The algorithm for calculating water density, ρw(TK), is Eq. (2.6) of Wagner and Pruess (2002) and employed only within the temperature range 273.15 K ≤TK≤ 383.15 K (≡TK,max). At temperatures greater than TK,max, then ρw(TK)=ρw(TK,max). This approach yields a range for ρw(TK) of 950 kg m-3<ρw(TK)< 1000 kg m-3, which represents a compromise between the fact that the density of (free saturated liquid) water continues to decrease with increasing temperatures (Yaws, 1995) and the hypothetical possibility that in a bound state a mono-layer of liquid water ρw(TK) may reach values as high as ≈ 5000 kg m-3 (Danielewicz-Ferchmin and Mickiewicz, 1996). dρw/dT is computed from the analytical expression derived by differentiating the expression for ρw(TK) and dρw/dT=0 for TK>TK,max.

The enthalpy of vaporization of water, Hv=Hv(TK,ψ) [J mol-1], is Eq. (5) of Somayajulu (1988) augmented by the soil moisture potential, ψ, (Massman, 2012; Campbell et al., 1995) and is expressed as follows: Hv=H1Tcrit-TKTK+H2Tcrit-TKTcrit38+H3Tcrit-TKTcrit94-Mwψ, where H1=13.405538 kJ mol-1, H2=54.188028 kJ mol-1, H3=-58.822461 kJ mol-1, and Tcrit=647.096 K is the critical temperature for water. Note that hv=hv(TK) [J mol-1] will denote the enthalpy of vaporization without the additional -Mwψ term, i.e., hv=hv(TK)=H1Tcrit-TK/TK+H2Tcrit-TK/Tcrit38+H3Tcrit-TK/Tcrit94. The present formulation differs from Massman (2012) because here hv(TK≥Tcrit)=0; whereas the equivalent from Massman (2012) (and Campbell et al., 1995) hv was a linear approximation of the present hv, which yielded hv(TK≥Tcrit)≫0. This distinction will become important when discussing the water vapor source term, Sv. Note that because Lv=Hv/Mw it also employs Eq. ().

The formulations for thermal conductivity of water vapor, λv=λv(TK) [W m-1 K-1], and liquid water, λw=λw(TK,ρw) [W m-1 K-1], are taken from Huber et al. (2012). For water vapor their Eq. (4) is used and for liquid water the product of their Eqs. (4) and (5) is used. The formulations for viscosity of water vapor and liquid water are taken from Huber et al. (2009) and are similar algorithmically to thermal conductivity. For water vapor, μv=μv(TK) [kg m-1 s-1≡ Pa s], their Eq. (11) is used and for liquid water, μw=μw(TK,ρw) [Pa s], their Eq. (36) is used. For liquid water both these formulations include a dependence on the density of water. Consequently, once soil temperature exceeds TK,max both λw and μw are assigned a fixed value determined at TK,max. On the other hand, λv and μv increase continually with increasing temperatures.

The formulation for the thermal conductivity of dry air, λd=λd(TK) [W m-1 K-1], is Eq. (5a) of Kadoya et al. (1985) and for the viscosity of dry air, μd=μd(TK) [Pa s], Eq. (3a) of Kadoya et al. (1985) is used. The model of the thermal conductivity of soil atmosphere, λa=λa(λv,λd,μv,μd) [W m-1 K-1], is a nonlinear expression given by Eq. (28) of Tsilingiris (2008). The relative weights used in this formulation are determined using the mixing ratios for water vapor (χv [dimensionless]) and dry air (χd=1-χv): where χv=ev/(Pd+ev), and Pd [Pa] is the dry air pressure. Here Pd will be held constant and equal to the external ambient atmospheric pressure, Patmos (i.e., 92 kPa), during the laboratory experiments (see Massman, 2012; Campbell et al., 1995). The vapor pressure, ev, is obtained from ρv and TK using the ideal gas law.

The volumetric-specific heat for soil air, ρacpa [J m-3 K-1], is estimated for the soil atmosphere from ρacpa=cpvρv+cpdρd, where ρd=MdPatmos/(RTK) [kg m-3] is the dry air density; Md=0.02896 kg mol-1 is the molar mass of dry air; and the isobaric-specific heats for water vapor, cpv [J kg-1 K-1], and dry air, cpd [J kg-1 K-1], use Eq. (6) of Bücker et al. (2003).

The saturation vapor pressure, ev,sat=ev,sat(TK) [Pa], and its derivative, dev,sat/dT [Pa K-1], are modeled using Eqs. (2.5) and (2.5a) of Wagner and Pruess (2002). The saturation vapor density, ρv,sat [kg m-3], is modeled using Eq. (2.7) of Wagner and Pruess (2002). Following Massman (2012), these saturation curves are restricted to temperatures below that temperature, TK,sat, at which ev,sat=Patmos. For the present case, TK,sat=370.44 K was determined from the saturation temperature equation or “the Backward Equation”, Eq. (31) of IAPWS (2007). For TK≥TK,sat, the saturation quantities ev,sat and dev,sat/dT remain fixed at their values TK,sat, but ρv,sat is allowed to decrease with increasing temperatures, i.e., ρv,sat=ρv,sat(TK,sat)[TK,sat/TK][Patmos/PST], in accordance with Table 13.2 (p. 497) of Wagner and Pruess (2002), where PST=101325 Pa is the standard pressure. The present treatment of ρv,sat is different from Massman (2012), who assumed that ρv,sat(TK≥TK,sat)=ρv,sat(TK,sat).

Embedded in the hydraulic conductivities (KH and Kn of Eq. ) are the surface tension of water, σw(TK) [N m-1], and the static dielectric constant (or the relative permittivity) of water, ϵw(TK) [dimensionless]. These physical properties of water are integral to the hydraulic conductivity model from Zhang (2011) associated with film flow, which is incorporated into the present model and detailed in a later section. The surface tension of water is modeled following Eq. (1) of Vargaftik et al. (1983) and ϵw(TK) is taken from Eq. (36) of Fernández et al. (1997).

Dve is modeled as Dve=τ(η-θ)SFEDv1+ev/Patmos, where τ [m m-1] is the tortuosity of soil with τ=0.66[(η-θ)/η]3 after Moldrup et al. (1997); E [dimensionless] is the vapor flow enhancement factor and is discussed in Massman (2012); Dv/(1+ev/Patmos) [m2 s-1] is the molecular diffusivity of water vapor into the soil atmosphere, which will be taken as a mixture of both dry air and (potentially large amounts of) water vapor; and SF [dimensionless] is Stefan correction or mass flow factor. Externalizing the 1/(1+ev/Patmos) term of the vapor diffusivity and combining it with SF allows for the following approximation for SF/(1+ev/Patmos)=1/(1-ev2/Patmos2)≈1+ev/Patmos; where the correct form for SF is 1/(1-ev/Patmos). The reason for this approximation is to avoid dividing by 0 as ev→Patmos. Massman (2012) and Campbell et al. (1995) avoided this issue by limiting SF to a maximum value of 10/3. This newer approximation for SF/(1+ev/Patmos) is an improvement over their approach. It is relatively slower to enhance the vapor transport by diffusion than the original approach, but this is only of any real significance when SF≥1. On the other hand because the linear form is not limited to any preset maximum value, it compensates for these underestimations when SF>10/3.

Mindful of the externalization of 1/(1+ev/Patmos), Dv is estimated from the diffusivity of water vapor in dry air, Dvd=Dvd(TK) [m2 s-1] and the self-diffusivity of water vapor in water vapor, Dvv=Dvv(TK) [m2 s-1], where Dvd=DvdSTPSTPatmosTKTSTαvd and Dvv=DvvSTPSTPatmosTKTSTαvv and TST=273.15 K is the standard temperature, DvdST=2.12×10-5 m2 s-1, αvd=7/4, DvvST=1.39×10-5 m2 s-1, and αvv=9/4. The parameters DvvST and αvv relate to the self-diffusion of water vapor and their numerical values were determined from a synthesis of results from Hellmann et al. (2009) and Yoshida et al. (2006, 2007). The uncertainty associated with this value for DvvST is at least ±15% and possibly more, e.g., Miles et al. (2012). Blanc's law (Marrero and Mason, 1972) combines Dvd and Dvv to yield the following expressions for Dv=Dv(TK,ρv): 1Dv(TK,ρv)=1-χvDvd+χvDvvorDv(TK,ρv)=DvdDvv(1-χv)Dvv+χvDvd.

The present model for the advective velocity associated with the volatilization of water, uvl, is taken from Ki et al. (2005) and is non-equilibrium equivalent to that used by Massman (2012) in his equilibrium model. Here ∂uvl∂z=Sv(η-θ)ρv, where the basic assumptions are that both liquid water and vapor are Newtonian fluids and that only incompressible effects are being modeled. In essence Eq. () assumes that the vaporization of soil moisture acts as a steady-state (and rapidly expanding or “exploding”) volume source term, which yields a 1-D advective velocity associated with volatilization of liquid water. For an equilibrium model of soil moisture evaporation that does not include water movement (i.e., KH≡0, Kn≡0, and Vθ,surf≡0), then Sv≡-ρw∂θ/∂t (from Eq.  above), which demonstrates the connection between present model of uvl with that used by Massman (2012). But unlike Massman (2012), the present model does not require any numerical adjustments to Eq. () in order to maintain numerical stability.

The functional parameterization of Sv follows from the non-equilibrium assumption, i.e., Sv∝(ρve-ρv), where ρve=awρv,sat(TK) [kg m-3] is the equilibrium vapor density and aw=eMwψ*RTKψn [dimensionless] is the water activity, modeled here with the Kelvin equation. The difficult part is how to construct the proportionality coefficient. Nevertheless, there are at least a two ways to go about this: (a) largely empirically (e.g., Smits et al., 2011, and related approaches referenced therein); or (b) assume that Sv=AwaJv (e.g., Skopp, 1985, or Novak, 2012), where Awa [m-1] is the volume-normalized soil water–air interfacial surface area and Jv [kg m-2 s-1] is the flux to/from that interfacial surface. This second approach allows for a more physically based parameterization of the flux, viz., Jv=Rv(ρve-ρv), where Rv [ms-1] is the interfacial surface transfer coefficient. For example, Novak (2012) proposed that the flux be driven be diffusion, so that Rv=Dv/rep, where rep [m] is the equivalent pore radius and Dv is the diffusivity of water vapor in soil air. After a bit of algebra and some simple geometrically based assumptions concerning the relationships between rep, a spherical pore volume, and Awa, one arrives at the Novak model of the source term: Sv[N]=S*[N]Awa2Dv(ρve-ρv), where S*[N] [dimensionless] is an adjustable model parameter.

But there is another way to model the vapor flux, Jv, which is also used in the present study. This second approach is based on the Hertz–Knudsen equation, which has its origins in the kinetic theory of gases and describes the net flux of a gas that is simultaneously condensing on and evaporating from a surface. A general expression for the Hertz–Knudsen flux is Jv=RTK/Mw(Keρve-Kcρv), where Ke [dimensionless] is the mass accommodation (or evaporation) coefficient and Kc=Kc(TK,ψn) [dimensionless] is the thermal accommodation (or condensation) coefficient. For the present purposes Ke≡1 can be assumed. This model of Jv yields the following model for Sv: Sv[M]=S*[M]AwaRTKMw(ρve-Kcρv), where S*[M] [dimensionless] is an adjustable model parameter, to be determined by “tuning” it as necessary to ensure model stability. This model for Sv[M] is now more or less complete, but the model for Sv[N] is neither quite complete nor precisely comparable to Sv[M]. This is now remedied by introducing Kc into Sv[N] and subsuming a factor of Awa into S*[N], yielding: Sv[N]=S*[N]AwaDv(ρve-Kcρv), where S*[N] [m-1] now has physical dimensions, but otherwise remains an adjustable parameter that will be scaled such that Sv[N]≈O(Sv[M]). In this form these two models for Sv can be used to test the sensitivity of the model's solution to different temperature forcing, because Sv[M]∝TK, whereas Sv[N]∝TKα, where α≥2.

Concluding the development of Sv requires models of Kc and Awa=Awa(θ). Kc is parameterized as Kc(TK,ψn)=eEav-MwψR1TK-1TK,in, where TK,in [K] is the initial temperature of the laboratory experiments and Eav-Mwψ [J mol-1] is an empirically determined surface condensation/evaporation activation energy. (Note that (a) TK≥TK,in, valid most of the time for any simulation, guarantees Kc≤1. (b) The enthalpy of vaporization, hv(TK), is a logical choice for Eav; but, model performance was significantly enhanced by simply assigning a constant value for Eav≈ 30–40 kJ mol-1 rather than assigning Eav≡hv.) The present formulation ensures that ∂Kc/∂TK<0, in accordance with experimental and theoretical studies (Tsuruta and Nagayama, 2004; Kon et al., 2014). Mathematically this present formulation of Kc largely eliminates model instabilities by suppressing condensation relative to evaporation throughout the experiment and will be discussed in greater detail in a later section.

Awa is parameterized as a parabolic function to simulate the conceptual model of Awa proposed by Constanza-Robinson and Brusseau (2002; see their Fig. 1b): Awa(θ)=Sw(1-Sw)a1+a2Sw(1-Sw)a3, where Sw=θ/η is the soil water saturation and a1=40, a2=0.003, and a3=1/8. This particular functional form ensures that Awa=0 when the soil is completely dry, θ=0, and when fully saturated, Sw=1. This particular parameter value for a1 was chosen so that the maximum value of Awa occurs at Sw=0.025 (i.e., 1/a1) in accordance with the model of Constanza-Robinson and Brusseau (2002).

The model for Cs(T,θ) is taken from Massman (2012): Cs(θ,T)=cs(T)ρb+Cw(T)θ, where ρb [kg m-3] is the soil bulk density; cs(T)=cs0+cs1T [J kg-1 K-1] is the specific heat capacity of soil; Cw(T)=Cw0+Cw1T+Cw2T2 [J m-3 K-1] is the volumetric heat capacity of water; and the parameterization constants cs0,cs1,Cw0,Cw1,Cw2 are given by Massman (2012). Note that the present Cs(θ,T) results from approximating Cw(T)≡cpw(T)ρw(TK) by cpw(T)ρw(TST), where cpw(T) [J kg-1 K-1] is the isobaric-specific heat capacity of water, and ρw(TST)=1000 kg m-3. This substitution for ρw(TK) is only made for Cw(T).

The present formulation for isobaric heat capacity of water, cpw(T), was developed from Yaws (1995) and confirmed by comparing to Wagner and Pruess (2002). In general cpw(T) is also a function of pressure (e.g., Wagner and Pruess, 2002), but this dependency can be ignored for the present purposes. Other parameterization of cpw(T) (i.e., Sato, 1990; Jovanović et al., 2009; Kozlowski, 2012) were also examined, but proved unsatisfactory. Finally, Kozlowski (2012) reported numerical values for the dry soil parameters cs0 and cs1 that are similar to those discussed in Massman (2012) and used with the present model.

The model of soil thermal conductivity, λs, is the sum of two terms. The first, λs[1](θ,TK,ρv), is taken principally from Campbell et al. (1994) and the second, λs[2](θ,TK), is taken from Bauer (1993). This second term incorporates the effects of high-temperature thermal (infrared) radiant energy transfer within the soil pore space, which may be significant for certain soils and high enough temperatures (e.g., Durany et al., 2010). λs[1] is summarized first and repeated here to emphasize the difference between the present model's functional parameterizations and those used in Massman (2012).

λs[1] is modeled as λs[1](θ,TK,ρv)=kwθλw(TK,ρw)+ka[η-θ]λa*(θ,TK,ρv)+km[1-η]λmkwθ+ka[η-θ]+km[1-η], where λa*(θ,TK,ρv)=λa(TK)+λv*(θ,TK,ρv) is the apparent thermal conductivity of the soil air and is the sum of the thermal conductivity of moist soil air, λa, and λv*, which incorporates the effects of latent heat transfer; λm is the thermal conductivity of the mineral component of the soil, which is assumed to be independent of temperature and soil moisture; and kw, ka, and km [dimensionless] are generalized formulations of the de Vries weighting factors (de Vries, 1963). Campbell et al. (1994) formulated λv* as proportional to the product of the enthalpy of vaporization (hv), the vapor diffusivity (Dv), the Stefan factor (SF), the slope of the saturation vapor pressure (dev,sat/dT), and the parameter fw(θ,T). For the present model λv* is λv*(θ,TK,ρv)=ρ^SThvfwSFDv[dev,sat/dT]Patmos, where ρ^ST=44.65 mol m-3 is the molar density of the standard atmosphere. Equations () and () are the same as those used in Massman (2012), but numerically they yield quite different results due to the different formulations for hv, SF, Dv, and ev,sat=ev,sat(TK). Otherwise the de Vries (1963) shape factors, the parameter fw, and all related parameters are the same as in Massman (2012).

λs[2] is modeled as λs[2](θ,TK)=3.8σN2RpTK3, where σ=5.670×10-8 W m-2 K-4 is the Stefan–Boltzmann constant; N=N(θ)=1+θ/(3η) is the medium's [dimensionless] index of refraction; Rp [m] is the soil's pore space volumetric radius; and the factor of 3.8 subsumes a numerical factor of 4, a [dimensionless] pore shape factor (i.e., 1 for spherical particles), and the [dimensionless] emissivity of the medium ≈0.95 (by assumption). In general Rp is considered to be a model parameter and it will be varied to evaluate the model's sensitivity and performance relative to λs[2] and unless otherwise indicated, Rp=10-3 m is the nominal or default value. In the present context variations in Rp are purely model driven, but in reality such variations are likely to be most strongly associated with (or proportional to) changes in the soil particle dimensions and secondarily with other soil characteristics (e.g., Leij et al., 2002).

In general a WRC is a functional relationship between soil moisture and soil moisture potential and temperature, i.e., θ=θ(ψ,T), although the temperature dependency is often ignored and was of little consequence to the model of Massman (2012). The three WRCs tested in the present study have been adapted to include a residual soil moisture, θr=θr(ψ,T) [m3 m-3], which is an atypical parameterization for both θr and the soil moisture's temperature dependency. Under more normal soil environmental conditions θr is assumed to be bound so securely to soil mineral surfaces that it is normally taken as a fixed constant. For the present purposes, the principal WRC is adapted from Massman (2012) and is θ(ψn,TK)=-θlαlln⁡ψn+θh-θr(ψn,TK)1+αhψn4-1p+θr(ψn,TK), where θr(ψn,TK)=θr*eb1Eav(1-b2ψn)R1TK-1TK,in and θl [m3 m-3] is the extrapolated value of the water content when ψ=ψl=-1 J kg-1; αl=ln⁡(ψ*/ψl)=13.8155106; and θh [m3 m-3], αh [dimensionless] and p [dimensionless] are parameters obtained from Campbell and Shiozawa (1992); θr* [m3 m-3] is a constant soil-specific parameter, such that θr*≤0.03 is to be expected; and b1 [dimensionless] and b2 [dimensionless] are adjustable parameters, which are expected to satisfy b1>0 and 0≤b2<1. Note that a further discussion concerning the original version of Eq. () can be found in Massman (2012).

There is a simple and physically intuitive argument for the parameterization of θr(ψn,TK) in Eq. (). First, under more normal soil environmental conditions, i.e., TK≈TK,in and at least ψn<1 (if not ψn≪1), then it is reasonable to expect that θr≈θr* and nearly constant throughout (what might be expected to be relatively small variations in) those conditions. But as the temperature increases, it is also reasonable to assume that the increasing amounts of thermal energy will begin to overcome the forces holding the bound water to the soil mineral surfaces and that θr will decrease. Mathematically, one might therefore expect that ∂θr/∂TK<0. Massman (2012; see his discussion of ψT) made similar arguments when he included the temperature dependency in his version of the same basic WRC. Consequently, the Eqs. () and () above offer a different approach to including temperature effects on the WRC that maintains the temperature dependent properties of WRCs outlined by Massman (2012). Second, as the temperature increases and the soil moisture (including θr) begins to decrease, the soil moisture potential ψn will begin to increase (or ψ<0 will decrease in absolute terms while increasing in magnitude), which in turn (it is hypothesized) will tend to strengthen the forces holding the bound water. Therefore, one might expect that as the soil dries out ∂θr/∂ψn>0, which will oppose, but not dominate the temperature effects (because b2<1). Equation () is designed to capture these two opposing influences, assuming of course that TK>TK,in. But, it is itself not intended to be a fully physically based dynamical theory of the residual soil moisture. Such a theory is beyond the intent of the present study. The sole intent here is to test and evaluate whether a dynamical θr can improve the model's performance. In so far as it may succeed at doing so, it will also indicate the value and need for a more detailed physically based dynamical model of θr.

The present study also includes similar adaptations to two other WRCs so as to test the model's sensitivity to different WRCs. These WRCs, which will not be shown here, are taken from Groenevelt and Grant (2004) and Fredlund and Xing (1994).

The hydraulic conductivity functions, Kn(TK,ψn,θ) and KH(TK,ψn,θ), are given as follows: Kn=KIKRρwμwψ*andKH=KIKRρwμwg, where g = 9.81 m s-2 is the acceleration due to gravity; KI [m2] is the intrinsic permeability of the soil (a constant for any given soil); and KR=KR(θ,θr,ψn,TK) [dimensionless] is the hydraulic conductivity function (HCF), which for the present study is taken as the sum of KRcap (associated with capillary flow) and KRfilm (associated with film flow). The model for intrinsic permeability, which is taken from Bear (1972), is KI=(6.17×10-4)dg2; where dg [m] is the mean or “effective” soil particle diameter. For the soils used in the present work (Campbell et al., 1995; Massman, 2012) dg was estimated from Shiozawa and Campbell (1991) and Campbell and Shiozawa (1992) or simply assigned a reasonable value if no other information was available.

For the present study, five difference parameterizations for KRcap were tested: two were from Grant et al. (2010), i.e., their Eq. (18) (Burdine) and Eq. (19) (Mualem); the Van Genuchten and Nielson (1985) model, their Eq. (22) with the mathematical constraints imposed as suggested by Assouline and Or (2013); the Brooks and Corey (1964) model; and Eq. (18) of Assouline (2001). The reason for testing several models of the HCF is to determine how different formulations for the HCF might impact the model's performance when comparing to the laboratory observations. The following HCF is the model of Assouline (2001), which is a relatively simple formulation for the HCF and serves as the reference HCF for the model simulations. KRcap(θ,θr)=1-1-θ-θr(ψn,TK)η1mmn where for the present application 0<m<1, and n>1.

KRfilm(TK,ψn) is taken from Zhang (2011) and is expressed, in the present notation, as KRfilm(TK,ψn)=Rw(TK)6.17×10-42σw2σw-ρwdgψ*ψn32 and Rw(TK)=2π2(1-η)Fwϵ0ϵw2σwdg12kBTKza3, where (Sect. 2.2.1) σw(TK) is the surface tension of water and ϵw(TK) is the static dielectric constant or relative permittivity of water; ϵ0=8.85×10-12 C2 J-1 m-1 is the permittivity of free space; kB=1.308568×10-23 J K-1 is the Boltzmann constant; a=1.6021773×10-19 C is the electron charge; z [dimensionless] is the ion charge, for which z=1 can be assumed; and Fw [dimensionless] is a soil-specific parameter, for which Zhang (2011) found that (roughly) 10<Fw<104.

The term ρwVθ,surf in Eq. () represents the soil moisture movement caused by water molecules “hopping” or “skipping” along the surface of the water films due to a temperature gradient (e.g., Medveď and Černý, 2011). The present model for Vθ,surf is adapted from the model of Gawin et al. (1999) and is given as Vθ,surf=-Dθs∂θ∂z=-DθsDθψ∂ψn∂z-DθsDθT∂T∂z, where Dθs=Dθs(TK,θ) [m2 s-1] is the surface diffusivity and is parameterized as Dθs=Dθs0exp⁡-2θθbβTSTTK, where Dθs0≈10-10 m2 s-1; θb≈0.02; and β=1/4 when θ≥θb or otherwise β=1 when θ<θb.

By expressing Vθ,surf in terms of the gradient of the “normalized” soil moisture potential, ψn, in Eq. (), Kn∗ and Km, used in Eqs. () and (), can be identified as Kn∗=Kn+DθsDθψ and Km=DθsDθT, where DθsDθψ will be defined as Knsurf, where Knsurf≤0∀TK and θ.