14 Soil Temperature Regime Graeme D. Buchan Lincoln University, Canterbury, New Zealand I. INTRODUCTION Temperature has a fundamental control on almost all processes in the environ- ment. In cool climates, it demarcates growing and ‘‘nongrowing’’ seasons. Storage and release of heat in soil control the temperature of both the soil and the lower atmosphere, thus affecting the whole terrestrial biosphere. Yet soil temperature and its effects were traditionally poorly researched, greater attention being given to water, mainly because, with adequate temperature established within the grow- ing season, it becomes the major and often erratic determinant of growth, while being more controllable via irrigation or drainage. More recently, a wider need has arisen to either measure or model the soil temperature regime, defined here to include the depth and time variations of both temperature and heat flux. Thus the literature shows increased attention to effects of soil temperature on soil bio- logical processes, nutrient and fertilizer transformations, physical processes in- cluding solute transport, and environmental issues such as soil–atmosphere gas exchanges, the global carbon budget, and the transformations and transport of contaminants. Also, crop growth and evapotranspiration models require improved submodels or measurements of soil temperature regime. Climate modeling and remote sensing require more accurate data, for both heat flow and soil (especially surface) temperature. Recent decades have seen significant advances in (1) theory: the analysis of coupled flows of heat and water, and of flow and phase-change processes in freez- ing soils; (2) applications, including (a) more realistic modeling of heat flow, or simultaneous heat and water flows, by inclusion of the surface energy balance as the governing boundary condition; (b) measurement and recording techniques for Copyright © 2000 Marcel Dekker, Inc. temperature, heat flux, and thermal properties; (c) engineering applications, e.g. ground heat pumps, and particularly (d) more intensive investigation of soil tem- perature as a key controller of biosphere processes including soil–atmosphere gas exchanges, transport and reactivity of solutes, and the fate of contaminants. The basic mechanisms of coupled heat and water flows in soil were first described by Philip and de Vries (1957). Despite this, the potentially large impact of this coupling is not yet fully appreciated. While models of simultaneous flows in field soils have correctly incorporated the coupled flow equations, in the design of experimental techniques and interpretation of field measurements, the assump- tion is often made that the heat flow equation can be viewed as ‘‘uncoupled’’ from the moisture flow equation (i.e., that heat flow in soils is ‘‘conductive,’’ and equal to a thermal conductivity l times a temperature gradient, where l implicitly con- tains the thermal vapor flux driven by the temperature gradient). While this as- sumption is valid in a uniformly moist soil, it can fail badly in the presence of a strong moisture (i.e., water potential) gradient, which drives an isothermal vapor flux. This both contributes to the total soil heat flux and implies latent heat demand at the sites of vaporization. This occurs in drying soils, where much of the total soil evaporation can derive from ‘‘subsurface evaporation,’’ which exerts a strong influence on heat flux and the temperature profile. Neglecting such effects can lead to large errors in measurements of heat flux and thermal properties (de Vries and Philip, 1986). This chapter therefore has a dual role. First, it reviews underlying theory and experimental methods. Second, as many of these methods assume that heat flow is purely conductive, it clarifies the potentially large effects of coupled flows on field measurements. The vital concept is the correct interpretation of the soil heat flux, including its surface value G 0 appearing in the energy balance equation. A review of solutions of the uncoupled conduction equation includes peri- odic solutions and Fourier methods; basic characteristics of the diurnal and annual waves, and noncyclic effects; ‘‘transient’’ solutions from Laplace transform and other methods; and numerical methods. The calculation of thermal properties from physical composition is described. A brief section reviews theories of freezing soil. The measurement section reviews (a) techniques of measuring temperature, heat flux, and thermal properties, and (b) sampling criteria and data smoothing. There is a remarkable dearth of works on soil temperature regime, with a few exceptions (Gilman, 1977; Farouki, 1986), notably in the Soviet literature (Chudnovskii, 1962; Shul’gin, 1965), though several texts devote sections to basic aspects (e.g., Hillel, 1980; Jury et al., 1991). This chapter should help to remedy this deficiency and to correct some prevalent misconceptions. Because the theory and measurement are so intimately related, Sec. II below concerns the theory underlying measurements, and its extension to modeling of soil temperature regime. Thus the reader concerned solely with field measure- ments may go straight to Sec. III. However, to understand the principles and po- 540 Buchan Copyright © 2000 Marcel Dekker, Inc. tential pitfalls of measuring soil heat flux and thermal properties, as well as the use of measurements in modeling, the theory of Section II is necessary. II. THEORY A. Surface Energy Balance The most powerful models of soil heat flow incorporate its fundamental driving mechanism, the energy balance at the soil surface. The net radiation R n received per unit area of the soil surface is R ϭ (1 Ϫ a)R ϩ eL Ϫ L (1a) nsdu where R s and L d are incident solar and longwave radiation, and a and e are the shortwave reflection coefficient and longwave emissivity of the soil surface, re- spectively. L u (the longwave emission) ϭ where s is the Stefan–Boltzmann 4 esT , 0 constant. This longwave emission is detected during infrared thermometry of the surface temperature T 0 (Huband, 1985). R n is partitioned at the soil surface ac- cording to the energy balance equation R ϭ H ϩ LEϩ G (1b) nv0 where H is the sensible heat flux from soil to air, L v E is the latent (evaporative) heat flux (L v ϭ the latent heat of vaporization), and G 0 is the heat flux into the soil. For vegetated soil, L d ‘‘seen’’ by the surface will include plant as well as sky emissions, H will include a small stem heat conduction term as well as convection, and E ,thesoil evaporation, will be only a portion of total evapotranspiration (Main, 1996). Note that ‘‘sensible’’ implies heat flow causing a local change of temperature. Thus most of G 0 produces sensible heat (i.e., temperature) change, but in a drying soil some supplies the latent heat required for evaporation within the bulk of the soil. The dominant solar term R s in Eq. 1a, with its diurnal and annual cycles, drives similar cycles in surface temperature T 0 and air temperature T a , while L v E, H, and L d are controlled by atmospheric temperature and vapor pressure. Thus Eq. 1b mechanistically relates soil temperature to meteorological variables and could help explain empirical relationships, e.g., between soil and air temperature (e.g., Hasfurther and Burman, 1973; Gupta et al., 1984), though under vegetation complex modeling of intracanopy exchanges would be required. Equation 1b also enables mechanistic understanding of practical alteration of temperature regime, e.g., by mulching. 1. Components of the Total Soil Heat Flux, G tot In practice the ‘‘surface’’ for the energy exchanges in Eqs. 1a and 1b will be a thin layer, with thickness controlled by the surface microprofile, but typically several Soil Temperature Regime 541 Copyright © 2000 Marcel Dekker, Inc. mm for a crumb-structured surface. However this layer is not necessarily the site of total soil evaporation, E tot . In drying soils the evaporation sites retreat, at least partially, into subsurface layers (de Vries and Philip, 1986). This is critical for interpretation of both Eq. 1b and the soil heat flux G(z, t), a function of soil depth z, with surface value G 0 . As shown in Fig.1a, E tot is partitioned as E ϭ E ϩ E (2) tot 0 s0 ϱ E ϭ ͵ E (z) dz (3) s0 s 0 542 Buchan Fig. 1 (a) Partitioning of total evaporation E tot ϭ E 0 ϩ E s0 at the surface of a drying soil. (b) The two possible interpretations of the terms in Eq. 1b, shown under typical daytime conditions. At night the direction of G T will usually reverse. Copyright © 2000 Marcel Dekker, Inc. Here E 0 is the evaporation sourced at the surface (replaced by liquid flow from below), and E s0 derives from subsurface evaporation. E s (z) (kg m Ϫ3 s Ϫ1 )isthe vapor source strength per unit volume at depth z, contributing to upwards vapor flowdrivenbythemoisture gradient. (Vapor distillation induced by the tempera- ture gradient is included in the effective thermal conductivity; see Sec. II.C). E s0 will be dominant in a soil with a dry surface. Equation 1b may then be interpreted in two ways; see Fig. 1b. First, if E ϭ E 0 , then G 0 ϭ G T (0, t ) (i.e., the surface value of the conductive or thermally driven heat flux, G T (z, t ); see Sec. II.C). Divergence in G T (z, t ) (i.e., variation of G T with depth) within the soil will then result from both changes in temperature and the subsurface phase change E s (z), corresponding to evaporation or condensation at depth z. Second, if, as is normally assumed, E ϭ E tot , then G 0 must be reduced by an amount L v E s0 , corresponding to the subsurface evaporative energy demand. Then G 0 becomes the surface value of the total soil heat flux G tot (see Sec. II.C) given by G ϭ G ϩ G (4) tot T vp The term ‘‘isothermal latent heat flux’’ is introduced here for G vp (ϭϪL v E s0 ) i.e., the latent heat carried from evaporating subsurface layers by the isothermal vapor flux (i.e., driven by a moisture gradient). For example, during daytime heat- ing of a drying soil, G T at the surface will be positive (into the soil), but G tot ϭ G T ϩ G vp will be reduced by the negative G vp . Then divergence in G tot is required to fuel only changes in soil temperature. Thus in the customary use of Eq. 1 to calculate total soil evaporation E tot , it is vital to identify G 0 with G tot . However, G 0 is often erroneously identified with the ‘‘thermal soil heat flux’’ G T , which (Sec. III.C) is the heat flux obtained by methods detecting the temperature gradi- ent (e.g., the heat flux plate). B. Heat Conduction: Uncoupled Equations Conduction of heat down a temperature gradient dT/dz is governed by the Fourier equation dT G ϭϪl (5) T dz where the thermal conductivity l (W m Ϫ1 K Ϫ1 ) includes a vapor distillation term (Sec. II.D). Divergence in G T causes heat changes, both sensible and latent, and so obeys energy conservation: ץT ץG T C ϭϪ ϩS(z, t ) (6) ץt ץz Soil Temperature Regime 543 Copyright © 2000 Marcel Dekker, Inc. where C (J m Ϫ3 K Ϫ1 ) is the volumetric heat capacity. S (W m Ϫ3 ) represents local heat sinks or sources, i.e., usually phase changes of water (Secs. II.C, II.F). Ne- glecting S (considered below) and spatial variations in l, Eqs. 5 and 6 give the simple uncoupled heat diffusion equation 2 ץT ץ T Ϫ1 k ϭ (7a) 2 ץt ץz 2 ץ T ץT Ϫ1 ϭϩr (7b) 2 ץr ץr where k ϭ l/C (m 2 s Ϫ1 ) is the thermal diffusivity. Equation 7b in cylindrical coordinates applies to the use of cylindrically symmetric probes (Sec. II.E). Equa- tion 7 is uncoupled in the sense that, with the thermal vapor flux implicit in l,it can be solved independently of the moisture flow equation. Its use implies a no- coupling assumption, invalid in soil undergoing aqueous phase changes, in par- ticular subsurface evaporation. The thermal properties l, C, and k are (a) functions of physical composi- tion and hence both position and time, so that analytic solutions require simpli- fying assumptions (Sec. II.E), and (b) relatively weak functions of T itself, so that Eqs. 7a and 7b are, strictly, weakly nonlinear. Equation 7 in three-dimensional form has ץ 2 T/ץz 2 replaced by ٌ 2 T. C. Heat Flow: Moisture Coupling Heat and water flows can interact strongly in soil. This interaction is small in soil close to absolute dryness or saturation, but important at intermediate states of wetness. The main coupling of flows is by two mechanisms: (a) the influence of gradients of temperature on water flow, in the liquid phase by its effect on surface tension, and more importantly in the vapor phase by its much stronger effect on vapor pressure (i.e., thermally driven water flow); and conversely (b) the influence of gradients of water potential, driving liquid and vapor flow, on the flow of heat (i.e., water potential driven heat flow). The interaction of heat and liquid water flow is often negligible (de Vries, 1975), with a few important exceptions. Ex- amples corresponding to mechanisms (a) and (b) are the often rapid migration of liquid water under temperature gradients towards a freezing front, possibly lead- ing to frost heave or formation of ‘‘ice lenses’’; and heat convection by intense infiltration of water. By contrast, heat and vapor flows may be strongly coupled, so conduction may be accompanied by a large latent heat flux. The source of this coupling is apparent in the one-dimensional (vertical) vapor flux J v (Bristow et al., 1986), the sum of the thermal (J vT ) and isothermal (J vp ) vapor fluxes. 544 Buchan Copyright © 2000 Marcel Dekker, Inc. de J ϭϪD ϭ J ϩ J (8) v v vT vp dz dT J ϭϪhDhs (9) ͫͬ vT v dz dh J ϭϪDe(T) (10) ͫͬ vp v s dz Here, e is the actual vapor pressure in the air phase, e s (T) is the saturation vapor pressure (svp), s ϭ de s /dT is the slope of the svp curve, and h ϭ e/e s is the relative humidity. D v ϭ au a nD va is the apparent vapor diffusivity (kg m Ϫ1 s Ϫ1 Pa Ϫ1 )in soil air, where D va is the diffusivity in bulk, still air, u a is air-filled porosity, and a is a pore space tortuosity factor. The mass flow factor n ϭ p/( p Ϫ e) ഠ 1 (where p is the total air pressure in soil) accounts for a small mass flow contribution to vapor transfer (Philip and de Vries, 1957). In Eq. 9, the added enhancement factor h is required to give the effective thermal vapor diffusivity hD v (Philip and de Vries, 1957; Cass et al., 1984; Bristow et al., 1986). Thus the vapor flux, Eq. 8, has two components. The thermal vapor flux J vT (Kimball et al., 1976) represents thermally driven vapor transfer. This carries la- tent heat from hotter (higher e s ) to cooler (lower e s ) regions, contributing to the effective thermal conductivity, l. Conversely, the isothermal vapor flux J vp repre- sents a water-potential-driven latent heat transfer, L v J vp . Thus, neglecting osmotic effects, a moisture gradient controls humidity h in Eq. 10 according to c M mw h ϭ exp (11) ͫͬ RT where c m (J kg Ϫ1 ) is the matric potential and M w ϭ 18.016 ϫ 10 Ϫ3 kg mol Ϫ1 is the molecular weight of water. Equation 11 implies h Ͼ 0.99 for c m ϾϪ13 bar. Thus J vp will typically be relatively small in soils wetter than the wilting point. Then only J vT (already inherent in l) need be considered. However J vp is signifi- cant under strong moisture gradients, e.g., in the upper layers of drying soils. Following Eq. 8, we may define a total soil heat flux G tot G ϭ G ϩ G ϩ G ϭ G ϩ LJ ϩ LJ (12) tot c vT vp c v vT v vp containing a ‘‘pure’’ conduction component G c , a ‘‘thermal latent heat flux’’ G vT , and an ‘‘isothermal latent heat flux’’ G vp . In reality, pure conduction and thermal distillation (G vT ) are intertwined as complex series–parallel processes, and so are not strictly additive. However, both processes are proportional to ϪdT/dz, and may be combined into a single ‘‘thermal soil heat flux’’ G ϭ G ϩ G TcvT dT ϭϪl (13) dz Soil Temperature Regime 545 Copyright © 2000 Marcel Dekker, Inc. where l is the apparent thermal conductivity (i.e., as calculated by the Philip– de Vries model discussed below). The uncoupled heat diffusion Eq. 6 then becomes the coupled equation (Philip and de Vries, 1957) ץT ץG tot C ϭϪ ץt ץz ץ(lץT/ץz) ץJ vp ϭϪL (14) v ץz ץz where the last term accounts for phase change induced by a moisture gradient. Divergence in J vp represents a heat sink (a site of net evaporation) or source (a site of net condensation). In field soils undergoing subsurface evaporation, the heat sink effect will tend to increase divergence in G T , and hence the curvature of the temperature profile. We will return to the practical impact of this on heat flux measurement in Sec. III.C. The concept of an effective thermal conductivity, enhanced by thermal va- por distillation, can be treated theoretically in two distinct ways. The first method solves simultaneously the coupled flow equations (e.g., Milly, 1982; Bristow et al., 1986). Thus Eq. 14 is the heat transfer equation. However this method, while more comprehensive and accurate, requires complex numerical modeling. The second method (Philip and de Vries, 1957) essentially builds the ther- mal vapor flux, Eq. 9, into the de Vries (1963) thermal conductivity model, which calculates l from the conductivities of individual soil components (see next sec- tion). As vapor transfer occurs in the air filled pores, with net distillation from warm to cold ends, the air phase conductivity becomes l ϭ l ϩ hl (15) av a vs Here l a is the conductivity of still air and de s l ϭ L nD (16) vs v va dT is the vapor distillation term for saturated air, n is the mass flow factor discussed below Eq. 10. Eq. 16 is essentially the same thermal vapor flux effect as Eq. 9 but contains the simple bulk air diffusion coefficient rather than an effective one for a complex pore space. The latent heat term hl vs can be ‘‘very effective in increas- ing the thermal conductivity of soils, since it multiplies the conductivity of the air- filled pores by a factor ranging from 2 at 0Њ C to 20 near 60ЊC’’ (de Vries, 1975). The advantage of this second method, albeit more approximate, is that it incorpo- rates thermal vapor transfer into a single macroscopic conductivity, l, effectively decoupling the heat and moisture flow equations. It does not, of course, account for heat transfer induced by a moisture gradient. 546 Buchan Copyright © 2000 Marcel Dekker, Inc. The theory of coupled flows in porous media can be approached more ab- stractly using irreversible thermodynamics (de Vries, 1975; Raats, 1975; Sidi- ropoulos and Tzimopoulos, 1983). Essentially this provides only an overlying formalism for the above coupled-flow approach. Phenomenological transport co- efficients are introduced, but they still need to be derived using the mechanistic ideas of that approach. Flow coupling can accumulate to visible level under prolonged steady-state heat flow. This can lead to marked thermally induced redistribution of moisture (e.g., around underground cables or pipes, or in laboratory determination of l (Sect. III.D). D. Calculation of Thermal Properties Soil thermal conductivity and heat capacity depend on physical composition, es- pecially moisture content, so single measurements are of limited use. Theory to predict the variation with moisture content is thus required. 1. Volumetric Heat Capacity, C The heat capacity C of a unit volume of soil is, simply and exactly, the sum of the heat capacities of its phases (de Vries, 1975): C ϭ xC ϩ xC ϩ xC mm oo ww 6 Ϫ3 Ϫ1 ϭ 4.18 ϫ 10 (0.46x ϩ 0.60x ϩ x ) J m K (17) mow where x denotes the volume fraction and C the volumetric heat capacity of a phase, with subscripts m, o, and w indicating mineral solids, organic matter, and liquid water, respectively. Air (moist) makes a negligible contribution. Table 1 shows thermal properties. Soil Temperature Regime 547 Table 1 Thermal Properties of the Principal Soil Phases (Solids at 10ЊC, Ice at 0ЊC) Material Volumetric heat capacity, C (MJ m Ϫ 3 K Ϫ 1 ) Thermal conductivity (W m Ϫ 1 K Ϫ 1 ) Quartz 2.0 8.8 Clay minerals 2.0 2.9 Organic matter 2.5 0.25 Water 4.2 0.552 ϩ 2.34 ϫ 10 Ϫ 3 T Ϫ 1.10 ϫ 10 Ϫ 5 T 2 Ice 1.9 2.2 Air 1.25 ϫ 10 Ϫ 3 0.0237 ϩ 0.000064T a a T in degrees Celsius. Source: de Vries (1975); Hopmans and Dane (1986a). Copyright © 2000 Marcel Dekker, Inc. 2. Thermal Conductivity, l The macroscopic conductivity l of Eq. 5 summarizes a heat flow that is spatially averaged over microscopically complex paths and so cannot be calculated exactly. An approximate ‘‘dielectric analog’’ model was developed by de Vries (1963), by application to a granular medium of ‘‘potential theory,’’ which treats systems in which an induced response (here, a flow of heat) at any point is proportional to the local gradient of a potential (here temperature). Figure 2 shows typical varia- tions of l with water content for sand, loam, and peat soils. The model views soil as a continuous medium (subscript c, either liquid water in moist soil, or air in drier soil), with volume fraction x c and conductivity l c , in which are dispersed regularly shaped ‘‘granules’’ of the other four compo- nents (either air or water, plus quartz, clay, and organic matter). The overall con- ductivity is then a weighted mean of the component conductivities (Table 1), x l ϩ͚kxl cc jjj l ϭ (18) x ϩ͚kx cjj 548 Buchan Fig. 2 Variation of soil thermal conductivity (solid curves) and diffusivity (broken curves) with volumetric water content u for (1) quartz sand (x m ϭ 0.55); (2) loam (x m ϩ x 0 ϭ 0.50); (3) peat (x 0 ϭ 0.20). (From de Vries, 1975, Courtesy of Hemisphere Publ. Corp.) Copyright © 2000 Marcel Dekker, Inc. [...]... range) are for a specific, good commercial sensor, and may vary for other probes a Liquid-in-glass Liquid-in-glass thermometers remain the standard soil probes in the meteorological services of many countries Spirit-in-glass types placed just above the surface (e.g., grass, bare soil) measure minima (e.g., grass minimum) In -soil probes are invariably mercury-in-glass, with two main types (Meteorological Office,... is 0 .14 for T0 but 0.24 for R s For the smoother annual wave (Fig 6), a two-harmonic fit is adequate for both soil (van Wijk and de Vries, 1963; Persaud and Chang, 1985) and air (Tabony, 1984) temperature In soil, A 2 /A 1 (typically 0.12 to 0.15; van Wijk and de Copyright © 2000 Marcel Dekker, Inc Soil Temperature Regime 557 Fig 6 Annual wave of soil temperature at 30 cm depth, Aberdeen, Scotland... secondary reference mercury-in-glass thermometer in a closely controlled temperature bath should suffice B Sampling and Smoothing Here we consider space and time sampling of soil temperature, and methods of data smoothing Heat flux is considered in Sec III.C A discussion of electrical/ electronic aspects of sampling (i.e., screening and grounding to reduce noise and interference, and sensor-recorder interfacing)... with the factors of 1 and 2 in Eq 19) emerges from averaging over random 3 3 granule orientation For both sand and clay soils, de Vries (1963) deduced representative averages n ϭ 5 and g 1 ϭ 0.125 for the soil particles The model, summarized as follows, subdivides the entire moisture range into four regions (Hopmans and Dane, 1986a) a Dry Soil Here air is the continuous medium, and large ratios l j... of soil water It is given by (Miller, 1980) L f DT c ϩp P ϭ m Ϫ i 273.15 r1 ri (35) where L f ϭ 3.33 ϫ 10 5 J kg Ϫ1 is the latent heat of fusion of ice, c m and p are the matric and osmotic components of the liquid water potential, r 1 and r i are the densities of liquid water and ice, and Pi is the ice pressure For soil with low heave pressure, or unsaturated soil (Fuchs et al., 1978), Pi ϭ 0, and. .. a ϭ 2.29 m From Eq 22, the conductive soil heat flux G T ϭ ϪlץT/ץz is G T (z, t ) A ͙lCnv expͩϪ Dz ͪsinͩnv t ϩ f M ϭ n 1 nϭ1 Copyright © 2000 Marcel Dekker, Inc 1 n n Ϫ ͪ z p ϩ Dn 4 (23) 554 Buchan Fig 4 Soil surface climate: 15-day average diurnal variations of bare soil surface temperature, T0 , showing measured data and one- and two-harmonic fits to data, and solar radiation, R s Note: Period... Freezing and Frozen Soil Soil water freezes either as polycrystalline ice within the soil matrix or as separate ice lens inclusions that accrete when water migrates towards a slowly moving freezing front Freezing brings large reduction in hydraulic conductivity and large increase in soil strength Frost heave, which can lift soil, roots, and overlying structures, occurs only at or close to saturation, and. .. (de Wit and van Keulen, 1972; Gerald and Wheatley, 1985) There are three main numerical methods, differing in the ways they divide the space-time grid into discrete elements, attribute variables (to either nodes or elements), and refine the time-integration They are 1 Finite difference, which assumes that node and time spacings are so small that parameters within them can be considered constant, and differentials... robustness, and high temperature coefficient (ca 10 times that of a PRT) They come in a variety of sizes suitable for soil use, down to Ͻ1 mm diameter catheter-types, excellent for point and surface contact measurement (Buchan, 1982b) Their large resistance minimizes interference and thermal emf errors, and it swamps connector and cable resistances, with cable errors typically only ca 0.001Њ C m Ϫ1 Self-heating... Fig 8) It can be handled using linearizing bridge circuitry, e.g., for the thermocouple (Woodward and Sheehy, 1983) or the strongly nonlinear thermistor (Fritschen and Gay, 1979) Table 3 Typical Error Requirements and Suitable Sensors Objective Plant response and function (Woodward and Sheehy, 1983) Validation of soil T prediction models Temperature gradients, spatial variability, physical management . of and in Eq. 19) emerges from averaging over random 12 33 granule orientation. For both sand and clay soils, de Vries (1963) deduced representative aver- ages n ϭ 5 and g 1 ϭ 0.125 for the soil. of coupled flows of heat and water, and of flow and phase-change processes in freez- ing soils; (2) applications, including (a) more realistic modeling of heat flow, or simultaneous heat and water flows,. the diurnal and annual waves, and noncyclic effects; ‘‘transient’’ solutions from Laplace transform and other methods; and numerical methods. The calculation of thermal properties from physical