DIVISION S-l-SOIL PHYSICS

Temperature and Water Profiles During Diurnal Soil Freezing and Thawing: FieldMeasurements and Simulation

J. L. Pikul, Jr.,* L. Boersma, and R. W. Rickman

ABSTRACTAgricultural soils of the inland Pacific Northwest undergo fre-

quent freezing and thawing during the winter and early spring.Repeated freezing and thawing of the soil surface may acceleratebreakdown of soil aggregates, resulting in decreased water infiltra-tion, and erosion resistance. The objective of this research was tosimulate soil temperature and water distributions in a field soil dur-ing freezing and thawing. A finite difference numerical model wasused to simulate soil temperature, depth of freezing, and water move-ment. Field measurements of soil temperature, ice and water content,and freezing depth in a bare surface treatment were used to validatethe simulation model. The model couples heat and water flow equa-tions by the change in soil ice content. An experimental relationshipbetween unfrozen water content and degrees below freezing was usedto estimate the change in soil ice content. Soil heat and water fluxduring 7 h of freezing followed by 12 h of thawing were simulatedfor two different days when the soil froze to a depth of 1.5 cm, Test1, and 0.75 cm, Test 2. Simulated maximum frost depths were 1.4and 0.8 cm for the respective tests. Simulated ice and water contentof the 0- to 1-cm soil layer, at the time of maximum frost penetrationwas 0.46 and 0.41 cm3/cm3, which compares to measured values of0.49 and 0.49 cm'/cm3 for the respective tests. Correlation of mea-sured soil temperature and simulated soil temperature at the 1-cmdepth was 0.99 for both tests. Standard deviation of the differencesbetween measured and simulated temperature at the 1-cm depth was1.04 and 0.29 °C for the respective tests. These results support thevalidity of this modeling approach for diurnal simulations of heatand water flux, near the surface, in freezing and thawing soil.

NIGHTTIME SOIL FREEZING and daytime thawingcycles are numerous in the Pacific Northwest

(Hershfield, 1974), especially in late fall and again inlate winter. These diurnal freezing and thawing cycleshave been termed radiation frosts (Oke, 1978). Duringthe nighttime freezing, water migrates upward to thefreezing front and is held there as ice. During the sub-sequent daytime thaw, near-saturated conditions arepresent in the soil at the surface and evaporative waterloss is high (Pikul and Allmaras, 1985). Repeated sat-uration and subsequent drying of the soil surface, asoccurs during diurnal freeze and thaw cycles, maybreak down soil aggregates resulting in decreases inwater infiltration (Moore, 1981).

Soil temperature gradients in freezing soil inducewater migration from the warm subsoil to the freezingfront. As the surface cools, a fraction of the water inthe soil pores freezes. Ice particles remain separatedJ.L. Pikul, Jr., and R.W. Rickman, USDA-ARS, Columbia PlateauConserv. Res. Ctr., P.O. Box 370, Pendleton, OR 97801; and L.Boersma, Dep. of Soil Science, Oregon State Univ., Corvallis, OR97331. Joint contribution of the USDA-ARS and Oregon State Univ.Oregon State Univ. Agric. Exp. Stn. Technical Paper no. 8388. Re-ceived 30 Nov. 1987. *Corresponding author.

Published in Soil Sci. Soc. Am. J. 53:3-10 (1989).

from the soil by a thin water film; the thickness of theunfrozen water film depends upon temperature, poresize distribution, solutes in the pore water, as well asfreezing and thawing history (Anderson and Tice,1972). Freezing of water effectually dries the soil inthe region of ice formation thereby decreasing the ma-tric potential. Water from the wetter subsoil flows up-ward toward the region of low matric potential.

Heat and water flow in freezing soil have been mod-eled using several approaches. A class of models,termed hydrodynamic, solves heat and water flowequations for heaving (Taylor and Luthin, 1978) andnonheaving (Jame and Norum, 1980) systems wherecoupled heat and water and phase change takes place.Taylor and Luthin (1978) and Jame and Norum (1980)simulated unsaturated water flow, in a laboratorystudy, using a water flow equation formulated on soilwater diffusivity. Under unsaturated conditions, waterflow is driven by both the gradient in unfrozen watercontent, which implicitly represents the matric poten-tial gradient, and an appropriate soil water sink, orsource, term that is based on the rate of change of icecontent within the freezing soil. An experimental orderived relationship of unfrozen water content to sub-zero temperature is used to couple the heat and waterflow equations.

A second class of model is the secondary frost heavemodel (Miller and Koslow, 1980). This model extendsthe hydrodynamic model to include stress conditionsand overburden pressures, and is considered the mostdetailed frost heave model (O'Neill, 1983) yet intract-ably complex (Gilpin, 1980). Smith (1985) concludesthat neither the hydrodynamic nor the secondary frostheave model can be applied in the field because ofinformation requirements and natural variability insoils. Recently, Gary (1987) presented a new methodfor calculating frost heave, including solute effects.

The objective of this research is to simulate soiltemperature and water distributions in a field soil dur-ing freezing and thawing of the soil. The model usedfor the simulation follows the hydrodynamic ap-proach proposed by Jame and Norum (1980) for cou-pled heat and mass transfer processes in a freezingnon-ice-lensing soil.

MATERIALS AND METHODSPlot Preparation

Field studies were conducted at the Pendleton ExperimentStation located 15 km northeast of Pendleton, OR. Averageannual precipitation at the station is 400 mm, occurringmostly as rain during October through June. Soil at the siteis a Walla Walla silt loam (coarse-silty, mixed, mesic TypicHaploxerolls). The terrain, at the Experiment Station, isnearly flat with no appreciable slope. Winter wheat (Triti-

SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989

cum aestivum L.) was harvested in August 1981, leaving astraw residue of 13 Mg/ha. A 40- by 40-m site was burnedto free it of all harvest residue and is called the bare-surfacetreatment.

Soil TemperatureSoil temperature at each depth was sensed by duplicate

temperature probes each of which electronically averagedthe signal from three thermocouples. Soil temperature wasrecorded on the 1/2 h. Depths of thermocouples were 1-cmincrements from 1 to 5 cm; 2-cm increments from 7 to 25cm; 30, 35, 40, 50, and 60 cm. Details on temperature probeconstruction are described by Pikul and Allmaras (1984).Soil temperature near the surface was measured with ther-mocouples placed at the 1-mm depth.

Bulk Density, Water Content, and Frost MeasurementsSoil water content in the surface to 11.0-cm soil layer was

measured at about 1-h intervals. Gravimetric samples werea composite of three subsamples taken at 1-cm depth incre-ments. A soil sampling device described by Pikul et al. (1979)was used to obtain the 1-cm increment samples.

Depth of frozen soil was visually determined at the timeof gravimetric water sampling. The term frozen soil indicatesa discernible mass of soil. The line of demarcation betweenunfrozen soil and frozen soil was determined by the resis-tance of the soil to cutting.

When the soil was unfrozen, bulk density (pb) was mea-sured in 2.0-cm increments to a depth of 40.0 cm with atube sampler. Three replications of eight composited 2.0-cmsegments, for a total of 24 segments, were taken to describepb at each 2.0-cm increment. This method has been de-scribed by Allmaras et al. (1988).

Soil Thermal ConductivityHeat is transferred in soils mainly by conduction through

solid particles, ice, water, and air, listed in descending orderof importance. Thermal conductivity of soil, X, is influencedby all of the soil components and the physical properties ofthe components (de Vries, 1963). The Walla Walla soil wasassumed to be composed of four components, resulting inthe following equation for apparent soil thermal conductiv-ity

cp = [0.460

X = kses(W/cm °C)

where 0W, 0S, 0a, and 0f are the volumetric fractions of water,solid, air, and ice, respectively, with each a thermal con-ductivity Xw, Xs, Xav, and \f, respectively. The weighting func-tions; ks, k3, kf, depend upon the thermal conductivity andgeometry of each component. Water is assumed to be thecontinuous medium for 0W > 0.10 (de Vries, 1963) and hasa weighting function, km equal to unity. Calculated soil con-ductivity includes an expression for conduction due to thetransport of latent heat in gas-filled pores; thus, the termapparent soil thermal conductivity is used. The apparentincrease in heat conduction is that due to heat conductionin dry air, Xa, plus that due to vapor movement, Xv, writtenas Xav. Pikul and Allmaras (1984) provide additional detailson the calculation of X for unfrozen soil. For frozen soil, kfis calculated using Eq. [6] of Pikul and Allmaras (1984) bysubstituting kf for Ksij and Xf for Xs. Thermal conductivity ofice (Xf), for -10 °C < T < 0.0 °C, is reported by van Wijkand de Vries (1963).

Soil Heat CapacityVolumetric heat capacity, cp, of the soil, water, and ice

system was calculated using the equation proposed by deVries (1963)

0.6000

(J/cm30.450f + 0W] 4.19

where 0m is the volumetric fraction of soil mineral matterand 00 is the volumetric fraction of soil organic matter. Theconstant of 4.19 converts cal/cm3 °C to J/cm3 °C. The av-erage fraction of organic matter in the plow layer was 0.02kg/kg. Soil organic C determinations were made at the samedepths as those described for pb sampling (Pikul and All-maras, 1986).Isothermal Water Diffusivity

Soil water diffusivity, D, is denned as the ratio of hy-draulic conductivity to the specific water capacity

Z>(0W) = (cm2/s) .Methods used to obtain the soil water characteristic curveand unsaturated hydraulic conductivity were described byPikul and Allmaras (1986). Soil water diffusivity as a func-tion of water content is shown in Fig. 1.Unfrozen Water Content of Frozen Soil

The freezing of water in fine-grained soils takes place overa range of subzero temperature. The greatest freezing pointdepression is thought to occur in water nearest the soil par-ticles (Williams, 1964a). Unfrozen water content, at a givensubzero temperature, is the mass ratio of unfrozen water todry soil. The relationship of unfrozen water content to sub-zero temperature will be called the soil freezing characteristic(SFC) and is used in the simulation to estimate the changein soil ice content per unit time.

Various computational and experimental techniques areused to determine the amount of unfrozen water at tem-peratures below zero (Williams, 1964a,b; Anderson and Tice,1972; Tice and Oliphant, 1984). Unfrozen water content asa function of temperature was experimentally determinedwith nuclear magnetic resonance (NMR) analysis of soilsamples obtained from the 0- to 5-cm and 25- to 60-cm soildepth and wetted to prefrozen water contents of 0.31 kg/kgand 0.41 kg/kg. Tice et al. (1978) have reported that un-frozen water content can vary with total sample water con-tent. However, for these samples there was little differencein the SFC determined for the 0- to 5-cm and 25- to 60-cmsoil depth, or in the SFC determined at 0.31 and 0.41 kg/kg water content. Initial water content at each node in thefinite difference scheme lies between 0.31 and 0.41 kg/kg.These experimental results simplified the need for an ex-perimentally determined SFC at each soil depth used in thesimulation.

V)

<M

Eo

i "6(7>

-11

-160.400.00 0.10 0.20 0.30

WATER CONTENT I.Fig. 1. Soil water diffusivity for the Walla Walla silt loam.

O.SO

PIKUL ET AL.: TEMPERATURE AND WATER PROFILES DURING DIURNAL SOIL FREEZING AND THAWING

Experimental cooling and warming SFCs based on a pre-frozen water content of 0.31 kg/kg are shown in Fig. 2. Expe-rimental determinations were made at temperatures rangingfrom about -0.3 to -10.0 °C. The dashed line in Fig. 2,shown as an example, is a linear extrapolation of the ex-perimental data to an initial water content of 0.30 kg/kg at0.0 °C. Each node in the finite difference scheme was as-signed a unique SFC that was extrapolated to the water con-tent at the node at the time freezing occurred at the node.Mathematical Model

The energy balance for a thin layer of soil where there isphase change between liquid and ice is written as

dT[1]

(i) (ii) (iii)where x is the position coordinate (cm), \ is apparent soilthermal conductivity (W/cm °C), T is temperature (°C), cpis volumetric specific heat (J/cm3 °C), t is time (s), L( islatent heat of fusion (J/g), pf is density of ice (g/cm3), and 0fis volumetric ice fraction (cm3/cm3).

Term (i) in Eq. [1] represents the rate change in energy ofthe soil layer due to a change in temperature; (ii) is a sinkor source that is based on the rate change of ice within thefreezing soil; and (iii) is the soil heat flux by conduction.Nixon (1975) and Taylor and Luthin (1976) have shown thatheat flow by conduction is two to three orders of magnitudegreater than convective transport. Fuchs et al. (1978) dem-onstrated that heat transported in the vapor phase in frozensoils is small compared to conduction transport.

The simplifying assumptions made in the use of Eq. [1]are: (a) heat transport by the movement of water and vaporare both negligible, (b) local fluid and solid temperatures areequal, and (c) heat flow is one dimensional.

The mass balance for a thin layer of freezing soil is writtenas

pf [2]

(i) (ii) (iii)where Z)(0W) is isothermal liquid diffusivity (cm2/s), 0W isvolumetric water content (cm3/cm3), and p, is density of water(g/cm3).

Term (i) in Eq. [2] is the rate change of the soil volumetricwater content; (ii) is the water sink or source term, whereby

0.30

CD

0.20

Oo.0.10

0.00

ooooo Experimental Cooling '***** Experimental Warming /

-10.00 -5.00TEMPERATURE (C)

0.00

Fig. 2. Experimental freezing characteristic curves for the WallaWalla silt loam. Experimental curves were determined using NMRtechniques on soil samples obtained from the field research site.

water is added or removed from storage based on the ratechange of ice content in the freezing soil; and (iii) is the fluxof water into and out of the soil layer due to the water con-tent gradient. In respect to term (iii) the water content gra-dient implicitly represents the matric potential gradient,which is the true driving force for water flow.

The simplifying assumptions made in the use of Eq. [2]are: (a) thermal liquid, thermal vapor, isothermal vapor, andgravitational fluxes of water are small compared to iso-thermal liquid flux; (b) Darcy's law applies to water in par-tially frozen unsaturated soil; (c) the functional relationshipof isothermal liquid diffusivity to liquid water applies topartially frozen unsaturated soil; and (d) water flux is in thevertical direction only.

At temperatures below freezing, the heat and water flowequations given by Eq. [1] and [2] are linked by the changein volumetric ice content with time, 88(/dt, which is esti-mated indirectly from temperature and the SFC. The nu-merical procedure that is used to estimate d8(/dt is describedin the next subsection. For these simulations soil heaving isnot considered and as such accumulated volumetric frac-tions of ice and water are restricted to be less than the porevolume. This model constraint simplified the numeric sim-ulation because the node system remained fixed.

Numerical SolutionSolution to Eq. [1] and [2] was obtained numerically using

a Crank-Nicholspn finite difference scheme. The followingboundary conditions were used for the heat flow solution

T! = FI (t) x = 0.0 cm; t > 0TX = F30 (t) x = 17.0 cm; t > 0

where r, and Tx are specified temperatures at Nodes 1 and30 that are at the soil surface and 17.0-cm depth, respec-tively. Temperatures T, and T30 are functions of time Ft(t)and -F?0(0, respectively.

Initial conditions wereT = F(x) 0 < x < 17.0 cm; t = 0

d8f/dt = 0.0 0 < x < 17.0 cm; t = 0where F(x) indicates a relation between T and soil depth.

The one-dimensional space domain was subdivided using0.1-cm increments from 0 to 1 cm; 0.2-cm increments from1.2 to 2.0 cm; 0.5-cm increments from 2.5 to 5.0 cm; 1.0-cm increments from 6.0 to 9.0 cm; and 2.0-cm incrementsfrom 11 to 17 cm. The Crank-Nicholson finite differencescheme for Eq. [1] is given, following Jame and Norum(1980), as:07VLY + (-a -0 -7) 77+1 + aT&V =

(a - aT,*+l -where

a =XF

(xr1

v [3]

(J/cm3 °C),

(J/cm3 °C),

= (cpf+t cpf)/2 (J/cm3

andA0f,, = Off1 — 6£i (dimensionless).

Subscript i and superscript n denote space and time, re-spectively. Finite difference equations for Eq. [3] were de-rived using the schematic shown in Fig. 3. The resultingsystem of 28 equations and 28 unknowns formed a tridi-agonal matrix that was solved using the Thomas algorithm(Wang and Anderson, 1982).

SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989

a29,/?29.Y29.A0f29,AX29

Ti x = 0.0cm Ti = Specified temperature

T2 x = 0.1cm

Ts x = 0.2cm

28 unknowns

T2e x = 13.0cm

T29 X=15.0Cm

TM x = 17.0cm Tso = Specified temperatureFig. 3. Schematic for deriving the finite difference equations for heat

flow. Specified temperatures are at T, and T30.

Boundary conditions for the water flow solution werespecified as zero flux at the top and the bottom of the col-umn during both freezing and thawing.

ao— - = 0 x = 0.0 cm; t > 0oX

— - = 0 x = 17.0 cm; t> 0.dx

Initial conditions were specified as0W = G(x) 0 < x < 17.0 cm; t = 0

— f = 0.0 0 < x < 17.0 cm; t = 0at

where G(x) describes the relation of 0W and soil depth.The Crank-Nicholson finite difference scheme for Eq. [2]

is given, following Jame and Norum (1980), as

= -c0a,/-i + (A + c-i)0a,,-P{

W.l+1 ^

where-1-1 _|_ ry»+l _L- TV -1._! 1- /-»,• -f l-'i-i-i T

[4]

A =

(dimensionless), and

(Df+1 - DfC =

(dimensionless).Finite difference equations for Eq. [4] were derived using

the schematic shown in Fig. 4. Reflective nodes (Taylor andLuthin, 1978) at the top and bottom of the column wereused to estimate the zero flux condition. These nodes havealso been referred to as fictitious or imaginary nodes (Geraldand Wheatley, 1984; Wang and Anderson, 1982). Sink orsource terms, A0f, are present at Nodes 2 through 29 in boththe heat and water flux solutions (Fig. 3 and 4). The resultingsystem of 30 equations and 30 unknowns formed a tridi-agonal matrix that was solved using the Thomas algorithm(Wang and Anderson, 1982).

The finite difference form of the heat flow equation givenby Eq. [3] was solved treating only T as an unknown at t +A?. Both X and cp are weakly temperature dependent andduring the small time steps it is assumed that X" — \"+' andcp" = cp"+1. The approach used to evaluate A0f will be dis-cussed later, because it cannot be assumed that Of = 0?+1.

A,,Ci,Axi

A2,C2,Ax2,A0f2

A29, C29, A/29,

8*3

fictitious node to simulate zero flux

x = 0.0cm

x = 0.1cm

x = 0.2cm

30 unknowns

x = 13.0cm

x = 15.0cm

x = 17.0cm

0*31 fictitious node to simulate zero fluxFig. 4. Schematic for deriving the finite difference equations for

water flow. Fictitious nodes at 0Wo and 0W3I are used to simulatezero flux at the boundaries.

The finite difference form of the water flow equation givenby Eq. [4] was solved treating only 0W as an unknown at timet + AJ. However, water diffusivity is dependent upon 0W(Fig. 1). Small time and space steps were used to linearizethe water flow solution.

Previous investigators have each employed different strat-egies for estimating A0f. Harlan (1973) suggests that an op-timization procedure be incorporated into the computa-tional scheme that iterates between the heat transfer equationand the mass transfer equation. Taylor and Luthin (1978)propose a coefficient whose magnitude is adjusted betweentime steps so that calculated water content agrees with thatgiven by the SFC. Taylor and Luthin (1978) report resultswith no iteration between the heat and water flow equation.Jame and Norum (1980) first estimate A0f by the heat trans-fer equation using apparent soil heat capacity, which in-cludes latent heat of fusion. Iteration was then carried outbetween the heat and water flow equation, but the criteriaused to stop the iteration was not discussed.

Change in ice content, A0f, in Eq. [3] and [4] was estimatedusing the following procedure. The flow diagram (Fig. 5)shows where the estimation procedure fits into the overallheat and water flow model. The only variable that changesduring this estimation procedure is A0f. That is, the tridi-agonal coefficient matrices set up using Eq. [3] and [4] re-main unchanged during the estimate of A0f. Similarly, thecomponents on the right-hand sides of Eq. [3] and [4] re-main unchanged with the exception of A0f. When the timestep is advanced, the coefficient matrices and the right-handsides of Eq. [3] and [4] are updated using new values for T,0W, 8{, \, cp, and D.

1. Sink or source terms, A0f, at Nodes 2 through 29 areset to zero.

2. Solve for T at time t + A? using Eq. [3] and A0f = 0.3. A0f is estimated at appropriate nodes whenever T <

0. The first determination of A0f is made by estimatingthe heat change during the time step and equating theheat loss/gain to a volumetric fraction of ice loss/gain.

4. Solve for Tat time t + A? using Eq. [3] and estimatedA0f.

5. Solve for 0W at time t + A? using Eq. [4] and estimatedA0f.

6. Temperature is determined as a function of unfrozenwater content (Fig. 2) using 0W and the relationship:Unfrozen water content = 0w/pb.

7. If the absolute difference of T and T (0W) > 0.1 °Cthe estimate of A0f at the node is increased or de-creased. A0f is positive when ice is forming providinga sink for 0W, and negative when ice is melting pro-viding a source for 0W.

PIKUL ET AL.: TEMPERATURE AND WATER PROFILES DURING DIURNAL SOIL FREEZING AND THAWING

Fig. 5. Flow diagram for the heat and water flux simulation.

8. Solve for T and 0W at time t + M using Eq. [3] and[4] and successive estimates of A0f.

9. Calculate the amount of ice, 6{, at the node. d( is in-creased or decreased based upon the sign of A0f.

10. If the sum of 0f and 0W exceeds the nodal pore spaceratio, PSR,, then the nodal value of A0f. is set to zeroand T and 0W are recalculated. Pore space ratio is de-nned as 1 — Pb/Pp, where pp is the density of soilsolids.

11. Advance the time by A/.

RESULTS AND DISCUSSIONStability

The simulation model was tested using various timestep, At, values in Eq. [3] and [4]. During a test, Atwas held constant even though the Crank-Nicholsonfinite difference scheme allows for variable time steps.Simulations were run, using A/ values that ranged from20 to 600 s, on a PRIME 2250' computer (PRIME,Natick, MA). Processing times for a 68 400-s simu-

1 Trademarks and company names are included for the benefit ofthe reader and do not imply endorsement or preferential treatmentof the product by the USDA.

23.0

20.0X

)'17.0

1*14.0

Ld

LJ5.0

2.0

-1.00.0 2.0 4.0 6.0 8J3 10.0 12.0 14.0 16.0 18.0 20.0

TIME IN HOURS FROM 2300Fig. 6. Predicted soil temperature, for the bare-surface treatment

during 21 and 22 Mar. 1982, for the case where soil water dif-fusivity was multiplied by 0.1. Measured temperature is shownfor the surface and 17.0-cm depth.

lation, using time steps of 60 and 600 s, were 3480and 600 s, respectively. There were little differencesin the overall simulation results obtained using thevarious time steps tested. However, during some por-tions of the freezing and thawing legs of the simula-tion, large A? values of 300 and 600 s resulted in anoverestimation/underestimation of A0f causing an os-cillation of predicted temperature and water content.For example, predicted soil temperature is shown inFig. 6 using At = 300 s. Between the hours of 6 and7 soil temperature at the 0.4-cm depth oscillated withan amplitude of about 0.1 °C, when compared to thetemperature predicted using At = 60 s. There was nooscillation of temperature and water content when At= 60 s was used in the simulation. All results reportedhere were obtained using At = 300 s.

Soil Water Diffusivity EffectsPrevious investigators have employed different

strategies to estimate mass transport parameters in soilfreezing studies. Jame and Norum (1980) selected asoil water diffusivity function that provided agree-ment between the numerical simulation and experi-mental observations for unfrozen conditions. For fro-zen conditions an impedance factor, that was assumedto be a function of total ice content, was introducedto reduce the water diffusivity value. Recently, Gary(1987) presented a new method that simulates coupledheat, water, and solute flow in unsaturated freezingsoil. Gary estimated the functional relationship be-tween liquid water content and hydraulic conductivityin an ice, water, and soil system using the methods ofCampbell (1974), where hydraulic conductivity is es-timated from moisture retention data. Gary found thatthe hydraulic conductivity function determined forunfrozen soil did a fairly good job of describing thefreezing system.

In this study the functional relationship between soilwater diffusivity and liquid water content was deter-mined on unfrozen soil. This functional relationshipwas assumed to be the same for both frozen and un-

8 SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989

frozen soil. Water content, especially in the frozenlayer, was overpredicted using experimentally deter-mined soil water diffusivity values (Fig. 1). Reason-ably good agreement between predicted and measuredwater content, for both freezing and thawing portionsof the simulations, was obtained after measured soilwater diffusivity (Fig. 1) was multiplied by 0.10.

For a closed system simulation, where AJ = 300 sand soil water diffusivity was decreased by a factor of0.1, final water content of the 0- to 17-cm profile was5.454 cm, which compares to a starting value of 5.528cm. This difference represents an error of about 1%,which suggests that the numerical method and thebookkeeping of the mass of ice and water are generallyaccurate.

Simulation Test 1Soil temperature and water content was simulated

for 21 and 22 Mar. 1982, when the soil froze to a depthof 1.5 cm. Measured soil temperature at 0.1 (surfacetemperature) and 17.0 cm provided upper and lowerboundary temperatures, respectively. Boundary tem-peratures and predicted temperatures are shown in Fig.6. Maximum predicted soil temperature at 1.0, 2.0,and 9.0 cm occurred at 13.5, 13.6, and 15.8 h, respec-tively. By comparison, maximum measured soil tem-perature, not shown, at these same depths occurred at14.0, 14.0, and 16.3 h, respectively. Correlation ofmeasured soil temperature and predicted soil temper-ature at the 1- and 2-cm depth was 0.99. Standarddeviation of the differences between predicted andmeasured temperature at the 1- and 2-cm depth was1.04 and 0.68 °C, respectively.

Predicted and measured ice and water content ofthe 0- to 1-cm soil layer are shown in Fig. 7 for 21and 22 Mar. 1982. Spatial standard deviation of themeasured water content of individual soil samples atthe 0- to 1-cm depth was 0.022 cm3/cm3. Results areshown for simulations where the measured soil waterdiffusivity, Fig. 1, was multiplied by 0.1 (DIFF*. 1) and0.01 (DIFF*.01). Predicted ice and water content, for

0.55

0.252.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

TIME IN HOURS FROM 2300Fig. 7. Comparison of simulated and measured ice and water content

of the 0- to 1-cm soil layer for the bare-surface treatment during21 and 22 Mar. 1982. Simulation results are shown where themeasured soil water diffusivity was multiplied by 0.01, DIFF*.01,and 0.1, DIFF*.l. Spatial standard deviation of the measuredwater content at this depth was 0.022 cmVcm3.

the DIFF*.l test, increases sharply at about 2 h co-inciding with the onset of subzero soil temperature.The relationship between subzero temperature andunfrozen water content was shown in Fig. 2. Predictedsubzero temperature is used to estimate the change inice content A0f. The strength of the sink/source termin the water flow equation is determined by A0f. Alarge decrease in subzero soil temperature during atime step results in a large sink term, and, conse-quently, a large increase in 0f at the node.

During soil freezing, sinks for 0W are set at appro-priate nodes, and 0W decreases because water is re-moved from the system and stored as ice. A decreasein 0W at a node creates a gradient whereby water willflow upward from the wetter subsoil. Water flow tothe sink location is determined by both the gradientin 0W and soil water diffusivity (Eq. [2] and Fig. 1). Inthe test case where soil water diffusivity was decreasedby a factor of 0.01, water flow to the surface was lim-ited by small water diffusivity values even though rel-atively large 0W gradients existed. For example, at 7 h

-2.000.0 2.0 4.0 6.0 8.0

TIME IN HOURS FROM 230010.0

Fig. 8. Comparison of simulated and measured frost depth for thebare-surface treatment during 21 and 22 Mar. 1982. Simulationresults are shown where the measured soil water diffusivity wasmultiplied by 0.01, DIFF*.01, and 0.10, DIFFMO.

14.0

8.0 10.0 12.0 14.0 16.0 18.0 20.0TIME IN HOURS FROM 2300

Fig. 9. Predicted soil temperature for the bare-surface treatmentduring 7 and 8 Mar. 1982, for the case where soil water diffusivitywas multiplied by 0.1. Measured temperature is shown for thesurface and 17.0-cm depth.

PIKUL ET AL.: TEMPERATURE AND WATER PROFILES DURING DIURNAL SOIL FREEZING AND THAWING

the average ice and water content of the 0- to 1.0-cmlayer was 0.384 cm3/cm3. The 0W at soil depths of 0.1,0.2, 0.3, 0.4, 0.6, 0.8, and 1.0 cm was 0.223, 0.226,0.232, 0.242, 0.261, 0.278, and 0.291 cm3/cm3, re-spectively. At these same depths, 0f was 0.357, 0.333,0.125, 0.087, 0.023, 0.008, and 0.006 cm3/cm3.

The neglect of soil water evaporative sinks may bethe cause of the poor agreement between measuredand predicted water content during nonfreezing con-ditions after 12 h (Fig. 7). No flow boundary condi-tions were specified at the surface and 17-cm depth.However, the bare surface plot lost 2.4 mm of waterfrom the 0- to 17-cm profile during 22 Mar. 1982. Thiswater loss was not attributed to drainage (Pikul andAllmaras, 1985). The inclusion of evaporative sinkswould serve to dry the soil and improve the agreementbetween predicted and measured water content in the0- to 1-cm layer.

Predicted depth of soil frost for the bare-surface14.0

O1.0o

C£ 8.0

C££ 5.0

LJ

2.0

-1.0

DIFFMO' MEASURED

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0TIME IN HOURS FROM 2300

Fig. 10. Comparison of predicted and measured soil temperature atthe 1.0-cm depth for the bare-surface treatment during 7 and 8Mar. 1982. Simulation results are for the case where soil waterdiffusivity was multiplied by 0.1.

0.55

2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0TIME IN HOURS FROM 2300

Fig. 11. Comparison of simulated and measured ice and water con-tent of the 0- to 1-cm soil layer for the bare-surface treatmentduring 7 and 8 Mar. 1982. Simulation results are for the casewhere soil water diffusivity was multiplied by 0.1. Spatial standarddeviation of the measured water content at this depth was 0.022cmVcm3.

treatment during 21 and 22 Mar. 1982 compared fa-vorably with the measured depth of soil frost (Fig. 8).Results are shown for simulations where the measuredsoil water diffusivity was multiplied by 0.1 and 0.01.Predicted depth of frost was estimated from the sim-ulated soil temperature. The soil was considered fro-zen when the temperature at a node was < 0.0 °C.Freezing point depression of the soil water solutionwas estimated to be —0.016 °C. Maximum predicteddepth of soil freezing was 1.2 and 1.4 cm for theDIFF*.01 and DIFF*.l simulations, respectively.Complete soil thawing occurred at 8.75 and 9.08 h forthe respective simulations. By comparison, measureddepth of frost was 1.5 cm and complete thawing oc-curred at about 9.5 h. The slightly longer time requiredfor complete soil thawing for the DIFF*.l simulationis a consequence of greater ice accumulation duringthe DIFF*.l simulation compared to the DIFF*.01simulation. On the thawing leg, ice is a sink for heatand soil temperature remains below zero until 0f be-comes zero.

Simulation Test 2Soil temperatures for 7 and 8 Mar. 1982, when the

soil froze to a depth of 0.75 cm, was simulated. Sim-ulation results are for DIFF*.l and Af = 300 s. Mea-sured soil temperature at 0.1 and 17.0 cm providedupper and lower boundary temperatures, respectively.Boundary temperatures and predicted temperaturesare shown in Fig. 9. Correlation of measured soil tem-perature and predicted soil temperature at the 1- and2-cm depth was 0.99. Standard deviation of the dif-ferences between predicted and measured temperatureat the 1- and 2-cm depth was 0.29 and 0.21 °C, re-spectively. A comparison of measured and predictedtemperature at the 1-cm depth is shown in Fig. 10.

Predicted and measured ice and water content ofthe 0- to 1-cm soil layer are shown in Fig. 11 for 7and 8 Mar. 1982. Ice and water content are overpre-dicted at the onset of soil freezing, and underpredictedat the time of maximum frost depth, as occurred inthe 21 and 22 Mar. 1982 simulation. Simulated iceand water content of the 0- to 1-cm soil layer, at the

-0.20

-0.50

; -o.so

-1.10Ulo

-1.40

-1.70

-2.00

• - - - DIFFMO***** MEASURED

0.0 2.0 4.0 6.0 8.0TIME IN HOURS FROM 2300

10.0

Fig. 12. Comparison of simulated and measured frost depth for thebare-surface treatment during 7 and 8 Mar. 1982. Simulation re-sults are for the case where soil water diffusivity was multipliedby 0.1.

10 SOIL SCI. SOC. AM. J., VOL. 53, JANUARY-FEBRUARY 1989

time of maximum frost penetration, was 0.41, whichcompares to a measured value of 0.49 cm3/cm3. Spa-tial standard deviation of the measured water contentof individual soil samples at the 0- to 1-cm depth was0.022 cm3/cm3.

Predicted and measured frost depth for 7 and 8 Mar.1982 are shown in Fig. 12. Maximum predicted frostdepth was 0.8 cm, which compares to a measured depthof 0.75 cm. Simulated complete thawing occurred at9.17 h, which compares to a measured time of about9 h. There were no field observations made betweenthe hours of 6.75 and 8, and consequently it is notcertain at what time complete thawing occurred or ifthe triangular shape of the measured frost depth rep-resents the field conditions during the thaw.

SUMMARY AND CONCLUSIONSSoil temperature and water content during freezing

and thawing of a field soil were measured and simu-lated. Finite difference methods were used to solve thepartial differential equations of heat and water flow.An experimental relationship between unfrozen watercontent and subzero temperature was used to estimatethe change in volumetric ice content with respect totime A0f. The change in volumetric ice content couplesthe heat and water flow equations by providing anestimate of heat and water sinks or sources. Measuredtemperature at the surface and 17-cm depth were usedto drive the simulation model. Boundary conditionsfor the water flow solution were specified as zero fluxat the surface and 17-cm depth. Soil water diffusivity,as a function of water content, was determined ex-perimentally on unfrozen soil. The functional rela-tionship between soil water diffusivity and liquid watercontent was assumed to be the same for frozen andunfrozen soil. Soil heat capacity and thermal conduc-tivity were calculated using mechanistic models thatinclude ice in the system.

Results from the temperature simulation comparefavorably with field measured soil temperature andfrost depth. The good agreement between measuredand predicted temperature during freezing and thaw-ing also suggests that sinks and sources of heat arebeing correctly estimated. Heat sinks or sources areestimated using A6{. Reasonably good agreement be-tween predicted and measured water content was ob-tained only after measured soil water difrusivity wasmultiplied by 0.10. Generally ice and water contentwere overpredicted at the onset of soil freezing, andunderpredicted at the time of maximum frost depth.Additional work is needed to estimate soil water dif-fusivity in both frozen and unfrozen soil. This studysupports the usefulness of this modeling approach fordiurnal simulations of soil heat and water flux nearthe surface during freezing and thawing.

ACKNOWLEDGMENTSThe authors wish to thank Alien Tice, located at the U.S.

Army Corps of Engineers Cold Regions Research and En-gineering Laboratory, Hanover, NH, for determining un-frozen water content of the test soil. Appreciation is alsoextended to Sue Waldman, mathematician, and John Zuzel,

hydrologist, for help in developing the computer programused to simulate soil water content and temperature.