A model of freezing soils is developed that accounts for the dependence of the frost heave rate on particle trapping. At suﬃciently low cooling rates the soil experiences primary frost heave with a single growing ice lens that rejects all soil particles. At higher cooling rates ice lenses start to engulf the largest soil particles and the rate of segregation heave is reduced. At the highest freezing rates all particles are engulfed by the ice and the pore water freezes in situ. A new kinetic expression for the segregation potential of the soil is obtained that accounts for particle trapping. Using this expression a simple transient frost heave model is developed and compared with experimental data. 1 Introduction Frost heave is the uplift of the soil surface as it freezes, and generally involves two components: an in situheave caused by the 9% density diﬀerence between water and ice, and a segregation heave caused by the formation of discrete ice lenses in the soil [1,2]. While many aspects of frost heave have been explained in recent years, one intriguing phenomenon that has received less attention is the dependence of the rate of heave on the rate at which the soil is frozen. Beskow [3] showed that the heave rate is under certain conditions independent of the freezing rate; upon changing the surface temperature of a soil sample from 2C to10C, it was observed that the rate of heaving remained constant [3, p.79]. He noted, however, that the independence held only as long as the freezing front was descending into the soil. Once the freezing front became stationary (relative to the laboratory frame), a single ﬁnal ice lens formed and the rate of heave was proportional to the rate of freezing [3]. Horiguchi [4], Penner [5] and Konrad [6] observed similar results, while also showing that at very high rates of freezing the frost heave rate begins to decrease; at the highest freezing rates no segregation heave occurs at all, and the only heave is due to the volume expansion of the pore water as it freezes. Related to these observations are the experiments of Corte [7 –9]. He found that when freezing a soil with a particle size distribution, at very low freezing rates a single ice lens pushes ahead all particles (segregation freezing). At higher rates the ice lens begins to engulf the largest particles, while pushing the smaller particles ahead, leading to a sorting process at the ice lens surface. As the freezing rate increases further the ice lens engulfs ever smaller particles, until all particles are engulfed beyond a critical freezing rate ( in situfreezing). Similar observations have been made by Jackson and Uhlmann [10], Branson [11] and Matsuoka et al[12]. Konrad and Morgenstern [13] found that at very low freezing rates in a transient frost heave experiment, near the time when the ﬁnal ice lens forms, the rate of segregation heave is proportional to the temperature gradient at the ice lens. The constant of proportionality, which they called the segregation potential (SP), is a soil parameter independent of the freezing path. They deﬁned the segregation potential as SP=vw=GT; (1) wherevwis the measured water intake rate at the initiation of the ﬁnal ice lens and GTis the temperature gradient in the frozen fringe region. Here, as in Konrad and Morgenstern [13], the frozen fringe is deﬁned as the region of soil between the warmest ice lens and the 0C isotherm. This deﬁnition is convenient as it applies whether or not the frozen fringe contains pore ice [14,15]. For God so loved the world, that He gave His only begotten Son, that whosoever believeth in Him should not perish, but have everlasting life. For God sent not His Son into the world to condemn the world; but that the world through Him might be saved. John 3:16-17 1 2 zTc; Pob Tw; pwlh(t) 0 X(t) SoilIce lens SoilEngulfed soil particle & Ice lens Figure 1: Schematic of a column of soil freezing from the top down, with an ice lens at the surface. Because of particle trapping the ice lens is not pure, but contains a volume fraction iof engulfed soil particles. The horizontal dashed line shows the original unheaved surface of the soil. The total amount of frost heave h(t)increases because of the expansion of water as it freezes and because of excess water drawn to the ice lens from the water table. As the ice lens engulfs particles, the freezing front X(t)moves downward into the soil. Konrad and Morgenstern tabulated SPas a function of soil properties, overburden pressure and salinity for a variety of soils [16,17], and several studies by them and others have demonstrated its usefulness in the prediction of frost heave rates in ﬁeld situations [18 –24]. However, a shortcoming of frost heave models based on the SPconcept is a tendency to overpredict the rate of frost heave at early times in step freezing tests, when the freezing rate is high [16,19,20,25 –27]. As noted by Konrad and Morgenstern [6,28], van Gassen and Sego et al.[25] and Svec [26], at very high freezing rates SPis no longer a unique function of soil properties but depends also on the rate of freezing. This is in part because at very high freezing rates the soil water freezes in situand the water intake rate vwapproaches zero, or can even be negative (water expulsion) [17,29]. Recently a theoretical expression for the segregation potential in saline soils has been obtained by considering the linear stability of an ice lens during primary frost heave [30]. Here a similar approach is taken that also accounts for the eﬀects of particle trapping. This leads to a new kinetic expression for the segregation potential that depends on the rate of freezing. In Section 3 the basic equations describing conservation of mass and energy at an ice lens are described, and conditions for the onset of hydraulic supercooling obtained in the presence of particle trapping. In Section 4 a constitutive equation for the particle distribution coeﬃcient at the ice lens surface is obtained and compared with experimental data. In Section 5 an expression for the kinetic segregation potential that accounts for particle trapping is obtained, and employed in Section 6 to predict the heave rate in a ﬁeld frost heave experiment and above a buried natural gas pipeline. 2 Eﬀect of particle trapping on frost heave The system to be studied is illustrated in ﬁgure 1. There is a single stable ice lens growing within a water-saturated soil under an overburden pressure Pob, with surface temperature Tc<0C. At the base of the soil the temperature is held at Tw>0C and the pore pressure is maintained at 3 atmospheric pwvia a ﬂuid reservoir or water table. The total frost heave rate is _h=_hs+_hi; (2) where _hsis the segregation heave rate and _hiis the in situheave rate. Because the ice lens is engulﬁng soil particles, the freezing front X(t)is moving downward into the soil. The dashed line in ﬁgure 1 shows the original unheaved surface z= 0of the soil before freezing began. The upper heaved surface of the soil is increasing in height both because of the expansion of the pore water as it freezes ( _hi) and because excess water is drawn to the freezing front from the reservoir ( _hs). In general obtaining expressions for _hsand _hias functions of the freezing conditions is a challenging task. Here, following Konrad and Morgenstern [16] it is assumed that for ﬁeld situations the segregation and in situheave rates are given by the expressions _hs=w ivwand _hi=w i1 n_X; (3) wherenis the porosity of the unfrozen soil, _X=dX=dtis the rate at which the freezing front moves into the soil, and wandiare the densities of water and ice [16,18]. As noted by Nixon [31], the above equations do not account for the water expulsion ( vw<0) that occurs in some frost heave experiments at early times and high overburden pressures [32,33]. However, the eﬀect is not typically important in ﬁeld frost heave situations [16,29] and is neglected here for simplicity. Because of particle trapping the ice lens is not pure, but contains a volume fraction iof soil particles equal to i=kp ; (4) where = 1nis the soil particle volume fraction in the unfrozen soil and kpis the particle distribution coeﬃcient at the ice lens surface, described in more detail below. When kp= 0;the ice lens is pure and the freezing front is stationary ( _X= 0). Whenkp= 1;the ice lens engulfs all particles and the soil freezes in situwith no ice segregation. Neglecting the eﬀect of the density diﬀerence between water and ice on the segregated ice growth rate, an expression for _hsas a function of kpcan be obtained as follows. As the ice lens engulfs particles, the freezing front X(t)moves downward into the unfrozen soil at rate _X. The total freezing rate of the soil is V=_X+_hs: (5) Conservation of soil particle number at the freezing front (lower ice lens surface) can be written as iV= _X; (6) which ensures that the particle ﬂux iVentering the ice lens is the same as the ﬂux _Xleaving the soil. Combining (4)–(6) then gives _X=kpVand _hs= (1kp)V: (7a,b) With (7) the total heave rate _hcan be written as _h=_hs+_hi= (1kp)V+w i1 nkpV: (8) At low freezing rates V Vc;whereVcis the critical trapping speed of the soil particles [34,35], the ice lens rejects all the particles and kp= 0:In this primary frost heave case the ice lens is pure, the freezing front is stationary ( _X= 0) and the rate of frost heave is _h=_hs=V:At relatively high freezing rates V Vc;the ice engulfs all of the particles ( kp= 1) and the pore water freezes in situwith _hs= 0:The only frost heave in this case is the volume expansion as the pore water converts to ice and _h=_hi= 0:09nV:1 1During in situfreezing ﬂow in premelted ﬁlms can also occur and contribute to the segregation frost heave rate (secondary frost heave) [36,37]; however, there is uncertainty in the literature as to the magnitude of secondary frost heave and the pore-scale experiments of Watanabe et al.[14,38] failed to detect a frozen fringe. Here it is assumed that the frost heave rate is determined mainly by the primary frost heave occurring at the warmest ice lens, and that _hs= 0during in situfreezing. 4 3 Conservation equations 3.1 Water balance at the ice lens Conservation of mass for the water at the freezing front z=X+can be written as ivi=wvw; (9) wherevi=_hsis the segregated ice growth rate. The water volume velocity vwin the unfrozen soil adjacent to the ice lens is given by Darcy’s law in the form vw=K wg@p @z X+; (10) whereKis the hydraulic conductivity of the soil at the ice lens surface z=X+andgis the acceleration of gravity. The hydraulic conductivity at the base of the ice lens is much less than that of the unfrozen soil because of the resistance to ﬂow in the premelted ﬁlms separating the soil particles from the ice [39,40]. The overburden pressure Pobreduces the thickness of the premelted ﬁlms and further decreases the hydraulic conductivity. Here as in previous work [21,30,41] it is assumed that Kis given as a function of Pobby the exponential equation K=K0ebPob; (11) whereK0is the zero-overburden hydraulic conductivity at the ice lens and bis a constant. 3.1.1 Eﬀect of particle trapping on the Darcy ﬂux With (3 a) and (7) Darcy’s law (10) can be written in terms of kpandVas vw=i w(1kp)V=K wg@p @z X+: (12) The factor (1kp)in (12) accounts for the eﬀect of particle trapping on ﬂow to the ice lens. When kp= 0equation (12) reduces to Darcy’s law for a pure ice lens [10,30]. When kp= 1; @p=@z = 0 and there is no ﬂow of water to the ice lens; in this case only in situfreezing occurs. 3.2 Heat balance at the ice lens The boundary condition expressing conservation of energy at the ice lens surface z=X(t)can be written as f@T @z Xu@T @z X+=iviLf+iLfn_X; (13) wherefanduare the thermal conductivities of the frozen and unfrozen soil and Lfis the latent heat per unit mass of ice [6,42,43]. The ﬁrst term on the right hand side of (13) accounts for the latent heat released by the water ﬂowing from the reservoir as it freezes onto the ice lens, while the second term accounts for the latent heat released as the pore water freezes. 3.3 Hydraulic supercooling and multiple ice lenses The temperature at the ice lens surface z=Xis given by the Clapeyron equation T=TCl=TmTm wLf(pip) (z=X); (14) wherepi=Pobis the pressure on the ice lens and pis the local ﬂuid pressure in the premelted ﬁlms separating the ice from the soil particles [15,39,44]. The temperature gradient GT= (@T=@z )X+ at the position of the ice lens is not in general equal to the gradient of the Clapeyron temperature GTCl= (@TCl=@z)X+:When the temperature gradient is less than the Clapeyron temperature gradient, GT<GTCl; (15) 5 the unfrozen soil below the ice lens is hydraulically supercooled [30,45,46]. As noted by Palmer [45] and Style et al.[46], hydraulic supercooling is a necessary condition for the formation of multiple ice lenses. A linear stability analysis indicates that the supercooled ice lens experiences unstable growth, leading to the formation of a mushy layer composed of a network of vertical ice veins and multiple ice lenses [30,47 –49]. The segregated ice fraction in the mushy layer increases until the hydraulic supercooling has been removed, similar to what occurs during the formation of mushy layers in aqueous solutions and colloidal suspensions [30,50–52]. Taking the gradient of the Clapeyron equation (14) gives, assuming piis constant, GTCl= (Tm=wLf)(@p=@z )X+. Combining this with (12) and inserting into (15) shows that a mushy layer forms when V >GTwLfK Tmig(1kp): (16) Equation(16)generalizestheconditionforhydraulicsupercoolinginsoilsobtainedpreviously[30,46] to allow for particle trapping, which tends to stabilize the ice lens; as kp!1;larger freezing rates Vare required to initiate hydraulic supercooling. When kp= 1there is no ﬂow to the ice lens (@p=@z = 0); the soil water freezes in situand hydraulic supercooling does not occur. The ice lens can also be stabilized by increasing the temperature gradient GTor by increasing the hydraulic conductivity Kat the ice lens surface, for example by modifying the soil chemistry to increase the premelted ﬁlm thickness [4,53,54]. 3.3.1 Mushy layer model Similarly to previous work [30,51], a simple mushy layer model can be used to determine the segregated ice fraction and the average segregated ice growth rate vi=_hsin the mushy layer. Darcy’s law at the position of the warmest ice lens can be written as i w(1kp)V=K wg@p @z : (17) Combining (17) with the Clapeyron equation (14) and the marginal equilibrium condition GT= GTClthen gives  =KwLfGT igTm(1kp)Vand _hs=w ivw=GTwLfK iTmg: (18) Equation (18) indicates that increasing the temperature gradient GTor hydraulic conductivity Kwill increase the segregated ice fraction and the rate of frost heave _hsin the mushy layer regime. The model predicts that during mushy layer growth the segregated ice lens growth rate _hs is independent of both the freezing rate Vand the particle distribution coeﬃcient kp, which may explain in part the observations of Beskow described in the Introduction. This is in contrast to the case of a stable ice lens, for which _hs= (1kp)V(equation (7)). It should be noted that the simple model developed here ignores the eﬀects of regelation in the mushy layer, which could become important in systems with a large temperature gradient [55,56], and could lead to a dependence of _hson the rate of freezing (Section 4.1). 4 The particle distribution coeﬃcient In order to determine quantitatively the eﬀect of particle trapping on freezing soils, a constitutive equation for the particle distribution coeﬃcient kpis required. Expressions for kp(V)for spherical particles at an ice surface have been obtained previously by Elliott and Peppin [57] and Sobolev [58,59]. Here a relatively simple equation of the form kp=(V=Vd (1V2=V2c)+V=Vd;(VVc) 1; (V >Vc):(19) is used, where Vdis the freezing speed at which particle trapping ﬁrst starts to occur in the soil andVcis the critical freezing rate above which all particles are trapped [58]. Experiments and 6 00.20.40.60.81.0 _hs (m/s)Stable ice lens (pure ice) A B  Stable lens (trapping) Unstable (multiple lenses) # 0 1.0 2.0 3.0 V(m/s)(a) 00.20.40.60.81.0 _h (m/s)Pure ice lens  In situ freezing # 0 1.0 2.0 3.0 V(m/s)(b) Figure 2: (a) Segregated ice growth rate _hsversusfreezing rate Vduring unidirectional solidiﬁcation experiments on a glass powder with average radius R= 1:1m [63]. The dotted (black) curve is the case of a pure ice lens with _hs=V;while the solid curve accounts for particle trapping via equation (20). The horizontal dashed line is the predicted rate of heave at which the ice lens becomes unstable forming a mushy layer. (b) The same data and curves as in (a), with the addition of the in situheave rate _hi=:09n_X;where _X=V_hsis the rate of particle engulfment, giving the total heave rate _h=_hs+_hi. theory show that Vcdepends in general on the particle radius R, the temperature gradient GT, the overburden pressure Pob;the pore pressure pand the surface charge and chemical nature of the soil particles [8,10,39,40,60 –62]. Here it is assumed for simplicity that the critical freezing rate varies inversely with the particle radius R, such that Vc=a=R;whereais a constant [35,60]. As mentioned in Section 3.1, the present work assumes that the permeability of the unfrozen soil is much larger than the permeability of the premelted ﬁlms separating the soil from the ice lens, so thatVcis controlled mainly by ﬁlm ﬂow [13,40]. For soils composed of colloidal or nano-sized particles, there is also a dependence of Vcon the permeability of the soil, as demonstrated by Saint-Michel et al.[61] and You et al.[62]. Here the focus is on soils with a larger size distribution and increased permeability, for which the pressure drop across the soil is small compared with that across the premelted ﬁlms [40]. 4.1 Steady-state freezing experiments With (7) and (19), the segregation frost heave rate _hsof a stable planar ice lens with particle trapping can be written as a function of the freezing rate Vas _hs=( V 1V=Vd (1V2=V2c)+V=Vd ;(VVc) 0; (V >Vc):(20) The solid curve in ﬁgure 2a is from equation (20) with Vd= 1:0m/s andVc= 2:0m/s chosen as ﬁtting parameters. The curve represents the predicted rate of frost heave of a stable planar ice lens as the freezing rate Vis increased. At low freezing rates V Vcequation (20) gives _hs=V; while at higher rates _hs<V:Similar to experiments by Horiguchi [4], Penner [5] and Konrad [6], the segregation heave rate shows a maximum at an intermediate freezing rate before going to zero above the critical rate Vc: The horizontal dashed line in ﬁgure 2a shows the predicted heave rate at the onset of hydraulic supercooling obtained from equation (16) with K= 0:7109m/s andGT= 190C/m [30,63]. Once hydraulic supercooling occurs (point A) a mushy layer containing multiple ice lenses forms, and the heave rate no longer obeys (20) but is a constant given by equation (18). At higher Vparticle trapping eventually restabilizes the freezing interface (point B) and the heave rate is predicted to again follow equation (20). The experimental data in ﬁgure 2a are from Watanabe [63] using a glass powder medium with average particle radius R= 1:1m and temperature gradient GT= 190C/m, showing 7 0204060 _h (mm/day)Stable freezing (pure ice)  Stable freezing (trapping). In situ freezing # 0 4 8 12 _q(103calcm2s1)(a) 0204060 _h (mm/day)Stable freezing (trapping) .R= 24mUnstable freezing (multiple lenses) #R= 2m 0 4 8 12 _q(103calcm2s1)(b) Figure 3: Total frost heave rate _hversusthe heat extraction rate _q:The data in (a) are from Horiguchi [4] using R= 20m zeolite particles, while the data in (b) are from [4] using carborundum particles with R= 24m (crosses) and R= 2m (stars). The curves are from equation (26). qualitative agreement with the basic features of the model. In particular, Watanabe noted that in his experiments the ice lens growth rate _hswas equal to Vup untilV0:19m/s; then _hs was approximately constant and equal to 0:24m/s up until V= 2m/s, above which only in situfreezing occurred. The experimental ice lens growth rate varies somewhat in the mushy layer regime between A and B, especially at lower freezing speeds. A potential explanation is regelation, which was neglected in the simple mushy layer model developed in Section 3.3.1. An alternative explanation is a dependence of the permeability of the premelted ﬁlm on the temperature of the ice lens [40]. Figure 2b plots the same data as in (a), but showing the total heave rate _h=_hs+_hi; with the in situheave rate calculated as _hi= 0:09n_X= 0:09nkpV;wheren= 0:4is the porosity. 4.2 Transient freezing experiments WhileVis a natural measure of the freezing rate in steady-state unidirectional freezing experiments, during transient step-freezing tests Vis diﬃcult to control. An alternative measure of the freezing rate that has been used is the net rate of heat extraction _q[4,6], deﬁned as _q=f@T @z Xu@T @z X+: (21) Equations (7 a), (9), (12) and (13) can be used to write _q=iLfV[1kp(1n)];showing that _q is closely related to V; that is, _q=(iLf)!VasV!0and _q=(iLf)!nVasV!1. Horiguchi [4] performed an experiment in which 20 m radius zeolite particles were frozen at diﬀerent heat extraction rates and the total rate of frost heave _hwas measured as a function of _q, with data shown in ﬁgure 3. Horiguchi’s experiment can be modelled via a similar approach to Section 4.1 by assuming, similar to Arakawa [43], that the heat balance at the freezing front can be separated into two terms, such that (1kp) _q=iLfviandkp_q=iLfn_X: (22a,b) That is, a fraction (1kp)_qof the heat extraction rate is employed to freeze the excess water volume added to the ice lens from the reservoir, while kp_qis used to freeze the in situpore water. The quantity 1kpin (22) is equivalent to Arakawa’s ice segregation eﬃciency ratio E[5,43], deﬁned as ELf=_q= 1kp; (23) where=iviis the rate of mass accumulation of segregated ice. Summing (22 a) and (22 b) gives the overall heat balance (13). Equations (22), (9) and (3) then give _hs= (1kp)_q iLfand _hi= 0:09kp_q iLf: (24a,b) 8 A constitutive equation for kp( _q)in a form similar to equation (19) is kp=(_q=_qd (1_q2=_q2c)+ _q=_qd;( _q_qc) 1; ( _q>_qc)(25) where _qdis the heat extraction rate at which particle trapping ﬁrst starts to occur and _qcis the critical rate above which all particles are trapped. The total frost heave rate _h=_hs+_hican then be written as _h=(_q iLf 1_q=_qd (1_q2=_q2c)+ _q=_qd +_hi;( _q_qc) _hi; ( _q>_qc);(26) where _hi= 0:09kp_q=(iLf). Equation (26) is plotted on ﬁgure 3a as the solid curve using the best-ﬁt values _qd= 5:7 103calcm2s1and _qc= 9:5103calcm2s1, and porosity n= 0:4, giving a good match to Horiguchi’sdata(crosses). Similarlytoﬁgure2, atlowfreezingrates _h=_q=iLf=V;corresponding to the growth of a pure ice lens (primary frost heave), while at high freezing rates _h=_hi(in situ frost heave). In contrast to ﬁgure 2, however, Horiguchi’s data appears to follow the trapping curve (26) at all freezing rates. This suggests that the primary ice lens remained stable in his system, and a mushy layer composed of multiple lenses did not form. Horiguchi used a relatively large particle size ( R= 20m) so that particle trapping may have stabilized the interface at all _qand hydraulic supercooling did not occur. This possibility is supported by other experiments in which Horiguchi varied the average particle size of the soil [4]. Figure 3b shows data obtained using carborundum particles with radius R= 24m (crosses) and R= 2m (stars). The solid blue curve for the larger particles is from equation (26) with best-ﬁt values _qd= 3:5103calcm2s1and _qc= 7:3103calcm2s1; the red curve for the smaller particles assumes an R1dependence giving values for _qcand _qd that are 12 times larger. For the larger particles the results are similar to ﬁgure 3a, with the data following the trapping curve at all _q. For the smaller particles, however, the data ﬂattens at a critical heat ﬂux _q2103calcm2s1and the heave rate becomes approximately independent of the freezing rate, similar to the glass powder results in ﬁgure 2. Using i= 0:91gcm3and Lf= 80cal/g the critical heat ﬂux corresponds to a freezing rate of V_q=(iLf) = 0:27m/s, which is close to the value 0:19m/s obtained by Watanabe [63] for R= 1:1m silica particles (Section 4.1). This suggests that hydraulic supercooling leading to a mushy layer with multiple ice lenses may have occurred in Horiguchi’s experiment with the 2m particles, while particle trapping stabilized the primary ice lens in the system with 24m particles. 5 Kinetic segregation potential As mentioned in the Introduction, Konrad and Morgenstern [13] deﬁned the segregation potential SPas the water ﬂow rate vwto the warmest ice lens divided by the temperature gradient GT at the onset of the ﬁnal stable ice lens in a transient frost heave experiment. On the basis of experimental data they obtained the correlation SP=SP0ebPob; (27) whereSP0andbare constants. Assuming that the onset of the ﬁnal ice lens corresponds to a state of marginal stability of the ice interface, a linear stability analysis leads to the expression (for non-saline soils) [30] SP=LfK Tmg: (28) A similar equation for SPwas obtained earlier by Horiguchi [64]. Comparison of (28), (27) and (11) givesK0=SP0Tmg=Lf:Equations (28) and (27) imply that SPis independent of the freezing rate; however, experiments by Konrad and Morgenstern [17,28], Svec [26] and Kim [65] show that during the early stages of transient frost heave experiments the measured segregation potential SP=vw=GTdepends on the freezing rate and can be much smaller than predicted by (27) or (28). 9 Here, motivated by equation (7), the dependence of SPon the freezing rate is accounted for by using a kinetic segregation potential of the form SPkin= (SP)E=LfK Tmg(1kp); (29) whereE= 1kpis Arakawa’s ice segregation eﬃciency ratio [43] described in Section 4.2. Equation (29) with (19) or (25) for kpshows that SPkin!0during the early rapid freezing stage of a transient frost heave experiment, while SPkin!SPat later stages involving slower freezing rates. In the next section equation (29) is used to model the frost heave rate during the early stages of a ﬁeld frost heave experiment and above a buried natural gas pipeline. 6 Field frost heave experiments As noted by Konrad and Morgenstern [16] and Nixon [18], robust estimates of the temperature proﬁle and depth of freezing in many ﬁeld situations are known either from experimental data or from models such as the modiﬁed Berggren equation [66,67]. Assuming the depth of freezing X(t)is known, along with the temperature gradient GT(t)at the freezing front, Konrad and Morgenstern [16] suggest that the total frost heave rate h(t)can be approximated by the equation h(t) = 1:09Zt 0vwdt+ 0:09nX(t); (30) where vw=SPGT (31) is the water intake rate. The ﬁrst term on the right-hand side of (30) accounts for the segregated ice lens growth while the second term accounts for the 9% volume expansion as the pore water freezes. Konrad and Morgenstern [16] suggest that in many ﬁeld situations X(t)andGT(t)obey power laws of the form atb;whereaandbare constants. Equations (30) and (31) have been used successfully to model frost heave by several researchers, including Konrad and Morgenstern [21], Konrad and Shen [22], Nixon [18], Carlson and Nixon [20], Kim et al.[68] and Yu et al.[69]. However, as acknowledged by Konrad and Morgenstern [16] and noted subsequently by Fukuda and Kinosita [19], Nixon [31,70] and Ming and Li [27], the model breaks down at high freezing rates, when the water intake rate vwapproaches zero (or can even be negative with water expulsion). Here a similar approach to Konrad and Morgenstern [16] and Nixon [18] is taken, but with a water ﬂux of the form vw=SPkinGT=SPGT(1kp): (32) Given that neither the total freezing rate Vnor the heat extraction rate _qare typically known in ﬁeld situations, it is proposed to use for kpa simpler constitutive equation of the form kp=_X=Vd 1 + _X=Vd; (33) where _X=dX=dtis the rate at which the freezing front moves into the soil and Vdis the freezing rate at which particle trapping ﬁrst starts to occur in the soil. Equation (33) is similar in form to the continuous growth model of the distribution coeﬃcient in rapid alloy solidiﬁcation [58,71]. It is simpler than (19) and (25) while still capturing the expectation that kp!1during the early rapid-freezing stages of soil freezing, while kp!0at later slower stages. 6.1 Circular plate experiment As a test of equations (30)–(33), in this section they are applied to ﬁeld frost heave data obtained by Nixon et al.[18,72], shown on ﬁgures 4 and 5. They conducted frost heave experiments on a soil (Calgary silt, porosity n= 0:38[21]) below a 0.76m diameter circular cold plate held at the constant temperature Tc=4C. Figure 4 shows measured data for the depth of freezing X(t) 10 0100200300400500 X (mm) 0 20 40 60 80 100 t(days)(a) 010203040 GT (K/m) 0 20 40 60 80 100 t(days)(b) 04080120 P0 (kPa) 0 20 40 60 80 100 t(days)(c) 101102103 SP 0 40 80 120 160 200105mm2/sK Pob(kPa)(d) Figure 4: (a) Data for the position of the 0C isotherm X(t)in soil (Calgary silt) below a 0.76m diameter plate cooled to -4C. (b) Temperature gradient measurements at the position of the freezing front X(t):(c) Soil pressure P0(t)measured at the level of the cooling plate. The data in (a-c) were obtained by Nixon et al.[18,72], while the curves are from equations (34). (d) Segregation potential data for Calgary silt obtained by Konrad and Morgenstern [21]. 11 04080120160 h (mm) 0 50 100 150 200 t(days)(a) 04080120 h (mm) In situ freezing regimeSegregation freezing regime hi(t) # 0 200 400 600 X(mm)(b) Figure 5: (a) Total heave h(t)data (stars) measured at the surface of a soil coolied by a 0:76m diameter copper plate held at -4C [18]. The dashed curve is the predicted heave using the segregation potential SPmodel given by (27) and (36). The solid curve is the predicted heave using the kinetic segregation potential SPkin=SP(1kp);withkpgiven by (33). The thin solid curves use the lower and upper bound segregation potential curves from ﬁgure 4d: SP0= 180105mm2=(C s)(lower bound) and SP0= 300105mm2=(C s)(upper bound) [21]. (b) Plot of total heave h(t)versusfreezing depth X(t), illustrating an in situfreezing regime at early times, in which the measured heave rate is very close to the predicted in situheavehi(t)(black dotted line). A segregation freezing regime involving multiple segregated ice lenses occurs at later times [72]. The blue curves are the same as in (a) but plotted versus X(t): below the plate, the temperature gradient GT(t)at the freezing front and the soil pressure P0(t) at the level of the plate, along with the power law correlations X= 104t0:31(mm); GT= 0:034t0:36(C/mm )andP0= 47t0:23(kPa);(34a,b,c) which provide good ﬁts to the data up to t= 100days. Given the depth of freezing X(t)and the approximate frozen soil density 1g/cm3[20], the total overburden pressure at the freezing front can be estimated as Pob=P0+gXP0(t) + 0:01X(t); (35) withP0(t)in kPa and X(t) in mm [20]. For the Calgary silt used by Nixon et al.[72], Konrad and Morgenstern [21] obtained data for the segregation potential SP, shown in ﬁgure 4d along with the correlation (27) with best-ﬁt parameters SP0= 230105mm2=(C s)andb= 0:0095kPa1: (36) Figure 4d also shows the lower and upper bounds on SPsuggested by Konrad and Morgenstern [21], withSP0= 180105mm2=(C s)(lower bound) and SP0= 300105mm2=(C s)(upper bound). With (34)–(36) and either (31) or (32) for vw, equation (30) can be integrated numerically, giving the results shown in ﬁgure 5a. The dashed curve is the predicted total heave h(t)of the soil using the segregation potential model (31). As noted by Konrad and Morgenstern [28], the model gives an accurate prediction of the heave rate (slope of the curve) at later times, but tends to overpredict h(t)at early times when the freezing rate is high. Accounting for particle trapping via equation (32) (solid curves) leads to a closer match to the early time data. Here a value Vd= 9mm/day0:1m/sfor the trapping speed has been used, which is close to the critical trapping speed measured by Cisse and Bolling [35] for glass spheres with radius 100m. This value corresponds to the largest 510%of particles in Calgary silt [72], supporting the interpretation of Vdas the freezing speed at which particle trapping starts to become signiﬁcant within the soil. Figure 5b shows the same data and curves plotted as the frost heave h(t)versusthe freezing depthX(t). In this form two distinct freezing regimes can be discerned – an in situregime at early times during which h=hi;and a segregation regime at later times when h=hi+hs[72]. The lower dotted line shows the calculated in situheavehi(t) = 0:09nX(t);in good agreement with the early time data. 12 0123 X (m) 0 400 800 1200 1600 t(days)(a) 02468 GT (K/m) 0 400 800 1200 1600 t(days)(b) 101102103 SP 0 50 100 150 200 250105mm2/s K Pob(kPa)(c) Figure 6: (a) Position of the 0C isotherm X(t)in soil below a buried natural gas pipeline cooled to -8.5C [72]. (b) Temperature gradient measurements at the position of the freezing front X(t)[72]. (c) Segregation potential data for Calgary silt obtained by Carlson and Nixon [20]. 6.2 Buried pipeline experiment As a further test of the model, ﬁgures 6 and 7 show frost heave data obtained by Carlson et al.[73] below a buried pipeline held at Tc=8:5C. Figure 6 shows data for the depth of freezing X(t)and temperature gradient GT(t)at the freezing front, which are approximated by power law expressions of the form X= 100t0:47(mm)andGT= 0:025t0:36(C/mm ); (37a,b) withtin days. In this experiment a similar soil (Calgary silt) was used, but at a lower depth. Segregation potential SPdata from Carlson and Nixon [20] is shown in ﬁgure 6c, giving the best-ﬁt parameters SP0= 240105mm2=(C s)andb= 0:05kPa1: (38) The initial overburden pressure at the level of the pipe was estimated by Carlson and Nixon [20] as 48kPa, and at t= 420days a 30kPa berm was added to the surface, giving the total overburden pressure at the freezing front X(t)as Pob=( 48 + 0:01X(t);0<t< 420days 78 + 0:01X(t); t420days;(39) withX(t)in mm and Pobin kPa [20]. With (37)–(39) equation (30) can be integrated to determine h(t), as shown by the blue curves in ﬁgure 7. Again the SPmodel (dashed curve) tends to overestimate the rate of frost heave at early times. The model with particle trapping gives a closer match to the early time data, using the same value for the trapping speed Vd= 9mm/day as in Section 6.1. 6.3 Saline soils Many soils contain a signiﬁcant salt content [74], in which case the segregation potential can be inﬂuenced by the salt diﬀusivity Das well as the hydraulic conductivity K[17,30]. In saline soils the segregation potential can be written in the form SP=Tmg LfK+mc0(1ks) Dks1 ; (40) wheremis the slope of the solute freezing temperature curve (for NaCl m= 0:065CL/g),c0is the solute concentration of the pore ﬂuid, DD0ebPOBis the solute diﬀusivity in the premelted ﬁlms andksis the solute distribution coeﬃcient at the ice lens [30]. With particle trapping the kinetic segregation potential in saline soils can be written as SPkin= (SP)E=1kp Tmg LfK+mc1(1ks) Dks: (41) 13 02004006008001000 h (mm)0:35g/L 3:5g/L 35g/L 0 400 800 1200 1600 2000 t(days) Figure 7: Total heave h(t)measured in the buried pipeline experiment (deep burial section) [18]. The dashed curve is the predicted heave using the segregation potential SPgiven by (27). The solid curve uses the kinetic segregation potential SPkin=SP(1kp);withkpgiven by (33). The dotted green curves show the predicted eﬀect of increasing the salinity c0of the soil, with SPkin given by equation (41). Using (41) with (30), (32) and (33) allows the eﬀect of soil salinity on frost heave in ﬁeld situations to be predicted. The green dotted curves in ﬁgure 7 show the eﬀect of soil salinity on the rate of frost heave in the buried pipeline experiment for the case ks= 0:4; D0= 109m2/s and various c0;with other parameters the same as in Section 6.2. A moderate salt concentration of 3:5g/L signiﬁcantly decreases the rate of frost heave, in qualitative agreement with the observations of Sheeran and Young [75,76], Chamberlain [47], Velsovskij et al.[77] and Liu et al.[78]. In equation (41) the salt distribution coeﬃcient ksat the ice lens surface has been treated for simplicity as an averaged constant; more generally ksalso depends on the freezing rate [79 –81]. Qualitatively similar results can be obtained by replacing ksin equation (41) with a kinetic salt distribution coeﬃcient of the form kv=ks=[ks+ (1ks)e _X][80]. This equation requires knowledge of the additional parameter :For saline sands Baker and Osterkamp [80] obtained ks= 0:34and = 0:076days/mm; experiments by Konrad and McCammon [81] suggest that ks and are of similar magnitude for silty clay soils. 7 Discussion In this work eﬀects such as regelation within the mushy layer and consolidation have been neglected. These eﬀects may be important in some systems, as discussed brieﬂy in Section 4.1. However, in ﬁeld situations involving relatively small temperature gradients and high freezing rates, in soils with a broad size distribution, regelation will be less of an issue (at least at relatively short time scales), and it is anticipated that particle trapping will have a more signiﬁcant eﬀect on the segregation potentialSP:Developing a model of the eﬀects of regelation and consolidation on SPis left to future work. The present model also assumes that the depth of freezing X(t)and temperature gradientGT(t)are known, either experimentally or from alternative models. A complete model should predict these quantities, along with the average segregated ice fraction in the mushy region. A preliminary approach to modelling mushy layers in colloidal suspensions, including the eﬀects of regelation, has been obtained recently [56]; it may be possible to extend this work to the present system, yielding a partial diﬀerential equation for the ice fraction (z;t), and this will be explored in future work. 8 Conclusions A model of frost heaving soils has been developed that takes account of particle trapping eﬀects. An expression for the particle distribution coeﬃcient at the ice lens surface has been obtained, and compared with experimental data on a range of model soils. Particle trapping tends to stabilize the ice lens and reduce the amount of frost heave. Based on these results a new kinetic expression for 14 the segregation potential has been developed that accounts for particle trapping and depends on the freezing rate. The new expression is the product of Konrad and Morgenstern’s [13] segregation potentialSPand Arakawa’s [43] ice segregation eﬃciency ratio E. This leads to a kinetic version of Konrad and Morgenstern’s ﬁeld frost heave model that simulates the in situfreezing that occurs at early times in step freezing tests. The model predictions have been compared to frost heave data obtained on soil below a cooled circular plate and below a chilled natural gas pipeline. For ﬁeld situations in which the depth of freezing and temperature gradient at the freezing front are known from independent measurements or models, the present method provides a simple approach to predict the total frost heave rate as a function of the freezing rate and overburden pressure in both saline and non-saline soils. Acknowledgements This work was made possible by the love and support of Jesus, Lord of heaven and earth and Saviour of all who believe. Believe on the Lord Jesus Christ and thou shalt be saved, and thy house. Acts 16:31. The heaven, even the heavens, are the LORD’S: but the earth hath He given to the children of men Psalm 115:16.