Corresponding author: K. M. Smits, Center for Experimental Study of Subsurface Environmental Processes (CESEP), Colorado School of Mines, Golden, CO 80401, USA. (firstname.lastname@example.org)
 Bare soil evaporation is a key process for water exchange between the land and the atmosphere and an important component of the water balance. However, there is no agreement on the best modeling methodology to determine evaporation under different atmospheric boundary conditions. Also, there is a lack of directly measured soil evaporation data for model validation to compare these methods to establish the validity of their mathematical formulations. Thus, a need exists to systematically compare evaporation estimates using existing methods to experimental observations. The goal of this work is to test different conceptual and mathematical formulations that are used to estimate evaporation from bare soils to critically investigate various formulations and surface boundary conditions. Such a comparison required the development of a numerical model that has the ability to incorporate these boundary conditions. For this model, we modified a previously developed theory that allows nonequilibrium liquid/gas phase change with gas phase vapor diffusion to better account for dry soil conditions. Precision data under well-controlled transient heat and wind boundary conditions were generated, and results from numerical simulations were compared with experimental data. Results demonstrate that the approaches based on different boundary conditions varied in their ability to capture different stages of evaporation. All approaches have benefits and limitations, and no one approach can be deemed most appropriate for every scenario. Comparisons of different formulations of the surface boundary condition validate the need for further research on heat and vapor transport processes in soil for better modeling accuracy.
 Soil water evaporation is a key component of the water balance, especially in semiarid and arid regions and areas subjected to desertification. Evaporation is affected by atmospheric conditions (e.g., humidity, temperature, radiation, and wind velocity) and soil thermal and hydraulic properties (thermal and hydraulic conductivity, porosity), all of which are strongly coupled. The strong coupling between processes leads to highly dynamic interactions between the transport in the porous medium and boundary conditions, resulting in dynamic evaporative behavior [Sakai et al., 2011]. Many studies have shown that the accurate prediction of land surface heat fluxes, surface temperature, and soil moisture is necessary for predicting evaporation and ultimately climatic conditions at a variety of temporal and spatial scales [e.g., Walker and Rowntree, 1977; Shukla and Mintz, 1982; Avissar and Pielke, 1989; Garratt, 1993]. However, the dynamic interaction of energy and mass transfer processes at the land-atmospheric interface are rarely considered in most current models or in practical application due to the complexity of the problem in field settings and the scarcity of field or laboratory data capable of testing and refining energy and mass transfer theories. Models that do incorporate energy and mass transfer processes often resort to the use of fitting parameters such as enhanced vapor transport [e.g.,Saito et al., 2006; Bittelli et al., 2008; Smits et al., 2011; Sakai et al., 2011]. The models are then tested with limited field or laboratory data [e.g., Saito et al., 2006; Bittelli et al., 2008; Novak, 2010; Sakai et al., 2011]. Controlled testing in laboratory scale systems provides a practical and cost-effective way to generate accurate data under controlled, yet transient conditions that can be expected in the field. This allows for the degree of control needed to simulate various scenarios that are of interest but not necessarily feasible in field settings (e.g., infiltration, water table variations, and soil heterogeneities). Use of the most complete form of the energy and mass transfer equations coupled with highly controlled experiments capable of testing and refining theory allows for the investigation of dominant mechanisms without any preliminary assumptions about the terms that are made in the model formulation and can provide guidance for the improvement of simplified parameterizations.
 In the early stages of evaporation from initially saturated porous media, liquid water is supplied from the underlying soil layers to the surface by capillary action. This early stage is characterized by a high and relatively constant drying rate and the formation of a drying front that recedes vertically into the soil profile. As the soil near the land surface begins to dry, evaporation at the interface is inhibited due to a decrease in the soil water supplied from the underlying soil, resulting in evaporation that is sustained primarily by vapor diffusion [Hillel, 1998]. This so-called second stage of evaporation is characterized by overall lower evaporation rates and the formation of a new vaporization plane or front below the soil surface. The new vaporization plane is often referred to as the secondary drying front or dry surface layer and defined as the interface between the wet and dry zones [Shokri et al., 2009]. Moisture transfer occurs in the vapor phase in the completely dry soil layer above the vaporization plane. This layer typically has a narrow range of depths (3–20 mm) [Novak, 2010; Shokri and Or, 2011]. Ultimately, when the soil becomes very dry, the soil physical characteristics control the evaporation rate [Bittelli et al., 2008].
 Another factor that affects the computation of vapor flow is the parameterization of the soil thermal properties, specifically the soil thermal conductivity. Peters-Lidard et al.  showed that the sensible and latent heat fluxes in the land surface energy balance and surface temperatures are very sensitive to the parameterization of the soil thermal conductivity. Most thermal conductivity/soil water content models have been developed based on easily measured parameters [e.g., de Vries, 1963; Johansen, 1975; Campbell et al., 1994; Côté and Konrad, 2005, 2009]. However, these models are often implemented into multiphase heat and mass transfer models without calibrating the fitting parameters using experimental data [e.g., Bittelli et al., 2008; Saito et al., 2006; Grifoll et al., 2005; Sakai et al., 2009], resulting in uncertainty in the effects of the thermal properties on the evaporation rate.
 Under dry soil conditions, liquid film flow has been found to significantly enhance the drying rate [e.g., Laurindo and Prat, 1998; Yiotis et al., 2003] and may play an important role in the evaporation from soil. According to Prat , the incorporation of film effects in pore network models leads to a much better agreement with experimental data. Adsorbed water films strongly affect the residual water content and hydraulic conductivities in porous media at low saturations. However, the common conceptual models for describing unsaturated flow often neglect film flow, only accounting for capillary flow, thus providing an oversimplified representation of flow in porous media. In many cases, such as dry soil conditions and infiltration conditions where the initial soil water content is less than the residual, accounting for changes in hydraulic conductivity due to film flow is needed. Several models accounting for both capillary flow and film flow have been recently proposed to describe soil hydraulic conductivity for the unsaturated porous media [e.g., Peters and Durner, 2008; Tokunaga, 2009; Lebeau and Konrad, 2010].
 Selection of the best method to estimate evaporation from bare soils remains a difficult task, as the models that are used make different assumptions, specifically at the land surface. Using a fully coupled land-atmospheric model that implicitly couples the soil to the atmosphere without any assumptions at the land-soil interface is ideal; however, these models are at the early stages of development. Hence, it is beneficial to understand and evaluate how the assumed boundary conditions affect the evaporation estimates. This knowledge is expected to help better formulate the land-atmospheric coupling by focusing on the heat and mass transfer processes at the boundary. The goal of this work is to test different conceptual and mathematical formulations that are used to estimate evaporation from bare soils to critically investigate various formulations and surface boundary conditions. In this study, we modified a fully coupled numerical model that solves for heat, liquid water, and water vapor flux previously developed bySmits et al. that allows nonequilibrium liquid/gas phase change with gas-phase vapor diffusion to better account for dry soil conditions often encountered in arid and semiarid environments. We then used this theory to evaluate different formulations of the soil surface boundary. To fundamentally understand the physical processes and to test numerical formulations, we performed an experiment using test sands for which the hydraulic and thermal properties were well characterized [Smits et al., 2010]. The experiment was conducted in a unique two-dimensional bench scale tank apparatus equipped with a network of sensors for automated and continuous monitoring of soil moisture, temperature, and relative humidity, allowing us to generate precision data under well-controlled transient heat and wind boundary conditions. The objectives of this work are to (1) critically evaluate how approaches that use different boundary conditions are able to capture the evaporation rate and cumulative evaporation of the system and (2) overcome the use of fitting parameters in modeling efforts by properly accounting for physical processes in the model. In addition to better understanding climatic conditions, this research is applicable to many engineering and industrial applications such as soil remediation, drying behavior for wood, paper, and food processing [Tietz and Schlunder, 1993], water management in fuel tanks, and storage of water in soils for agricultural purposes and numerous construction activities [Lehmann et al., 2008].
2. Material and Methods
2.1. Sand Material
 Uniform specialty silica sand, Accusand #30/40 (identified by the effective sieve number), was used (Unimin Corp., Ottawa, Minn.) for this experiment. Based on the technical sheet provided by the manufacturer, the uniformity coefficient is approximately 1.2, the grain density is 2.66 g cm−3, the grain shape is classified as rounded, and its composition is 99.8% quartz. Selected important properties of the test sand are summarized in Table 1. The capillary pressure (Pc)-water content (θ) relationship and thermal conductivity (λ)-water content (θ) relationship as a function of temperature for the sands under tight and loose packing conditions were measured separately using a small tank with a network of sensors that continuously monitored soil moisture, temperature, capillary pressure, and soil thermal properties [Smits et al., 2010]. A detailed description of the measurements of the Pc-θ and λ-θ relationships is given by Smits et al. .
Estimated from sieve data provided by the manufacturer.
Measured in a separate column test.
Measured in a separate hydraulic conductivity test.
van Genuchten model parameters estimated using the computer code RETC.
2.2. Development of Experimental Apparatus
 Laboratory testing was conducted in a two-dimensional bench scale tank constructed using plexiglass (height = 55 cm, length = 25 cm, inner width = 9 cm, wall thickness = 1.25 cm, specific heat = 1464 J kg−1 K−1, thermal conductivity = 0.2 W m−1 K−1, and density = 1150 kg m−3), as seen in Figure 1. Water content and temperature distributions within the tank were continuously monitored using dielectric soil moisture sensors (ECH2O EC-5, prong length = 5.5 cm, measurement frequency = 70 MHz, Decagon Devices, Inc.) and temperature probes (EC-T, Decagon Devices Inc.), respectively. The soil moisture sensors were calibrated using the method proposed bySakaki et al. . Relative humidity and temperature were monitored 5 cm above the soil surface, at the soil surface, and 1 cm under the soil surface and ambient conditions using relative humidity probes [EHT relative humidity (RH)/temperature sensor], Decagon Devices Inc., accuracy ±2% from 5% to 90% RH, ±3% from 90% to 100% RH. The relative humidity sensor at the soil surface was placed directly in contact with the soil grains so that the readings could better reflect that of the soil surface and not the surrounding air. Figure 1bshows a schematic of the placement of soil moisture and temperature sensors within the tank. Sensors were installed in the first 25 cm depth of the tank, horizontally through the plexiglass walls of the tank to avoid water channeling along any wires within the tank. A total of 25 each of soil moisture and temperature sensors were installed. In addition to the locations within the tank, temperature was also monitored at the soil surface, 5 cm above the surface, along the tank side boundaries, and at predetermined locations outside of the plexiglass wall of the tank to monitor the temperature gradient and estimate heat loss (which was very small) out of the plexiglass walls of the tank in the model. The tank was placed on a scale (model 11209-95, Sartorius, range = 65 kg, resolution = ±1 g) to continuously monitor the tank weight and hence cumulative water loss/evaporation rates over the course of the experiment.
2.3. Experimental Procedure
 The sand was carefully wet packed into the tank in 2 cm incremental depths in an effort to achieve a uniform bulk density. Maximum densities were assumed to be achieved by the repeated and thorough tapping of the tank walls [e.g., Sakaki and Illangasekare, 2007; Smits et al., 2011]. To enhance the visibility of the evaporation front, deionized water was dyed with a diluted solution of blue dye (0.2 g L−1 enVision, Fischer Scientific S92507) that does not alter the physical behavior of the water (Figure 2).
 The water table was initially established at the top surface of the tank, and the top boundary was initially covered with a plastic sheet to avoid evaporation. The bottom boundary was kept under no-flow conditions. The upper boundary temperature was controlled as a periodic time-dependent function by supplementing thermal irradiation to the soil surface. An infrared (IR) heating element (model HSE 500-120, Salamander), together with a reflector, was positioned 7 cm directly above the tank soil surface to obtain a nearly uniform heat flux distribution. The soil surface temperature was monitored with an IR noncontact temperature sensor (model IRt/c.03, Exergen Corporation) connected to a temperature control system (model 2104, Chromalox). The IR sensor provided input back to the temperature control system to ensure that the soil surface temperature remained constant. With this additional heating source, surface temperature amplitudes were in the range of 24°C–27°C (typical temperature in the laboratory = 22°C ± 1°C). At the start of the experiment, the plastic sheet was removed from the top boundary of the tank, allowing for evaporation, and temperature variations were induced at the soil surface using the IR heater. Water content, temperature, and relative humidity were continuously monitored at every 10 min intervals.
 For this experiment, a fan was used to induce wind at the soil surface. Sheet metal was placed along the sides of the tank to channel the wind in a linear fashion. An anemometer (Davis Anemometer, Decagon Devices, accuracy ±5%) was used to record wind speed and direction. The experiment was run for 14 days.
3. Numerical Model Formulation
 The model described by Smits et al. [2011, 2012a]that accounts for nonequilibrium phase change and simulates coupled heat, water vapor, and liquid water flux through soil was amended for this numerical experiment to better account for dry soil conditions and avoid the use of fitting parameters. The model was then tested with precision data obtained from laboratory experiments with well-controlled boundary conditions in an attempt to both calibrate and test calculations of coupled heat and water transfer. The reader is referred toSmits et al.  for the full model description; however, the governing equations for the transport of heat, liquid water, and water vapor are presented here.
3.1. NonIsothermal Multiphase Flow
 Multiphase flow in porous media is defined on the basis of Darcy's law [Bear, 1972]:
where θs is the total porosity or saturated volume fraction (m3 m−3); Sw and Sa are the saturations (–) of the water and gas phases, respectively; t is the time (s); kint is the intrinsic permeability tensor of the porous medium (m2); kr is the relative permeability function for a given fluid (–), ηw and ηa are the dynamic viscosities of water and gas as a function of temperature, respectively (kg m−1 s−1) [Lide, 2001]; p is the pressure (Pa); ρw is the fluid density as a function of temperature (kg m−3) [Lide, 2001; Grifoll et al., 2005]; g is the gravitational acceleration (m2 s−1); z is the coordinate of vertical elevation (m); and is the phase change rate between the water and vapor due to evaporation or condensation (mass of water transfer from one phase to another per unit volume of soil per unit time) [Smits et al., 2011].
3.2. Heat Transfer
 The energy balance equation in the soil tank considers heat transfer through convection, conduction, and latent heat due to phase change and can be expressed as follows:
where T is the temperature (K); Ceq is the effective heat capacity per unit volume of soil (J m−3 K−1); Cw and Ca are the specific heat capacities of water and gas, respectively (J m−3 K−1); uw and ua are the Darcy velocities of the water and gas, respectively; λ denotes the effective thermal conductivity (W m−1 K−1) [Campbell et al., 1994]; Q is the heat loss term (W m−3); L is the latent heat of water vaporization (J kg−1) [Monteith and Unworth, 1990]; and Lfvw is the latent heat due to phase change in the soil (W m−3).
3.3. Water Vapor Transport
 The transient transport of water vapor in the dry soil layer is given by the following equation:
where Cv is the water vapor concentration in the gas phase (kg m−3), ug is the air velocity in the dry soil layer (m s−1), DL is the hydrodynamic dispersion coefficient (kg m−2 s−1) [e.g., Biggar and Nielsen, 1976; Grifoll et al., 2005], and θg is the volumetric air content (m3 m−3). The formulation of phase change fvw based on the chemical potential gradient as (e.g., Zhang and Datta, 2004; Smits et al., 2011) follows:
where c is an empirical parameter, Mw is the molar mass of water (kg mol−1), Cvs is the saturated vapor concentration (kg m−3) [Campbell, 1985], and Hr is the relative humidity (–) calculated using Kelvin's equation [Philip and de Vries, 1957].
3.4. Soil Water Retention
 Several empirical soil water retention curve models such as the functions developed by van Genuchten  and Brooks and Corey  are widely used to describe water retention at intermediate soil water contents. Previous investigations have shown that these functions describe soil hydraulic properties well at high and moderate water contents. However, they commonly fail to accurately capture soil water retention at low water contents, i.e., when the water content is less than the residual water content [e.g., Ross et al., 1991; Silva and Grifoll, 2007]. A common practice among researchers is to use the residual water content shown in van Genuchten  as a fitting parameter [Groenevelt and Grant, 2004].
where θa (m3 m−3) is a curve-fitting parameter representing the volumetric water content whenPc = 1, is the adsorption of water on the solid phase, and α (m−1) and n are empirical fitting parameters expressed in the van Genuchten  model.
 The term χ in the adsorption of water on soil is described as follows:
where Pc is the capillary pressure (m) and Pcm is the capillary pressure in which water content approaches to zero (m). According to Fayer and Simmons , is generally taken to be 107 cm and higher for fine textured soils. And, the same value of was also used in this work.
3.5. Hydraulic Conductivity for the Full Range Saturation
 As mentioned in section 1, film flow may contribute to unsaturated flow under dry conditions. In this work, both capillary flow and film flow are considered, where the hydraulic conductivity for the full range of saturation is the sum of the contributions due to capillary flow and film flow [Zhang, 2010]:
where Kcap is the hydraulic conductivity due to the capillary flow (m s−1) and Kfilm is the hydraulic conductivity due to film flow (m s−1).
 Hydraulic conductivity due to the capillary flow is traditionally described by using the van Genuchten  model as given by the following equation:
where Ks is the saturated hydraulic conductivity (m s−1); Sew is the effective water saturation; and L and m are the fitting parameters and commonly equal to 0.5 and 1 − 1/n, respectively [van Genuchten, 1980].
 The model developed by Tokunaga  was implemented to describe the portion of the hydraulic conductivity due to film flow. This model does not require empirical fitting parameters and is given as follows:
where (m s−1) is the saturated hydraulic conductivity corresponding to Pc = 0 due to film flow, φ is the porosity (–), dg is the mean grain diameter (m), ηw is the fluid viscosity (kg m−1s−1), σ is the surface tension (N m−1), ε is the relative permittivity of water and equal to 78.54, ε0 is the permittivity of free space and equal to 8.85 × 10−12 (C2 J−1 m−1), kb is the Boltzmann constant (1.381 × 10−23 J K−1), T is the absolute temperature (K), i is the ion charge, and e is the electron charge and equal to 1.602 × 10−19i.
3.6. Formulation Options for the Upper Boundary Condition of Water Vapor Transport
 In this work, three different approaches were used to determine the boundary condition at the top of the soil tank. For approaches 1 and 2, evaporation from the soil system is assumed to equal the diffusive flux of water vapor from the depth of the evaporative front to the atmosphere, using the following form:
where ρe is the water vapor density at the depth of the evaporative front (kg m−3), ρa (kg m−3) is the water vapor density at the reference height ha (m), ra (s m−1) is the aerodynamic resistance of water vapor from the soil surface to the height ha, and rs (s m−1) is the soil surface resistance of water vapor from the depth of the evaporative front to the soil surface. Equation (13) is a kinetic mass transfer expression that approximates net vapor flux through the interface and is applicable for transient conditions.
 The soil surface resistance of water vapor, rs from equation (13), is computed using two different approaches (hereafter referred to as approaches 1 and 2). Approaches 1 and 2 are formulated with the assumption that the phase change may occur within the soil and that water vapor diffusion from the evaporation plane to the soil surface is considered a transport limiting step. This limiting transport step may be “suspect” by the soil surface resistance. Therefore, the soil surface resistance is considered a key parameter to compute evaporation of water from the soil to the atmosphere for this type of model formulation.
 The only difference in approaches 1 and 2 is in the determination of rs. In approach 1, the soil surface resistance is determined through knowledge of the depth of the evaporative front. The soil surface resistance of water vapor in the dry soil layer, rs1, can be determined by the following equation [e.g., Yamanaka et al., 1997; Shokri et al., 2009]:
where Le(t) is the depth of the evaporative front (m; obtained from blue dye deposition from experimental data), Da is the diffusivity coefficient of water vapor in the free air (m2 s−1), is the volumetric air content in the dry soil layer (m3 m−3), and τ is the tortuosity that is determined by the relationship [Penman, 1940]. Equation (14) (approach 1) is mainly based on the knowledge of the depth of the evaporative front (Le), which is measurable in this case in the laboratory but could prove to be difficult to measure under field conditions.
 In approach 2, the soil surface resistance of water vapor transport, rs2, is determined based on the relationship established by van de Griend and Owe :
where θr is the residual water content, θtop is the water content in the top 1 cm layer, and the constant (0.3563) is dimensionless. Previously developed models often use empirical relationships to describe the soil surface resistance term [e.g., Bittelli et al., 2008; Zeng et al., 2011; Smits et al., 2011]. Bittelli et al.  evaluated different formulations of the soil surface resistance term and reported that the one used for approach 2 herein was the best to compute soil water dynamics and therefore chosen for this work. The estimation of rs based on equation (15) (approach 2) is applied in various works such as Bittelli et al. , Smits et al. , and Zeng et al. . Although this approach has shown merit and is widely used, the determination of θtop is difficult and uncertain in most modeling scenarios. The dependence of the soil surface resistance on the soil water moisture at a constant depth of soil (e.g., 0–1 cm) as seen in approach 2 implies that the drying soil layer is artificially kept at a constant depth [Saravanapavan and Salvucci, 2000]. Moreover, the accuracy of the estimation of soil water content in the top soil layer controls the accuracy of this formulation. Furthermore, the use of the empirical relationship in the soil surface resistance term can also lead to problems of physical interpretation [Saravanapavan and Salvucci, 2000].
 For approaches 1 and 2, the aerodynamic resistance to the transfer of water vapor from the soil surface to the height, ha, is given as follows [Monin and Obukhov, 1954]:
where k is the Karman's constant (=0.41), u* is the friction velocity (m s−1), ha is the height above that soil surface where humidity and wind speed were measured (m), d0is the zero plane-displacement height (d0 = 0 for bare soil), zov is the water vapor roughness length (m), and is the stability correction function for water vapor transfer.
where ua is the wind speed (m s−1), zom is the momentum roughness length (m), and is the stability correction function for momentum transfer. For bare soils, zov and zom are assumed to be equal to 10−3 m [Mahfouf and Noilhan, 1991; van de Griend and Owe, 1994]. The stability correction functions for momentum transfer can be estimated using the formulation developed by Webb which relies on the calculation of the Monin-Obukhov stability parameter [Monin and Obukhov, 1954]. The stability parameter includes a sensible heat flux term. For this work, it was assumed that the sensible heat flux and hence the stability correction factor were constant.
 Approach 3, with a third type of boundary conditions, was also tested. In this approach, time-dependent vapor concentration was assigned as the top boundary condition based on knowledge of the relative humidity at the soil surface:
where cvs is the vapor concentration at the soil surface, Hr is the relative humidity at the soil surface, and ρvs is the vapor density at the soil surface. To compute , experimental measurements of relative humidity and temperature at the soil surface were used (relative humidity varies between 0.98 and 0.87 and temperature varies between 19 and 28.3°C). Approach 3 could be useful in situations where the relative humidity boundary condition is known as opposed to the evaporation rate. However, in field situations where the atmospheric boundary condition is constantly changing, other environmental factors must also be considered, which may further complicate the boundary condition [Smits et al., 2011].
3.7. Simulation of the Experiment
 This test problem considers a two-dimensional unsteady state problem of heat, liquid water, and water vapor transfer under diurnal temperature and relative humidity fluctuations in the porous medium. The dimensions of the porous medium are the same as the previously discussed experimental setup for direct comparison. Three different numerical approaches were considered to compare different methods to calculate the surface boundary condition. In approach 1, the method in which the knowledge of the depth of the evaporative surface was considered (equations (13) and (14)), while in approach 2 the empirical relationship developed by van de Griend and Owe  was used (equations (13) and (15)). In the third approach, time-dependent vapor concentration was assigned based on knowledge of the relative humidity at the soil surface (equation (18)). Simulations were then compared to experimental data. Simulations were also performed comparing the contribution of capillary flow and film flow. In this work, it is important to highlight that a vapor enhancement factor which was previously included in the model developed by Smits et al.  was not included.
3.8. Initial and Boundary Conditions and Modeling Procedures
 During the numerical simulation of the tank experiment, initial water pressure was assumed to be hydrostatic and no water flow at the bottom boundary was assigned. The total gas phase pressure was assumed to be atmospheric at the top boundary, and no gas flow boundary was set at the bottom boundary. A time-dependent boundary condition for temperature and relative humidity was assigned at the top boundary based on measured temperatures (varies ∼19°C–28.3°C) and relative humidity (varies ∼0.98–0.87) conditions as seen inFigure 3. Figure 4 shows the surface wind velocity throughout the experiment that was used to determine the friction velocity (equation (17)). The experimental variation in the wind velocity (varied between approximately 2.5 and 3.15 m s−1) was due to the stepwise nature of the signal. At the bottom boundary, temperature was assigned based on the measured temperature (varies ∼18.7°C–23.1°C) conditions, and no flux of vapor was given. The simulations were run for 14 days. After the implementation of the initial and boundary conditions, the system of partial differential equations for the nonequilibrium model in a two-dimensional domain was simultaneously solved using the COMSOL Multiphysics software that is based on the finite element method. The number of two-dimensional elements in the models was 16,948, consisting of 8704 mesh points. The average mesh size was about 5 mm, and typically smaller elements were chosen close to the top soil surface.
4. Results and Discussion
4.1. Comparison of Numerical and Experimental Results for Approach 1
Figure 5 presents a comparison between simulated and measured saturations at selected locations within the tank. Generally, we found that the model overestimated saturation in the wet range and underestimated saturation in the dry range. In analyzing the integrated behavior by comparing the statistical results, the modified index of agreement (MIA) values ranged from 0.832 to 0.998, presented in Table 2. For the layer located 7.5 cm below the soil surface (Figure 5, sensors 6 and 7), the predicted and measured saturations for intermediate times did not agree as well. The observed saturations were lower than the model predicted, demonstrating that the tank dried faster than the model predicted for these times. The model predicted drying rate could have been improved to the point of matching experimental data by including vapor enhancement through the use of a vapor enhancement factor. However, as mentioned earlier, the goal of this work is not to fit experimental and modeling data but rather to investigate soil evaporation formulated with different land surface boundary conditions. In that attempt, we did not want to introduce additional empirical fitting parameters, such as an enhancement factor, even though they are widely used in numerical modeling efforts of unsaturated soils. Although the model captured the saturation over time over the entire range of saturations reasonably well, the overestimation and underestimation of saturation in the wet and dry regimes, respectively, demonstrate that there are still some limitations in the model's ability to capture all the physical processes.
Table 2. MIA Between Observed and Simulated Saturations at Different Locations Corresponding to Different Sensorsa
Figure 6 shows the comparison between predicted and measured temperatures at different locations within the test tank. The observed and modeled temperatures agreed well for all locations over the entire duration of the experiment. This is statistically confirmed with the MIA values (range from 0.950 to 0.988). The largest deviations between simulated and measured temperatures come from thermistors 1 and 5 located near the walls of the tank, implying wall effects, as represented by the lowest corresponding MIA values of 0.950 and 0.961, respectively (Table 3). The deviations between simulated and measured temperatures may, in part, be due to the accuracy and resolution of the thermistors compared to the model.
Table 3. MIA Between Observed and Simulated Temperatures at Different Locations Corresponding to Different Thermistor Sensorsa
 The thermal properties (e.g., effective thermal conductivity (λ), thermal diffusivity (D), and specific heat (C)) of partially saturated soil are expected to contribute strongly to heat transfer in porous media, playing an important role in determining the temperature distribution within unsaturated soil. Thermal properties are known to vary with both saturation and temperature [de Vries, 1963; Sepaskhah and Boersma, 1979; Hopmans and Dane, 1986; Campbell et al., 1994; Heitman et al., 2007; Smits et al., 2010, 2012b]. λ, for example, can be reduced by a factor of 10 when it is dry compared to when it is wet. In terms of temperature, as temperature increases, the thermal properties of soils also increase [e.g., Sepaskhah and Boersma, 1979; Campbell et al., 1994; Hiraiwa and Kasubuchi, 2000; Tarnawski and Gori, 2002; Heitman et al., 2007], especially at intermediate water contents. For the sand used in this work, Smits et al. [2012b] experimentally demonstrated that at high temperatures (e.g., 50°C–70°C), the thermal conductivity of the soil was nearly 2 times that of the 30°C tests for intermediate water contents. In this work, we accounted for λ of the sand as a function of θ and T using the model proposed by Campbell et al. . Prior to implementing the Campbell et al.  model, experimental data from Smits et al. [2010, 2012b] was used to match the fitting parameters within the model. Measured D values from Smits et al.  were also used. C (MJ m−3 K−1) can be determined from λ and D as C = λ/D. As Wang and Bou-Zeid  point out, thermal diffusivity is a “weak function of soil water content and can be approximated as constant with reasonable accuracy.” This is possible for the sand used here, except at very low soil moisture values (at or below residual water contents) where D can be five times smaller than at intermediate water contents. It is noteworthy that Smits et al. [2012b] showed that the thermal diffusivity is also significantly enhanced under high temperatures for intermediate water contents. For this work, it was our intent to use as many known parameters as possible. Therefore, we put much effort into characterizing both the thermal properties of the soils as to not use them as fitting parameters.
4.2. Comparisons Between Three Different Approaches to Compute Evaporation
 Three different simulations were performed, corresponding to approaches 1–3 to critically evaluate how approaches that use different boundary conditions are able to capture the evaporation rate and cumulative evaporation of the system. Figure 7a shows the observed and model simulated evaporation rates from the soil. The observed soil evaporation rate indicates the stages (referred to as stages 1 and 2) of drying similar to that of many previous laboratory studies [e.g., Fisher, 1923; Penman, 1940; Hide, 1954; Philip, 1967; Idso et al., 1974]. During the first stage of evaporation (stage 1), the observed evaporation rates are relatively high (e.g., the hourly average evaporation rate is about 1.15 mm h–1) due to high atmospheric demands (i.e., high temperatures and wind velocity at the soil surface and low ambient relative humidity), controlled by the amount of energy available to vaporize moisture in the upper soil layer and by the vapor pressure deficit between the soil surface and the atmosphere [Bittelli et al., 2008]. After approximately 1.5 days, liquid water cannot be transmitted to the soil surface fast enough to meet evaporative demand; thus, evaporation decreases significantly and the second stage of evaporation (stage 2), sustained by vapor diffusion, begins.
 As seen in Figure 7a, all three of the modeling approaches underestimated the evaporation rates for the first 1.5 days (stage 1) of the experiment. In addition, all three models could not capture the transition between stage 1 and 2 evaporation. As mentioned previously, approach 1 requires knowledge of the depth of the dry surface layer, Le(t) (equation (14)), which can be measured from the blue dye deposition. However, during stage 1 evaporation, which lasts until approximately 1.5 days for this work, the vaporization plane is not visible as the blue dye deposition does not occur until t = 1.5 days. At t= 1.5 days, the vaporization plane could be observed, and its depth ranged from 1 to 10.8 mm over the course of the experiment. After day 1.5, the simulated and measured evaporation rates were in good agreement. Approach 1 assumes fully diffusion-driven transport controlled by the soil properties, thus predicting stage 2 evaporation well. This finding is consistent with the work done byShokri et al.  who measured the evaporative front by using the deposition of a blue dye solution, similar to the approach presented here. This method may be easily applicable for laboratory experiments during stage 2 evaporation. However, it is difficult to apply this approach for stage 1 evaporation when the dry soil layer does not appear clearly. Therefore, the upscaling of this method may be difficult.
 Model approach 2 underestimates stage 1 evaporation as well. The underestimation of evaporation using approach 2 could, in part, be due to disagreement between the experimental setup and the assumptions used in the formulation of the aerodynamic resistance equation (equation (16)). Equation (16) assumes a flat homogeneous soil surface in which transport processes are in equilibrium (uninterrupted airflow). For the experimental setup here, the airflow conditions were somewhat turbulent as there was not enough distance to establish uniform airflow. This, in effect, may have contributed to the underestimation of the evaporation.
 As mentioned earlier, approaches 1 and 2 are different in terms of the formulation of the soil surface resistance term. Some experimental works have been carried out to determine the soil surface resistance [e.g., Yamanaka et al., 1997; Aluwihare and Watanabe, 2003], but, generally, it is difficult to determine. Established relationships between the soil surface resistance and estimated evaporating surface depth or soil moisture in the top layer could not be applied herein due to our experimental conditions being much different from previous works [e.g., Yamanaka et al., 1997; Aluwihare and Watanabe, 2003].
 Approach 3 underestimates stage 1 evaporation as well. In approach 3, time-dependent vapor concentration was assigned as the top boundary condition based on knowledge of the relative humidity at the soil surface. This approach is convenient if knowledge of the relative humidity is known as opposed to the evaporation rate (e.g., in field situations). However, this approach does not consider the aerodynamic resistance which may further complicate the boundary conditions, and the accuracy of the relative humidity measurements at the soil surface requires verification.
Figure 7b shows the simulated and observed cumulative soil evaporation from the experiment using all three modeling approaches. Because the bulk of water evaporated from the soil system is mainly in the first 3 days (stage 1 and the beginning of stage 2 evaporation), all three approaches underestimated the cumulative evaporation as compared to experimental data. For approach 1, the lack of agreement in the cumulative evaporation is due to the approach's inability to capture stage 1 evaporation. However, as seen in Figure 8, if we only consider stage 2 evaporation, disregarding the behavior of stage 1 (the first 1.5 days of the experiment), modeling approach 1 agrees well with experimental observations for the first portion of stage 2 and overestimates evaporation as stage 2 progresses (MIA = 0.968). Modeling approaches 2 and 3 still underestimate the cumulative evaporation. Although approach 1 agreed well for stage 2 evaporation, it is important to highlight the sensitivity of approach 1 to the depth of the evaporative surface that has to either be estimated or measured. A separate simulation was run in which the values for the depth of the evaporative surface were decreased by 25% for the entire experiment (i.e., the maximum depth of the evaporative surface went from the actual value of 10.8 mm to 8.1 mm). As a result, the cumulative evaporation increased from a maximum value of 0.69–0.98 kg, due to the 2.7 mm difference in the evaporative front. Although these small differences are measurable in the laboratory, they would prove difficult to estimate in field settings.
 In this work, water vapor enhancement was not included. However, separate simulations (results not shown) were performed in which a water vapor enhancement factor, η, was employed. The empirical relationship derived by Cass et al.  and Campbell  and implemented in Smits et al.  was used:
where a is an empirical constant to be fitted in this study and fc is mass fraction of clay that is equal to zero for this study. Results demonstrated that a slightly smaller empirical constant than was originally presented by Smits et al.  was needed for approaches 2 and 3 (“a” parameter from Smits et al.  = 16.1 for approach 2 and a= 17.3 for approach 3). However, the need for an enhancement factor in approaches 2 and 3 demonstrates that the models fail to capture all of the physical behaviors that are occurring or there is some uncertainty in the data, suggesting that more experimental work is needed that focuses on small-scale hydraulic and thermal properties to better understand vapor diffusion near the surface [Smits et al., 2011].
4.3. Contributions of Capillary Flow and Film Flow to Hydraulic Conductivity
Figure 9 shows the simulated profiles of the saturation and relative hydraulic conductivity of the capillary flow and film flow, respectively, along cross section AA′ (Figure 1) at the end of the experiment (day 14). At the end of the experiment, the drying front reached a depth of about 13.5 cm below the soil surface as seen in Figure 9a. Figure 9b shows that the relative hydraulic conductivity due to capillary flow decreased rapidly for the narrow layer just above the drying front and then remained constant and low. Moreover, the relative hydraulic conductivity (Kw/Ks) of film flow was almost constant along section AA′. In the soil layers above the drying front, the relative hydraulic conductivity of film flow is about 2.5 times higher than of capillary flow for the layer above the drying front. As discussed in section 3, we assume that the total hydraulic conductivity is the sum of the contribution of the hydraulic conductivity of both capillary flow and film flow. This shows that the contribution of film flow to hydraulic conductivity for the layer above the drying front is dominant compared to that of capillary flow. A separate simulation was run in which film flow was not included, keeping everything else the same, to determine the contribution of film flow to the cumulative evaporation and the saturation profile. For this simulation, not including film flow had a very small effect on the cumulative evaporation and saturation profile. As seen in equation (10), the contribution of the hydraulic conductivity due to film flow is a function of dg, the mean grain diameter. As dgdecreases, the contribution due to film flow increases. In this experimental and modeling scenario, we used relatively coarse-grained sand with a largedg value (0.52 mm). In situations where the mean grain size is small, the contribution of the hydraulic conductivity due to film flow is expected to become more significant.
 Selection of the best method to estimate evaporation from bare soils remains a difficult task, as the models that are used make different assumptions, specifically at the land surface. The most accurate way to estimate evaporation is the use of integrated land-atmospheric models that implicitly couple the soil to the atmosphere without making any assumptions of boundary conditions. Recently, numerical advances have been made in the coupling of free flow (Navier-Stokes) with porous media flow (Darcy flow) [e.g.,Nield, 2009; Shavit, 2009; Chidyagwai and Riviere, 2011; Mosthaf et al., 2011]. As such models are at their early stages of development, it is appropriate to evaluate how the assumed boundary conditions affect the evaporation estimates. This knowledge is expected to help better formulate the land-atmospheric coupling by focusing on the most important processes.
 This article reported a study where three different model formulations that use different boundary conditions were critically evaluated using accurate experimental data generated in the laboratory. All three modeling approaches underestimated the evaporation rate, specifically for stage 1 evaporation. Model approach 1, in which the soil surface resistance is determined through knowledge of the depth of the evaporative front, predicted stage 2 evaporation well. Comparison between the two approaches of evaporation formulation showed that using the approach based on knowledge of the depth of evaporative front (vaporization plane) to estimate the soil surface resistance for water vapor transport (approach 1) allows for better prediction of the drying rate than the approach based on the knowledge of water content of the upper soil layer (approach 2). One of the advances of this work is that the developed model using approach 1 overcomes the arbitrary of using the empirical relationship of the soil surface resistance. However, model approach 1 is very sensitive to the depth of the evaporative surface, a property which may prove to be difficult to determine in field settings.
 Approach 3 may be useful when the relative humidity boundary condition is known as opposed to the evaporation rate. However, in field situations, other factors must also be considered, which may further complicate the boundary condition. All three approaches have various limitations and no one approach can be deemed most appropriate for every scenario. Future work focusing on the land/atmospheric interface, properly incorporating the complex interactions between the land and the atmospheric boundary layer, is needed to increase our understanding of the fundamental processes that control the shallow subsurface soil moisture.
 This research was funded by the U.S. Army Research Office Award W911NF-04-1-0169 and the Engineering Research and Development Center (ERDC) and the Air Force Office of Scientific Research Award FA9559-10-1-0139. The authors also acknowledge Russell Harmon from the Army Research Office and Stacy Howington, John Peters, and Matthew Farthing from ERDC for support and technical contributions.