Y.-X. Tao and D. M. Gray Di~~ision of Hydrology, · Y.-X. TAO AND D. M. GRAY Local Averaging Vdume.V I (b) (c) Q Soil Particles a Uquld EJ Ice 0 Moist Air Figure 1. Infiltration into

PREDICTION OF SNOWMELT INFILTRATION INTO FROZEN SOILS

Y.-X. Tao and D. M. Gray Di~~ision of Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0 WO

A numerical model is presented, based on the local volume auemghg formulation of transport p h e n o m e ~ in porous media, for simulating meItwater infiltration into unsohc- rated, frozen soil. With the dejined flow and freezing boundnry conditions at the snow-soil int@ace, using the concept of a sugace local averaging volume, the time variation in profiles of temperature, liquid / ice content, infilhation /percolation rates, and mte of phase change in upper soil layers are predicted. In addition to a partimetric analysis, model estimutes of infiltratratron are compared with qunntities calculated from fieId measurements of soil moisture changes and temperature dwing snow cover ablation, showing a reasonable agreement.

INTRODUCTION

Water flow through snow and into underlying soil is an important physical process that is of interest to researchers working on a variety of engineering applications, such as hydrologic forecasting and the impact of global warming on excess snow runoff. Reliable estimates of the apportionment of the meltwater to runoff and soil moisture depend to a large extent on the accuracy of predicting infiltration through snow and soil.

Generated near the snow cover surface as results of thermal energy reception from solar radiation and ambient convection, meltwater infiltrates snow cover, causing a structural change of snow. A strong interaction between water flow and phase change (freezing and melting) occurs, especially in a domain near the water flow front, and results in an energy exchange between involved phases. Upon reaching the interface between snow and soil, water could further percolate into unsaturated, frozen soils. This percolation process is also governed by the interaction between heat and mass transfer, flow and phase change. Therefore, the amount of infiltration and the distribution of this water within the soil are strongly coupled with temperature distributions in both snow and soil layers.

Received 17 November 1993; accepted 10 February 1994. The authors acknowledge the assistance of D. Bayne and T. Brown, Division of Hydrology,

University of Saskatchewan, in the collection, tabulation, and processing of field data. The support provided by the Strategic Grants Program, Natural Sciences and Engineering Research Council of Canada, is greatly appreciated.

Address correspondence to Y.-X. Tao, Division of Hydrology, University of Saskatchewan, Room 2A20, Saskatoon, Saskatchewan, Canada, S7N OWO.

Numerical Heat Transfer, Part A, 26:643-665,1994 Copyright O 1994 Taylor & Francis

1040-7782/94 $10.00 + .OO

644 Y.-X. TAO AND D. M. GRAY

NOMENCLATURE

a effective thermal diffusivity, m2/s P density b pore-size distribution index e porosity c solute concentration, mol/kg 0, volume fraction of j phase C~ heat capacity at constant pressure ( j = i , l ) d snow grain size, m 0, irreducible liquid volume fraction D effective vapor diffusivity, m2/s T tortuosity Fo Fourier number (= a,, ,,t*/L2) # dimensionless rate of water g acceleration due to gravity, m/s2 accumulation at z = 0 hi^ enthalpy of fusion, J/kg @ time average of # h iv enthalpy of sublimation, J/kg # air entry potential, m ~ I V enthalpy of vaporization, J/kg INF* infiltration, m Subscripts k thermal conductivity K permeability, m2 e ff effective 1 length scale for the local D dispersion

averaging volume, m g gaseous phase ,

L depth of soil, m i ice m rate of phase change j jth phase (Table 1) m, n empirical constants in Eq. (19) 1 liquid phase P pressure m total moisture

4 dimensionless properties o at time = 0; reference point ( j = 1,2,. . . ), defined in Table 1 (Table I)

Pe Piclet number (= uL/ao, ,ff) ref reference Q heat flux, w /m2 s soil particle r position vector T thermal wave R gas constant, J/(kg K) v water vapor s total moisture saturation w wetting

[= (8, + el ) /@] 0 a t z = O S normalized liquid saturation 1 in the snow part of the surface

[= (8, - 8,)/(e - si - 8,)l local averaging volume t time 2 in the soil part of the surface T temperature local averaging volume u Darcy velocity, m/s Y arbitrary variable Superscripts z coordinate axis P empirical constant in Eq. (19), m-' * - dimensional AT = Tzf - T,*, K time average

Various studies based on field measurements on infiltration in frozen soils have been reported [1,2]. During the last two decades, many numerical studies have been reported to predict physical processes related to snowmelt infiltration. Some of the studies dealt with the redistribution of water in soil and temperature profiles due to freezing under controlled boundary conditions and without external water flow [3-51. Others concentrated on the snow domain only [6,7]. Some models aiming at the long-term forecast of seasonal hydrological processes included both snow cover and soil sublayers [8,9]. The analysis of the complex physical effects was limited. For example, a semi-empirical relation for water infiltration in soil was used in Ref. [a] as an alternative to the continuity equation, which is not physically rigorous. The model in Ref. [9] used an energy flux balance approach without

SNOWMELT INFILTRATION INTO FROZEN SOILS 645

solving the differential energy equation numerically. This simple approach is apparently less accurate. In both models, snow was treated as a lumped system and soil as coarsely meshed layers. In the models treating wet snow as a distributed system, two distinct approaches were adopted. One approach is to assume the snow is a homogeneous porous medium [6]; the other includes the phenomenon of water front fingering [7], which is an important part of the flow process in wet snow. The process after the wetting front reaches the soil surface, i.e., infiltration into frozen soil, is rarely analyzed, regardless of the large number of investigations of heat and mass flow in porous materials reported in the literature. This analysis is a prerequisite of modeling the overall snowmelt infiltration process. For example, on the Canadian prairies, meltwater infiltration during snow cover ablation is made up of a number of daily infiltration events. Each short-term event is very dynamic and usually follows a diurnal cycle. Typically, melting and infiltration occur during the daytime hours, and refreezing of the snow cover and upper layers of the soil mantle occur during the night.

This paper describes a rigorous, numerical model for simulating infiltration into unsaturated, frozen soils. It is considered a first step toward linking all the transport phenomena in both snow cover and soil regimes. The one-dimensional, unsteady model is based on a local volume averaging (macroscopic) formulation of transport phenomena in porous media, which was derived for all the phases from a microscopic (pore) level. Within this model is presented the complete formulation of conduction, convection, liquid flow (capillary and gravity flows), phase changes (freezingphawing, evaporation/condensation, and sublimation/ablimation), and vapor diffusion. The physical phenomena occurring at the snow-soil interface are examined by defining a surface local averaging volume (SLAV), part of whose borders coincides with the snow-soil interface. The boundary conditions imposed on the SLAV consider the discontinuity in the liquid Darcy velocity at the snow-soil interface due to phase change and mass accumulation of ice in the SLAV, i.e., the boundary conditions for flow and phase change are coupled. Examples are given of the temporal and spatial distributions of temperature, liquid/ice content, percolation velocity, and rate of phase change predicted by the model for developed boundary conditions and initial temperature and moisture conditions. Due to the coupling between heat transfer and flow by phase change, a number of constitutive equations are required to obtain the appropriate thermal and hydraulic properties. A parametric study is therefore performed to examine the sensitivity of main properties to the results. The performance of the model in predicting infiltration is evaluated through comparisons with infiltration estimates and temperature monitored at field sites.

WATER MOVEMENT IN FROZEN SOIL

Model Definition

Consider a case of meltwater infiltration into a frozen soil in combination with the penetration of a temperature wave. Assume the soil layer is homogeneous and unsaturated with a characteristic depth L (see Figure la). The physical

Y.-X. TAO AND D. M. GRAY

Local Averaging Vdume. V

I

( b ) (c)

Q Soil Particles a Uquld E J Ice 0 Moist Air

Figure 1. Infiltration into a frozen, homogeneous soil: ( a ) coordinates and boundary conditions, ( b ) local averging volume in the soil, and ( c ) concept of the SLAV with one boundary coinciding with the snow-soil interface.

mechanisms responsible for all transport phenomena occurring within the layer from a microscopic level (size of a pore) include (see Figure lb)

heat conduction through all the constituents (soil particle, ice, liquid, and gas mixture with water vapor and dry air); heat convection due to liquid flow that is driven by capillarity and gravity; phase changes for water: solid-liquid, vapor-liquid, and solid-vapor; and molecular diffusion in the vapor-air mixture.

The analysis excludes gaseous phase movement, heaving phenomena, and solute transport because they are not considered to be important factors influenc- ing the infiltration process. The equations describing macroscopic transport are derived from the fundamental conservation equations for the individual phases using the local volume averaging theory [lo, 111. This theory has been used successfully to model transport phenomena in porous media such as porous insulation and frost [12-141. At any location in space, z, a quantity y is said to be spatially averaged when it is defined as


where V is the value of the local averaging volume (see Figure lb). In this paper, V contains all three phases for water in a porous matrix where liquid movement prevails. A quantity in phase /3 is said to be intrinsic phase averaged when it is defined as

where T/B is the volume of the /3 phase in V. With respect to this complexity, the following assumptions are used:

1. The total gas phase pressure is constant and equal to the atmospheric pressure.

2. In the local averaging volume, all the involved phases are near thermal equilibrium states, defined as

( T ) g = (T ) ' = ( T ) ~ = (T)S = ( T )

3. The phase change between ice and vapor, or between liquid and vapor, is less significant than the phase change between liquid and ice.

4. Within the local averaging volume, there coexist liquid-ice, liquid-vapor, and ice-vapor interfaces (see Figure lb), and ice is immobile. The length scales of the liquid-vapor and ice-vapor interfaces are proportional to the liquid and ice volume fractions, respectively.

5. Liquid movement in frozen soil is governed by the relationship between capillary pressure and liquid saturation for the soil in the unfrozen state. In using the relation, ice content is treated as part of the solid matrix.

Assumption (1) is consistent with the observation that no significant air movement occurs within homogeneous soils. Assumption (2) is an important foundation for the local volume averaged total energy equation and is justified for cases where phase change and flow within porous media are moderate. This usually implies that the timescale required to reach the local thermal equilibrium is much shorter than the macroscopic timescale required for global transport phenomena. However, this assumption may break down if a significant phase change is coupled with a relatively high flow rate in the porous matrix. Assumption (2) is justified for the typical cases within the scope of this study [15]. Assumption (3) is justified for infiltration in frozen soil (see discussion below). Assumption (4) is reasonably proposed to construct the necessary constraints used in Eqs. (8) and (13) (see below) to close the formulation. Although little experimental evidence (if not none) is available, assumption (5 ) has been used in many studies [3,51 that deal with freezing and thawing in soils without boundary infiltration. The reader is referred to Ref. [lo] for other assumptions associated with the application of averaging theory to porous media.


Governing and Supplemental Equations

The following dimensionless equations, together with the dimensionless variables and parameters given in Table 1, describe the physical problem. The averaging symbols are omitted for the sake of convenience. The variables of T , Pe, m, etc., should otherwise be denoted as ( T ) , (Pe), (m) , etc.

Energy equation

dT dT pc - + p,c, Pe- + Plmlv + P,miv + P3mi, = * dt d z dz

Ice phase continuity equation

Liquid phase continuity equation

Liquid phase momentum equation

Gas phase dimsion equation

Thermodynamic relations


Volumetric constraint

Phase change constraint

Freezing point depression equation [8]

This relation is semi-empirical and is for thermal equilibrium conditions. Accord- ing to assumption 2 above, it is reasonable to apply this relation in the case where liquid is mobile. The air-entry potential i,!~ and the pore-size distribution index b can be derived from the soil texture analysis [16].

In the equations above there are twelve unknowns: T, Pe, m,, h i" , IjZlv, pv, pa, pv, pa, di, Og. Equations (3)-(14) are used to solve for these unknowns, provided that the following relations for thermodynamic, thermophysical, or hydraulic properties are specified.

EfSective thermal conductivity [lo]

where r k and k , are functions considering the tortuosity and thermal dispersion effects in porous media [lo].

Volumetric specific heat [lo]

Effective vapor mass diBsiuity [I31

where D , is a function accounting for mass dispersion effects.

Liquid permeability [17]

SNOWMELT INFILTRATION INTO FROZEN SOILS

Capillary pressure [I81

where

Note the definition of the normalized liquid saturation, S, in which the ice phase is treated as part of the porous matrix. Therefore, S is a function of both liquid and ice volume fractions.

Of the above expressions, Eqs. (14), (18), and (19) are empirical or semi- empirical, while the remaining equations are theoretical, although some of the parameters may be determined by experiment.

INFILTRATION AND PHASE CHANGE AT THE SNOW-SOIL INTERFACE

Surface Local Averaging Volume

The appropriate boundary conditions describing infiltration phenomena at the snow-soil interface are essential to solve for the temperature and moisture (liquid and ice) distributions in the frozen soil domain. Since the temperature of wet snow is approximately 0" C [6,7], it is reasonable to separate the temperature boundary condition from the flow and phase change boundary conditions. Consider an SLAV, V(r,), the center of which is denoted by a position vector, ro, as shown in Figure lc. The part of the boundary of this SLAV coincides with the interface between the snow cover and the soil. The dimensionless continuity equation for liquid water is this SLAV is

where the rates of evaporation and sublimation are neglected from the dimensional analysis. Because of the differences of porosity, permeability, and the relation between capillary pressure and liquid saturation in snow and soil, there exists the discontinuity of (Pe) at z = 0. Integrating Eq. (21) along the length scale of the SLAV, I , we have


where (Pel) characterizes the water release rate (per unit area) from snow and (Pe,) is the dimensionless infiltration rate of the soil (see Figure lc). They are related to the liquid saturation in the snow and soil, respectively, by

(Pe,) = K , - + P 6 p , [ : I.. in which K , and K , are liquid permeabilities (dimensionless) in snow and soil, respectively.- The wet snow permeability is evaluated by the following expression [19,20]:

" f - = s:{0.077d2 exp [-7.N p,/p,)l) [m2 I

Furthermore, the continuity equation for the ice phase in the SLAV is

doiz (mil,,> -+-=O a t r ,

d t p4

The complete solutions for governing equations, outlined in the previous section, require the specification of temperature, liquid volume fraction (or Darcy velocity), ice volume fraction, and vapor pressure at the snow-soil interface. Since the surface temperature is known (0" C), the vapor pressure can be found from the Clausius-Clapeyron equation, Eq. (8). For other variables, there are four equations, Eqs. (22), (23), (24), and (26), and seven unknowns, O,,, O,,, Oil, Biz, (Pel), (Pe,), and (mi,,,). The variables in the snow domain, denoted by the subscript 1, require the solution for the distributions of liquid saturation and ice content in snow. Since this paper focuses only on infiltration into soils, the complete formulation of flow and phase change in wet snow is not attempted. In the following, we specify either (Pel) or the total moisture content at the soil surface. Thus, the number of variables is reduced to four.

Initial and Boundary Conditions

In this study, two sets of boundary conditions describing snowmelt infiltration into frozen soils are considered: case one, in which the rate of water release from snow (characterized by (Pel )) is specified for performing a parametric study, and case two, in which the surface water saturation level is specified in order to compare with the field data.

Case one: specifying (Pe,). Consider that the initial subfreezing temperature, liquid volume fraction, calculated from the freezing depression equation, Eq. (14), and total water saturation of a frozen soil are uniform with depth. In


addition to Eqs. (22), (24), and (26), the following equation describing the dimensionless cumulative infiltration, INF, is used:

INF = (Pel)Fo = / (27) 0

The above equation implies that INF equals the net change of total water content, since no other sources of water flow exist. Equations (22), (241, (26), and (27) now give the solutions for el,, Oi2, (mil.0), and (Pe,) for specified (Pel).

The boundary conditions discussed above are applicable to the transient period, when there is a difference between (Pe,) and (Pe, ). A quasi-steady state surface liquid flow can be reached when (Pe, ) = (Pe,). If this condition is enforced, Eq. (22) is reduced to

Case two: for comparison with field data. When field data for the soil surface moisture saturation are available, s, may be expressed as a function of time and used as one of the boundary conditions. The diurnal variation of temperature at z = 0, To, is derived from the measured air temperature data, which are corrected based on the estimated thermal resistances in the air boundary layer and snowpack. The initial soil moisture and temperature are also obtained from measurements with a two-probe gamma ray system and temperature probes (thermistors). Infiltration is assumed to occur when T,* = O" C. Equation (27) now has the following form:

d INF -=

d

dt ( ~ e , ) ( t ) = -/ ' (el d t o + 0,") FO d~

Equations (22), (24), (26), and (29) are then used to calculate (Pe, ), (Pe, ), (mil, ), and 8,. We also have

Solution Method

The finite difference forms for governing equations and appropriate boundary conditions are derived using the implicit scheme. The central difference form is used for internal nodes, and the backward or forward difference is used for boundary nodes. The convection term in Eq. (3) is discretized using the first-order difference (upwind scheme) to stabilize the iteration procedure. The system of equations to be solved is highly nonlinear and coupled and has no steady state solutions for most variables (except for Pe,). It is very difficult to check the


accuracy of all the solutions by a "rule of thumb," that is, independence of grid size and time step. Following a common practice, we choose the time step and grid size, with the insurance of numerical stability, such that the most important variables obey the rule of thumb. A uniform grid size of 0.02 for 0 I z i 1 and an equal time step based on A ~ / ( A z ) ~ < 0.05 are used. Further reduction in grid size and time steps ensures a change of less than 5% in the heat flux and total mass flux (ice and liquid) at the snow-soil interface. The underrelaxation scheme is used in the iteration for each time step. The solution is considered to be converged when the root-mean-square deviation of all the variables from the last iterated values is less than

RESULTS AND DISCUSSION

Parametric Study (Case One)

Colbeck [201 plotted the capillary pressure in wet snow as a function of liquid water saturation for pT = 550 - 590 kg/m3. On the basis of these data, it can be inferred that the effect of the capillary potential on water movement can be neglected when the liquid water saturation of snow is greater than 0.1 (e.g., dp:/ds = 6.3 kPa). When the liquid saturation is below 0.1, however, capillary flow dominates the movement process. For example, at s = 0.06, dpT/ds =: 340 kPa, which is about 54 times that for s > 0.1. Using this relation as a guide, numerical calculations were performed for each case defined in the previous section. A listing of parameters used in the analyses is given in Table 2. The soil properties are those for the silty clay used in the infiltration studies.

Distributions of T, Q, 0,. 0,. s, Pe, mil. Figure 2 shows typical distributions of temperature, heat flux (= -k,,, dT/dz), liquid volume fraction, total moisture saturation, local Pkclet number, and rate of phase change at various Fourier numbers for the boundary conditions assuming the gravity-dominant flow in snow. The simulation proceeds until Fo = 0.02, equivalent to 12 hours if the

Table 2. Physical Values for the Soil Used in Case Studies

Unit Value


* v - - - = : - - 8

- . I

# - /- .

0 ).).

H . / 1 ,<I ,' ,/- I !

I /. I !

/' I Fo

I - I 0.0(3135

I I --. 0.0085

--- Id- I, , ( 8 ) 0 . m

Figure 2. Typical distributions of T, Q, O , , s, Pe, and mi, at different elapsed times for case one: (Pe,) = 0.3.

characteristic depth of soil is 1 m (see Table 2). The Piclet number ((Pe,)) of 0.3 corresponds to a snowmelt release rate of 0.5 mm/h. The data in Figure 2 show that at Fo = 0.02, the penetration depth for the temperature wave, z,,: is about 0.42 (Figure 2a), whereas the depth to which liquid is noticeably mobile, z,, is constrained to about 0.17 by the supercooling temperature (Figure 2c). The depth, above which there is a significant increase in moisture due to infiltration, z,, is even less, about 0.065 (Figure 2c). This is due to the higher local percolation rate (characterized by the local PCclet number) near z = 0 than at z > z, (Figure 2d). For z, < z < z,, the total water content remains relatively unchanged, and only the ratio of ice to liquid contents changes. This indicates that z, marks the lower limit in the depth of soil that is influenced by external liquid flow, while z, - z, defines the soil layer where liquid and ice redistribution is governed by the phase change due to propagation of the thermal wave. The variation in the rate of phase change between liquid and ice, mi,, with depth (Figure 2e) demonstrates that during the early transient period (e.g., Fo = 0.0005), strong melting (mi, being


Figure 3. Time variations of (a ) Q2, (mil ,0); ( f ~ ) INF, (Pe2); ( c ) s,, B, , , and I I , , for case one: (Pe,) = 0.3.

positive) occurs in the soil layers near the surface as a result of the combined effects of the high flux, Q (Figure 2b), and near-surface liquid flow. As the temperature wave penetrates deeper into the soil, freezing occurs in the region 0 < z < z,, while melting always occurs near the thermal wave front, z,. For Fo > 0.0005 the heat flux is maximum in soil layers near the surface (Figure 2b). Strong freezing effects occur in these layers (Figure 2e), causing their ice content to increase with time and their liquid content to first increase, then later to decrease.

The time variations of the surface heat flux, rate of freezing/melting, soil surface P6clet number, cumulative infiltration, soil liquid and ice volume fractions, and total moisture saturation, all defined at z = 0 (except for INF), are shown in Figure 3. During the initial transient period, the high heat flux causes thawing ((mi,.,) being positive), which coupled with water infiltration, results in an increase


in (Pe,) and 8,,. After a steady flow is reached, marked by Fo,, = 0.004 in Figure 3b, i.e., (Pe,) = (Pel), (h i , , , ) becomes negative, indicating a freezing condition. As a result, 8, increases and 8,, decreases (Figure 3c). It is worth mentioning that the increase in Oi, results in a reduction of pore space available for liquid movement, i.e., the effective porosity of soil is decreased. Although 8,, decreases as time elapses, the liquid saturation still increases as a result of a constant (Pe,) (see Eq. (20)). Figure 3c also indicates that the total soil moisture saturation, s,, increases with time. This is again due to the constant (Pe,) during the quasi-steady state period. The data in Figure 3b also allow the comparison of INF with a surface water accumulation term @ Fo, which is defined as

INF = /FY((~e , ) + 4 ) dt = [(F) + @(FO)]FO 0

where @(Fo) = l/Fo/f"rj dt is the average rate of water accumulation at z = 0 from t = 0 to Fo. During the transient period, the snowmelt infiltration is basically attributed to the increase in the water content in the SLAV. Significant infiltration only occurs after Fo,, is reached when @ Fo remains unchanged.

Effects of the snowmelt release rate (Pe,). When the capillary flow dominates liquid movement in snow during the transient period, the phase change and flow in the SLAV present a slightly different pattern. Figure 4 shows the variations of the variables with time for this condition in which the specified (Pe,) = 0.15 is one half of that in Figures 2 and 3. The decrease in (Pel) results in a longer transient period in this case than in the previous figures (Fo,, = 0.0097). Surface freezing occurs long before the steady state flow boundary condition is reached. It is also shown that as (Pel) decreases (from 0.3 in Figure 3 to 0.15 in Figure 4), the percentage of @ Fo in INF increases, indicating that a large percentage of water release from snow accumulates at the snow-soil interface for low (Pe,) cases. In Figure 5a the increase in (Pe,) does not increase the depths within which the liquid volume fraction, the ice volume fraction, and the saturation level undergo a significant change. These findings are in agreement with field observations reported in Ref. [2] that show that the depth of penetration of an infiltrating meltwater into a completely frozen soil that does not contain macropores is relatively insensitive to the infiltration rate. An increase in the snowmelt release rate, in addition to increasing the water saturation of the soil layers near the surface, affects the temperature and conduction heat flux in these layers. Figures 5b and 5c show that at z = 0.02, an increase in (Pe, ) results in a decrease in Q and an increase in temperature. This proves the significance of convection heat transfer and latent heat near the surface.

Sensitivity of the absolute permeability K. Figure 6 shows an increase in K causes a decrease in the liquid content at z = 0. This results because a higher K requires a smaller gradient in liquid content to maintain the same (Pe,) (see Eqs. (6) and (19)). As a result, s, decreases as K increases. For the same specified (Pe,), the increase in K causes further penetration of infiltration; as seen from Figure 6 , z , increases. The most significant change resulting from an increase in K is in (Pe,). In Figure 6c, the trends in (Pe,) indicate that the increase in K causes a shorter transient period.

Y,-X. TAO AND D. M. GRAY

Figure 4. Time variations of ( a ) Q,, (mi,,,); ( b ) INF, (Pe,); (c) s2, O i 2 , and O, , for case one: (Pe,) = 0.15.

Effects of the initial soil temperature, To. Different initial frozen soil temperatures mean different equilibrium liquid contents in the soil, according to the freezing depression equation, Eq. (14). The decrease in To (from - 3 to - 8" C, as in Figure 7) results in a faster penetration of temperature wave (Figure 7a), a negligible effect on quasi-steady state value for s (Figure 7b), and a delay for (Pe,) in reaching steady state (Figure 7c).

Effects of the initial soil moisture content so. Figure 8a shows that a change in the initial saturation level from so = 0.25 to so = 0.5 produces no appreciable effect on the temperature distribution for given (Pe,). Conversely, the increase in so significantly influences the liquid and ice distributions in soil near the surface and the infiltration depth (Figure 8b). A low initial moisture content provides a higher capillary potential for liquid movement and, consequently, higher increase in the ice content after freezing (see Eq. (19)). As shown in Figure 8c,


( a )

Figure 5. (a) Effects of (Pe,) on the distributions of the liquid content, ice content, and total water saturation at Fo = 0.013, and the time variations in (h ) surface conduction heat flux and (c) temperature at z = 0.02 for various values of (Pe, ).

(Pe,) is generally higher for low so except for so = 0.50, where (Pe,) is higher during a short, initial transient period. The trend in the total moisture change, s - so, shows that for given (Pe,), an increase in so causes water to percolate deeper. It should be noted that the initial liquid content distribution also depends on so. In general, the unfrozen portion of water content increases with the total soil moisture [21]. The results shown in Figure 8 therefore contain the combined effects from both liquid and ice phases.


Figure 6. ( a ) Profiles (for Fo = 0.013) and ( b ) time variations (at z = 0) of 0,, O,, and s, and (c) time variation of (Pe,) for different values of the absolute permeability K: (Pe, ) = 0.15.

Comparison with the Field Data (Case Two)

The performance of the model in estimating infiltration was tested against field data collected in a silty clay soil during the snowmelt period, March 16-21, 1979. The field estimates were calculated from measurements of soil moisture changes monitored with a twin probe soil densitometer at 2-cm increments to 80 cm and at 4-cm increments below 80 cm. Soil temperature profiles were measured with thermal probes equipped with thermistors located at 2.5, 5-, lo-, 20-, 40-, 80-, and 160-cm depths. The correlated diurnal variations of To and so are


Figure 7. Effects of initial soil temperature: (a) profiles of T - To, (b) O , , O,, and s at Fo = 0.0195, and (c) the time variation in (Pez ): (Pe, ) = 0.15.

shown in Figure 9a, About five diurnal cycles are simulated, and the typical results are shown in Figures 9 and 10. For the first two cycles (Figure 9b), the infiltration rate (characterized by (Pe,)) during the infiltration process (which is daytime) shows a trend similar to that in case one. In the next two cycles, (Pe,) has a step change pattern during infiltration due to the step change of s,. (During the fifth cycle, no infiltration occurs, since the surface temperature is always less than O" C.) Although the trend in (Pe,) is a result of the specified s, correlation, it indicates that case one gives a reasonable representation of the realistic infiltration in field conditions. In Figure 9c the predicted INF is compared reasonably with the measured data. Measured and simulated profiles of soil temperature and total moisture content are shown in Figure 10. The discrepancies between the field and simulated values are attributed mainly to uncertainties in the quality of the surface moisture and temperature data used as input to the model. While the simulation


Figure 8. Effects of initial total water saturation on (a) profiles of T, (6) O,, Oi, and s - so at Fo = 0.0195, and ( c ) the time variation in (Pe,): (Pe,) = 0.15.

catches the detailed information near the snow-soil interface, the field data cannot reflect these variations in a smaller scale of both space and time. In addition, since the onset of infiltration is set at the time when the surface temperature is O" C in the present stage of the model, the uncertainty in To has a significant effect on the cumulative infiltration. In reality, the start of infiltration is governed by the interaction of heat transfer, phase change, and liquid flow at the snow-soil interface, which is coupled with transport processes taking place within the snow cover. Further development of the model to simulate dynamic processes in both snow and soil is the next step of this study.

CONCLUSIONS

This study presents a rigorous numerical model, based on a local volume averaging formulation of transport phenomena in porous media, for simulating


1 -10

0 0.05 0.1 0.15 0.2 06:W. Mar. 16.1979 1 Fo

( a )

Figure 9. (a) Total water saturation and temperature at the soil surface obtained from the field data from March 16-21, 1979, Saskatoon, Canada, and time variations of ( b ) Q,, (Pe,) and (c) INF using the data shown in Figure 9a as boundary conditions.

infiltration into an unsaturated, frozen soil. An analysis, based on a surface local averaging volume (SLAV), for flow and freezing/melting boundary conditions at the snow-soil interface, is performed for various flow conditions in wet snow and infiltration in frozen soil. The numerical results obtained from simultaneous heat transfer, flow, and phase change in frozen soil under such boundary conditions lead to the following conclusions.

1. For a given water release rate from snow, the most influential variables affecting the distribution of soil moisture in a frozen soil are the soil permeability, the freezing depression characteristics, and the degree of saturation of the soil at the start of infiltration. The process is less


t8 ,.I' oms - o 0.080 A 0.m 0

Figure 10. Measured and predicted profiles of (a) temperature and ( b ) total water saturation for daytime hours.

sensitive to the initial temperature of soil, provided a soil does not contain large macropores and flow is driven only by capillarity and gravity.

2. Due to the structural nature of porous media, the concept of the SLAV is necessary to account for the microscopic volumetric behavior (such as phase change, mass accumulation) at the macroscopic interface of two distinct porous media, which are snow and soil in this study. It is therefore desirable to distinguish the snowmelt release rate from the soil infiltration rate at such an interface. This distinction is especially important during the initial stage of the infiltration process.

3. In environments subject to strong diurnal cycling of temperature that cause melting and refreezing of water in the snow and soil, the total seasonal meltwater infiltration is the sum of the contributions from daily events. Accurate prediction of the onset time and duration of each infiltration event, which is affected by heat transfer, is essential to the reliable prediction of cumulative infiltration. This is especially true for soil undergoing a diurnal freezing-melting cycle.

REFERENCES

1. D. M. Gray, P. G. Landine, and R. J. Granger, Simulating Infiltration into Frozen Prairie Soils in Streamflow Models, Can. J . Earth Sci., vol. 22, pp. 464-474, 1985.


2. R. J. Granger, D. M. Gray, and G. E. Dyck, Snowmelt Infiltration to Frozen Prairie Soils, Can. J. Earth Sci., vol. 21, pp. 669-677, 1984.

3. Y. W. Jame and D. I. Norum, Heat and Mass Transfer in Freezing Unsaturated Porous Media, Water Resour. Res., vol. 16, pp. 811-819, 1980.

4. G. S. Taylor and J. N. Luthin, A Model for Coupled Heat and Moisture Transfer during Soil Freezing, Can. Geotech. J., vol. 15, pp. 548-555, 1978.

5. R. L. Harlan, Analysis of Coupled Heat-Fluid Transport in Partially Frozen Soil, Water Resour. Res., vol. 9, pp. 1314-1323, 1973.

6. T. H. Illangasekare, R. J. Walter, M. F. Meier, and W. T. Pfeffer, Modeling of Meltwater Infiltration in Subfreezing Snow, Water Resour. Res., vol. 26, pp. 1001-1012, 1990.

7. P. Marsh and M. K. Woo, Wetting Front Advance and Freezing of Meltwater within a Snow Cover, 2: A Simulation Model, Water Resur. Res., vol. 20, pp. 1865-1874, 1984.

8. G. N. Flerchinger, Simultaneous Heat and Water Model of a Snow-Residue-Soil System, Ph.D. thesis, Washington State University, Pullman, Washington, 1989.

9. R. F. Grant, Dynamic Simulation of Phase Changes in Snowpacks and Soils, Soil Sci. Soc. Am. J., vol. 56, pp. 1051-1062, 1992.

10. S. Whitaker, Simultaneous Heat, Mass and Momentum Transfer in Porous Media: A Theory of Drying, in J. P. Hartnett and T. F. Irvine, Jr. (ed.), Advances in Heat Transfer, vol. 13, pp. 119-203, Academic, San Diego, Calif., 1977.

11. J. C. Slattery, Momentum, Energy, Mass Transfer in Continua, 2nd ed., R. F. Krieger, Melbourne, Florida, 1981.

12. K. Vafai and S. Whitaker Simultaneous Heat and Mass Transfer Accompanied by Phase Change in Porous Insulation, J. Heat Transfer, vol. 108, pp. 132-140, 1986.

13. Y.-X. Tao, R. W. Besant, and K. S. Rezkallah, Unsteady Heat and Mass Transfer with Phase Change in an Insulation Slab: Frosting Effects, Int. J. Heat Mass Transfer, vol. 34, pp. 1593-1603, 1991.

14. Y.-X. Tao, R. W. Besant, and K S. Rezkallah, A Mathematical Model for Predicting the Densification and Growth of Frost on a Flat Plate, Inf. J. Heat Mass Transfer, vol. 36, pp. 353-363, 1993.

15. Y.-X. Tao and D. M. Gray, Validation of Local Thermal Equilibrium in Porous Media with Simultaneous Flow and Freezing, Int. Commun. Heat Mass Transfer, vol. 20, pp. 323-332, 1993.

16. M. Fuchs, G. S. Campbell, and R. I. Papendick, An Analysis of Sensible and Latent Heat Flow in a Partially Frozen Unsaturated Soil, Soil Sci. Soc. Am. J., vol. 42, pp. 379-385, 1978.

17. Y. Mualem, A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media, Water Resour. Res., vol. 12, pp. 513-522, 1976.

18. M. Th. van Genuchten, A Closed-Form Equation for Predicting the Hydraulic Conduc- tivity of Unsaturated Soils, Soil Sci. Soc. Am. J., vol. 44, pp. 892-898, 1980.

19. H. Shimizu, Air Permeability of Deposited Snow, Inst. Low Temperature Sci., Ser. A , vol. 22, pp. 1-32, 1970.

20. S. C. Colbeck, Theory of Metamorphism of Wet Snow, U.S. Army Cold Regions Res. and Eng. Lab. Res. Rep. 313, Hanover, New Hampshire, 1973.

21. R. N. Yong, Soil Suction Effects on Partial Soil Freezing, Highway Res. Rec., vol. 68, pp. 31-42, 1965.

Documents

Y.-X. Tao and D. M. Gray Di~~ision of Hydrology, · Y.-X. TAO AND D. M. GRAY Local Averaging Vdume.V I (b) (c) Q Soil Particles a Uquld EJ Ice 0 Moist Air Figure 1. Infiltration into