present the WRCs of the imbibition and drainage simulations applied to the 5 selected snow samples presented in Table . A first result is that the WRCs of the snow samples present generally significant hysteresis, as they differ between imbibition and drainage (see also Fig. b). Overall, a steeper WRC is found for imbibition compared to drainage, and h values are higher for drainage compared to imbibition.

The influence of the snow geometrical properties can be observed. Snow samples presenting small grains (0.06–0.15 mm), such as the samples fr (PP), 0A (RG), and grad3 (DH), show higher pore pressures at a given water content than the samples NH2 (MF) and NH5 (MF) presenting coarse grains (0.53–0.87 mm). Snow density also has a direct influence on the WRCs, as it limits the maximum water content that can be reached. More subtly, the WRCs tend to show sharper transitions for the most evolved snow microstructures.

For each microstructure, we also present the related VG fit (Eq. ) that best reproduces the simulated WRC. In other words, the VG parameters αvg, nvg, and θwr were optimized to best fit the simulated WRC of each snow sample for imbibition and for drainage. The fitting is rated in terms of MAE (mean absolute error) on θw and expressed as a percentage. The fitted WRCs show overall good agreement with the simulated WRCs, as illustrated for the five selected snow samples in Fig. a and b. The fits show slightly better results for drainage than for imbibition, and for MF samples compared to the other snow types. The model fits the data points better in the wet part compared to the dry part for snow types such as RG and PP. This might be due to (i) the fact that mvg=1-1/nvg constrains the shape of the fit, and that (ii) the data points are not evenly distributed in this θw, as the simulation step is the pore radius and not the water content. In future studies, the VG formulation could be refined using more parameters, such as mvg, to enable fitting both the left and right inflection points of the WRCs, especially for drainage.

Figure  presents the VG parameters αvg, nvg and θwr obtained by fitting the VG model to our simulated WRCs for all our dataset. The parameters, obtained for imbibition and drainage, are expressed as a function of the term ρ/d or ρ, following , and compared to the regressions suggested by (black solid curves in Fig. ). In addition, the measurement data of αvg and nvg from and are also shown. They correspond to values derived from WRCs obtained from laboratory experiments of imbibition and drainage. As specified in Sect. , in all our computations, d was estimated based on the SSA, following the formula d=2×res=6/(SSA×ρi) with ρi = 917 kgm-3 the ice density.

Overall, the parameter αvg decreases with increasing ρ/d, so when density increases and/or grain size decreases. This trend is in overall consistent with the measurements of , , , and . Moreover, αvg values from the imbibition simulations are systematically larger than the ones from the drainage simulations. This is in agreement with the hysteresis reported by and with the high αvg values obtained from imbibition experiments by . The αvg values from our simulations and the regression of are overall in good agreement for drainage, with a MAE of 25 %, a little less so for imbibition, with a MAE of 38 %.

Following the formulation of , we proposed two new regressions of αvg, which allow reproducing the different behaviors between imbibition and drainage. Their expressions are provided in Table . They are shown in Fig. a and b, and have a MAE of 17 % when compared to our data. The proposed regressions were derived from snow samples with ρ/d values comprised between 0.25 × 10⁶ and 1.3 × 10⁶ kgm-4, and do not cover the lowest values of ρ/d (< 0.25 × 10⁶ kgm-4), for which a steep evolution of αvg was reported . Still, we derived our regressions with an exponential form to be consistent with these observations and following .

We use the αvg parameter to quantify the degree of hysteresis of the WRCs, based on the ratio αvg from imbibition over αvg from drainage. For our set of images, this ratio ranges from 1.19 to 1.42, with an average value of 1.28. It is consistent with the ratios measured by , for which values of 1.46, 1.52, and 1.38 are found for the S, M, and L samples, respectively. No correlation was found between this ratio and grain type, grain size, or density. Our ratios and the ones of tend to confirm hysteresis ratios around 1.5 for snow, which is lower than the classical value of 2 used for soils .

The results obtained from the simulations are more surprising regarding the parameter nvg. For imbibition, the nvg values show little variation for the whole range of ρ/d and are around 4.7 (Fig. d). These results are rather consistent with the nvg values of but not with those of . The latter authors report a correlation of the nvg with ρ/d that follows the regression of based on drainage experiments, not observed in our work and the one of .

The nvg values for drainage are overall much higher compared to imbibition, as opposed to the findings of for snow or many soils observations , which mention similar values of nvg for drainage and imbibition. Our results are more consistent with the ones of for snow or with granular materials for which nvg values in imbibition can be much smaller than the ones in drainage.

For drainage, our nvg values are spread and show little correlation with ρ/d. Looking at the experimental data, a large spread is also observed in , , and for the rounded grains samples of . Our estimated nvg values overall do not follow the regression of , although the nvg values of the melt forms samples are closer to this regression compared to the other snow types. Let us recall that presented a regression based on drainage experiments on sieved melt forms only (black filled dots in Fig. d), which was in good agreement with nvg values estimated on other samples of natural melt forms (black circles), but in poor agreement for samples of natural rounded grains (black stars), for which nvg appear to be independent of ρ/d. attributed these different behaviors between snow samples to differences in pore-size uniformity, which seems coherent with the definition of nvg.

To test the hypothesis that nvg could depend on the pore-size uniformity, we characterized pore size variations by using the distributions of mean curvature of the snow microstructure, which is a standard parameter used for snow e.g. see. The mean curvature was calculated at each point on the surface of the 3D images and represented as a statistical distribution (see , and the Supplement for additional information). Figure  shows nvg as a function of the interquartile range of the mean curvature IQRMC of each 3D snow image for both imbibition and drainage, thus describing the relationship between nvg and the mean curvature uniformity for both processes. A trend can be observed, so that low IQRMC values tend to be correlated to large nvg values, following the form of an inverse function. The trend is overall similar for imbibition and drainage, with lower values for imbibition. The observed trend is consistent with the fact that, during drainage, water leaves the pores more or less all at once for snow with rather uniform pore sizes, such as melt forms (in red), as showed by very sharp WRCs and modeled by large nvg values (see Fig. b). The lower limit of IQRMC corresponds to a material with uniform pore sizes and higher values of nvg (step function). For snow types showing large pore size variability, such as fresh snow (in light green), nvg values are smaller, and the resulting WRC shows a smoother transition as a function of the water content, i.e., the drainage is more gradual. The same considerations apply to imbibition, while the impact is less prominent, due to the rather large snow porosity (at least greater than 0.4 for each sample of our dataset). The imbibition is thus less dependent on the local microstructure than the drainage process.

In conclusion, to estimate nvg, our results do not support the use of the ρ/d ratio but rather of more refined parameters, such as the mean curvature distribution (see Fig.  and Table  for the detailed regression proposed). This can be seen as a limitation for larger-scale modeling as this parameter can currently only be derived from 3D images.

The last parameter required for our use of the VG model is the residual water content θwr. As already mentioned, this parameter was set to 0 for imbibition as simulations were performed on fully dry snow images, and was only determined for drainage as the minimum value of water content reached during the drainage simulations. Two distinct groups are observed (Fig e). Samples with density below 450 kgm-3 show values centered around 0.046, which slightly increase with density from about 0.036 to 0.061. Samples of melt forms with density above 450 kgm-3 show smaller values around 0.029, including the NH2 and NH5 samples. This division is probably due to the fact that the denser snow samples are composed of MF that show large and uniform pores compared to the other samples. The latter could be subjected to more disconnections during drainage due to the small and complex throats, resulting in higher residual water content. All the values of θwr from our simulations are larger than the value of 0.02 proposed in . Following our results, we propose to approximate θwr by two constant values depending on the snow type: 0.029 for melt forms and 0.046 for the other snow types (Fig.  and Table ).

To complete the picture, it seems worth discussing the saturated water content θws. This parameter is here approximated by the snow porosity ϕ, as opposed to observations from the snow imbibition and drainage experiments which report θws ranging from 0.6ϕ to 0.9ϕ with the experimental challenge of filling complex geometries with melt and freezing processes occurring during the imbibition. In porous media such as soils and sands, imbibition and drainage experiments also show a large range between 0.3ϕ to around ϕ. Low values of θws are either seen as underestimated due to the experimental limits linked to the challenge of filling complex geometries, or as reflecting the real physical processes at stake e.g.,. Therefore, two approaches are commonly used in the literature, either the approximation θws=ϕ is taken or θws is adjusted to experimental data. The experiments of and have shown that having θws smaller than ϕ generally implies greater αvg values, but has no significant impact on the nvg values. For snow purposes, the actual values of the saturated water content still remain unclear and should be further investigated to refine the VG parameters.

The regressions of are in good agreement with our simulations for αvg, especially for drainage, but not for nvg for both imbibition and drainage, and not for θwr. We recall that the regressions of are based on drainage experiments only and realized on a limited number of snow types, mainly composed of dense MF, which may explain some of the observed discrepancies. For αvg, we proposed new regressions based on ρ/d for both imbibition and drainage, for a wide range of ρ/d values. For nvg, a constant value was suggested for imbibition, but no estimates for drainage could be proposed using ρ/d. A parameter that captures the pore size distribution of snow, such as the interquartile range of the mean curvature, seems to be required. For θwr, two mean values were provided depending on the snow type for drainage.

Here, we applied different VG models to predict the WRCs for drainage and imbibition for two imaginary snow samples. The properties of those samples have been chosen to be representative of melt forms (sample 1: d = 1.5 mm, ρ = 450 kgm-3, IQRMC = 5 mm-1) and precipitation particles (sample 2: d = 0.1 mm, ρ = 130 kgm-3, IQRMC = 15 mm-1). We present predictions based on the regressions of the shape parameters αvg, nvg, and θwr from , , , and from this study for both imbibition and drainage. These regressions of the shape parameters are provided in Table . and present a regression of αvg and nvg based on grain size, while include both grain size and snow density, using the variable ρ/d, with d the mean grain diameter. We recall that the latter three regressions have been developed based on drainage measurements only.

Figure  presents the WRCs for the two representative samples for the different VG models. These curves are expressed as a function of the effective saturation e.g.,, defined in Eq. (). Our VG models are closer to the VG model of for both samples, the one of and being consistently above and below our estimates, respectively. Besides, the difference of WRCs estimated for imbibition and drainage on the same sample is lower than the difference between models. A larger spread of the curves can be observed for sample 2, which tends to show the highest sensitivity of the VG shape parameters for low-density and small-grain snow.

First, we study the effective water permeability Kwu. To compare all the samples together, we use the relative water permeability Kwr(θw)=Kwu(θw)/K, with K the intrinsic permeability of the saturated media. The evolution of the relative permeability with the effective saturation is shown in Fig. a. The relationship describes an exponential increase, which tends, for all samples, to merge into a single curve. This shows that the water permeability is at first order driven by the water content and the snow density, and that other dependencies with other microstructural parameters are, if any, of lesser strength. We compare our results with the van Genuchten–Mualem (VGM) model of relative water permeability , described in the introduction and in Eq. (). For that, the shape factors of the VG model nvg, αvg, and mvg=1-1/nvg were taken using the values fitted to the WRC of each image. In addition, the VGM model includes the parameter τvg, which describes the effects of the connectivity and tortuosity of the flow paths. Here we assume that τvg=1/2, which is the default value suggested in . Note that this parameter can vary depending on the porous material e.g. and its value for snow could be refined in future works. For the 5 reference snow samples presented in Table , good agreements are found between the VGM model and our numerical computations (Fig. b).

From the relative water permeability, the unsaturated hydraulic conductivity Kwu can be obtained by the following equation: 8Kwu(θw)=Kwsat×Kwr(θw) with 9Kwsat=Kρwg/μw K depends on the snow microstructure and is estimated using the parameterization of based on the density and the spherical equivalent radius of snow. Figure  presents the unsaturated hydraulic conductivity Kwu as a function of the effective saturation, showing a non-linear increase with increasing saturation. As Kwu contains a K factor as compared to the relative permeability Kwr, an effect of the microstructure can here be observed: for a given water saturation, Kwu varies with density and the spherical equivalent radius, such as lighter and/or larger grain snow samples show larger Kwu values (Fig a and b). To estimate the unsaturated hydraulic conductivity, the VGM model is here used with the shape parameters estimated from the regressions proposed in this study for drainage (given in Table ), and combined with the parameterization of . The estimates are overall in good agreement with the computed data, as shown in Fig. b. The estimates of the unsaturated hydraulic conductivity from the VGM model using the shape parameters from (Table ) are also provided for comparison (dashed lines). Both VGM models are overall fairly close. For the melt forms samples, a slight improvement is found using our regressions, with MAE values around 10 % for and around 9 % for our model. These similarities can be related to the fact that, on one hand, the regression of provides slightly better estimates of αvg for this snow type, compared to our regression (Fig. b); on the other hand, nvg values are better estimates from our regression (Figs. d and ). For the other snow types, the VGM model using our shape parameter estimates provides better predictions of unsaturated hydraulic conductivity, showing MAE values around 15 % compared to the VGM model using the estimates of , with MAEs around 22 %. Indeed, for all the snow types excluding melt forms, both shape parameters αvg and nvg are overall better estimated using our regression (as optimized to best match our numerical simulations). This highlights the advantage of considering a large diversity of snow to develop the regressions of the shape parameter, so that the regressions can be applied more widely. However, we point out that the VGM model based on the shape parameter estimates of , which only required the knowledge of density and grain size, still allows for fair estimates of the unsaturated hydraulic conductivity, with MAEs ranging from 10 % to 20 %, when compared to the simulations on our five samples.

The unsaturated effective thermal conductivity of wet snow ku, which accounts for heat conduction in the ice, air, and liquid water, is presented for different saturation levels in Fig. . As expected, thermal conductivity increases with increasing saturation, as liquid water conducts better than air. Values of thermal conductivity of fully saturated snow are increased by about 0.5 to 0.6 Wm-1K-1 compared to the ones of dry snow, which means multiplying by 6 the thermal conductivity of a fresh snow sample or by 2 that of a melt form sample. The major impact of snow density is also shown, as already reported for dry snow e.g.,. Density also influences the steepness of the linear conductivity-saturation relationship, such as dense snow shows less steep slopes than light snow. Indeed, the pore space available for a conductivity gain due to an increase in water content is smaller for denser snow. To represent the evolution of thermal conductivity with both density and liquid water content, we propose the following regression based on our data: 10ku(ρ,θw)=kCalonnedry(ρ)+θw1.68×10-3ρ+kw-ka11with kCalonnedry(ρ)=ka+ρ2.5×10-6ρ-1.23×10-4 where kCalonnedry(ρ) is the parameterization of thermal conductivity for dry snow of and ρ is the dry snow density. The choice of the regression form was motivated by a concern for simplicity and to respect the two extreme cases: a volume fully made of liquid water (ρ = 0 kgm-3 and θw = 1) and a volume fully made of air (ρ = 0 kgm-3 and θw = 0). The case of a volume fully made of ice is not considered, as the regression of is only valid for densities corresponding to snow (ρ≤ 550 kgm-3). As illustrated in Fig. a and b, the proposed regression reproduces well the computed data. In Fig. , the self-consistent (SC) estimate of thermal conductivity for a 3-phase composite aggregate of spherical inclusions is included for comparison. This analytical model is derived by solving the polynomial expression 12∑α={i,a,w}θαkα-kukα+2ku=0 where θα and kα are the volume fraction and thermal conductivity of the phase α, respectively. The SC model prediction is consistent with the numerical values computed on the 3D images.

For several models of the literature, the thermal conductivity of snow as a function of the water content is needed, including snowpack models, such as the Crocus model , or detailed models of wet snow processes, such as the one of . They often rely on simple approximations to include the effect of water on the effective snow thermal conductivity. In the current version of Crocus, the thermal conductivity estimate from kYen1981, developed for dry snow, is applied to wet snow by simply accounting for the volume fraction of both ice and liquid water, as: 13kCrocusu(ρ,θw)=kYen1981dry(ρ+θw×ρw)=ki×ρ+θw×ρwρw1.88

The model of relies on a weighted average of the thermal conductivity of water and of dry snow, such as: 14kLeroux2017u(ρ,θw)=kCalonnedry(ρ)×(1-θw)+kw×θw.

To evaluate these two approaches, Fig.  presents a comparison with our data from computations and our proposed regression, for 3 snow samples. Major differences between the estimates are observed for each sample. The approximation used in Crocus largely overestimates the thermal conductivity of wet snow, up to a factor of 3 for the case of light snow at full saturation. The formulation of leads overall to large underestimations compared to our data. This comparison shows that the water distribution in the pore space plays an important role in the thermal conductivity of wet snow and that considering the bulk water content only is not sufficient. This motivates further studies to improve the modeling of wet snow conductivity and test the regression proposed here.

Extending the description of water vapor diffusion of and to the wet snow problem using a similar method as for the heat transport and the liquid water flow of , would involve the effective unsaturated water vapor diffusion Du. As done for the water permeability, the unsaturated diffusivity Du is normalized, here by the value of the dry effective vapor diffusivity D using the numerical estimate from .

Figure  shows Du/D as a function of the water saturation Sw=θw/ϕ, with ϕ the porosity of dry snow. Very similar evolutions of the relative unsaturated water vapor diffusivity with the term θw/ϕ are shown by all the samples, which highlights the strong correlation of the unsaturated diffusivity with both porosity and liquid water content. Diffusivity decreases exponentially as the proportion of liquid water increases, and for a given proportion of liquid water, diffusivity is higher for low-density snow, and inversely. Looking now at the maximum value of θw/ϕ reached for each sample, we observe that this value varies, as suggested by Fig. b. The domain of definition of Du as a function of θw/ϕ is determined by the value of water content at which the pore space is obstructed by liquid water, and so when Du can no longer be estimated. This specific value of water content is referred to here as the closed pore water content θwCP. It is similar to the close-off density at which the air can no longer diffuse in the pore space, as used in for dry snow and firn samples. Values of θwCP for all our snow samples are presented as a function of the dry porosity in Fig. . The closed pore water content θwCP is significantly smaller than the dry porosity, on average 17 % smaller. This means that vapor diffusion in wet snow is no longer possible long before saturation is reached. θwCP increases with increasing porosity (decreasing density), but does not seem specifically impacted by the type of snow considered. A fit, shown in black in Fig. , is estimated here as: 15θwCP=ϕ-0.17

Based on our data, a simple regression is proposed to estimate Du from the liquid water content and the dry snow porosity: 16Du(ϕ,θw)=DSC×1-θwϕ3=DSC×1-θw1-ρ/ρi3θw≤θwCP0θw>θwCP17DSC=Dv(3ϕ-1)/2 with DSC the self-consistent estimate of effective vapor diffusivity as used in . This relationship follows the general form of estimates of Du/Ddry=(1-θw/ϕ)a classically used for soils with a=10/3 in and a=5/2 in . The regression is shown in Fig.  by the black line, along with the soil models of and that are displayed with gray dotted and dashed lines. The latter two models fairly represent the behavior of Du and bound the regression for snow with higher and lower values. Both are used for predicting Du in structureless natural soils.

The analysis of the effective properties based on the images obtained from the drainage simulations showed interesting results. As already mentioned, it should be kept in mind that many of these images are likely far from natural snow microstructures, as the ice structures remain unchanged in the presence of liquid water in the simulations. Yet, they offer new insights on how the liquid water distribution in snow could influence the effective transport properties for a variety of microstructures. They also enable a comparison between the current parameterizations of the transport properties and suggestions for new estimates. Especially, we showed that the classic VGM parameterization for the relative water permeability and the unsaturated hydraulic conductivity reproduces well our simulated data and seems thus a good choice for both properties. For the effective unsaturated vapor diffusivity and the effective unsaturated thermal conductivity, we proposed new regressions, which, for the conductivity, differ strongly from some of the current parameterizations used in snow modeling. Finally, as all the above effective properties were computed on images from drainage simulations only, it could be interesting to compare those estimates to the imbibition case.