Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing

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1 J. Jpn. Soc. Soil Phys. No. +*0, p.,+-,,**1 ** *Jiri S imu nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru SAKAI* and Jiri S IMU NEK** + Philip and de Vries +3/1,**0 (q l (t + (r nq a ( (h r + (t (z Klh (z Klh (T (h (T K lt (z Knh (z KnT (z q l q a m - m - r l r n kg m - h m T t s z m K lh ms + K lt m, s + + K nh ms + K nt m, s + +,**0 (T (r nq a C p (t Ls ( (t (z l (T (z (T (T (r nq n C l q l (z Cn qn (z Ls (z R (c (t D (, c (c, n (z (z C p, C l, C n Jm - K + l Wm + K + q l, q n ms + L s,./*++* 0,-03.,T Jkg + D m, s + n ms + c moll + R * - * Watanabe and Mizoguchi,,**, Harlan +31- Generalized form of ClausiusClapeyron equation, GCC Fukuda et al. +32* Newman and Wilson +331 Hansson et al.,**. GCC HYDRUS-+D S imu nek et al.,,**/ GCC * /+.2/*1 +/11 ** Department of Environmental Sciences University of California, Riverside Riverside, CA 3,/,+, USA :

2 22 +*0,**1 HYDRUS-+D,,+ (q l ri (q i (t r + (t (T (h K lhk lt (z Knh (t + r l (r nq a ( (h (z Klh (z (T (z KnT (z q i m - m - r i 3-+ kg m - C p (T (t Lfri (q i (r nq a (t Ls (t (T (T (r nq n C l q l (z Cn qn (z Ls (z ( (z l (T (z L f -.-.+* / Jkg + q lh q i q it,, h q l + a Dash et al., +33/ T T + b Williams, +30. ; Koopmans and Miller, +300 P al P il T P alp il Koopman and Miller +300 C p (T (t Lfri (q i (r nq a (t Ls (t C p (T (t dq i (T L fr i dt (t dr n (T (q a Lsqa dt (t Ls rn (t dq i dr n (T (q a CpLfri dt Lsqa dt (t Ls rn (t dq i dr n C ac pl fr i dt Lsqa dt (T C a (t ( (z l (T (T (z Cl ql (z (T (q n r n (q a C nq n (z Ls L s r n (z (t C a J m - K Fig. + a b Schematic illustration of liquid water geometry in soil pores : (a) drying with the airliquid water interface under room temperature ; (b) freezing with the ice-liquid water interface in a saturated soil.

3 : 23 Black and Tice +323,- GCC dp l dp i n l dt ni dt Lf T P l P i Pa n l *.**+ m - kg + n i m - kg + T K *, dp l Lf dt n lt GCC r l ghp l DTT mt K h T m,1-.+/ K h Lf g ln TmDT T m, * T*.+ h+,,.2 cm T+ h+,,/** cm * Schofield +3-2 pf..+logt,. q l T dq ldt dq idtdq ldt,. + van Genuchten +32* S eh qlhqr q sq r +ah n m q s, q r S e a m + n, m GCC T h * h* + + h+,,/** cm *.*+ q r +* h+* / cm * - + Table + - van Genuchten Parameter values of van Genuchten model for three di#erent soils., Fig., GCC T h. Unfrozen water pressure head in a watersaturated frozen soil h, based on the temperature T with the generalized form of Clausius-Clapeyron equation. q s q r a n m l *4/-/ *4*/ *4* *4, *4/ *4./ *4*01 *4*, n *4/ *4.+ *4*0/ *4*1/ n *4/

4 24 +*0,**1 Jame and Norum +32* K flh+* Q K lh - Fig. - GCC - Unfrozen water curves for three di#erent water-saturated frozen soils, based on the soil water characteristic curves described with Eq. and the GCCE K lh m s + van Genuchten, +32* K lhk ss el ++S +m e m, K s ms + l K lt m, s + K + Noborio et al., +330 K ltk lh hg + dg g * dt G. 2 + Nimmo and Miller, +320 g g1/.0*.+.,/t,.-2+*. T,, g *,/1+.23 gs, K nh, K nt Scanlon et al.,**- Kq l f 2,*-* Gosink et al., +322 Q. a b + - * G1. K flh K flt * K flh+* +* cm s + -, C n, C o, C air, C l, C i Jm - K + C p C pq nc nq oc oq ac airq lc lq ic i q n q o. c - C p C a T C n+.3,+* 0 kg s, m +, C o,./++* 0 kg s, m +, C air+,,/* kg s, m +, C l..+2 +* 0 kg s, m +, C i+.3/-+* 0 kg s, m + q l, q i - q n, q o, * C a C p - C a -- Campbell +32/ l Wm + K + lc +C,q lc +C.expC -q l C /

5 : 25. Fig.. a K flh, b K flt, c C p C a, d l. Temperature dependence of physical properties of frozen soils : (a) isothermal hydraulic conductivities K flh ;(b) thermal hydraulic conductivities K flt ;(c) soil heat capacity C p and apparent heat capacity C a ;(d) thermal conductivity l., Table, - Thermal parameter values for three di#erent soils. q n q o C + C, C - C. * * * *42. *42. *42. C i i+ / C /. l Kennedy and Sharratt, +332 Hansson et al.,**. lc +C,qFq ic +C.exp C -qfq i C / ,1 qq l, F ++-.*/ F,+.*0 q i - *4- *4// *4/1 *4/- *41, +4/2 *4*3- *4*3- *4*3- F+F +q F, i q F. d -, * * q r

6 26 +*0,**1 -,* cm. T init/ h init+** +,*** cm, c init+* m mol L + T L/ T R/ R+ L+ cm, D **.+ cm, day + *.*, cm, *.*. h init+** cm., / a 0 a * 0 / / c * *./, mm h + / *./ mm h + 2* *.-- cm + + cm + / a / b / c +* T init+/ T L/ T R+/ +*,. / Fig. / a b c * c +* +/ / +/ (a) Temperature and (b) thermal conductivity profiles in a freezing loam soil ; (c) Progress of the freezing front (* isotherm) for T init/, T L/, T R/ and the +* isotherm for T init+/, T L/, T R+/. +* - / b -.

7 : a 0 b 0 c 0 a +*,*** cm 0 b Fig. 0 0 a b c d e c l, f qc l. d Profiiles of (a) temperature, (b) pressure head, (c) water flux, (d) water contents, (e) solute concentration c l and (f) amount of solute in a unit soil qc l in a freezing loam soil. The dashed line indicates the freezing front. In figure (d), the di#erence between the total water content (solid line) and the liquid (unfrozen) water content (dotted line) gives the ice content.

8 28 +*0,**1 0 c 0 b 0 c. a 0 d Fukuda et al., +32* ; Newman and Wilson, +331 GCC - *.. 0 e 0 a c l 0 d mm 0 f 0 d q l 0 e c l q lc l c l 0 e q l 0 d c l 0 f Konrad and Mc- Cammon, +33*./ 0 h init+,*** cm 1 a / h init+** cm +,*** cm 1 b c K lh h init +** cm h init+** cm 2 +** cm +,*** cm /. q lh, q lt, q nh, q nt q lh

9 : 29 2 Fig. 2 / q lh, q nh, q lt, q nt. Isothermal liquid (q lh) and vapor (q nh) fluxes and thermal liquid (q lt) and vapor (q nt) fluxes in a freezing loam soil after / hours for different initial water pressure heads. 1 Fig. 1 / a b c a Profiles of (a) water and (b) pressure head and (c) isothermal hydraulic conductivity in a freezing loam soil after / hours for di#erent initial water pressure heads. In figure (a), the solid and dotted lines indicate total and unfrozen water contents, respectively. 2 h init+** cm q lh 3/ q lt / 3/ q nt 2 a K flh h init+,*** cm q lh 2- q lt ++ q nt 0 q lh q lh 2 b 1 c K flh New-

10 30 +*0,**1 man and Wilson q init*.-- h init/-. cm h init+** cm q init*.+, h init+** cm 3 - / q i *.,/ *., /. HYDRUS 3 Fig. 3 - / Water content profiles for three di#erent freezing soils after / hours. The solid and dotted lines indicate total and unfrozen water contents, respectively. Campbell, G.S. (+32/) : Soil physics with BASIC, Elsevier, New York. Dash, J.G., Fu, H. and Wettlaufer. J.S. (+33/) : The premelting of ice and its environmental consequences. Rep. Prog. Phys., /2 : ++/+01. Black, P.B. and Tice, A.R. (+323) : Comparison of soil

11 : 31 freezing and soil water curve data for Windsor sandy loam. Water Resour. Res.,/ :,,*/,,+*. Harlan, R.L. (+31-) : Analysis of coupled heat-fluid transport in partially frozen soil. Water Resour. Res., 3 : ,-. Hansson, K., S imu nek, J., Mizoguchi, M., Lundin, L.C. and van Genuchten, M.Th. (,**.) : Water flow and heat transport in frozen soil : Numerical solution and freezeothaw applications. Vadose Zone Journal, - : 03-1*.. Kay, B.D.Sheppard, M.I :.3 ; /.0+. Fukuda, M., Orhun, A. and Luthin, J.N. (+32*) : Experimental studies of coupled heat and moisture transfer in soils during freezing. Cold Reg. Sci. Technol., - :,,-,-,. Gosink, J.P., Kawasaki, K. Osterkamp, T.E. and Holty, J. (+322) : Heat and moisture transport during annual freezing and thawing. In Proceedings of /th International Conference on Permafrost, Trondheim, Norway. -//-0*. Jame, Y.W. and Norum, D.I. (+32*) : Heat and mass transfer in freezing unsaturated porous media. Water Resour. Res., +0 : Kennedy, I. and Sharratt, B. (+332) : Model comparisons to simulate soil frost depth. Soil Sci. +0- : 0-00./. Konrad, J.M.and McCammon, A.W. (+33*) : Solute partitioning in freezing soils. Can. Geotech. J., 01 : 1,01-0. Koopmans, R.W.R. and Miller, R.D. (+300) : Soil freezing and soil water characteristic curves. Soil Sci. Soc. Am. Proc., -* : 02*02/. Newman, G.P. and Wilson, G.W. (,**.) : Heatand mass transfer in unsaturated soils during freezing. Can. Geotech. J., -. : 0-1*. Nimmo, J.R. and Miller, E.E. (+320) : The temperature dependence of isothermal moisture vs. potential characteristics of soils. Soil Sci. Soc. Am. J., /* : ++*/+++-. Noborio, K., McInners, K. J. and Heilman, J.L. (+330) : Two-dimensional model for water, heat and solute transport in furrow-irrigated soil : I. Theory. Soil Sci. Soc. Am. J. 0* : +**++**3. Philip, J.R. and de Vries, D.A. (+3/1) : Moisture movement in porous materials under temperature gradients. Trans. AGU, -2 (,) :,,,,-,.,**0 : J.R. Philip and D.A de Vries +*- : +*/++,. Scanlon, B., Keese, K., Reedy, R.C. S imu nek, J. and Andraski, B. (,**-) : Variations in flow and transport in thick desert vadose zones in response to paleoclimatic forcing (*3* kyr) : Monitoring, modeling, and uncertainties., Water Resour. Res., No. 1, ++13, doi : +*.+*,3/,**,WR**+0*., Schofield, R.K. and Bothlho da Costa, J.V. (+3-2) : The measurement of pf in soil by freezing point. Agric. Science,,2 : 0./0/-. S imu nek, J., van Genuchten, M.Th. and Sejina, M. (,**/) : The HYDRUS-+D software package for simulating the one-dimensional movement of water, heat and multiple solutes in variably-saturated media. Version -.*.HYDRUS Software Series +, Dep. of Environmental Sciences, Univ. of California Riverside, Reverside, CA.,**0 : van Genuchten, M. Th (+32*) : A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Soc. Am. J.,.. : 23,232 Watanabe, K. and Mizoguchi, M. (,**,) : Amount of unfrozen water in frozen porous media saturated with solution. Cold Reg. Sci. Tech., -. : +*-++*. Williams, P. (+30.) : Unfrozen water content of frozen soils and soil moisture suction. Geotechnique, +. :,-+,.0.

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