Volume 56, Issue 9 e2020WR027466
Research Article
Free Access

Simulation of Preferential Flow in Snow With a 2-D Non-Equilibrium Richards Model and Evaluation Against Laboratory Data

Nicolas R. Leroux

Corresponding Author

Nicolas R. Leroux

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Correspondence to:

N. R. Leroux,

[email protected]

Contribution: Conceptualization, Software, Formal analysis, Writing - original draft

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Christopher B. Marsh

Christopher B. Marsh

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Contribution: Software, Writing - review & editing

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John W. Pomeroy

John W. Pomeroy

Centre for Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Contribution: Conceptualization, Formal analysis, Writing - review & editing

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First published: 10 August 2020
Citations: 4

Abstract

Recent studies of water flow through dry porous media have shown progress in simulating preferential flow propagation. However, current methods applied to snowpacks have neglected the dynamic nature of the capillary pressure, such as conditions for capillary pressure overshoot, resulting in a rather limited representation of the water flow patterns through snowpacks observed in laboratory and field experiments. Indeed, previous snowmelt models using a water entry pressure to simulate preferential flow paths do not work for natural snowpack conditions where snow densities are less than 380 kg m−3. Because preferential flow in snowpacks greatly alters the flow velocity and the timing of delivery of meltwater to the base of a snowpack early in the melt season, a better understanding of this process would aid hydrological predictions. This study presents a 2-D water flow through snow model that solves the non-equilibrium Richards equation. This model, coupled with random perturbations of snow properties, can represent realistic preferential flow patterns. Using 1-D laboratory data, two model parameters were linked to snow properties and model boundary conditions. Parameterizations of these model parameters were evaluated against 2-D snowpack observations from a laboratory experiment, and the resulting model sensitivity to varying inputs and boundary conditions was calculated. The model advances both the physical understanding of and ability to simulate water flow through snowpacks and can be used in the future to parameterize 1-D snowmelt models to incorporate flow variations due to preferential flow path formation.

Key Points

  • Preferential flow is simulated combining the 2-D non-equilibrium Richards equation with a random field of snow permeability
  • The model is parameterized for future application with a broader range of snow conditions
  • The model was evaluated against snow laboratory data

1 Introduction

Recent research on the physics of water flow through snowpacks has improved the understanding of snowmelt processes. Snowmelt runoff is the primary source of freshwater in cold regions (e.g., Gray, 1970; Mankin et al., 2015), and improved predictions of the magnitude and timing of snowmelt runoff are of interest to hydrologists and water managers. Melt is typically produced at the snow surface, and the timing and magnitude of meltwater delivery to the bottom of the snowpack where it can form runoff is greatly influenced by the formation of preferential flow paths (e.g., Colbeck, 1979). Better understanding water flow through snow is also important for wet snow avalanche prediction (e.g., Kattelmann, 1984; Wever, Vera Valero, et al., 2016). Many field experiments have demonstrated the occurrence of preferential flow in snow in different environments and under differing snowpack conditions such as the High Canadian Arctic (Marsh & Woo, 1984), the Sierra Nevada Mountains of California (McGurk & Marsh, 1995), and northern Japan (Yamaguchi et al., 2018). Laboratory experiments have helped to better understand fine-scale water flow processes in snow, such as the formation of capillary barriers (Avanzi et al., 2016; Waldner et al., 2004), the occurrence of capillary overshoot during infiltration into dry snow (Katsushima et al., 2013), and the importance of wet snow metamorphism on the transition from preferential flow to matrix flow (Hirashima et al., 2019). Laboratory snow experiments also provide valuable data for evaluating physically based water flow through snow models, such as the models of Hirashima et al. (2014, 2017) and Leroux and Pomeroy (2017, 2019). Katsushima et al. (2017) conducted laboratory experiments in thin snow samples and observed the evolution of the wetting front through time without the formation of preferential flow. This data set allows the evaluation of snow models in 1-D.

Previous snow laboratory studies have shown the similarity between water infiltration into dry snow and dry soil. In both cases, capillary and saturation overshoots (i.e., the build-up of liquid water at the wetting front) occur, and preferential flow is observed. Originally, capillary overshoot in snow was considered to be caused by water entry pressure in dry snow, that is, a threshold in the capillary pressure that the flowing water phase needs to reach before percolating a dry snow area (Hirashima et al., 2014; Katsushima et al., 2013; Leroux & Pomeroy, 2017). Leroux and Pomeroy (2017) showed through modeling that capillary overshoot in dry snow is more likely related to dynamic effects at pore scale. This finding is consistent with the results of water flow through soil research (DiCarlo, 2010). In addition, Leroux (2018) showed that the models of Hirashima et al. (2014) and Leroux and Pomeroy (2017) that incorporate a water entry pressure for dry snow in Richards equation are not suitable for simulating preferential flow in natural snowpacks or when snow densities are lower than 380 kg m−3. Leroux (2018) concluded that the parametric equation of Katsushima et al. (2013) relating the water entry pressure in snow with snow grain size was not applicable to snow properties found under natural conditions; this was also identified in Hirashima et al. (2019). The water entry pressure was also included in the model SNOWPACK to simulate preferential flow in 1-D using a dual domain approach (Wever, Würzer, et al., 2016; Würzer et al., 2017). By resolving the 1-D non-equilibrium Richards equation (NERE), which includes a dynamic capillary pressure instead of a water-entry pressure, Leroux and Pomeroy (2019) reproduced the capillary pressure overshoot observed by Katsushima et al. (2013) better than similar studies using models that only included a water entry pressure for dry snow. In addition, the 1-D NERE model included hysteresis in the water retention curve, coupled with water entrapment within the pore space. Two parameters controlling the magnitude of the dynamic capillary pressure and the main wetting water retention curve were introduced in the NERE model and were determined through manual calibration against laboratory experimental observations. These parameters varied with snow properties and the input flux, but no parametric equation was established. This lack of parametric descriptions of the behavior of these parameters makes it difficult to apply the NERE snow model to different snow properties without prior calibration of the parameters.

Hirashima et al. (2014) simulated preferential flow propagation in snow by generating random fields of snow properties (snow grain size and density), resulting in randomized snow permeability and the formation of an unstable wetting front. This random field in snow permeability was necessary to simulate preferential flow paths with the Richards equation accounting for capillary entry suction. Cueto-Felgueroso and Juanes (2009) also applied a random permeability field to simulate preferential flow in soil using a phase field model. The ability of the NERE to simulate preferential flow formation in a heterogeneous porous medium has yet to be shown.

In this study, a parametric function to relate the two parameters in the 1-D NERE model of Leroux and Pomeroy (2019) with snow properties and the input water flux is determined through evaluation of the simulated wetting front propagation with 1-D laboratory experiment data. The NERE model using the newly developed parametric equation is extended to 2-D and is compared to another set of 2-D laboratory data of water content profiles. Finally, the 2-D model is coupled with random fields in snow properties to simulate preferential flow in snow.

2 Method

2.1 Model Theory

The NERE (Equation 1) model with capillary hysteresis and water entrapment within the pore space introduced in Leroux and Pomeroy (2019) is applied in 2-D in this study:
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0001(1)
where θ is the volumetric water content (m3 m−3), kr(θ) is the relative permeability (−) estimated from the Mualem-van Genuchten model (Mualem, 1974; van Genuchten, 1980), Ks is the saturated hydraulic conductivity (m s−1) (Equation 2), Pdyn(θ) is the dynamic capillary pressure (m) (Equation 3), and z is the vertical direction (m). The saturated hydraulic conductivity is estimated from the dry snow density and grain size (Calonne et al., 2012):
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0002(2)
where g is the gravitational acceleration (m s−2), μw is the dynamic viscosity of water (Pa s), ropt is the optical grain radius (m) (assumed equal to the mean grain size, hereafter denoted r), and ρds is the dry snow density (kg m−3).
The NERE model implements the parametric equations commonly applied in snow models to estimate the van Genuchten parameters of the main drying water retention curve (αd and nd) (Equation 3) (Yamaguchi et al., 2012). A parameter is used to scale the van Genuchten parameter of the main wetting curve in snow from the of the boundary drying curve (Kool & Parker, 1987), that is, . The irreducible water content of the main drainage curve is fixed at 0.02 m3 m−3 and the saturated water content at 90% of the porosity (Yamaguchi et al., 2012).
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0003
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0004(3)
The dynamic capillary pressure is related to the static capillary pressure following Equation 4 (Leroux & Pomeroy, 2019). A relaxation parameter τ (m s−1), which symbolizes the time needed for water to reach equilibrium within the pore space, is introduced.
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0005(4)
where Pstatic is the static capillary pressures (m). The classical Richards equation is obtained for τ = 0 m s−1, that is, for Pdyn = Pstatic.

In Leroux and Pomeroy (2019), the two parameters γ and τ were manually calibrated against observations of dynamic capillary pressure in three snow samples under three different input rates, but they were not related to snow properties.

Leroux and Pomeroy (2019) included capillary pressure hysteresis in the NERE model, which was essential to properly represent observed capillary pressure overshoots; however, including capillary pressure hysteresis also resulted in model instabilities; thus, small numerical time steps were required. In this study, a simplified capillary pressure hysteresis model is implemented, in which only the main wetting, main drainage, and first order scanning curves are accounted for.

2.2 Numerical Setup

The spatial derivatives of the NERE equation were discretized using the cell-centered finite difference method, and the explicit Heun scheme, a modified Euler's method (e.g., Ascher & Petzold, 1998), was applied to solve for the ODE time component of the 2-D NERE. An adaptive time stepping scheme is implemented to improve the computational expense of the model based on the work of Kavetski et al. (2002). For all simulations, a time step of 0.01 s was used. The domain to solve the discretized equation contains Nx by Nz cells. The unsaturated hydraulic conductivity at the interface of two numerical cells was calculated using the geometric mean of the hydraulic conductivity of the two numerical cells. Using the geometric mean allowed model convergence for coarser grid resolutions than using the arithmetic mean when solving the NERE, as in Chapwanya and Stockie (2010). Using the geometric mean for the interblock hydraulic conductivity, which is the estimated hydraulic conductivity at the boundary of two adjacent numerical cells, is further discussed in section 4. Periodic boundary conditions are chosen for the lateral boundaries, and a free-flow boundary condition is selected at the bottom of the snowpack (the interaction of a soil layer below the snowpack is not accounted for).

To simulate preferential flow in the 2-D model, an unstable wetting front has to form (Hill & Parlange, 1972). Previous snow studies (Hirashima et al., 2014; Leroux & Pomeroy, 2017) showed that incorporating spatial heterogeneities in the snow properties resulted in unstable wetting. The present study follows the same procedure in representing a snowpack as Hirashima et al. (2014), which is the following:
  1. uniform mean snow properties (grain size and density) are chosen for each snow layer;
  2. a numerical mesh is created;
  3. the snow properties are varied cell by cell following a uniform distribution with the mean chosen in Step 1 and a standard deviation σ.

This results in a random field in snow grain size and density in used for the numerical simulations.

2.3 Model Setup for the Parameterization

The 1-D water flow through isothermal snow data presented in Katsushima et al. (2017) are used in this study to parameterize the model. The reader is referred to Katsushima et al. (2017) for additional information about the laboratory experiment. The measured depths of the wetting front through time were used to evaluate the model in 1-D and to parameterize τ and . Table 1 summarizes the property of the snow samples (density and grain size), which were used as inputs in the parameterization. Three different input fluxes for each snow sample were applied at the upper boundary: ~150, ~500, and ~850 mm h−1. Such large fluxes are unrealistic but were utilized in the experiment due to the narrow snow tubes. The 200 mm high snow samples were vertically discretized into 200 cells.

Table 1. Summary of Snow Sample Properties Used for Model Parameterization (Section 3.1)
Snow sample Snow density (kg m−3) Snow grain size ( ) (mm)
Fine snow (FS) 483 0.406
Medium snow (MS) 491 1.054
Coarse snow (CS) 455 1.463

To parameterize the two unknown parameters τ and γ, a star-based sampling strategy was used to generate an ensemble of {τ, } from the factor space of each parameter (Razavi & Gupta, 2016a, 2016b). From a first manual calibration, the factor space of τ was determined to be between 0 and 1 m s−1 and between 1.5 and 2.5 for . The VARS toolbox (Razavi & Gupta, 2016a, 2016b) was used to create an input file containing 1,170 combinations of {τ, } used in the parameterization. The model was then run for each combination of {τ, } within the ensemble created, and results for each combination were compared to the laboratory observations of Katsushima et al. (2017) presented in Table 1, that is, the fine, medium, and coarse snow (CS) samples, for three different input fluxes. For the finest snow sample (FS) with the highest input flux (~850 mm h−1), model instabilities were present, and thus, these model results were disregarded for the parameterization.

For each simulated wetting front propagation for each combination of {τ, }, the normalized root mean square error (NRMSE) between the simulated and observed wetting fronts depths through time were calculated. The results within the lowest 2% NRMSE were retained for the parameterization. The 2% threshold was used, as it provided enough points for the stepwise regression (cf. section 3.1), and a visual verification was also done to ensure that the simulations were close to the observations. In addition, the water content profiles from the results within the lowest 2% RMSE that were unstable were disregarded.

2.4 Model Setup for Evaluation of the Parametric Equation

The 3-D laboratory data of liquid water content distributions within two-layered snow samples of Avanzi et al. (2016) were used for the evaluation. The reader is referred to the original publication for additional details about the experiment. For the model evaluation (section 3.2), the experimental observations for the medium-over-coarse layered snow sample were used (cf. Avanzi et al., 2016). The snow sample was 200 mm high and 50 mm wide and discretized with a grid of 20 horizontal cells and 80 vertical cells (each cell is 2.5 mm × 2.5 mm). The upper layer in the snow sample had a mean grain size of 1.4 mm, the lower layer had a mean grain size of 2.9 mm, and the mean dry snow density of the layers was similar (equal to 472 and 487 kg m−3 for the upper and lower layers, respectively). Snow density and grain size were perturbed cell by cell following a normal distribution with a standard deviation of 10% of the mean values (Leroux & Pomeroy, 2017). A water input flux of 10 mm hr−1 was applied at the upper boundary. The simulated water content was compared with the observed water content at a vertical resolution of 2 cm.

2.5 Model Setup for 2-D Model of Preferential Flow

Two-dimensional models run were conducted for a grain size of 1 mm and a dry snow density of 300 kg m−3, with an input flux of 5 mm hr−1; this requires a heat input flux of ~463 W m−2 at the snow surface. Unless specified otherwise, is fixed to 2. The snowpack dimensions are 0.5 m high and 0.5 m long and were represented by 200 × 200 numerical cells. The grain size and dry snow density were disturbed cell by cell from the mean dry snow density and grain size following a normal distribution with a standard deviation of 5% by default. This resulted in a random field in snow permeability. Two-dimensional simulations are run for different snow densities and grain sizes, as well as for different standard deviations in the random fields in snow properties.

Water flow through snow was then simulated in a fine-over-coarse layered snowpack for an input flux of 5 mm hr−1. The snowpack was 1 m deep and 0.5 m wide. The layer interface is at a height of 0.5 m, and the grain size of the upper layer is 0.5 mm while the grain size of the lower layer is 1 mm; both layers have the same density equal to 300 kg m−3.

3 Results

3.1 Parameterization of τ and γ With Snow Properties

Figure 1 shows model simulations of wetting front depth through time for the lowest 2% NRMSE (black lines) against the observations of Katsushima et al. (2017) (dots) for the CS sample with the smallest influx. Using the classical Richards equation (τ = 0) with  = 2 to solve for the flow of water through the snow sample (blue line in Figure 1), the modeled infiltration was faster than the observations. For the sets of {τ, } with the lowest 2% NRMSE, the model represented the observations well. Figure 2 shows the simulated water content profiles at 5 min for the CS sample under the smallest input flux (150 mm hr−1). The water content profiles were similar, and only small differences can be observed at the saturation overshoot (wetting front).

Details are in the caption following the image
Simulated wetting front depths through time (black lines) for the sets {τ, } that provided 2% or lower RMSE in the CS sample for an input flux of 150 mm hr−1. The blue line shows the wetting front propagation from the classical Richards equation (τ = 0 and  = 2).
Details are in the caption following the image
Water content profiles after 5 min of infiltration for the CS sample under the smallest input flux (≈150 mm hr−1) for the simulations resulting in the 2% or lower RMSE (see Figure 1).

Figure 3 shows the sets {τ, urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0023} that provided the lowest 2% NRMSE for all the snow samples for the three different input fluxes. For a given , τ increased with grain size (different colors in Figure 3) and decreased with increasing input flux (different symbols in Figure 3). No results were obtained for the FS under the highest input flux, as the simulated water content profiles were oscillatory. From these results, it is clear that τ depends on , the input flux (flow velocity), and snowpack properties.

Details are in the caption following the image
Sets {τ, } that gave the lowest 2% RMSE for the three snow samples (represented by the colors, fine snow [FS], medium snow [MS], and coarse snow [CS]; cf. Table 1) and three different input fluxes (represented by the symbols).
To find a relationship among τ, , the input flux, and snow properties, a stepwise regression was used on the results of Figure 3. Because the snow samples have different grain sizes but similar snow densities, only snow grain size was used in the quadratic relationship. The Matlab (The MathWorks, 2012) function “stepwiselm” was applied to determine the quadratic relationship. The quadratic relationship takes the form
urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0006(5)
with a, b, c, d, e, and f are constant coefficients (Table 2), dg is the grain size ( urn:x-wiley:00431397:media:wrcr24809:wrcr24809-math-0028) (m), and qin the input flux at the top of the snowpack (m s−1). The RMSE for the regression was 0.071 m s−1, and R2 was 0.92, making it a suitable parameterization of the relationship for modeling purposes.
Table 2. Values of the Constant Coefficients of Equation 4
a (m s) b (m s−1) c (m s−2) d (s) e (s) f (s2 m−1)
0.487 0.387 −6,411 704 −536 2.18 × 106

3.2 Evaluation of the Model Parameterization in 2-D

The simulated 2-D water content was next compared to the 3-D laboratory data of Avanzi et al. (2016) (section 3.2) to evaluate the parametric equation (Equation 5). Figure 4 shows the comparison between simulated and measured water contents at 2 cm intervals (resolution of the experiment data). The simulation results of Leroux and Pomeroy (2017) are also shown in Figure 4 (dashed line) for comparison with the model introduced in the present study. The model was run for different values of , varying from 1.5 to 2.5 with a step of 0.25, and τ was estimated from Equation 4 using the mean layer grain size values. The model shows good performance in simulating ponding liquid water at the interface of the two layers and in the upper layer, particularly in comparison to the model of Leroux and Pomeroy (2017), which underestimated the water content in the snow sample. The model performance was similar to the model of Leroux and Pomeroy (2017) for the two other samples in Avanzi et al. (2016) (not shown), that is, for a fine-over-coarse and fine-over-medium snow samples. In both models, the water content of the ponding layer was simulated well (see Leroux & Pomeroy, 2017). The model performance in the lower layer was poorer; this is caused in part by liquid water accumulating at the bottom of the artificial snow samples in the experiments likely due to a capillary barrier between the artificial snow and the air below the snow sample. For all the simulations with the different , the time for liquid water to reach the bottom of the snow sample was similar (differing at most by 4 min).

Details are in the caption following the image
Comparison between simulated water content distributions (lines) from the 2-D NERE (averaged horizontally) and vertical measurements of water content (dots) from the laboratory experiment of Avanzi et al. (2016) for the medium-over-coarse snow sample, with an input flux of 10 mm hr−1. The dashed line shows the simulation results from Leroux and Pomeroy (2017).

3.3 Two-Dimensional Simulations of Preferential Flow Through Snow

Figures 5 and 6 show 2-D simulated water content distributions for different snow properties and model inputs and parameters. The simulated preferential flow differed in length, width, and number for the different cases. The top row in Figure 5 shows simulations of water content percolation after 1 hr of infiltration for different values of standard deviation (σ) in the random fields in both grain size and snow density for = 2.0, and the snow properties were set to the reference case. As expected, for a uniform snowpack (σ = 0%), no preferential flow paths formed in the simulation, and saturation overshoot was simulated at the wetting front. For increasing heterogeneities in snow density and grain size (increasing σ), the number of preferential flow paths increases but with decreasing width. The bottom row in Figure 5 presents the simulation of preferential flow paths in snow for three different values of (1.5, 2.0, and 2.5 for the graphs from left to right) after 1 hr of infiltration. Greater numbers per unit area but thinner preferential flow paths with higher water contents were simulated when increased. The timing of water reaching the bottom of the snowpack also varies with ; this time varied from 90 min for = 1.5, to 75 min for = 2.0, to 72 min for = 2.5.

Details are in the caption following the image
Two-dimensional water content distributions in uniform snow for three different values of σ (top row) and different values of (bottom row), for an input flux of 5 mm hr−1, a snow density of 300 kg m−3, and a grain size of 1 mm after 1 hr of infiltration.
Details are in the caption following the image
Two-dimensional water content distributions in uniform snow for three different input fluxes at different times (top row), three different snowpack densities after 1 hr of infiltration (middle row), and three different grain sizes (bottom row) after 1 hr of infiltration, for an input flux of 5 mm hr−1. For the reference case, the snow density is 300 kg m−3, and the grain size is 1 mm.

Figure 6 (top row) shows simulated water content distributions for different water input fluxes at different times. Increasing infiltration rate (or heat input at the surface) resulted in an increasing number of thinner preferential flow paths, and a faster delivery of the liquid water at the base of the snowpack. Figure 6 (middle row) shows different simulated liquid water content distributions for different mean dry snow densities (150, 300, and 450 kg m−3) with a mean grain size of 1 mm, and the bottom row for different mean grain sizes (0.5, 1, and 2 mm) with a mean dry snow density of 300 kg m−3 after 1 hr of infiltration. The number of preferential flow paths, their thicknesses, and their water content demonstrate strong sensitivity to snow permeability, and thus to grain size and snowpack density. For increasing snowpack density (or decreasing grain size), snow permeability decreased, resulting in a higher simulated water content and more, thinner preferential flow paths. For a lower snow permeability (lower snow density or higher grain size), the opposite was simulated. During the melt of a snowpack, both the density and grain size increase; thus, these compensating effects would result in preferential flow path patterns not changing considerably as the snowpack ripens.

Figure 7 shows the simulated water content distribution after 3 hr. Liquid water ponded at the interface of the two layers due to the existence of a capillary barrier. A similar result is simulated if the density of the two layers is different (lower density in the upper layer than in the lower layer) due to the formation of a hydraulic barrier (not shown).

Details are in the caption following the image
Two-dimensional water content distributions in a fine-over-coarse layered snowpack for an input flux of 5 mm hr−1 after 3 hr. The black line shows the layer interface.

4 Discussion

The 2-D NERE model was able to reproduce realistic preferential flow patterns observed in previous field and laboratory experiments (Marsh & Woo, 1984; McGurk & Marsh, 1995; Waldner et al., 2004). As stated in DiCarlo (2010) for soil, using the NERE to simulate preferential flow in porous media and capillary and saturation overshoots is more appropriate than using a water entry pressure as previously done in snow (e.g., Hirashima et al., 2014, 2017; Leroux & Pomeroy, 2017, 2019). In addition, Leroux (2018) showed that including the water entry pressure parameterization of Katsushima et al. (2013) in a numerical flow through snow model based on Richards equation (as done in Hirashima et al., 2014, 2017; Leroux & Pomeroy, 2019; Würzer et al., 2017) does not allow for the formation of preferential flow paths in snow when parameterized using natural values of snow properties (e.g., for snow densities ≤380 kg m−3). The limitations of using the water entry pressure of Katsushima et al. (2017) to simulate preferential flow paths in snow models were recently pointed out by Hirashima et al. (2019), in which the parameterization for the water entry pressure had to be adjusted. This shows a lack of transferability of the water entry pressure method to different snow conditions for the simulation of preferential flow paths. In this study, despite using laboratory snow samples with high densities (≥450 kg m−3) to parameterize the model, the parameterization permitted NERE to successfully simulate preferential flow in snow for densities lower than 400 kg m−3.

It is important to note that snow grain metamorphism through temperature gradients and due to the presence of liquid water plays an important role in controlling the persistency of preferential flow paths through time. Thus, our findings on flow through preferential flow paths are relevant only early in the melt season when the snowpack is characterized by different snow layers with distinctive properties.

Lower numbers per unit plan area of preferential flow paths were simulated with increasing grain size (Figure 6). This is consistent with findings from Avanzi et al. (2016), but this trend is opposite to the experiments of Katsushima et al. (2013). This might be caused by the small samples used in Katsushima et al. (2013) (5 cm diameter tubes), leading to large effects of the sample walls on the flow.

In previous soil studies, the 2-D NERE was able to reproduce realistic preferential flow patterns (e.g., Chapwanya & Stockie, 2010), but periodic input fluxes were applied to introduce an unstable wetting front and thus to develop propagating preferential flow paths. In this study, coupling the 2-D NERE model with random fields in snow properties (grain size and density) resulted in the formation of preferential flow paths. This finding is relevant to preferential flow simulation through soil as well.

Additional research is needed to measure the water retention curves for wetting snowpacks, as well as to determine the scanning curves when the flow processes switches from wetting to drying or drying to wetting. In this study, the well-used method of Kool and Parker (1987) was applied to determine the wetting curve from the drying curve. This method added one extra parameter in the model ( ) and introduced additional uncertainties. From this study and previous snow studies including hysteresis in the water retention curve, the parameter should be close to 2.0 in snow. However, it was assumed that the parameters nd and nw of the van Genuchten parameterizations of the drying and wetting curves were the same, which can be unrealistic (for instance, see van Genuchten parameters for soil in DiCarlo, 2004, and in Zhuang et al., 2017). A simplified version of the hysteresis model in the capillary pressure of Leroux and Pomeroy (2019) was used in this study. This simplified model resulted in more numerically stable simulations by neglecting the second-order (and higher-order) scanning curves; no notable differences were observed for 1-D and 2-D water flow infiltration in dry snow (not shown), because only continuous water infiltration was simulated. In the case of melt cycles, the 2-D NERE model with simplified hysteresis would likely differ only slightly from the 2-D NERE model with full hysteresis because of the difference pressure and water relationships on the second- and higher-order scanning curves in the full hysteresis model.

The model parameter τ that controls the strength of the dynamic capillary pressure and symbolizes the time needed for water to redistribute within the pore space was parameterized in this study. In this study, it was assumed that τ does not depend on the water content, contrary to laboratory observations by Das and Mirzaei (2012). This assumption can be justified by the small influence of a τ-saturation relationship on simulated capillary overshoots in snow (Leroux & Pomeroy, 2019). It was found that τ increases with both decreasing grain size and decreasing input flux. This is consistent with other findings in snow (Leroux & Pomeroy, 2019) and experimental soil studies (Camps-Roach et al., 2010; Das & Mirzaei, 2012). The dynamic coefficient τ was also related to the parameter used to estimate the wetting boundary curve from the drying boundary curve. It is intuitive that these two parameters are related as they are both used to estimate the strength of the dynamic capillary pressure ( through the static wetting boundary curve and τ through the strength of the dynamic effects). τ was also linked to the input flux at the upper boundary. Due to limited data, τ was not related to the internal water flux within the snowpack; however, due to the parametrization of the model in 1-D, the internal flux is related to the input flux at the surface. Including the input flux in the parametrization of τ allows the application of this research model to different input conditions. This phenomenon was not previously directly observed in soil but was pointed out by Leroux and Pomeroy (2019) through numerical calibration with capillary overshoot measurements. In soil, it was observed that capillary pressure overshoot, which is a direct result of dynamic effects in capillary pressure, depends on the input flux (DiCarlo, 2007); thus, it is natural to assume that τ also depends on the input flux.

The random fields in snow properties (grain size and density) were drawn from uniform distributions with set means and standard deviations from 0% to 20%. The distribution of the random fields plays an important role on the spatial distribution of preferential flow paths. However, little is known about the random fields in snowpack properties. Using micro-CT imaging technology or acoustic imaging could help determine the spatial correlation of snow properties and, thus, the prediction of preferential flow in dry snow in numerical models (Kinar & Pomeroy, 2015).

As stated in section 2, the geometric mean was used in the model to estimate the interblock hydraulic conductivity as done in Chapwanya and Stockie (2010). When used in the discretized cell-centered Richards equation (τ = 0 m s−1) to simulate mass flow through dry snow, a very fine grid resolution is required. For instance, Figure 8 shows 1-D simulated water content profiles within a 0.5 m deep snowpack at 30 min for a input water flux of 10 mm hr−1, snow density of 300 kg m−3, and a grain size of 1 mm for different vertical grid resolutions; for the coarser vertical grid discretization of 500 cells, the numerical solution with geometric mean presents a saturation overshoot at the wetting front, which is not present when the discretization is finer with 2,000 vertical cells (the solution has converged). This saturation overshoot for a grid resolution of 500 cells with the geometric mean is a numerical artifact; care should be taken when using the geometric mean to estimate the interblock hydraulic conductivity when simulating mass flow through dry porous media (e.g., as done in Wever et al., 2014; Wever, Würzer, et al., 2016; Würzer et al., 2017, in SNOWPACK to simulate preferential flow in snow using a dual-domain approach). Using the geometric mean for estimating the interblock hydraulic conductivity of the discretized NERE (τ > 0 m s−1), model convergence was achieved for a smaller grid spacing than using the arithmetic average (not shown); this is the contrary to the discretized Richards equation.

Details are in the caption following the image
One-dimensional simulated vertical water content distributions for different vertical grid resolutions (Ny), using either the geometric mean or arithmetic mean for estimating the interblock hydraulic conductivity in the discretized Richards equation (τ = 0 m s−1). An artificial saturation overshoot is simulated using the geometric mean with a grid resolution of 500 cells.

5 Conclusions

The 2-D NERE was solved to simulate preferential flow in snow. A random perturbation of permeability was used in the 2-D snow model and resulted in an unstable wetting front: a necessary condition for the formation of preferential flow. Using the NERE to simulate preferential flow in snow is preferred to other modified implementations of Richards equation that only use a capillary entry pressure in dry snow, such as in Hirashima et al. (2014, 2017) and Leroux and Pomeroy (2017); the NERE results in faster simulations and better representations of water content profiles but also provides a more physically accurate model.

In this study, model parameters were related to snow properties and rain-on-snow or meltwater fluxes at the snow surface. This makes the model more independent and applicable to a broader range of snow conditions. Additional snow laboratory work is needed to better constrain the model parameters through controlled experiments. In addition, more information on the length scales and spatial distributions of snow properties should be determined in the field as they influence the spacing and scale of preferential flow paths and so are important parameters for snowmelt models.

For hydrological purposes, this model could be included in 1-D snowmelt hydrology models. For example, this model could be used to parameterize a dual domain approach model that simulates both matrix and preferential flows through snow, such as the SNOWPACK model (Würzer et al., 2017).

Acknowledgments

Funding was received from the Global Water Futures Programme of the Canada First Research Excellence Fund, the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery Grants and NSERC Changing Cold Regions Network, and the Canada Research Chairs and the Canada Excellence Research Chairs programmes. The authors would like to thank Dr. M. Clark for useful discussions about the model and Dr. S. Katsushima and Dr. F. Avanzi for providing the data used in this study. The comments and suggestions of J. Staines are greatly appreciated.

    Data Availability Statement

    The data presented were originally presented in Katsushima et al. (2017) and Avanzi et al. (2016).