Modeling soil heating and moisture transport under extreme conditions: Forest fires and slash pile burns


Corresponding author: W. J. Massman, Rocky Mountain Research Station, Forest Service, USDA, 240 W. Prospect Rd., Fort Collins, CO 80526, USA. (


[1] Heating any soil during a sufficiently intense wildfire or prescribed burn can alter it irreversibly, causing many significant, long-term biological, chemical, and hydrological effects. Given the climate-change-driven increasing probability of wildfires and the increasing use of prescribed burns by land managers, it is important to better understand the dynamics of the coupled heat and moisture transport in soil during these extreme heating events. Furthermore, improved understanding and modeling of heat and mass transport during extreme conditions should provide insights into the associated transport mechanisms under more normal conditions. The present study describes a numerical model developed to simulate soil heat and moisture transport during fires where the surface heating often ranges between 10,000 and 100,000 W m−2 for several minutes to several hours. Basically, the model extends methods commonly used to model coupled heat flow and moisture evaporation at ambient conditions into regions of extreme dryness and heat. But it also incorporates some infrequently used formulations for temperature dependencies of the soil specific heat, thermal conductivity, and the water retention curve, as well as advective effects due to the large changes in volume that occur when liquid water is rapidly volatilized. Model performance is tested against laboratory measurements of soil temperature and moisture changes at several depths during controlled heating events. Qualitatively, the model agrees with the laboratory observations, namely, it simulates an increase in soil moisture ahead of the drying front (due to the condensation of evaporated soil water at the front) and a hiatus in the soil temperature rise during the strongly evaporative stage of the soil drying. Nevertheless, it is shown that the model is incapable of producing a physically realistic solution because it does not (and, in fact, cannot) represent the relationship between soil water potential and soil moisture at extremely low soil moisture contents (i.e., residual or bound water: θ < 0.01 m3 m−3, for example). Diagnosing the model's performance yields important insights into how to make progress on modeling soil evaporation and heating under conditions of high temperatures and very low soil moisture content.

1. Introduction

[2] Most studies of the coupled heat and moisture transport in soils have concentrated primarily on conditions that encompass “normal” ambient environmental conditions, i.e., those involving daily and seasonal variations in radiation, temperature, precipitation, etc. [e.g., Philip and De Vries, 1957; Milly, 1982; Novak, 2010; Smits et al., 2011]. But, a few studies have examined these processes under more extreme conditions, i.e., those occurring during wildfires and prescribed burns [e.g., Aston and Gill, 1976; Campbell et al., 1995; Durany et al., 2010]. Although the physical principles and the basic equations of all these models are more or less the same, the emphasis of the different modeling regimes (i.e., normal versus extreme conditions) is different. Models developed for the former tend to focus on the movement of soil moisture and evaporation; whereas models developed to describe the latter emphasize soil temperatures and the duration of the soil heating.

[3] Amount and duration of soil heating, which in a modeling context are provided by the upper boundary condition, determine the depth of heat penetration. Associated with this depth are certain critical temperature thresholds. For example, soil temperatures in the range of 60–80 C for short periods of time are lethal to plant seeds, plant roots, and plant tissue in general and at temperatures approaching 120–160 C range microbial life is extinguished [e.g., Levitt, 1980; Jimènez Esquilín et al., 2007; Choczynska and Johnson, 2009]. At higher temperatures often irreversible physical, chemical, mineral, and hydrologic changes begin to occur to the soil [e.g., DeBano et al., 1998; Neary et al., 2005; Cerdà and Robichaud, 2009; Massman et al., 2010].

[4] As a consequence of this historical emphasis on soil heating and temperatures during fires, there have been many modeling studies that have focused solely on soil temperatures and the associated heat flow during these extreme events [e.g., Pafford et al., 1985; Chinanzvanana et al., 1986; Steward et al., 1990; Oliveria et al., 1997; Preisler et al., 2000; Antilén et al., 2006]. But modeling heat and moisture transport simultaneously is a much more difficult task and although such models [i.e., Aston and Gill, 1976; Campbell et al., 1995; Durany et al., 2010] have had some success at modeling soil temperatures during severe heating events, they have yielded somewhat disappointing simulations of the coupled soil moisture dynamics. For example, Albini et al. [1996] review the models of Aston and Gill [1976] and Campbell et al. [1995] and found that the earlier model was prone to instabilities and that the later model did not provide a fully faithful simulation of soil moisture content; similar conclusions were reached by Campbell et al. [1995] themselves. Likewise, the model of Durany et al. [2010] provides reasonable model performance when simulating soil temperatures, but the model “retained” soil volumetric moisture contents as high as 0.15 despite heating the soils to over 500 C. The present study attempts to diagnose these systemic failures at modeling coupled heat and moisture transport during the extreme heating events that occur during fires through the development and evaluation of the performance of a new numerical model for simulating coupled heat and moisture transport during fires.

2. Physical Processes and Model Equations

[5] The present model is a one-dimensional model that uses some of the same parameterizations and employs some of the same basic assumptions as Campbell et al. [1995]; but it also differs from this older model in many important aspects as well. One key similarity between these two models is that neither explicitly allows for the vertical movement of water during a soil heating event. This seems reasonable for the laboratory studies used to evaluate both the older model and present model because they employed cylindrical containers with solid bottoms, which preclude the possibility of a draining soil column. Nonetheless, future research (models) should examine whether the extreme gradients in moisture potential created during a fire might be sufficient to cause capillary water rise toward the soil drying front. As with the older model, the present model also includes advective (mass flow) effects associated with the rapid volatilization (and associated gas expansion) of the soil moisture during the heating events. On the other hand, the present model does not include any induced advective currents that may be caused by external temperature or pressure (nonvolatilization) effects. But this is not to say that such advective effects are not important or significant. In fact, they can completely overwhelm the normal diffusive mass fluxes exiting of the soil by directly injecting large quantities of combustion products into the soil [Massman et al., 2010]. But as these in situ field observations of Massman et al. [2010] show, such effects are likely to be two-dimensional in nature (at least for a slash pile burn) and are likely to be driven by complex effects associated with the radiation and convection associated with the fire itself. Consequently these issues are not addressed with the present study because they are beyond its scope. (Note that the one-dimensional model of Durany et al. [2010] includes both vertical water movement and implicitly includes any possible one-dimensional (nonvolatilization) advective effects as well.)

[6] There are several ways the present model differs from that of Campbell et al. [1995]. First, the present model assumes that the soil volumetric specific heat is temperature dependent. Second, it also uses a different soil water retention curve, which better describes the moist end of the water retention curve than did the older model. Third, the present model also uses a different formulation for soil surface evaporative flux and also employs extrapolative lower boundary conditions rather than specifying them directly from observations. All these changes are intended to make the present model more physically realistic and easier to use for application to real fires.

2.1. Mass Conservation For Soil Moisture

[7] The total volumetric water content of a porous medium is math formula; where math formula (103 kg m−3) is the density of water, math formula (kg m−3) is the water vapor density, θ (m3 m−3) is volumetric soil moisture, math formula (m3 m−3) is the total porosity of the soil, and ( math formula is the air filled porosity.

[8] After adjusting for advection and excluding soil hydraulic conductivity, Milly's [1982] results yield the following one-dimensional equation of mass conservation for water (liquid plus vapor):

display math

where t (s) is time, z (m) is soil depth (positive downward), Dve (m2 s−1) is the (equivalent) molecular diffusivity associated with the diffusive transport of water vapor in the soil's air-filled pore space, and uvl (m s−1) is the advective velocity associated with volatilization, which is discussed in more detail in section 2.5. Dve is expressed as

display math

where τ (m m−1) is the tortuosity of soil, math formula (dimensionless) is the vapor flow enhancement factor, ev (Pa) is the soil vapor pressure, and Pa (Pa) is the ambient pressure, Dv (m2 s−1) is the molecular diffusivity of water vapor in air, and the math formula term in the denominator is the Stefan correction, also termed the mass flow factor. (Note that for modeling purposes the Stefan term is limited to a maximum value of 10/3 (i.e., math formula). This is similar to Campbell et al. [1995] and is required to minimize numerical problems caused by ev becoming large with increasing temperatures, which will be detailed below.)

[9] In turn, Dv is a function of temperature and pressure:

display math

where math formula m2 s−1, math formula is the standard pressure, TK (K) is the soil temperature, math formula is the standard temperature, and the exponent math formula. There are several possible models for τ, three of which are math formula [Penman, 1940a, 1940b], math formula [Du Plessis and Masliyah 1991], and math formula [Moldrup et al., 1997]. The present study uses the model of Du Plessis and Masliyah [1991]. The parameterization of the enhancement factor is taken directly from Campbell et al. [1995]: math formula, where the parameters fw and ka (discussed in section 2.4, which follows) are taken from the soil thermal conductivity model of Campbell et al. [1994]. But because math formula the term math formula in equation (1) can be approximated by math formula. Furthermore, math formula can also to be expected. Therefore, the conservation of mass for soil moisture for a nondraining soil, equation (1), can be reasonably approximated as follows:

display math

2.2. The Kelvin Equation and Saturation Vapor Density

[10] The Kelvin equation is used to model the relative humidity in the soil pore space, hs, as follows:

display math

where math formula (J kg−1) is the soil moisture potential, math formula is the molecular mass of water vapor, and R = 8.314 J mol−1 K−1 is the universal gas constant. This last relation is important in order to relate the soil moisture variables, math formula and ev to math formula. This is given next.

display math

where math formula (Pa) is the saturation vapor pressure, which is a function of TK. Following Campbell et al. [1995], the present study also uses Richards' [1971] formulation for esat, i.e.,

display math

where ϑ (dimensionless) is the reduced temperature, math formula, and math formula. Although Richards Equation was originally developed for temperatures between −50 C and 140 C, it nevertheless is accurate up to the critical temperature of water, math formula (or 647.096 K). Assuming that water vapor is an ideal gas, then it follows that the saturation vapor density, math formula (kg m−3), is given as math formula.

[11] It is important to recognize that equation (5) should be restricted to temperatures, not only below the critical temperature, Tcrit, but to all temperatures below that temperature, math formula, at which the saturation vapor pressure becomes comparable to the ambient pressure; i.e., math formula. For applications near one earth's atmosphere this yields: math formula K. For the present study because Pa = 92,000 Pa at the location of the laboratory where the fire heating experiments were performed math formula K. Unfortunately this lower temperature limit cannot be imposed on the present model without producing either an instability or a loss of convergence. The reason for this (and possible solutions to it and other model performance issues) can only be discussed in the context of other more fundamental model limitations and only after displaying and diagnosing the mathematical solution produced by the model. Therefore, for the time being the only limit imposed on equation (5) and math formula will be that math formula and math formula whenever math formula.

2.3. Model Water Retention Curve

[12] Mathematically, the water retention curve (WRC) is denoted most generally as math formula, i.e., soil moisture is a function of both moisture potential and temperature. Historically most of the functions developed to describe WRCs have focused solely on math formula and there have been a variety of functions developed over the past several decades for specifying this simpler relationship [e.g., Fredlund and Xing, 1994]. The model of math formula used by Campbell et al. [1995] is the following one-parameter water retention curve originally developed by Campbell et al. [1993].

display math

where math formula is the water potential for oven dry soil, math formula is the extrapolated value of the water content when math formula −1 J kg−1, and math formula. Although equation (6) may be adequate at the dry end of the water retention curve [Resurreccion et al., 2011], it is not necessarily adequate at the moist end [Campbell and Shiozawa, 1992].

[13] The present study uses the following relation, adapted from Campbell and Shiozawa [1992] and further augmented from the original by a temperature function math formula, to model the water retention curve as a function of both temperature and moisture potential:

display math

where math formula (m3 m−3), math formula (dimensionless), and m (dimensionless) are parameters obtained from Campbell and Shiozawa [1992]. The specific relationship between the parameters math formula and math formula of the present model and the equivalent parameters A and α of Campbell and Shiozawa [1992] are (1) math formula, where math formula (kg m−3) is the bulk density of dry soil, and (2) math formula, where g (9.81 m s−2) is the acceleration due to gravity. It should also be mentioned that the soil moisture variable coded into the model itself is not actually math formula, but rather math formula, which is mentioned here solely to explain the use of this ratio in equation (7).

[14] In this last equation math formula (dimensionless) results from the surface tension of water being temperature dependent [e.g., Vargaftik et al., 1983; Kestin et al., 1984] and for the present modeling purposes math formula is in essence the temperature function of Salager et al. [2010], which is given as

display math

where math formula (K) is a soil (or porous media) dependent parameter, which may range approximately between −500 K and −350 K [Bachmann and van der Ploeg, 2002] and Tref (K) is a reference temperature, assumed here to always be approximately 293.15 K = 20 C. For the present purposes math formula and because the initial soil temperatures during the laboratory studies of soil heating are always near 293.15 K, Tref is always assigned the same value as the initial soil temperature. One important requirement of math formula (or any other comparable function) is that it must (and in the present case does) ensure that math formula in accordance with observations [e.g., Olivella and Gens, 2000; Salager et al., 2010]. Because math formula needs to be fulfilled for all soil temperatures, this last requirement on math formula means that for any soil temperature TK, which is greater than some cutoff temperature, Tcut (where math formula), then math formula. For the present model Tcut is defined as that temperature such that math formula. In other words, to avoid possible numerical problems math formula is required during all model simulations.

2.4. Conservation of Energy and Heat Flow

[15] Including the evaporation of soil moisture in the soil heat flow equation yields the following conservation of energy for the coupled soil heating and evaporation model:

display math

where T (C) is soil temperature; math formula (J m−3 K−1) is the volumetric heat capacity of soil, which is a function of both soil temperature and soil moisture; math formula (J kg−1) is the latent heat of vaporization (which is taken from Campbell et al. [1995]); math formula (W m−1 K−1) is the thermal conductivity of the soil, which is also a function of both soil temperature and soil moisture; math formula (kg m−3) is the mass density of the soil air, also a function of temperature; and cpa (J kg−1 K−1) is specific heat capacity of ambient air, which is a function of both temperature and the vapor density. But for the present purposes math formula K−1 is a reasonable assumption.

[16] The model of soil thermal conductivity, math formula, is taken directly from Campbell et al. [1994] and is summarized as follows (by using functional notation, math formula, to specify the specific dependencies of the model variables on temperature and soil moisture).

display math

where math formula is the thermal conductivity of water; math formula is the apparent thermal conductivity of air and is the sum of the thermal conductivity of dry air, math formula, and water vapor, math formula, which incorporates the effects of latent heat transfer; math formula 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]. In general, Campbell et al. [1994] formulate math formula as a function of the latent heat of vaporization, the vapor diffusivity, Dv, the Stefan factor, the slope of the saturation vapor pressure, and the parameter math formula (mentioned earlier in regards to the evaporation enhancement factor and to be discussed shortly).

[17] The de Vries weighting factors all have the same general form, which is

display math

where the subscript asterisk refers to water, air, or mineral, in accordance with notation of equation (10), and ga is the de Vries [1963] shape factor, an empirically determined model parameter, for which, in general, math formula [Campbell et al., 1994; G. S. Campbell, personal communication, 2011]. math formula is a weighted mixture of the thermal conductivities of air and water, which is given as

display math
display math

where math formula is a soil-specific empirically determined model parameter, math formula is a soil-specific empirically determined model function, and math formula depending on the degree of soil wetness: math formula for a dry soil and math formula for a wet soil. Numerical values for these model constructs are likewise obtained from Campbell et al. [1994] and Campbell (personal communication, 2011).

[18] The model for Cs is

display math

where math formula (J kg−1 K−1) is the specific heat capacity of soil; math formula (J m−3 K−1) is the volumetric heat capacity of water; and math formula, math formula, math formula, math formula, and math formula are model constants, of which four ( math formula, math formula, math formula, and math formula) are relatively easy to specify and are given next: math formula K−1 [after Campbell et al., 1995], math formula K−1, math formula K−2, and math formula K−3. But math formula (J kg−1 K−2) has not been measured for many soils. The data and analysis of Kay and Goit [1975] suggest that math formula increases linearly with temperature with a slope, math formula, ranging between math formula K−2 and math formula K−2. But these data are valid only between −73 C and 27 C, so it would seem worthwhile to examine math formula at soil temperatures that occur during fires. Figure 1 presents such data and shows the variation in heat capacity of a dry soil as a function of temperature in the range 25 C to 300 C. These data were obtained in the laboratory as part of a larger study of the long-term impact of soil heating on soils at the Manitou Experimental Forest (central Colorado, USA) [Massman et al., 2008, 2010]. These data suggest that math formula may not increase linearly with increasing temperature. Nonetheless, approximating these data by a linear function yields a value of math formula K−2, which is somewhat less than found by Kay and Goit [1975]. For the present study neither the exact functional dependency (linear versus nonlinear) nor the precise value of the slope, math formula, is particularly significant. Consequently, math formula will be assigned a value of 2.5 J kg−1 K−2 for the soils used in this study.

Figure 1.

Heat capacity of a dry soil as a function of temperature. Shaded area shows range of variation of laboratory observations from soil samples obtained at Manitou Experimental Forest in the central Rocky Mountains of Colorado (see Massman et al. [2008] for further details). Solid line is a subjective estimate of the observed temperature dependency, where the relevant parameter for modeling studies is the slope.

2.5. Advection Associated With Volatilization

[19] Assuming that vaporization of soil moisture acts as a volume source term, the 1-D advective velocity associated with volatilization of water, uvl, has been modeled using a steady state continuity equation [Ki et al., 2005] as

display math

Here the basic assumptions are that both liquid water and vapor are Newtonian fluids and that only incompressible effects are being modeled. Although equation (12) is not included in the formal mathematical development of the [Campbell et al., 1995] model, it is briefly discussed in the paper and it is clearly included in the original computer code.

[20] Unfortunately equation (12) proved impossible to use directly in the present model, because the model would not converge (i.e., the solution is nonexistent). Consequently, the present model ensures convergence by a simple multiplicative adjustment to equation (12), denoted math formula in the following expression:

display math

where math formula was determined by trial and error to always allow convergence. (Note that math formula is reexamined along with issues regarding math formula in section 4.2.)

[21] Also note that to ensure that the advective fluxes have the proper sign, i.e., mass and energy are transported toward the free atmosphere above the soil surface, the modeled shear, equations (12) and (13), must satisfy math formula at all times and depths, which will only occur during evaporation, i.e., when math formula.

3. Numerical Implementation

[22] The numerical model as outlined above and detailed in this section is coded as MATLAB (The MathWorks Inc., Natick, MA, version R2011b) script files.

3.1. Newton-Raphson Method

[23] The Newton-Raphson method, which is well described by Campbell [1985] and Lynch [2005], is used to solve the following finite difference forms of equations (9) and (2). Employing the fully implicit (Euler's) finite differencing scheme yields the following heat (Hi) and moisture (Wi) balance functions:

display math
display math

where math formula; math formula; and equations (14) and (15) are second order in the spatial derivatives and center differenced in the gradient-advective terms, math formula and math formula. math formula = 2 s is the nominal integration time step (although in some model simulations it was decreased in order to ensure numerical stability), h = 0.001 m is spatial (vertical) resolution of the model, the superscript j is the time step index, and the subscript i refers to the spatial index with math formula referring to the surface node and math formula referring to the bottom node. Here the spatial half index math formula for the soil thermal conductivity and equivalent vapor diffusivity, Dve, refers to the harmonic mean of their respective values at the i and the math formula nodes, which is in accordance with the flux preserving methods of finite differencing [Lynch, 2005].

[24] Implementing the Newton-Raphson method requires (1) calculating the Jacobian, i.e., the partial derivatives math formula, math formula, math formula, and math formula, and (2) inverting the resulting block-tridiagonal matrix for every time step, math formula, of the simulation. The partial derivatives of the Jacobian are specifically coded in the model as analytical expressions. The block tridiagonal is inverted for each time step with the generalized Thomas algorithm [Karlqvist, 1952; Thomas, 1995; Mendes et al., 2002] and the solution at each time is iterated until the following global energy convergence criterion is met:

display math

where math formula and the partial derivative math formula is determined analytically from the water retention curve, equation (7). Most of the time convergence is usually achieved in three to ten iterations, but can take as many as fifty or more. The tridiagonal form of the solution assumes the application of the upper and lower boundary conditions, which are outlined next.

3.2. Upper Boundary Condition

[25] The upper boundary condition is formulated in terms of the surface energy balance.

display math

where the 0 subscript refers to the surface and the terms from left to right are: the incoming or downwelling radiant energy, math formula (W m−2), absorbed by the surface, which is partitioned into the four terms (fluxes) on the right side of the equation: the infrared radiation lost by the surface, the surface sensible or convective heat, the surface latent heat, and the surface soil heat flux. math formula is a function of time and is prescribed as an external condition as discussed a little later. The soil surface emissivity is math formula (dimensionless), which is a function of soil moisture and parameterized as math formula when math formula and math formula when math formula. math formula W m−2 K−4 is the Stefan-Boltzmann constant.

[26] For the surface sensible heat flux, math formula, math formula (kg m−3) is the mass density of the ambient air at the surface temperature, math formula (K). Here math formula. CH (m s−1) is the transfer coefficient for convective heating. For the present model math formula (m s−1) is assumed as an approximate average of the wildland fire observations of Butler [2010]. math formula (C) is the ambient temperature somewhere above the soil surface and like incoming radiation is also prescribed externally.

[27] Nonetheless, despite the care taken in the present treatment of the surface convective heat flux, it was still necessary for the laboratory simulations used in the present study to impose the condition that math formula C before allowing this term to contribute to the surface energy balance. This is because math formula as a boundary forcing function will always increase faster than T0, which might be expected because as a model response variable T0 always lags behind math formula. That is, in reality for most model simulations developed for the present study, math formula C, which then allowed the sensible heat flux to become a significant (and physically implausible) heat source to the soil surface (often resulting in a numerical instability).

[28] The surface evaporation rate, E0 (kg m−2 s−1), is parameterized as the sum of a nonadvective (traditional) component and an advective component:

display math

where CE (m s−1) and CU (dimensionless) are adjustable model parameters transfer coefficients, which were determined empirically to maximize E0 without destabilizing the model. math formula is associated with the jet of volitalized air emanating from the soil and math formula m s−1. The surface humidity, math formula, is obtained from the Kelvin equation, equation (3). Finally, in the case of the laboratory experiments of Campbell et al. [1995], math formula, along with math formula and math formula, is an external forcing function at the soil surface.

[29] For the unique laboratory conditions of Campbell et al. [1995] these boundary forcing functions take the following generic form:

display math

where Vin is the value of the function at the beginning of the soil heating experiment, Vf is the value of the function at the end of the experiment, and τ(s) is a time constant of the heating source. All these input boundary condition parameter values vary with each individual soil heating experiment. Typical values for math formula are math formula C and math formula C, for vapor pressure, math formula, (from which math formula is computed by combining with math formula and the ideal gas law) math formula Pa and math formula Pa, and 1 s math formula s. For the present study all Vf and Vin values, except math formula, are the same as in the original Campbell et al. [1995] model runs. Here math formula is adjusted to ensure that the uppermost soil temperature time series are well simulated by the model. This was done in order to compensate for the difference in soil heat capacities between the present study, equation (11), and Campbell et al. [1995], which did not include any temperature dependency. As a consequence, math formula for all simulations with the present model.

[30] Incorporating the boundary conditions on the balance equations math formula and math formula is straightforward and yields the following expressions:

display math
display math

These expressions for H0 and W0 are included here primarily for completeness and it should be noted that (1) the math formula index for the soil thermal conductivity, math formula, and the equivalent vapor diffusivity, Dve, refer to their respective surface values math formula and math formula and (2) math formula by the same convention.

3.3. Lower Boundary Condition and Initial Conditions

[31] A numerical lower boundary condition [Thomas, 1995] is used at math formula for the present model. This yields the following relations:

display math

Additionally math formula and math formula. These assigned values of math formula and Dve at the lower boundary are completely analogous to their counterparts at the upper boundary.

[32] The numerical boundary conditions above are equivalent to the analytical boundary condition math formula and math formula at math formula. They can be thought of as a “pass through” boundary condition, in which the model variables at the lower boundary become part of the numerical solution while the corresponding fluxes are transmitted through the lower boundary. This simplification is used in the present study because specifying the boundary conditions externally from data obtained during a wildfire or a prescribed burn is extraordinarily difficult or impossible to do. Consequently employing these numerical boundary conditions, rather than observationally based analytical boundary conditions, seems a more realistic test of the model.

[33] Nonetheless, the present lower boundary conditions are different than that used by Campbell et al. [1995], who used the (time-dependent) soil temperature and moisture measured near the bottom of a heated soil column as a analytical lower boundary condition, i.e., math formula and math formula. These soil columns were created by packing different (and well-described) soils into cylinders with highly insulated sides. These cylinders were 0.12 m in diameter and 0.155 m deep. For the present 1-D model the modeling domain is 0.20 m deep, which exceeds the depth of the soil column used in the laboratory in order to minimize any influence that the lower boundary condition could have on the model simulations.

[34] The differential equation that describes the advective velocity uvl, equation (12) or equation (13), also requires a lower boundary condition. This is

display math

which is apropos to the physical experiments, as developed and described by Campbell et al. [1995], that enclose the heated soil samples in a container with a solid impermeable bottom.

[35] Finally the initial conditions (soil temperature and moisture content), which assumed to be uniform throughout the soil column for each soil type and heating experiment, are taken directly from Campbell et al. [1995].

4. Results

4.1. Brief Description of the Experimental Setup

[36] The overall performance of the present model is assessed using the same observed soil temperature and moisture data that Campbell et al. [1995] used to evaluate the performance of their soil heating model. As such, all soil temperatures were measured with copper-constantan thermocouples at the sample surface and at 5, 15, 25, 35, 65, and 95 mm depth. Changes in soil moisture were obtained by gamma ray attenuation at the same depths (except the surface). The moisture detecting system was linearly calibrated for each experimental run between (a) the initial soil moisture amounts, which were determined gravimetrically beforehand, and (b) the point at which the sample was oven-dried (also determined before the heating experiment) where math formula is assumed. But oven-drying a soil will not necessarily remove all the liquid water from a soil, i.e., a soil can display residual water content after oven-drying. Consequently, the soil moisture data obtained by Campbell et al. [1995] show negative soil moistures at the time the soil dryness passes outside the ovendry range. Temperature and moisture data were recorded frequently and simultaneously, but at irregular intervals. This is because the gamma ray detection system required several minutes to complete a full profile of the sample. Campbell et al. [1995] provide further details concerning the laboratory setup, the gamma ray system, and the data.

[37] In addition to these laboratory experiments, soil heating and moisture data during an experimental slash pile burn [Massman et al., 2008, 2010] are also used to assess the significance of the volitalization-induced advective flow term to soil heating and vapor transport during wildfires and prescribed burns.

4.2. Model Performance

[38] Campbell et al. [1995] used several soils with differing initial conditions for their model validation study. But for the present purposes, it is only necessary to report the results for one soil type, i.e., Quincy sand with an initial volumetric soil moisture content of 0.14 m3 m−3.

[39] Figure 2 compares the measured (symbols) and modeled (lines) soil temperature during the Quincy sand heating experiment. The colors indicate the depths (mm) of the experimental and model data. The corresponding measured and modeled soil moisture is shown in Figure 3. Ostensibly, these Figures 2 and 3 indicate that the present model produces results that are similar to both the original Campbell et al. [1995] model and the observations [see Campbell et al., 1995, Figures 2C and 3C]. Both models faithfully simulate the hiatus or pause in the soil temperature rise during the evaporative (or drying) period and an increase in soil moisture ahead of the rapidly drying depths (presumably due to the transport of evaporated soil moisture to cooler regions deeper in the soil—note: the relative importance of diffusive and advective transport is discussed later in this section). But given the differences between the present model and the original Campbell model, precise agreement between the two model should not be expected. Nonetheless, Figures 2 and 3 (and other model simulations that are not shown here) are quite similar to many of the other model simulations shown by Campbell et al. [1995]. But, upon a closer inspection, especially of the soil moisture results, there are some glaring differences between the present model and the observed data, and the original model as well.

Figure 2.

Comparison of measured (symbols) and modeled (lines) soil temperature during the Quincy sand heating experiment. The initial soil moisture content is 0.14 m3 m−3 (see Campbell et al. [1995] for further details), and the various depths are color-coded with similar colors.

Figure 3.

Comparison of measured (symbols) and modeled (lines) soil moisture content during the Quincy sand heating experiment. The initial soil moisture content is 0.14 m3 m−3 (see Campbell et al. [1995] for further details), and the various depths are color-coded with similar colors.

[40] These differences are most obviously displayed in the next four figures. Figure 4 shows the observations plotted as temperature versus soil moisture for all the monitored depths. For the model this type of plot is termed a “solution space” or “ math formula ” plot, which for the present model simulation is shown in Figure 5. Figure 4 clearly indicates that the temperature at which the soil moisture evaporates most quickly (or the “evaporative front” temperature) is at about 90 C, which of course is also the temperature of the pause in the soil temperature rise. But the model results indicate that the evaporative front occurs at temperatures between about 100 and 180 C and that this evaporative front temperature also increases with the depth. Furthermore, Figure 5 suggests that the amount of evaporated moisture that is recondensing ahead of the evaporative front (i.e., deeper into the soil) is not only much greater than that observed (Figure 4), but that it also is increasing with depth as well. All this suggests that not enough of the modeled evaporated soil moisture is escaping the soil through the upper surface, but rather is being driven deeper into the soil. This is confirmed by Figures 6 and 7.

Figure 4.

Measured soil moisture versus measured soil temperatures for the Quincy sand heating experiment (see Figures 2 and 3).

Figure 5.

Modeled soil moisture versus measured soil temperatures for the Quincy sand heating experiment (see Figures 2 and 3). This is the solution space representation of the model solution, which is to be compared with the observations shown in Figure 4.

[41] Figure 6 compares the vertical profiles of the soil temperatures at the end of the laboratory experiment with those at the end of the numerical simulation, and Figure 7 makes a similar comparison for the volumetric soil moisture content. (Note that as was discussed above, the final vertical profiles obtained from the laboratory experiment are approximate only because the final measurements for any given depth are not precisely coincident in time with the measurements made at any other depth. Nevertheless, this is not a significant issue for the present purposes.) The temperature profiles indicate that the model underpredicts the temperatures in the upper third of the soil column and overpredicts them in the lower two thirds. This is mainly because the soil moisture recondensing ahead of the evaporative front (located at about 30 mm on Figure 7) increases the soil thermal conductivity just ahead of the front so much that the modeled heat being transported (conducted) from the upper part of the soil column to the lower part is much greater than would be expected from the observations.

Figure 6.

Comparison of the modeled and measured temperature profiles at the completion of the Quincy sand heating experiment. The data shown in the measured profile are not precisely coincident in time (see text for further details).

Figure 7.

Comparison of the modeled and measured temperature profiles at the completion of the Quincy sand heating experiment. The data shown in the measured profile are not precisely coincident in time (see text for further details). Note the large increase in soil moisture ahead of the drying front (located at about 30 mm depth).

[42] But an even more surprising result is obtained from comparing the total water lost during the laboratory experiment with that derived from the model simulations. The observed data suggest that about 35% of the original soil moisture has evaporated during the (approximate) 90 min experiment; whereas, virtually none of the original soil moisture has been lost in the model simulations. Rather than actually transporting the evaporated water out of the soil column, the model has “pushed” it deeper into the soil ahead of the evaporative front. This is clearly not something that could have easily been anticipated from the description of the performance of the original model of Campbell et al. [1995], who suggested that (at least for Quincy sand and the Bouldercreek soil samples) the model produced about a 25% loss. Consequently, this particular measure of model performance suggests that the present model produces a fundamentally different type of solution than the original model. In fact the present solution is somewhat reminiscent of what one might expect if the model had been posed in terms of a Stefan-like or moving boundary condition problem [e.g., Whitaker and Chou, 1983–1984; Liu et al., 2005]. Nevertheless, the specific reason for this difference in the solutions produced by the present model and the Campbell et al. [1995] model is not clear. But the reason for the present (nonphysical) model solution is clearly displayed in the next figure.

[43] Figure 8 shows the modeled profile of vapor density, math formula, at the end of the two 90 min model simulations: one with uvl (leftmost curve) and one without (rightmost curve). According to these model runs, the vapor density is quite high within the extremely hot and dry zone (i.e., the top 30 mm or so) and much lower in the cooler regions deeper in the model domain. In fact the maximum soil vapor density predicted by the model is between about 2.7 and 4.2 kg m−3, which is physically unrealistic when one considers that one ambient atmosphere at STP is approximately 1.3 kg m−3. Nevertheless, these results do explain the nature of the model solution, which is demonstrated by calculating of the soil humidity, hs, for a dry soil at several different temperatures.

Figure 8.

Modeled profiles of the vapor density at the end of the 90 min model simulation. The leftmost curve corresponds to the simulation that includes uvl, equation (13), and the rightmost curve does not include uvl. The vapor density in the dry region (0–30 mm) is physically unrealistic (atmospheric density of air at STP = 1.3 kg m−3) because of the inappropriate use of the Kelvin equation.

[44] In order to be consistent with the present model simulations the following calculations assume math formula = −1.25 MJ kg−1 for a completely dry soil. For TK = 300 K, 350 K, 650 K, and 850 K this yields: hs(300 K) = 0.012%; hs(350 K) = 0.044%; hs(650 K) = 1.6%; and hs(850 K) = 4.1%. In other words, the soil humidity increases with increasing temperature, i.e., math formula. Combining this result with math formula (at least up to the critical temperature of water) yields math formula, math formula C and math formula. So as the temperature rises the dry soil humidity rises, which when coupled with the exponential rise in saturation vapor density, autonomously creates soil vapor far in excess of what is physically realistic.

[45] But uvl also plays an important part in the modeled math formula. Through its ability to accelerate moisture loss, uvl produces a final model value of math formula that is slightly greater (−1.09 MJ kg−1) than that produced without it (−1.25 MJ kg−1). Consequently, both hs and math formula are higher at the same temperatures with uvl than without it (i.e., math formula).

[46] Dynamically then, the following picture of the model solution emerges. After an initial burst of the soil surface moisture, E0 is driven mostly by math formula, equation (17) (because math formula 0.013 kg m−3 at all times, so it is always relatively small compared to math formula), which in turn increases monotonically with temperature. In the region below the dry front the vapor gradient, math formula, is extremely large and positive so the evaporated soil moisture cannot escape through the soil surface and into the free atmosphere above the soil column, but can only be driven deeper into the soil (or deeper into the modeling domain). Consequently, in the dry zone the evaporative flux is simply removing the autonomously created vapor; while the diffusive flux, which is dominant in the moist zone ahead of the evaporative front transports evaporated soil moisture deeper into the soil.

[47] The model's ability to “push” moisture deeper into the soil ahead of the dry front is also the cause of model instabilities, which occur because as the soil moisture increases math formula. If soil moisture increases fast enough or becomes large enough, then instead of math formula remaining bounded by zero (i.e., math formula) math formula occurs due to a loss of numerical precision. As a result of math formula the corresponding value of math formula becomes an imaginary number and the model fails. In fact, the oscillations of the soil moisture near the evaporative front shown in Figure (7) indicates that for the present choice of parameters the model is close to an instability. The original Campbell et al. [1995] model also suffers from this same type of instability. On the other hand, this instability issue is alleviated somewhat in the present model by using WRC given by equation (7), which permits θ to be greater for an given ψ, than the original WRC, equation (6). Nonetheless, this instability will always be more likely for any soil that is relatively moist compared to the same soil with drier initial conditions.

[48] Clearly the roles of the math formula and the Kelvin equation are central to the model's failure, but in reality they are merely byproducts of the WRC's (equations (6) and (7)) inability to describe the relationship between soil moisture content and soil water potential for very dry soils (i.e., when math formula). These low moisture contents are more properly understood as “bound” or “residual” water, rather than “free” water. Therefore, adopting the intuitive notion that math formula represents the amount of physical work required to oven dry a soil, i.e., vaporize all the free water, then to completely dry out a soil must require greater physical work ( math formula) than math formula. Or more specifically math formula at least, but more likely math formula. Neither equation (6) or (7) is adequate for this situation, implying that this is also an issue with the original Campbell et al. [1995] model as well.

[49] On the other hand, if the current WRCs properly allowed for math formula, then hs would be significantly reduced, possibly to the point where math formula was, not only physically more realistic, but also numerically insignificant. Within limits this hypothesis is testable by evaluating model performance after artificially increasing math formula (to −4 MJ kg−1), while simultaneously limiting the vapor saturation calculations to math formula and increasing math formula, the adjustment parameter associated with the volitalization velocity uvl (equation (13)), to its nominal value of 1. Although the results were encouraging, they were not necessarily conclusive. The limiting or maximum value of math formula was much more realistic, i.e., the vapor density at the surface math formula 0.5 kg m−3, and uvl did not have to be artificially reduced. Unfortunately, the model ceased to converge after about 4 min of simulated heating. Nevertheless, the solution at this time was quite similar to that already discussed (Figures 2, 3, 5, 6, and 7). Likewise, the final vertical gradient of math formula was just a weaker version of those shown in Figure 8. In summary, model convergence and stability issues involving math formula and math formula and the nonphysical nature of the model solution and math formula, all appear at least in part to be linked to the structure of the WRC and its inability to capture the evaporative dynamics of bound water.

[50] There can be little doubt that improving modeling of soil heating and moisture transport during fires is likely to require improving understanding of the thermodynamics of bound water and better parameterizations of the WRC. But there are two other aspects of this issue that are worthy of discussion. First, it might be tempting to employ a WRC that allows math formula to be unbounded, i.e., math formula. To the author at least, this would seem an unlikely approach for improving model realism because it risks either (1) numerical instabilities as math formula (which will be driven by the heating dynamics) because math formula or (2) another physically unrealistic solution in which the model cannot evaporate all the soil moisture despite heating the soil to several hundred degrees Celsius. Second, a more useful approach to the present modeling dilemma might be to directly incorporate math formula as model variable. This has some potential to reduce the problems introduced by the Kelvin equation and the saturation vapor density, both of which are exponential in the other model variables, TK and math formula, and both of which amplify the weakness of WRC.

4.3. Additional Considerations Regarding uvl and math formula

[51] Given the importance of uvl to the performance of the present, and potentially any other, model there are two other aspects of the volitalization-induced advective flow that are worth discussing: first is its intrinsic significance to heat flow (equations (14) and (18)) and to mass flow (equations (15) and (19)), and second is its potential relevance to field application. First, with the present model math formula during the entire simulation and math formula during most of the simulation. For vapor flow math formula during the initial burst of soil surface moisture and otherwise decreasing to math formula during most of the rest of the simulation. Consequently, the uvl term can be neglected for heat flow without significantly compromising the model and that all impacts of uvl on soil temperatures occur as a result of its ability to influence evaporation and the resulting soil moisture distribution.

[52] Second, Figure 9 is a θ-T plot for a prescribed slash pile burn performed at Manitou Experimental Forest in April 2004 [Massman et al., 2008, 2010]. These data were obtained every half hour over a 5 day period at an 0.05 m depth under the center of a large slash pile burn. The time series shown in Figure 9 proceeds clockwise from the top left corner, which corresponds to the soil temperature and volumetric moisture at the time the burn was initiated. The rate of change in soil moisture during the evaporative phase in Figure 9 suggests that math formula m3 m−3 s−1. Whereas, Figure 4 and other lab experiments with different soil types suggests that math formula varies within math formula m3 m−3 s−1 (2 orders of magnitude greater than the field observations). This comparison indicates that uvl is likely to be negligible in field experiments. But these different values of math formula only reflect the differences in soil heating that can be achieved in the lab and the field. Unsurprisingly, the rate of forcing during the lab experiments, math formula, exceeds, by 2–3 orders of magnitude, the soil forcing, math formula, achieved during the Manitou slash pile burn. (Note that math formula is estimated from Figure 3 of Massman et al. [2010]). Although math formula may be valid in the field, it does not fundamentally alter the present model's performance or nonphysical behavior, which will always produce the same physically unrealistic solution shown in Figure 8.

Figure 9.

The θ-T (solution space) diagram at 0.05 m depth at the center of a prescribed slash-pile burn performed at Manitou Experimental Forest in April 2004 [Massman et al., 2008, 2010]. The red portion of the curve corresponds to the soil heating phase of the burn, and the blue portion corresponds to the cooling phase. The data points indicted by stars were recorded at half-hourly intervals. The total length of the time series is 5 days (240 half hours), with time proceeding clockwise from the top left corner, which corresponds to the soil temperature and volumetric moisture at the time the burn was initiated.

[53] Finally, it is important to note that math formula had very little impact on the present model's performance. Nonetheless, although its influence on moisture dynamics was fairly small at all times, overall it should be characterized as a small improvement to the model.

5. Summary and Recommendations

[54] The present model largely employs standard equations, concepts, and methods for modeling coupled heat and moisture flow in soils, but applies them to the extremes in temperatures and moisture contents that occur during the slash pile burns and wildfires. The model includes temperature dependencies that are not often employed in model formulations of the daily cycles of soil heating and evaporation and advective effects associated with the volume expansion of air that occurs during the rapid volatilization of the soil moisture. Comparing the model simulations with laboratory observations showed that the model produced a numerically stable, but physically unrealistic, solution. The model's shortcoming were traced to the inability of the water retention curve to adequately describe the relationship between soil moisture and soil water potential for very low soil moistures ( math formula, for example). This is a significant result because many present-day WRCs are modeled after the ones used in this study, suggesting that even under less extreme heating conditions many (if not most or all) present-day models of soil evaporation will fail at extremely low soil moistures.

[55] Although the weakness of the present model may limit its utility, it does offer insights into how to design a more physically realistic model. First, improved understanding and models of the thermodynamics of evaporation of bound water are a clear necessity. Field data shown in Figure 9, as well as the lab data shown in Figure 4, and all the other laboratory results shown in Campbell et al. [1995] that are not included in the present study, clearly indicate that soil moisture is retained in the soil until the soil has been heated to between 200 C and somewhere above 300 C (well above the temperature for oven drying a soil). These data obviously support the need for improving WRCs near the dry end and beyond the oven-dried state. Second, a dual-phase model, one in which θ and math formula are modeled separately, is likely to be required. Such a model could employ either an “equilibrium” or a “nonequilibrium” evaporation approach [e.g., Smits et al., 2011], which in either case would reduce the role that the Kelvin Equation and the saturation vapor density play in the numerical calculations of the present model. Third and final, it may be beneficial to include hydraulic conductivity in the model to permit liquid water movement in response to the extremely high moisture potential gradients that invariably occur during fires.


[56] I sincerely thank Gaylon Campbell for providing the laboratory data used in this study and for generously contributing his time, his thoughts, and his feedback during the innumerable discussions I had with him on virtually all aspects of this study. I would also like to thank James Thomas for his insights into and discussions of the mathematical and numerical issues I encountered during the development of this model. Finally, I thank John Selker and the anonymous reviewers for their many insights and contributions to this study.