Increased use of prescribed fire by land managers and the increasing
likelihood of wildfires due to climate change require an improved modeling
capability of extreme heating of soils during fires. This issue is addressed here by developing and testing the soil (heat–moisture–vapor) HMV-model, a 1-D
(one-dimensional) non-equilibrium (liquid–vapor phase change) model of soil
evaporation that simulates the coupled simultaneous transport of heat, soil
moisture, and water vapor. This model is intended for use with surface
forcing ranging from daily solar cycles to extreme conditions encountered
during fires. It employs a linearized Crank–Nicolson scheme for the
conservation equations of energy and mass and its performance is evaluated
against dynamic soil temperature and moisture observations, which were
obtained during laboratory experiments on soil samples exposed to surface
heat fluxes ranging between 10 000 and 50 000 W m
Since the development of the theory of Philip and de Vries (PdV model) almost
60 years ago (Philip and de Vries, 1957; de Vries, 1958), virtually all
models of evaporation and condensation in unsaturated soils have assumed that
soil water vapor at any particular depth into the soil is in equilibrium with
the liquid soil water (or soil moisture) at the same depth. (Note that such soil
evaporation models also assume thermal equilibrium, so that at any given
depth the mineral soil, the soil moisture, and the soil air and water vapor
within the pore space are also at the same temperature.) In essence, this
local equilibrium assumption means that whenever the soil moisture changes
phase it does so instantaneously. This assumption is quite apropos for its
original application, which was to describe the coupled heat and moisture
transport in soils (and soil evaporation in particular) under environmental
forcings associated with the daily and seasonal variations in radiation,
temperature, precipitation, etc. (e.g., Milly, 1982; Novak, 2010; Smits
et al., 2011). Under these conditions, assuming local equilibrium is
reasonable because the time required to achieve equilibrium after a change of
phase is “instantaneous” (short) relative to the timescale associated with
normal environmental forcing. The great benefit to the equilibrium assumption
is that for modeling purposes it is a significant simplification to the
equations that describe heat and moisture flow in soils because it eliminates
the need to include soil water vapor density,
Subsequent to the development of the original PdV model the equilibrium assumption has also been incorporated into models of heat and moisture transport (evaporation and condensation) in soils and other porous media under more extreme forcings associated with high temperatures and heat fluxes. For example, it has been applied to (i) soils during wildfires and prescribed burns (Aston and Gill, 1976; Campbell et al., 1995; Durany et al., 2010; Massman, 2012), (ii) drying of wood (Whitaker, 1977; di Blasi, 1997), (iii) drying and fracturing of concrete under high temperatures (Dayan, 1982; Dal Pont et al., 2011), (iv) high temperature sand–water–steam systems (e.g., Udell, 1983; Bridge et al., 2003), and (v) evaporation of wet porous thermal barriers under high heat fluxes (Costa et al., 2008).
Although the PdV model and the equilibrium assumption have certainly led to many insights into moisture and vapor transport and evaporation in porous media, they have, nonetheless, yielded somewhat disappointing simulations of the coupled soil moisture dynamics during fires (see Massman, 2012, for further details and general modeling review). Possibly the most interesting of these modeling “disappointments” is the soil/fire-heating model of Massman (2012), who found that as the soil moisture evaporated it just recondensed and accumulated ahead of the dry zone; consequently, no water actually escaped the soil at all, which, to say the least, seems physically implausible. He further traced the cause of this anomalous behavior to the inapplicability of the equilibrium evaporation assumption, which allowed the soil vapor gradient behind the drying front to become so small that the soil vapor could not escape (diffuse) out of the soil. Moreover, more fundamentally, the calculated vapor and its attendant gradient became largely meaningless because it is impossible for water vapor to be in equilibrium with liquid water if there is no liquid water. Of course, such extremely dry conditions are just about guaranteed during soil heating events like fires. Novak (2012) also recognized the inapplicability of the equilibrium assumption for very dry soils, but under more normal environmental forcing. On the other hand, even under normal (and much less extreme) soil moisture conditions both Smits et al. (2011) and Ouedraogo et al. (2013) suggest that non-equilibrium formulations of soil evaporation may actually improve model performance, which implies that the non-equilibrium assumption may really be a more appropriate description for soil evaporation and condensation than the equilibrium assumption. The present study is intended to provide the first test of the non-equilibrium hypothesis during extreme conditions.
Specifically, the present study develops and evaluates a non-equilibrium (liquid–vapor phase change) model for simulating coupled heat, moisture, and water vapor transport during extreme heating events. It also assumes thermal equilibrium between the soil solids, liquid, and vapor. It uses a systems-theoretic approach (e.g., Gupta and Nearing, 2014) focused more on physical processes than simply tuning model parameters; i.e., that whatever model or parameter “tuning” does occur, it is intended to keep the model numerically stable and as physically realistic as possible.
In addition, the present study (model) is a companion to Massman (2012). It uses much of the same notation as the earlier study. But, unlike its predecessor, this study allows for the possibility of liquid water movement (i.e., it includes a hydraulic conductivity function for capillary and film flow). It also improves on and corrects (where possible and as noted in the text) the mathematical expressions used in the previous paper to parameterize the high-temperature dependency of latent heat of vaporization, saturation vapor density, diffusivity of water vapor, soil thermal conductivity, water retention curve, etc. This is done in order to achieve the best representation of the physical properties of water (liquid and vapor) under high temperatures and pressures (see, e.g., Harvey and Friend, 2004). Lastly, in order to facilitate comparing the present model with the earlier companion model the present study displays all graphical results in a manner very similar to those of Massman (2012).
The present model is one-dimensional (1-D) (in the vertical) and is developed
from three coupled partial differential equations. It allows for the
possibility that the soil liquid and vapor concentrations are not necessarily
in local equilibrium during evaporation/condensation, but it does assume
local thermal equilibrium during any phase change. The present model has
three simulation variables: soil temperature (
The conservation of thermal energy is expressed as
The conservation of mass for liquid water is
This last equation can be simplified to
The conservation of mass for water vapor is
The final model equations are expressed in terms of the model variables (
Apropos to these last three equations (i)
The algorithm for calculating water density,
The enthalpy of vaporization of water,
The formulations for thermal conductivity of water vapor,
The formulation for the thermal conductivity of dry air,
The volumetric-specific heat for soil air,
The saturation vapor pressure,
Embedded in the hydraulic conductivities (
Mindful of the externalization of
The present model for the advective velocity associated with the
volatilization of water,
The functional parameterization of
But there is another way to model the vapor flux,
Concluding the development of
The model for
The present formulation for isobaric heat capacity of water,
The model of soil thermal conductivity,
In general a WRC is a functional relationship between soil moisture and soil
moisture potential and temperature, i.e.,
There is a simple and physically intuitive argument for the parameterization
of
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,
For the present study, five difference parameterizations for
The term
By expressing
The numerical model as outlined above and detailed in this section is coded as MATLAB (The MathWorks Inc., Natick, MA, Version R2013b) script files.
The linearized Crank–Nicolson method is used to solve Eqs. (
The upper-boundary condition on heat and vapor transfer is formulated in
terms of the surface energy balance and, except for the latent heat flux, is
identical to the upper-boundary condition in Massman (2012).
The surface evaporation rate,
where
The upper-boundary condition on soil water is
The boundary forcing functions
As with the companion model (Massman, 2012), a numerical (or extrapolative
or “pass-through”) lower-boundary condition (Thomas, 1995) is also used
for the present model. Analytically this is equivalent to assuming that the
second spatial derivative (
Except for the initial value of
In the original soil heating experiments of Campbell et al. (1995) soil
temperatures were measured with copper–constantan thermocouples at the sample
surface and at 5, 15, 25, 35, 65, and 95 mm depth and 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
Comparison of measured (symbols) and modeled (lines) soil
temperatures during the Quincy Sand heating experiment. Neither simulation
includes a dynamic residual soil moisture term,
The present model is evaluated against five of the
soil heating experiments from Campbell et al. (1995): (1) Quincy Sand, which has an initial soil moisture
content =
Comparison of measured (symbols) and modeled (lines) soil moisture
contents during the Quincy Sand heating experiment. Neither simulation
includes a dynamic residual soil moisture term,
Figure
Figure
Measured soil moisture vs measured soil temperatures for the Quincy Sand heating experiment (see previous two figures).
Modeled soil moisture contents vs modeled soil temperatures for the
Quincy Sand heating experiment (see Figs.
Figure
Comparison of the final modeled and measured temperature profiles at the completion of the 90 min Quincy Sand heating experiment. Because the data shown in the measured profile (black) are not precisely coincident in time, the full model results (solid red and blue lines) were sub-sampled in synchrony in time (and coincide in space) with the observations. These time-synchronized model profiles are shown as dashed red and blue lines. To compare with the equilibrium model, see Fig. 6 of Massman (2012).
The difference between the these two model simulations is less obvious with
Fig.
Comparison of the final modeled and measured moisture profiles at
the completion of the Quincy Sand heating experiment. Because the data shown
in the measured profile (black) are not precisely coincident in time, the
full model results (solid red and blue lines) were sub-sampled in synchrony
in time (and coincide in space) with the observations. These synchronized
model profiles are shown as dashed red and blue lines. The observed data
(black) suggest that the total water lost during the 90 min experiment was
31 % of the initial amount. The (red) model simulation indicated a
31.4 % loss and the corresponding (red) synchronized model yielded a
33.8 % loss. The (blue) model simulation indicated a 34.6 % loss and
the corresponding (blue) synchronized model yielded a 34.2 % loss. Note
there is very little recondensing soil moisture ahead of the drying front (at
about 40–50 mm depth), in agreement with Figs.
For the present Quincy Sand experiment this yields a fractional
Final modeled profiles of vapor density [
Figure
Final modeled profile of vapor pressure at the end of the 90 min
model simulation. The solid line is the model simulation with
If there is an implausibility with the present model it might be the soil
vapor pressure,
Figure
Unfortunately, there are no independent confirmations for any values of
Example of the hydraulic conductivity,
Although it is undeniably true that the present model is an improvement over
the equilibrium model, the inclusion of the HCF within this non-equilibrium
model (and its lack of inclusion in the equilibrium model) makes it difficult
to conclude unambiguously that the improvement over equilibrium model is the
sole consequence of the non-equilibrium assumption. But the non-equilibrium
model was tested in a mode that basically “turned off”
Finally it should also be pointed out that, unlike
Comparison of measured (symbols) and modeled (lines) soil
temperature during the Palousse B Wet heating experiment. The solid lines are
for a model simulation that does not include the dynamic residual soil
moisture,
Comparison of measured (symbols) and modeled (lines) soil moisture
content during the Palousse B Wet heating experiment. The solid lines are for
a model simulation that does not include the dynamic residual soil moisture,
Modeled soil moisture vs modeled soil temperatures for the
Palousse B Wet heating experiment (see Figs.
Observed and modeled soil moisture vs soil temperatures
(trajectories) for the Wet Palousse B heating experiment. Solid lines are
observed data and the dash-dot lines are from the model that includes the
dynamic residual soil moisture,
Figure
Comparison of the final modeled and measured temperature profiles at the completion of the 70 min Palousse B Wet heating experiment. Because the data shown in the measured profile (black) are not precisely coincident in time, the full model results (solid red and blue lines) were sub-sampled in synchrony in time (and coincide in space) with the observations. These time-synchronized model profiles are shown as dashed red and blue lines.
The Quincy Sand and Palousse B results in general confirm that the
non-equilibrium model's performance is enhanced (and quite significantly)
with the incorporation of liquid water transport (HCF) and that its
performance is sensitive to (and can be improved by) either or both (a) the
infrared thermal conductivity,
Comparison of the final modeled and measured moisture profiles at
the completion of the 70 min Palousse B Wet heating experiment. Because the
data shown in the measured profile (black) are not precisely coincident in
time, the full model results (solid red and blue lines) were sub-sampled in
synchrony in time (and coincide in space) with the observations. These
time-synchronized model profiles are shown as dashed red and blue lines. The
observed data (black) suggest that the total water lost during the 70 min
experiment was 28.8 % of the initial amount. The (red) model simulation
indicated a 14.7 % loss and the corresponding (red) synchronized model
yielded a 15.8 % loss. The (blue) model simulation indicated a 27.8 %
loss and the corresponding (blue) synchronized model yielded a 29.4 %
loss. Note there is very little recondensing soil moisture ahead of the
drying front (at about 35 mm depth), in agreement with
Figs.
The remainder of this section is a summary of various (secondary, but important) model sensitivity analyses performed with all soil heating experiments. The ultimate intent here is to shed light on which physical process are relatively more important and to provide some guidance for further research.
Central to the success of the present model, relative to the performance of
the equilibrium model of Massman (2012), is the functional parameterization
of the source term,
It should not be surprising that the model is sensitive to soil thermal
conductivity,
The most important parameter controlling surface evaporation rate is the
surface transfer coefficient
The two other WRCs tested for model performance were Groenevelt and
Grant (2004) (GG04) and Fredlund and Xing (1994) (FX94). But prior to
implementing them in the model they were both calibrated to be numerically
similar near the dry end (
Universal to all of HCFs tested here is
Of the remaining three heating experiments only Wet Bouldercreek, which had
an initial soil saturation level of about 50 %, showed anything
unexpected. In general, the model was able to capture the observed soil
temperatures and temperature dynamics extremely well, even better than shown
in either Fig.
Unlike with the companion model (Massman, 2012), the present model did not
require reducing the magnitude of
The point was made earlier that the present model was developed assuming the
mass form for the diffusive flux, i.e.,
A similar result is produced for
The final expression for
Comparing Eqs. (
None of the model solutions resulting from either
This study has developed and tested a non-equilibrium (liquid to vapor phase change) model for simulating heat and moisture flow in soils during fires, but the model does assume thermal equilibrium. By and large the simulations of soil temperature and moisture are not only credible but also often quite good. In general, all model results showed a significant improvement over all comparable results from the companion equilibrium model of Massman (2012).
The principal reason for the present model's success is the incorporation of
a dynamic condensation coefficient,
In general, the model simulates the observed soil temperatures quite well. It
is often slightly less precise for soil moisture and the best simulations
were usually a compromise between faithfully representing the observed soil
temperatures or the observed soil moisture. Nonetheless, the model does
capture reasonably well many observed features of the soil moisture dynamics,
viz., it simulates an increase in soil moisture ahead of the drying front
(due to the condensation of evaporated soil water at the front) and the
hiatus in the soil temperature rise during the strongly evaporative stage of
the soil drying. Furthermore, the model also captures the observed rapid
evaporation of soil moisture that occurs at relatively low temperatures
(50–90
Sensitivity analyses (SAs) were also performed with different formulations
for the water retention curve, soil hydraulic conductivity function, one
variant of the present evaporative source term,
I would like to thank Gaylon S. Campbell for providing the laboratory data used in this study, as well as James W. Thomas for his insights into and discussions of the mathematical and numerical issues I encountered during the development of this model and Marcia L. Huber and Allan H. Harvey for various discussions concerning the thermophysical properties of water and aid with the published resources used in parameterizing the self-diffusion of water vapor. Edited by: T. Kato