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Water Resources Research
RESEARCH ARTICLE
10.1002/2013WR015127
Key Points:
The interactions between
evaporation and salt precipitation
Numerical model simulating the
transport of water, solute, and heat
Efflorescence affects more
significantly evaporation than
subflorescence
Supporting Information:
Readme
Sensitivity analysis on the developed
numerical model
Correspondence to:
C. Zhang,
[email protected]
Citation:
Zhang, C., L. Li, and D. Lockington
(2014), Numerical study of
evaporation-induced salt accumulation
and precipitation in bare saline soils:
Mechanism and feedback, Water
Resour. Res., 50, 8084–8106,
doi:10.1002/2013WR015127.
Received 8 DEC 2013
Accepted 25 SEP 2014
Accepted article online 30 SEP 2014
Published online 18 OCT 2014
Numerical study of evaporation-induced salt accumulation and
precipitation in bare saline soils: Mechanism and feedback
Chenming Zhang1, Ling Li1,2, and David Lockington1
1
National Centre for Groundwater Research and Training, School of Civil Engineering, University of Queensland, Brisbane,
Queensland, Australia, 2State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University,
Nanjing, China
Abstract Evaporation from bare saline soils in coastal wetlands causes salt precipitation in the form of
efflorescence and subflorescence. However, it is not clear how much the precipitated salt in turn affects the
water transport in the soil and hence the evaporation rate. We hypothesized that efflorescence exerts a
mulching resistance to evaporation, while subflorescence reduces the pore space for water vapor to move
through the soil. A numerical model is developed to simulate the transport of water, solute, and heat in the
soil, and resulting evaporation and salt precipitation with the hypothesized feedback mechanism incorporated. The model was applied to simulate four evaporation experiments in soil columns with and without a
fixed shallow water table, and was found to replicate well the experimental observations. The simulated
results indicated that as long as the hydraulic connection between the near surface soil layer and the water
source in the interior soil layer exists, vaporization occurs near the surface, and salt precipitates exclusively
as efflorescence. When such hydraulic connection is absent, the vaporization plane develops downward
and salt precipitates as subflorescence. Being more substantial in quantity, efflorescent affects more significantly evaporation than subflorescence during the soil-drying process. Different evaporation stages based
on the location of the vaporization plane and the state of salt accumulation can be identified for characterizing the process of evaporation from bare saline soils with or without a fixed shallow water table.
1. Introduction
Evaporation from terrestrial surface is a fundamental component of the global hydrologic cycle and a key
element of energy exchange between the earth surface and atmosphere. Better understandings on the process of evaporation from bare soils and related heat exchange at the soil-air interface would assist in
improving the efficiency of irrigation in agricultural applications [Ritchie, 1972] and the measures to slow
down the process of degradation and desertification in some arid and semiarid areas [Xue, 1996]. In many
eco-hydrological systems, evaporation provides an important drive for transport of water (liquid and vapor),
solute, and heat, which underlie the behavior and functions of these systems [Hughes et al., 2001; Carter
et al., 2008; Moffett et al., 2010]. Mainly driven by the gradient of water vapor density between the near surface soil layer (defined as the soil layer closest to the soil surface where the soil properties are practically
measurable. It is usually on the milimeter scale and is abbreviated as NSL, hereafter) and atmosphere, the
evaporation process involves phase change of liquid water to water vapor and their movement in the soil
prior to vapor leaving the soil surface. When the NSL is fully saturated with freshwater, the evaporation rate
is largely determined by the solar radiation level, wind velocity, humidity, and temperature in the air above
the soil [Penman, 1948; Lehmann and Or, 2009, Ren et al., 2012]. When the NSL is not fully saturated, the
evaporation rate becomes dependent on local soil conditions, including soil water content (saturation), solute (salt) concentration, and temperature [e.g., Wilson et al., 1994; Fujimaki et al., 2006; Kollet et al., 2009].
Previous studies have identified that the evaporation processes from bare soils in the absence of salt are
rather dependent on the condition of water table below the evaporating soil surface [e.g., Shimojima et al.,
1990; Rose et al., 2005]. Various classifications on evaporation stages have been defined to describe the
behaviors of evaporation resulted from different water table conditions. For the cases with a fixed shallow
water table, a stable hydraulic connection is maintained between NSL and the water source in the interior
soil layer (defined as the soil layer beneath the NSL, and abbreviated as ISL hereafter); and so relatively high
water saturation is maintained in the NSL. The water loss due to evaporation gets compensated by liquid
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water transported from the water table, so that the evaporation process keeps at a stable stage with evaporation rate close to the potential rate [Nachshon et al., 2011b]. In the cases without a fixed shallow water
table, evaporation starts from an initially water-saturated soil with no external water supply. The soil-drying
process occurs in three stages, corresponding with the evaporation intensity [e.g., Idso et al., 1974; Wilson
et al., 1994; Rose et al., 2005; Smits et al., 2011]. The first stage has relatively high and constant evaporation
rate, as evaporation is only limited by the atmospheric demand. The second stage is featured by a sharply
falling evaporation rate since soil water becomes a limiting factor for evaporation. During both the first and
second stages, the vaporization plane, defined as the interface where intensive water vaporization takes
place, is located in the NSL. During the third stage, the vaporization plane moves into the ISL. A layer of dry
soil with liquid saturation nearly zero and liquid water flow completely ceased (defined as dry soil layer
hereafter), developed above the vaporization plane. The evaporation at the third stage has constantly low
rates as it is sustained by water vapor diffusing through the dry soil layer.
When evaporation occurs from soils with saline water, the behavior of evaporation becomes more complex as
it is not only affected by water saturation, but also salt concentration/precipitate in soils [e.g., Nachshon et al.,
2011a]. When the solute concentration is below the solubility, solute accumulates in the vaporization plane
with increased salt concentrations in the soil water. Once the increased salt concentration reaches the solubility, salt starts to precipitate in the vaporization plane. Many studies have proposed classifications to describe
the evaporation process subject to both water and salt change. For example, Fujimaki et al. [2006] and Nachshon et al. [2011b] investigated the evaporation process from a saline soil column with a fixed shallow water
table. They defined two stages to describe the entire evaporation process, i.e., a fast evaporation decline stage
due to falling osmotic potentials followed by a slow evaporation decrease stage with solid salt formed above
the soil surface as efflorescence. Based on their experiments in a saline soil column without a fixed shallow
water table, Nachshon et al. [2011a] described the evaporation process in three stages: the first stage with a
gradual decrease of the evaporation rate due to both reduced osmotic potential and liquid water saturation in
the NSL, the second stage with a sharp evaporation rate decline due to both low liquid water saturation in the
NSL and the formation of a salt crust on the soil surface, and the third stage with a constantly low evaporation
rate as it is hampered by the thickening of the dry soil layer and the continuous salt precipitation.
As the vaporization plane moves into the ISL, salt precipitates in between the soil grains as subflorescence.
The salt crystal developed in between soil grains changes the local soil porosity, and hence the soil hydraulic and thermodynamic properties, including permeability of liquid water, porosity, dispersivity, water retention curve, and heat conductivity. Many studies have been carried out to quantify the effects of changes in
soil hydraulic properties due to evaporation and dissolution, such as the statistical grain size analysis [Panda
and Lake, 1996], discrete description of mineral geometries [Steefel and Lasaga, 1994], and change of poresize distribution estimation [Baveye and Valocchi, 1989; Clement et al., 1996]. However, most of these analyses and methods apply only to saturated conditions. Recently Wissmeier and Barry [2009] developed a radius
shift model to correlate changes in mineral volume due to precipitation and dissolution to changes in
hydraulic properties, including porosity, volumetric water saturation, and permeability of liquid water. This
method involves a curve-fitting procedure to resolve the discontinuity of the soil water retention curve.
The above literature has exposed two major knowledge gaps. First, the classifications of evaporation stages
are found to be rather dependent on the understandings of the solute concentration and groundwater
table conditions in the evaporating soil, which may be unknown beforehand. Therefore, it is essential to
propose a unified classification of evaporation stages regardless of soil water salinity and the shallow water
table conditions. In this study, we systematically defined the evaporation stages according to the locations
of the vaporization plane as well as the state of salt accumulation as shown in Figure 1. Water stage 1 (W1)
is defined for the time period when the vaporization plane is in the NSL. During W1, if the salt solute concentration in the NSL is below the solubility, it continues to accumulate in that layer (defined as salt change
phase 1, and abbreviated as S1, hereafter). Once the solute concentration reaches the solubility, salt precipitates above the soil surface as efflorescence, inducing an extra resistance to evaporation (defined as salt
phase 2, and abbreviated as S2, hereafter). Water stage 2 (W2) and salt phase 3 (S3) generally occurs simultaneously, when the vaporization plane moves to the ISL and salt precipitation occurs below the NSL as subflorescence (Figure 1b).
Another knowledge gap lies on the quantifications of the efflorescence and subflorescence during the soildrying process, and the understanding of the influence of solid salt on the evaporation rate. Although the
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Figure 1. Schematic diagrams of the process of evaporation from bare soils. (a) During W1S2, salt precipitates above the soil surface as
efflorescence. Evaporation is sustained by water vapor being converted from liquid water in the vaporization plane below the salt crust
and moving to the atmosphere subjected to salt resistance and aerodynamic resistance sequentially at the soil-air interface. (b) During
W2S3, salt precipitates as subflorescence. Evaporation is maintained by water vapor at vaporization plane in the ISL moving through the
dry soil layer with flux qv and further to the atmosphere subjected to surface resistance rs, salt resistance rsc, and aerodynamic resistance ra
at the soil-air interface.
studies of Fujimaki et al., [2006] and Nachshon et al. [2011b] have found the efflorescence could reduce the
evaporation rate, their studies were only limited to the cases with a fixed shallow water table and did not
discuss how subflorescence would affect the evaporation. Although subflorescence changes the soil properties, it is not clear how significantly such disturbance in turn affects the evaporation process. However, in
the case without a fixed shallow water table, evaporation rate is high only when the vaporization plane
remains in the NSL, while becomes significantly low once the vaporization plane moves underneath the
NSL, even if there is no salt in the soil [Novak, 2010; Gran et al., 2011a]. Therefore, it is inferable that the salt
precipitaed near the soil surface would have more substantial effect on evaporation than the salt precipitated in the soil. Moreover, the location where salt precipitates as subflorescence is in the dry soil layer,
where the liquid water flow is completely ceased. This implies that although subflorescence may have the
potential to disturb the soil properties and so to the liquid water flow and evaporation, such disturbance
may be trivial during evaporation. If this is true, the mathematical descriptions on feedback effect of subflorescence to evaporation could be significantly simplified. To further investigate these issues, in this study,
following Fujimaki et al. [2006] and Nachshon et al. [2011b], we first hypothesize that salt precipitated in the
NSL does not affect the soil properties there, but develops above the soil surface as efflorescence, exerting
a mulching resistance to evaporation. We further hypothesized that the subflorescence would not affect
the liquid water flow to cause changes in the evaporation rate, but reduce the soil porosity and lead to
obstruction to the water vapor flow. To test the proposed classifications on the evaporation stages, and the
hypotheses on the feedback of efflorescence and subflorescence on the evaporation process, a numerical
model was developed to simulate the transport of water, solute, and heat in the soil, and resulting evaporation and salt precipitation. The model was applied to and validated by results from four previous laboratory
experiments in soil columns: two under an isothermal condition with a fixed shallow water table conducted
by Fujimaki et al. [2006] and two under a nonisothermal condition without a fixed shallow water table conducted by Gran et al. [2011a]. Combining results from numerical simulations and experiments, we aimed to
explore the evaporation processes and behaviors at the proposed different stages.
2. Methodology and Approach
2.1. Model Development
The numerical model considers the transport of liquid water, water vapor, solute, and heat in soils and
energy balance at the soil-air interface. The processes associated with evaporation and salt precipitation
with the hypothesized feedback mechanism are also incorporated.
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2.1.1. Water Transport
One-dimensional mass balance equations for liquid water and water vapor moving in the soil are, respectively [e.g., Bear, 1972]:
@ql hl
@q ql
52 l 2Se ql0
@t
@z
@qv qv
1Se ql0
052
@z
(1)
(2)
where ql , qv , and ql0 (kg m23) are the densities of liquid water, water vapor (qv 5qv hr , where hr is the relative humidity in the soil; qv (kg m23) is the saturated water vapor density), and freshwater (freshwater is the
water with zero solute concentration, ql0 5 1000 kg m23), respectively; hl (m3 m23) is the volumetric liquid
water content (hl 5hp Sl , where hp (m3 m23) is porosity and Sl is liquid water saturation); t (s) is the time;
z (m) is the vertical coordinate pointing upward; ql and qv (m s21) are liquid water and water vapor fluxes,
respectively; Se (s21) is the local vaporization (positive) or condensation (negative) rate per unit volume of
soil. Note that no storage change of local water vapor content is considered in equation (2). This approximation is based on the assumption that water vapor content is in equilibrium with the local liquid water content and the fact that water vapor content is negligible smaller than liquid water content. With the
approximation, the storage of liquid water content in equation (1) lumped both water in liquid and vapor
phase [e.g., Philip and De Vries, 1957; Fujimaki et al., 2006; Novak, 2010]. Combining equations (1) and (2)
leads to:
@ql hl
@q ql @q qv
52 l 2 v
@t
@z
@z
(3)
The liquid water flux ql is given by Darcy’s law [e.g., Bear, 1972]:
ks kr ql g @Wm
ql 52
11
@z
l
(4)
where ks (m2) is the intrinsic permeability; kr is the relative permeability of liquid water; g (9.8 m s22) is the
magnitude of the gravitational acceleration; l (kg m21 s21) is the dynamic viscosity of liquid water, and Wm
(m) is the matric potential. The effect of temperature on the liquid water flow is small and hence neglected
[Milly, 1984]. Driven by the gradient of water vapor density, the water vapor flux can be written based on
Fick’s law [e.g., Campbell, 1985], including an enhanced vapor flux term for intermediate water content [Philip and De Vries, 1957]:
qv 52Dv sha
@qv
@hr
@q
52Dv sha qv
2Dv sha hr g v
@z
@z
@z
(5)
where Dv (m2 s21) is the molecular diffusivity of water vapor in the air; ha (m3 m23) is the volumetric air content; s is the tortuosity factor (s5ha7=3 =h2p ) [e.g., Millington and Quirk, 1961; Saito et al., 2006; Sakai et al.,
2009]; and g is the vapor flux enhancement factor.
2.1.2. Solute Transport
The one-dimensional nonreactive solute (salt) transport is governed by [e.g., Bear and Cheng, 2010]:
@ ðql hl C Þ
@ql ql C @
@C
52f 2
1
q hl ðDi 1Dm Þ
@t
@z
@z l
@z
(6)
and
@M
5f
@t
(7)
where C (kg kg21) is the solute concentration in the liquid water calculated by the mass of solute per mass
of solution; Di (m2 s21) is the molecular diffusion coefficient; Dm (m2 s21 ) is the mechanical dispersion coefficient (Dm 5ajql =hp Sl j, where a (m) is longitudinal dispersivity); M (kg m23) is the mass of solid salt per unit
volume; f (kg m23 s21) is the precipitation (positive) and dissolution rate of salt per unit volume.
Equations (6) and (7) need to be solved iteratively to determine the solute concentration C and solid salt
mass per unit volume M, respectively. However, if allowing C to exceed the solubility Cs (kg kg21) and
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Salt mass per unit volumn of soil (kg m−3)
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10.1002/2013WR015127
70
Salt dissolved in liquid solvent when
solute concentration is below Cs
60
Salt dissolved in liquid solvent that
causes excessive density driven flow
50
Salt stored in the virtual media as
solid phase
40
30
20
10
0
0
0.05
0.1
0.15
0.2
−1
Solute Concentration (kg kg )
0.25 Cs
0.3
Figure 2. Salt mass (including solid and solute from) per unit volume of soil at different solute concentrations with the liquid water saturation Sl , initial porosity hp0 , and parameter m, respectively, fixed at 0.2, 0.4, and 1000.
assuming M as a function of C beyond the solubility to store the precipitated salt mass per unit volume, one
can combine these two equations and solve them simultaneously. For that purpose, M and C are related as
follows:
(
0
C Cs
M5
(8)
mðC2Cs Þ C > Cs
where m (kg m23) is a parameter representing the density of a virtual medium that stores precipitated salt.
Figure 2 shows the relationship between salt mass per unit volume of soil and solute concentration. When
C Cs , salt is completely dissolved in the solute phase within the liquid water. When C > Cs , m is set to a
large value so that the majority of the excess salt is stored in the virtual medium as the precipitated salt
with a small proportion stored in the liquid water (increase of C above Cs). The error associated with the
excess solute salt stored in the liquid water can be minimized by setting m to a very large value. With the
relation given by equation (8), equations (6) and (7) can be combined as:
@ ðql hl C1mC Þ
@ql ql C @
@C
52
1
q hl ðDi 1Dm Þ
(9)
@t
@z
@z l
@z
2.1.3. Heat Transport
To consider the effect of soil temperature on the water vapor flow and evaporation, heat transport in the
soil is simulated in the numerical model. The heat transport equation is written as [e.g., Smits et al., 2011]:
@
@
@T
@ qs cs T2T 0 1ql cl hl T2T 0 5
kT
2
L0 ql qv 1ql cv T2T 0 qv
@t
@z
@z
@z
(10)
@ 2
ql cl T2T 0 ql
@z
where qs (kg m23) is the density of soil solid grains; cs , cl , and cv (J kg21 K21) are, respectively, the specific
heat capacity of soil solid grains, liquid water (4200 J kg21 K21), and vapor (cv 5 2000 J kg21 K21); T is the
bulk soil temperature (K); T 0 is the initial soil temperature; kT (J m21 s21 K21) is the bulk thermal conductivity of soil matrix and fluid; and L0 (J kg21) is the latent heat required for vaporization
(L0 52:5013106 22369:2T). Note that minor energy generation and consumption due to soil drying and
wetting, salt precipitation and dissolution, viscose liquid movement have been neglected [e.g., Nassar et al.,
1992].
2.1.4. Liquid Water Transport Properties
Different types of soils were used in the four experimental cases that the model simulates The functions of
soil water retention and relative permeability were obtained from studies that conducted the experiments
(see Table 1). For Masa loamy sand, the following constitutive relation between matric potential and liquid
water saturation was used [Fujimaki and Inoue, 2003]:
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Table 1. Parameters Values Applied in the Experiment and Numerical Simulation for Each Evaporation Casea
Properties
Soil Hydraulic Properties
Density of soil solid grains qs (kg m23)
Initial porosity (no salt present) hp0 (m3 m23)
Intrinsic permeability ks (m2)
Dispersivity a (m)
Ionic diffusion coefficient Di (m2 s21)
Water retention curve equation
a1 (m21)
n1
a2 (m21)
n2
Slr
W0 (m)
w
Relative permeability equation
x1
ak
bk
Heat Conditions
Thermodynamic Properties
cs (J m23 s21)
R (W m2)
Solute Properties
Initial salt concentration C0 (kg kg21)
Solubility Cs (kg kg21)
Aerodynamic Conditions
Temperature in the air TRef (K)
Relative humidity in the air hrRef
Initial temperature in soil column T0 (K)
Matric potential at the bottom of the column WmBot (m)
ra5 rH (s m21)
rHSide (s m21)O
Numerical Parameters
Time step Dt (s)
Nodal spacing below the NSL Dz (m)
m (kg m23)
Depth of the NSL DzNSL (m)
Case ML Masa
Loamy Sand
Case TS
Toyoura Sand
Case SS
Silica Sand
Case AS
Aluminous Silt
2620a
0.43a
5.38 3 10213a
1.4 3 1023a
3.89 3 10210a
(11)a
24.21a
1.37a
2640a
0.44a
2.04 3 10211a
2.3 3 1023a
7.78 3 10211a
(12)a
22.65a
33.6a
22.14a
1.84a
2650b
0.4c
2.8 3 10211c
0.01c
5 3 10211c
(13a)c
24c
8.5c
3960b
0.4c
7.09 3 10213d
0.01
5 3 10211c
(13a)e
21.63d
1.37d
0.1c
25 3 104c
0.074d
25 3 104e
0.748a
(19)a
(20)c
(20)c
1.57a
0.023a
Isothermala
Nonisothermalb
Nonisothermalb
875c
750c
875
300
0.031a
21 3 105a
a
(18)
7.2a
Isothermala
0.003a
0.231a
0.003a
0.231a
0.007b
0.264b
0.020b
0.264b
298.15a
0.25a
298.15a
20.75a
90a
298.15a
0.4a
298.15a
20.4a
64a
297.15b
0.5b
297.15b
297.15b
0.5b
297.15b
60c
50c
60c
50c
10
0.001
1000
0.0005
10
0.001
1000
0.0005
5
0.002
1000
0.001
5
0.002
1000
0.001
a
The data with superscript a, b, c, d, and e are from Fujimaki et al. [2006], Gran et al. [2011a, 2011b], Carsel and Parrish [1988], and
Fayer and Simmons [1995], respectively.
Sl 5
(
12Slr
½11ða1 Wm Þn1 ð121=n1 Þ
1Slr
)
ln ð2Wm 11Þ 2
12
ln ð2W0 11Þ
(11)
where Slr is the residual liquid water saturation; a1 (m21) and n1 are fitting parameters; W0 (m) is the matric
potential when the liquid water saturation is nearly zero. A bimodal function was applied to describe the
soil water retention characteristic of the Toyoura sand [Fujimaki et al., 2006]:
Sl 5
w
½11ða1 Wm Þn1 ð121=n1 Þ
1
ð12w Þ
½11ða2 Wm Þn2 ð121=n2 Þ
(12)
where w, a1 (m21), a2 (m21), n1 , and n2 are fitting parameters. A modified van Genuchten water retention
equation was applied to describe the behavior of the Silica sand and Aluminous Silt [Gran et al., 2011a]:
Sl 5
12bSlr
½11ða1 Wm Þn1 b5
ð121=n1 Þ
1bSlr
ln ð2W0 Þ2ln ð2Wm Þ
ln ð2W0 Þ
(13a)
(13b)
where a1 (m21) and n1 are fitting parameters.
It should be noted that these three soil water retention equations were modified from the original equation
proposed by van Genuchten [1980]. For equations (11) and (13a), the modifications followed the approaches
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(a)
(b)
−10
10
Legend for (a) and (b)
4
Case ML
Case TS
Case SS
Case AS
−Matric Potential (m)
3
10
−12
10
Permeability (m2)
10
2
10
1
10
−14
10
0
10
−16
10
−1
10
−2
10
−18
0
10
0.1
0.2
0.3
0.4
0.5
Volumetric Water Content (m3 m−3)
0
0.1
0.2
0.3
0.4
Volumetric Water Content (m3 m−3)
Figure 3. Water retention curve and liquid water permeability of soils applied in the study.
introduce by Mehta et al. [1994] and Fayer and Simmons [1995], respectively, in order to extend the lower
saturation limit from the residual in the original equation to nearly zero (i.e., oven dry). This extension is
needed particularly for the dry soil layer, where the liquid water saturation drops below the residual. The
bimodal equation (12) is useful for describing the soil water retention relationship of a mixed-type soil
[Simůnek et al., 2003].
Based on the studies that conducted the laboratory experiment, three equations were applied to express
the relative permeability-liquid water saturation relationship for the four types of soil. The equation given
by Campbell [1974] was applied to the Masa loamy sand [Fujimaki et al., 2006]:
1
kr 5Sx
l
(14)
where x1 is a fitting parameter. For the Toyoura sand, the relative permeability data were fitted with the following equation [Fujimaki et al., 2006]:
kr 5
ak
ak
1
11bk Sl 1bk
(15)
where ak and bk are fitting parameters. According to Gran et al., [2011b], the relative permeability of the
Silica sand and Aluminous silt was calculated according to the pore-size distribution model of Mualem
[1976]:
Sl 2Slr
kr 5
12Slr
8
92
0:5 < " n #ðn21
n Þ=
Sl 2Slr ðn21Þ
12 12
:
;
12Slr
(16)
The water retention curves and the liquid water permeabilies (i.e., ks kr ) of the four soils are illustrated in Figure 3. The parameters involved in equations (11–16) are listed in Table 1.
The liquid water density ql varies with the concentration of the solute (salt) as follows [e.g., Hughes and Sanford, 2004; Voss and Provost, 2010]:
ql 5ql0 1
@ql
C
@C
(17)
l
for sodium chloride, @q
@C equals 702.24 based on the units used in density and salt concentrations. The effect
of temperature on the liquid water density is negligible. The liquid water viscosity is affected by the temperature, solute concentration, and air pressure [e.g., Kestin et al., 1981]. Assuming a constant air pressure and
linear solute concentration-viscosity relationship, the viscosity can be calculated according to Hughes and
Sanford [2004]:
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248:37
@l
l5239:4310ðT 2140Þ 1 C
@C
27
(18)
@l
where @C
is equal to 3.25 3 1023 [Kestin et al., 1981].
2.1.5. Water Vapor Transport Properties
The saturated water vapor density qv in equation (5) is calculated by the following equation based on the
ideal gas law:
qv 5
pv Mw
RT
(19)
where Mw (0.018 kg mol21) is the molecular weight of water; R (8.314 J mol21 K21) is the ideal gas constant.
The saturated water vapor pressure pv is a function of temperature alone [e.g., Murray, 1967], i.e.,
17:27ðT2273:15Þ
pv 5611 exp
(20)
T235:85
The relative humidity hr in the pore space can be evaluated based on the assumption of a thermodynamic
equilibrium between the liquid water and water vapor (i.e., Kelvin’s equation) [e.g., Philip and De Vries, 1957;
Bresler, 1981]:
Ww gMw
hr 5exp
(21)
RT
where Ww is the water potential, which can be calculated according to:
Ww 5Wm 1Wo
(22)
where Wo (m) is the osmotic potential, dependent on the solute concentration [e.g., Campbell, 1977]:
Wo 52
vvCMs RT
g
(23)
where v is the number of ions per molecule (2 for sodium chloride) and Ms (0.0585 kg mol21) is the molecular weight of sodium chloride. v (mol J21) is the osmotic coefficient and can be calculated as follows [Fujimaki et al., 2006]:
ax ql C
bx ql C px
v50:91
(24)
1
12C
12C
where ax , bx , px are fitting parameters. For sodium chloride, they equal 0.0026, 0.0026, and 0.053,
respectively.
The coefficient of water vapor diffusion in the air is temperature-dependent as follows [e.g., Kimball et al.,
1976]:
1:75
T
Dv 52:2931025
(25)
273:15
The empirical equation developed by Cass et al. [1984] was used to describe the enhancement factor g in
equation (5), which is possibly induced by the existence of liquid islands and increased temperature gradients in the gas phase [Philip and De Vries, 1957]:
( 4 )
2:6
g59:5 1 3Sl 2 8:5 exp 2 11 pffiffiffiffi Sl
(26)
fc
where fc (kg kg21) is the mass fraction of clay in the soil.
The volumetric air content in the soil where water vapor moves through is subjected to changes due to the
liquid water saturation as well as the salt precipitation in the soil:
ha 5ð12Sl Þhp 5ð12Sl Þ hp0 2hsalt
(27)
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where hsalt (m3 m23) is the volumetric solid salt content (hsalt 5M=qsalt , where qsalt (2165 kg m23) is the density of solid salt); hp0 (m3 m23) is the initial porosity of the soil without the presence of solid salt. Note that
the change of volumetric solid salt content and the subsequent change of volumetric air content only occur
beneath the NSL, where salt precipitates as subflorescence.
2.1.6. Soil Heat Transport Properties
The model of Chung and Horton [1987] is applied to describe the bulk thermal conductivity, i.e.,
kT 5 b1 1 b2 hl 1 b3 h0:5
l
(28)
where b1 , b2 , and b3 are fitting parameters and equal to 0.228, 22.406, and 4.909, respectively, according to
Chung and Horton [1987]. The assignments of these parameters by the literature are due to the fact that the
soil thermal properties are much more affected by water content than by the textures and that the thermoproperties of mineral components in the solid phase are almost in the same order of magnitude [Jury and
Horton, 2003; Saito et al., 2006].
2.1.7. Boundary Conditions
Assuming no heat storage, the energy exchange in the infinitesimal air layer just above the soil surface can
be expressed by the following surface energy balance equation [e.g., Saito et al., 2006]:
G 1 L0 Eql0 1 H 5 Rn
(29)
where Rn (W m22) is net radiation; G (W m22) is the heat flux density to the soil (positive downward); H (W m22)
is the specific sensible heat exchange between soil and atmosphere (positive upward); E (m s21) is the evaporation rate. With the assumption that the temperature of the infinitesimal air layer equals the temperature in the
NSL, the specific sensible heat flux from the infinitesimal air layer to the atmosphere, H, can be calculated from
the following equation [e.g., Saito et al., 2006]:
H5
ca qa
ðTNSL 2TRef Þ
rH
(30)
where ca (1013 J kg21 K21) is the specific heat capacity of the dry air; qa (kg m23) is the density of dry air
(qa 5pa Ma =RTa , where Ma (0.02897 kg mol21) is the molecular weight of dry air; pa (100,325 Pa) is the
atmospheric pressure; and Ta (K) is the dry air temperature); rH (s m21) is the aerodynamic resistance to the
heat transfer, TNSL (K) is the temperature in the NSL, TRef (K) is the temperature at the reference point in the
atmosphere. Note that equation (30) does not only apply at the soil surface at the nonisothermal cases, but
also on the side of the soil column to describe the sensible heat exchange between soil column and ambient environment.
With the effects of atmospheric and soil conditions included, the evaporation rate can be calculated by
[Camillo and Gurney, 1986; Fujimaki et al., 2006]:
E5
qv NSL 2qvRef
ql0 ðra 1rs 1rsc Þ
(31)
where the subscript NSL and Ref refer to, respectively, values in the NSL and at the reference point; ra 5rH (s m21)
is the aerodynamic resistance to water vapor transfer; rs (s m21) is the surface resistance; and rsc (s m21) is the
resistance due to salt precipitation as efflorescence.
Describing the variation of evaporation rate due to water availability in the evaporating soil, the surface
resistance is found to be one of the key parameters to determine the temporal change of evaporation rate,
particularly for the cases without the presences of a fixed shallow water table. The selection of surface
resistance equation is conducted by comparing the root mean square error of the predicted evaporation
rates by the models using different surface resistance equations (e.g., equations proposed by Sun [1982],
Camillo and Gurney [1986], Kondo and Saigusa [1990], and van de Griend and Owe [1994]) with regard to the
measured rates. The linear equation proposed by Camillo and Gurney [1986] was used in this study:
(32)
rs 52as 1bs hp0 2hlNSL
where as (s m21) and bs (s m21) are fitting parameters, and were found to be 2805 and 4140, respectively
[Camillo and Gurney, 1986].
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For the salt resistance, the empirical equation developed by Fujimaki et al. [2006] was adopted in the present model. Assuming that solid salt precipitated in the NSL is formed as effloresce above the soil surface,
we can calculate rsc by:
8
0
C<0
<
rsc 5
(33)
b
: ar ln 100C1exp 2 r 1br C 0
ar
where C (kg m22) is the specific mass accumulation (per unit area) of efflorescence (C5MNSL DzNSL , where
DzNSL (m) is the depth of the NSL and MNSL (kg m23) is the mass of solid salt precipitated in the NSL);
ar (s m21) and br (s m21) are fitting parameters, and equal 69 and 2104, respectively, for sodium chloride.
2.1.8. Model Integration and Solution
The mathematical model was integrated and solved numerically based on SUTRA [Voss and Provost, 2010],
an existing three-dimensional numerical model simulating coupled variably saturated density-dependent
groundwater flow and solute transport. The original SUTRA model solves the liquid water transport equation (1) and solute transport equation (6). To extend SUTRA’s capability for simulating the evaporation process, the water vapor transport equation (2) and the change of porosity due to salt precipitation was
incorporated into the liquid water transport equation (1) to form water transport equation (3), which was
solved by the finite element solver provided in the original model. The salt precipitation as modeled by
equation (9) can be readily simulated by SUTRA with the apparent absorption term (m) included in the solute transport equation. The heat transport equation (10) was solved by a fully implicit, central finite difference method [Zheng and Bennett, 2002; Bear and Cheng, 2010]. The energy balance equation (29) was
integrated as the boundary conditions for both heat and water transport equations.
2.2. Laboratory Experiments and Application of the Model to the Experiments
The developed model was tested through applications to simulate laboratory experiments conducted previously under various conditions (Table 1). The laboratory data for the cases with a fixed shallow water table
(cases ML and TS) were from Fujimaki et al. [2006], who conducted the evaporation study based on a cylindrical column under an isothermal condition. Masa loamy (ML) sand and Toyoura sand (TS) were used in
the two cases, respectively. The column had a size of 0.038 m in diameter and 0.052 m in depth. A watersaturated ceramic plate was placed at the bottom of the column. The ceramic plate was connected to a
Mariotte bottle for supplying saline water while maintaining certain negative matric potential at the bottom
of the column. The matric potentials at the bottom of the soil column were 20.75 and 20.4 m for cases ML
and TS, respectively. These conditions were equivalent to fixing the water table at 0.802 and 0.452 m below
soil surface. Before the experiment started, equilibrium between the matric potential and liquid water saturation was achieved in the soil column. Such equilibrium was maintained by the constant matric potential
applied at the bottom of the column. Evaporation was then intensified by blowing air over the surface of
the column. An isothermal condition was maintained at 298 K by regulating the upper heat source and
wrapping the side of the column with polystyrene foil. The evaporation rate was manually measured by
weighing the column using an electronic scale at a time interval of 12 h. Both the water content and solute
concentrations were measured over the depth during the experiments. Further details can be found in Fujimaki et al. [2006].
The cases without a fixed shallow water table (cases SS and AS) were from Gran et al. [2011a], who conducted the evaporation study under nonisothermal conditions. Silica sand (SS) and Aluminous silt (AS) were
used in the two cases, respectively (Table 1). The size of the cylindrical column used in the experiments was
0.144 m in diameter and 0.24 m in depth. The soil medium was initially saturated by saline solution of a different initial concentration in each case. As the heat was introduced at the top of the column through an
infrared lamp to accelerate the evaporation rate, the temperature profile in the column varied over time,
which in turn affected the evaporation rate. The evaporation rates were measured by weighing the column
using electrical scale. The liquid water saturation and solute concentration were obtained by setting up and
dismantling identical columns for measuring the profiles of both parameters at different stages of the
experiments. The temperature profiles were also measured before the dismantlement.
The numerical model has been configured to simulate the evaporation processes during the four experimental cases. The lengths of the calculation domain in all cases were based on the length of the soil
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(b)
0
10
10
20
20
30
30
Legend for Figure (a) (b) and (c)
Measurements at 11 days
Calc. right after started
Calculations at 3.5 days
Calculations at 11 days
40
50
Evaporation and groundwater
−1
uptake rate (mm day )
(c)
0.4
0.5
0.6
Saturation (−)
(d)
0.7
0
Depth (mm)
(a)
40
0.05 0.1 0.15 0.2
−1
Concentration (kg kg )
0
5
10
15
−1
Liquid water flux (mm day )
50
W1
S1
20
S2
3.5 days
16
0.1
Measured evaporation
Calculated evaporation rate (RMSE=1.0 mm/day)
Calculated groundwater uptake
Calculated solid salt depth on soil surface
12
11 days
0.08
0.06
8
0.04
4
0.02
0
0
2
4
6
Time (day)
8
10
Solid salt depth (mm)
Depth (mm)
0
10.1002/2013WR015127
0
12
Figure 4. Predicted and measured results for case ML: (a) liquid water saturation, (b) solute concentration, (c) liquid water flux across the
soil column, and (d) temporal changes of evaporation, groundwater uptake and solid salt depth on the soil surface. The evaporation stages
based on the predicted evaporation rate are indicated in the top of Figure 4d.
column. For the isothermal cases ML and TS, only transport equations of water and salt were solved. The
fixed matric potential at the bottom of the column was simulated using Dirichlet boundary condition. The
evaporation rate was calculated using equation (31). At the beginning of the simulation, the matric potential in the column was hydrostatic maintained by the fixed matric potential at the bottom of the column;
the temperature and solute concentration were uniform. For the nonisothermal cases SS and AS, the transport equations of water, salt, and heat are solved iteratively. No water fluxes were set on both the bottom
and the side of the column. Equation (30) was applied to calculate the exchange of heat on the side of the
soil column. Surface energy balance equation (29) was adopted on the top of the soil column, which calculates both the sensible and latent heat exchange of heat. Simulations were started from the hydrostatic condition where the soil column was water-saturated with uniform temperature and solute concentration. The
relevant parameters and their sources are listed in Table 1. The results were tested to be independent from
the numerical parameters.
3. Results and Analysis
Generally, the model predicted reasonably well the measurements made in the experiments for all four
cases. The results are presented below for each case with the focus on the depth profiles of liquid water saturation, temperature and salt concentrations, and the evaporation rates at times corresponding with the
measurements. Additional simulations results, including the depth profiles of the liquid water and water
vapor flux, thickness of effloresce, and change of porosity, are also analyzed to better elaborate the transport processes of water, solute, and heat underlying the evaporation behavior.
3.1. Evaporation From a Soil Column With a Fixed Shallow Water Table Under an Isothermal
Condition
Figure 4 shows the measured and simulated results for case ML. With a fixed shallow water table, the evaporation process remained at state W1 over the whole period of the experiment, with both salt accumulation
(S1) and precipitation (S2) occurred (Figure 4d). At the beginning of the process, the water saturation and
solute concentration over the column depth largely remained at the initial levels (solid lines, Figures 4a and
4b). The liquid water in the upper soil layer started to move upward to compensate the evaporating water
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(b)
0
10
10
20
20
30
30
Legend for Figure (a) (b) and (c)
Measurements at 11 days
Calc. right after started
Calculations at 3.5 days
Calculations at 11 days
40
50
Evaporation and groundwater
−1
uptake rate (mm day )
(c)
0
(d)
0.2
0.4
Saturation (−)
0.6
0
Depth (mm)
(a)
40
0.05 0.1 0.15 0.2
−1
Concentration (kg kg )
0
5
10
15
20
−1
Liquid water flux (mm day )
50
W1
S1
20
S2
16
12
0.1
Measured evaporation
Calculated evaporation rate (RMSE=0.8 mm/day)
Calculated groundwater uptake
Calculated solid salt depth on soil surface
3.5 days
0.08
11 days
0.06
8
0.04
4
0.02
0
0
2
4
6
Time (day)
8
10
Solid salt depth (mm)
Depth (mm)
0
10.1002/2013WR015127
0
12
Figure 5. Predicted and measured results for case TS: (a) liquid water saturation, (b) solute concentration, (c) liquid water flux across the
soil column at three time stages, and (d) temporal changes of evaporation, groundwater uptake and solid salt precipitation depth on the
soil surface. The evaporation stages based on the predicted evaporation rate are indicated in the top of Figure 5d.
loss with an intensified flux closer to the soil surface (solid line, Figure 4c). The highest evaporation rate
occurred at the beginning of the evaporation process due to the lowest solute concentration and the
absence of salt precipitation in the NSL (Figure 4d). As evaporation continued, the liquid water flow
(upward) developed with a relatively uniform rate over the whole column. Essentially this uniform flow rate
supplied the amount of water for evaporation and decreased gradually over time (Figure 4c) as the evaporation weakened slowly (Figure 4d). The presences of a shallow water table allowed the loss of water due to
evaporation to be readily compensated by water moving up from the source underneath. Therefore, there
was no significant change of the water saturation profile (Figure 4a).
Evaporation resulted in rising solute concentrations in the NSL during W1S1 (water stage W1 and salt stage
S1), which led to reduced osmotic potential and decline of the evaporation rate. The rise of the solute concentration also led to increase in the liquid density. This effect resulted in lower liquid water flux in the
upper soil layer of high solute concentrations than that in the area below (dashed line and dash-dotted line,
Figure 4c). The solute concentration reached the solubility in the middle of the second day with salt starting
to precipitate. Subsequently, the evaporation process entered W1S2 where the evaporation rate declined
gradually as the salt resistance increased slowly with the salt precipitation. The upward liquid water flux
profile remained uniform over the depth with the flux rate decreased slowly with the decline of the evaporation rate (Figure 4c). Since the diffusion and dispersion processes were slow, little salt was transported
downward despite high solute concentrations near the soil surface and low solute concentrations underneath (dashed line, Figure 4b). Over the stage of W1S2, the depth of efflorescence increased almost linearly
(Figure 4d) but with no salt precipitated as subflorescence.
The measured and simulated results for case TS are shown in Figure 5. The major difference between case
ML and TS lay on (1) the soil used in case TS had coarser soil grains, thus yielding a higher intrinsic permeability but lower liquid water saturation under the same matric potential; (2) the aerodynamic resistance in
case TS (64 m s21) was lower than that in case ML (90 m s21); (3) the ionic diffusion coefficient in case TS
(3.89 3 10210 m2 s21) was about a fifth of that in case ML (7.78 3 10211 m2 s21). Similar to case ML, with
the presence of a fixed shallow water table, the evaporation process remained at stage W1 throughout the
experiment/simulation, with initial accumulation of solute (S1) followed by salt precipitation (S2) in the NSL.
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Although the liquid water saturation in the NSL was lower than that in case ML due to a weaker capillary
force (Figure 5a), the liquid water flow (Figure 5c) was capable of delivering sufficient liquid water required
for evaporation. Therefore, there was little change of saturation profile during the evaporation process (Figure 5a). Due to relatively lower aerodynamic resistance in case TS, the initial evaporation rate was greater
than that in case ML (Figure 5d). This higher initial evaporation rate, together with lower diffusivity, resulted
in faster solute accumulation in the NSL and hence earlier commencement of W1S2 in case TS (day 1.2)
than case ML (day 1.7).
3.2. Evaporation From a Soil Column Without a Fixed Shallow Water Table Under a Nonisothermal
Condition
Figure 6 depicts the measured and simulated results for case SS. The evaporation process in this case covered all the stages shown in Figure 1: W1S1 from the beginning to middle day 3; W1S2 from middle day 3
to middle day 6; and W2S3 from middle day 6 to the end of the simulation/experiment (Figure 6h). Different
from the cases shown above, there was a short period of evaporation rise at the beginning of the process
(solid line, Figure 6h). Being consistent with experimental observations by Norouzi Rad and Shokri [2012]
and Nachshon et al. [2011a], this predicted evaporation rise was caused by the rise of soil temperature: at
the beginning of the experiment, the vaporization plane in the NSL was subjected to not only liquid water
desaturation (Figure 6a) and solute accumulation (Figure 6b) which hampered evaporation, but also rising
temperature (Figure 6c) which enhanced evaporation. The enhancing effect overwhelmed the hampering
effect in the early stage and led to the rise of the evaporation rate.
Also different from the cases with a fixed shallow water table, the loss of water due to evaporation in the
vaporization plane in this case was not adequately compensated by upward liquid and vapor flows from ISL
(Figures 6d and 6e), resulting in an immediate water table drawdown (Figure 6a). As the liquid water saturation decreased, the upward water flow became weakened, which in turn led to further reduction of the liquid water saturation as well as intensified solute accumulation (i.e., increased solute concentration). As a
result, the rising evaporation trend was reversed approximately 2 h after the start (solid line, Figure 6h).
Along with the decline of the evaporation rate, the soil temperature kept on rising (solid line, Figure 6c).
The reduced evaporation rate was still greater than the liquid water flow rate and thus the liquid water saturation in the NSL continued to decrease (solid line, Figure 6a). The liquid water desaturation vacated pore
space, which allowed the development of water vapor movement (solid line and dotted line, Figure 6e).
The solute concentration reached the solubility on middle day 3, marking the commencement of stage
W1S2. At this stage, the vaporization plane still remained in the NSL; thereby salt precipitated in the form of
efflorescence. Similar to the case with a fixed shallow water table, the depth of efflorescence increased linearly with time (dashed line, Figure 6h) and no salt precipitation occurred as subflorescence (dashed line, Figure 6f). Stage W1S2 ended on day 6 with the stop of efflorescence and an abrupt drop of the evaporation
rate (solid line, Figure 6h). This signified the start of stage W2S3 with the dry soil layer formed and vaporization plane moved into the ISL. Evaporation was then sustained by only water vapor converted from liquid
water in the moist vaporization plane beneath the dry soil layer and then moving upward to the
atmosphere.
At the beginning of stage W2S3, since the depth of the dry soil layer was thinnest, the water vapor flux
through the dry soil layer reached the highest rate (dashed line, Figure 6e). After that, the rate of the water
vapor flux decreased due to the thickening of the dry soil layer (dash-dotted line, Figure 6e). In contrast, the
liquid water flux in the dry soil layer ceased as the liquid water saturation there was nearly zero (dash-dotted line, Figures 6a and 6d). As the evaporation process continued, the liquid water saturation in the vaporization plane beneath the dry soil layer dropped below the residual, leading to the thickening (downward
expansion) of the dry soil layer. Since the evaporation rate during stage W2S3 was nearly constant, the dry
soil layer thickening process followed a linear trend over time (Figure 6g). By day 20, the dry soil layer
expanded to reach a thickness of 3 cm. During the dry soil layer thickening process, no salt precipitated as
efflorescence (dashed line, Figure 6h). Instead, the solute concentration in the vaporization plane beneath
the dry soil layer increased rapidly and reached the solubility, resulting in salt precipitation in the form of
subflorescence. The solid salt occupied the pore space in between the soil solid grains, reducing the porosity at the dry soil layer (Figure 6f). The extent of porosity reduction exhibited a decreasing trend with the
depth, corresponding with less salt precipitated locally as the vaporization plane moved downward. The
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(b)
0
50
50
100
100
Legend for Figure (a) (b) and (c)
Measurements at 74% sat.
Measurements at 50% sat.
Measurements at 32% sat.
Calculations at 74% sat.
Calculations at 50% sat.
Calculations at 32% sat.
150
200
0
0
Depth (mm)
(c)
0.5
Saturation (−)
1
(d)
0
0.1
0.2
−1
Concentration (kg kg )
150
200
0.3
30
35
40
45
50
Temperature ( ° C)
(e)
(f)
0
50
50
100
100
150
150
Legend for Figure (d) (e) and (f)
Calculations at 74% sat.
Calculations at 55% sat.
Calculations at 50% sat.
Calculation at 32% sat.
200
Depth (mm)
(a)
Depth (mm)
Depth (mm)
0
10.1002/2013WR015127
200
0
5
10 0
1
2
3 0.39
−1
−1
Liquid water flux (mm day )
Water vapor flux (mm day )
0.395
0.4
3 −3
Porosity (m m )
(g)
Depth (mm)
0
20
40
Legend for Figure (g)
Calculated location of the vaporization plane
2
4
6
8
Evaporation (mm day−1)
(h)
10
12
Time (day)
W1
S1
74% sat.
32
14
16
18
20
W2
S3
S2
55% sat.
50% sat.
32% sat.
24
0.08
0.06
Legend for Figure (h)
Measured evaporation rate
16
0.04
Calculated evaporation rate (RMSE=2.7 mm/day)
8
0
22
Calculated solid salt depth on surface
0
2
4
6
8
10
12
Time (day)
14
16
18
20
0.02
Solid salt depth (mm)
0
0
22
Figure 6. Predicted and measured results for case SS: (a) liquid water saturation, (b) solute concentration, (c) temperature, (d) liquid water
flux, (e) water vapor flux across the soil column, (f) porosity, (g) temporal change of vaporization plane location in the NSL, and (h) temporal changes of evaporation rate and solid salt depth on the soil surface. The evaporation stages based on the predicted evaporation rate
are indicated in the top of Figure 6h.
temperature profile over stage W2S3 did not change significantly as the equilibrium developed among the
net radiation, heat flux into the soil, sensible heat exchange from both the surface and side of the column
and latent heat released during evaporation.
There are three notable discrepancies between the simulated and measured results: (1) the over prediction
of the temperature profile on day 6 (Figure 6c); (2) the under prediction of dry soil layer and evaporation
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(Figure 6a); and (3) the under prediction of evaporation rate during stage W1 (Figure 6h). These discrepancies could be caused by the model not properly considering some key processes in the experiment. For
example, (1) the variation of axial temperature may exists [Smits et al., 2011]; (2) the neglect of the albedo
change on the soil surface due to efflorescence; (3) the model adopts a simple consideration of sensible
heat exchange on the side of the soil column (i.e., fixed ambient temperature and aerodynamic resistance
on the column wall); (4) the correlation between evaporation and liquid water saturation may vary among
soils [Shahraeeni et al., 2012], while the empirically based surface resistance (i.e., equation (32)) lacks such
physical insight. Albeit these discrepancies, the proposed model can still produce the temporal and spatial
changes of the liquid water saturation, solute concentration and temperature, and the temporal changes of
the evaporation rate with general agreement with the measured results. Solving these issues would be
essential to further improve the accuracy of the model.
Figure 7 displays the measured and simulated results for case AS. There are three major differences
between cases AS and SS in terms of input conditions (Table 1): (1) the aluminium silt applied in case AS
had finer soil grains, which gave lower intrinsic permeability but higher liquid water saturation at the same
matric potential (Figure 3); (2) the initial solute concentration was 0.02 in case AS, higher than 0.007 in case
SS; (3) the net radiation in case AS was 300 W m22, lower than 750 W m22 applied in case SS. The comparison of the results between case AS and SS can be served as sensitivity analysis of the model in terms of soil
types, initial solute concentration and heat input. Similar to case SS, the evaporation process of case AS covered all the stages shown in Figure 1, that is, W1S1 from the beginning to day 4 with an early evaporation
rise also due to increasing soil temperature; W1S2 from day 4 to day 12; and W2S3 from day 12 to the end
of the simulation/experiment. Although the initial solute concentrations differed, the spatial and temporal
variations of the solute concentration (Figure 7b) were similar.
One of the distinct differences in terms of the simulated results between the two cases was that the evaporation rate in case AS (Figure 7h) was lower than that in case SS. This is due to the variations of net radiation
input in two cases [Gran et al., 2011a]. With less heat absorbed by the soil, less energy became available for
evaporation in case AS, resulting in overall lower evaporation rates. Another major difference between the
two cases was that case AS had longer duration of W1 in case AS; in particular, the period of W1S2 (Figure
7g). Due to the prolonged W1S2 period and higher initial solute concentration, the entire depth of efflorescence above the soil surface, around 0.45 mm (Figure 7h), was 8 times thicker than that in case SS. One of
the reasons that case AS had prolonged stage W1S1 is that the evaporation rate is generally lower in case
AS. Another reason for prolonged stage W1S1 in case AS is that the aluminous silt could maintain higher liquid water saturation and higher relative permeability than the Silica sand in the unsaturated zone under
same matric potential (Figure 3). This led to water at the bottom of the column be more readily transported
upward for evaporation during stage W1 (Figure 7d) and more uniform liquid water saturation profiles (Figure 7a) in case AS. Comparing with case SS, the variation of soil properties, initial solute concentration and
heat input in case AS had not significantly affected the vapor transport regime (Figure 7e). However, even if
the potential evaporation rate is lower, case AS had faster development of dry soil layer than case SS as indicated by the rate that vaporization plane moving downward (the penetration rates of vaporization plane
during stage W2S3 in cases SS and AS were, respectively, 1.9 and 2.6 mm/d depicted in Figures 6g and 7g).
This is due to the fact that during the stages W2S3, the liquid water saturation below the vaporization plane
in case AS was generally lower than that in case SS, making the soil more easily to be dried (dash-dotted
lines, Figures 6a and 7a). Case AS also had greater shrinkage of porosity in the dry soil layer than case SS
(Figure 7f), while such shrinkage had yet significantly disrupted the vapor transport regime.
The simulated results in case AS agree generally well with the measured values, except about the early
emergence of the dry soil layer (Figure 7a) and the nonagreement on the temperature profile (Figure 7c) at
the time point with 32% water saturation. These discrepancies are similar to the ones observed in cases SS
and so can attribute to the same reasons listed above.
4. Discussions
The developed model was based on the hypothesis that salt precipitates in the NSL as efflorescence, which
does not change the local soil properties but exerts mulching resistance to the water vapor flow; and salt
precipitates in the ISL as subflorescence, the effect of which is mainly manifested in reduction of the soil
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(b)
0
50
50
100
100
150
150
Legend for Figure (a) (b) and (c)
Measurements at 32% sat.
Calculations at 81% sat.
Calculations at 32% sat.
Calculations at 21% sat.
200
0
0
Depth (mm)
(c)
0.5
Saturation (m3 m−3)
1
(d)
0
0.1
0.2
Concentration (kg kg−1)
Depth (mm)
(a)
200
0.3
(e)
30
35
40
45
°
Temperature ( C)
(f)
0
50
50
100
100
150
150
Depth (mm)
Depth (mm)
0
10.1002/2013WR015127
Legend for Figure (d) (e) and (f)
Calculations at 81% sat.
Calculations at 34% sat.
Calculations at 32% sat.
Calculations at 21% sat.
200
200
0
5
10
0
0.5
1
1.5
Liquid water flux (mm day−1)
Water vapour flux (mm day−1)
0.39
0.395
Porosity (m3 m−3)
0.4
(g)
Depth (mm)
0
20
40
Legend for Figure (g)
Calculated location of the vaporization front
0
5
10
20
W1
−1
Evaporation (mm day )
(h)
15
Time (day)
S1
W2
S2
81% sat.
10
25
S3
34% sat.
32% sat.
21% sat.
0.4
Legend for Figure (h)
Measured evaporation rate
5
Calculated evaporation rate (RMSE=0.8 mm/day)
0.2
Calculated solid salt depth on surface
0
0
5
10
15
Time (day)
20
25
Solid salt depth (mm)
60
0
Figure 7. Predicted and measured results for case AS: (a) liquid water saturation, (b) solute concentration, (c) temperature, (d) liquid water
flux, (e) water vapor flux across the soil column, (f) porosity, (g) temporal change of vaporization plane location in the NSL, and (h) temporal changes of evaporation rate and solid salt depth on the soil surface. The evaporation stages based on the predicted evaporation rate
are indicated in the top of Figure 7h.
porosity and hence slowing-down of vapor movement in the soil. Through the comparison between the
experimental measurements and simulated results, it is confirmed that this hypothesis worked well for two
experimental cases in isothermal environment with the presence of shallow water table, which differ each
other by soil types and atmospheric conditions. The model with hypothesis incorporated also shows
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Table 2. Mass Fractions of Water and Salt at the Beginning and End of Each Evaporation Stage for Each Casea
Case ML
Mass Fraction of
Case TS
Initial
W1S1
W1S2
Initial
W1S1
W1S2
88
12
0
88
12
0
0
64
12
24
66
34
0
0
0
12
88
0
50
50
0
94
6
0
93
7
0
0
74
6
20
76
24
0
0
0
6
94
0
28
72
0
Water in the reservoir (%)
Water in the column (%)
Water evaporated (%)
Salt in the reservoir as solute (%)
Salt as solute in the column (%)
Salt as efflorescence (%)
Salt as subflorescence (%)
Case SS
Mass Fraction of
Water in the column (%)
Water evaporated (%)
Salt as solute in the column (%)
Salt as efflorescence (%)
Salt as subflorescence (%)
Case AS
Initial
W1S1
W1S2
W2S2
Initial
W1S1
W1S2
W2S2
100
0
100
0
0
68
32
100
0
0
53
47
79
21
0
32
68
65
21
14
100
0
100
0
0
66
35
100
0
0
33
68
48
52
0
38
62
41
52
7
a
For case ML and TS, the cumulative mass of water is calculated as the sum of water evaporated and water remained in the column
at the end of the simulation; the cumulative mass of salt is calculated as the sum of salt (in both solute and solid form) at the end of simulation. For cases SS and AS, the cumulative mass of water (and salt) is the sum of water (solute) in the column at the beginning of the
simulation.
agreement with two experimental cases under nonisothermal condition without the presence of shallow
water table, which are distinct by soil types, initial solute concentration, and atmospheric conditions.
Table 2 lists the mass fractions of water and salt at the beginning and the end of each evaporation stage
during the experiments/simulations. The results show that as long as sufficient liquid water can be supplied
to the NSL, salt precipitates exclusively on the soil surface as efflorescence. For the cases with no fixed shallow water table, the ratios of efflorescence to the total salt at the end of the simulation/experiment are 52%
for case AS with finer soil grains and 21% for case SS with coarser soil grains. These ratios are higher than
ratios of subflorescence to the total salt, indicating the dominance of efflorescence in salt precipitation due
to evaporation. The salt resistance formulation to account for the effect of efflorescence on evaporation
was developed by regression analysis from a soil with intermediate saturation and relatively high potential
evaporation rate [Fujimaki et al., 2006], However, the effect of efflorescence on evaporation is not only
dependent on the thickness of the efflorescence, but also potential evaporation rates and water saturation
on the soil surface. For example, if water saturation in the NSL is relatively high, evaporation may not be sustained by vapor transporting through the mulching efflorescence, but by liquid water transporting through
the precipitated salt layer—i.e., vaporization occurs above the efflorescence rather than below it [Sghaier
and Prat, 2009]. Also it has been found that salt tends to precipitate as large crystals at high potential evaporation rates while small homogeneous crystals at low potential evaporation rates [Nachshon et al., 2011b].
How the different crystal forms affect the behavior of efflorescence and its resistance to evaporation
requires further investigation.
The ratio of subflorescence to the overall salt at the end of the simulation/experiment was 7% for case AS
with finer soil grains and 14% for case SS with coarser soil grains. This amount of subflorescence caused
only up to 2.5% reduction of porosity in the upper dry soil layer beneath the NSL (Figures 6f and 7f). It is
worth noting that in the cases without a fixed shallow water table, approximately a third of the water
remained in the column at the end of the simulation/experiment, which dissolved approximately 4–6 times
the amount of salt that precipitated as subflorescence at the end of the simulation/experiment (65% for
case SS and 41% for case AS, as list in Table 2). The solute would further precipitate as subflorescence if the
evaporation continued. Assuming that all of the remaining solute in the column at the end of simulation/
experiment precipitates as subflorescence within 5 mm depth of soil, the estimated reduction of porosity
can rise up to 10% and 17% relative to the original porosity, respectively, for case SS and AS. Given the fact
that the vaporization plane will be moving downward further if the drying process continues, the estimated
further reduction of porosity at the dry soil layer is still small and therefore the hypothesis would still hold.
Although the model with proposed hypothesis can properly describe the processes of the four evaporation
cases, it is still not clear how much the presence of salt in pore water would disrupt the evaporation
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(b)
(d)
0
50
50
100
100
150
150
200
200
0
0.5
Saturation (−)
1
0
0.1
0.2
0.3
Concentration (kg kg−1)
30
35
40
45
0.25 0.3 0.35 0.4
Porosity (m3 m−3)
Temperature ( ° C)
(e)
0
50
100
Evaporation rate (mm day−1)
Efflorescence depth (mm)
Vaporization plane location (mm)
(c)
0
Depth (mm)
Depth (mm)
(a)
10.1002/2013WR015127
10
0
5
10
15
20
Time (day)
25
30
35
40
5
10
15
20
Time (day)
25
30
35
40
5
10
15
20
Time (day)
25
30
35
40
(f)
5
0
0
10
(g)
5
0
0
0
Case AS−F, C =0
0
Case AS, C =0.02
0
Case AS−S, C =0.25
Figure 8. Predicted results for case AS-F, AS, and AS-S: (a) the profile of liquid water saturation, (b) solute concentration, (c) temperature,
(d) porosity at the end of the simulation, (e) the temporal change of vaporization plane location, (f) depth of efflorescence, and (g) evaporation rate. The circles in Figure 8g show the measured evaporation rate in case AS.
processes with other factors unchanged, and whether the hypothesis still holds when the initial solute concentration is extremely high. To investigate these two issues, we further simulated case AS with initial solute
concentrations C 0 (kg kg21) being 0 (named as case AS-F) and 0.25 (named as cased AS-S) in longer simulation time (i.e., 40 days). The simulated results of the two cases with that from original case AS (C 0 50:02) are
displayed in Figure 8. The deepening of vaporization plane (Figure 8e) identifies the duration of stage W2S3
(or W2 for case without saline soil water), while the thickening of efflorescence (Figure 8f) shows the period
of stage (W1S2). The proposed classification of evaporation stages can also properly describe the procedures of evaporation in cases AS-F and AS-S (specifically, for case AS-F, stage W1 was from the beginning to
day 8 and stage W2 lasted from day 8 to the end of simulation; for case AS-S, stage W1S1 was very short
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due to high initial solute concentration, stage W1S2 lasted from day 1 to day 23, stage W2S3 lasted from
day 23 to the end of simulation).
Although the variation of the initial solute concentration did not affect temperature profile significantly (Figure 8c), less water got vaporized from the more saline soil water cases at the end of the simulation (87%,
85%, and 78%, respectively, for cases AS-F, AS, and AS-S). Within the lost water, the majority gets vaporized
during stage W1 (including W1S1 and W1S2 for saline soil water cases), accounting for 73%, 77%, and 82%,
respectively, in cases AS-F, AS, and AS-S. Although the presence of salt in soil water lengthened the duration
of stage W1, the evaporation intensity at that stage was significantly reduced, as the average evaporation
rates in cases AS-F, AS, and AS-S were, respectively, 7.31, 5.08, and 3.13 mm/d. As a comparison, stage W2
(or W2S3 for saline soil water cases) only takes 27%, 23%, and 18% of all lost water, respectively, in cases
AS-F, AS, and AS-S, with average evaporation rates of 0.59, 0.54, and 0.63 mm/d. This suggests that the rise
of solute concentration and increasing efflorescence has more significant effect to disrupt evaporation than
subflorescence, which reduces the porosity and possibly affects the soil properties.
Similar to case AS, The majority of the precipitated salt in case AS-S is in the form of efflorescence, mulching
about 8 mm thickness above the soil surface (Figure 8f). With extremely high initial solute concentration,
case AS-S also had more salt precipitated as subflorescence than case AS, with the maximum shrinkage of
porosity down to 0.27, occurred in the soil below the NSL (Figure 8a). Such high porosity reduction,
appeared when soil water is extremely saline, has yet completely obstructed the water vapor flow that contributes to evaporation during stage W2S3. The location where subflorescence presents had liquid water
saturation almost zero, indicating no liquid water movement taking place. This again confirms the hypothesis that subflorescence had little effect on the liquid water flow during the soil-drying processes.
5. Concluding Remarks
In summary, a numerical model is developed to quantify the evaporation from soils saturated with saline
water, based on direct simulations of the transport of liquid water, water vapor, salt solute, and heat in soils.
The model incorporates the feedback of salt precipitation on the transport processes and evaporation.
Through applications to previous laboratory experiments, the model was shown to be capable of simulating
the effects of salt accumulation and precipitation on evaporation under different groundwater conditions
as controlled by the water table. In particular, the representations of salt precipitation in the forms of efflorescence and subflorescence, and resulting effects on the evaporation were validated by the experimental
results. The evaporation stages proposed in this study are systematically based on the location of the vaporization plane and salt accumulation. These stages mechanistically express the evaporation process from
saline soil under conditions with and without a fixed shallow water table. Apart from the potential improvement to the model discussed above, it is yet to be tested whether the model assumption still stands where
large porosity reduction occurs (e.g., intensive salt precipitation as subflorescence may result in blockage of
water vapor flow path). New experiments should be designed and conducted to further test the model with
potential needs to further expand its capacities. Further work can also focus on incorporating the shape of
solid salt and the mineral hygroscopic into the model, which is essential for describing the mixed geometry
of soil grains and solid salt, and associated effects on the transport of liquid water and water vapor in the
soil. The model should be further tested under conditions involving salt dissolution, which occurs in soils
subjected to flooding and infiltration of fresher water. Ultimately this model, upon further development
and validation, may become a useful tool for studying the transport and distribution of water and salt in
coastal wetlands, which underpins the services provided by these ecosystems.
Notation
C
W0
Wm
WmBot
Wo
ZHANG ET AL.
specific mass accumulation (per unit area) of efflorescence, kg m22.
matric potential when liquid water saturation is nearly zero, m.
matric potential, m.
matric potential applied at the bottom of the soil column for the cases with a fixed shallow water
table, m.
osmotic potential, m.
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Ww
a
a1
a2
b
ha
hl
hl NSL
hsalt
hp
hp0
g
kT
l
v
qa
ql
ql0
qs
qsalt
qv
qvRef
qvNSL
qv
s
v
x1
C
C0
Cs
Dv
Di
Dm
E
G
H
L0
M
MNSL
Ma
Ms
Mw
R
Rn
Sl
Slr
Se
T
T0
TNSL
TRef
Ta
ak
ZHANG ET AL.
10.1002/2013WR015127
water potential, m.
dispersivity, m.
fitting parameter in the water retention curve, m21.
fitting parameter in the water retention curve, m21.
a parameter in the water retention to establish a linear relationship between ln ð2Wm Þ and Sl
below Slr .
volumetric air content, m3 m23.
volumetric water content, m3 m23.
volumetric water content in the NSL, m3 m23.
volumetric solid salt content, m3 m23.
porosity, m3 m23.
initial porosity (no solid salt present), m3 m23.
enhancement factor of water vapor.
bulk thermal conductivity of soil matrix and fluid, J m21 s21 K21.
dynamic viscosity of liquid pore water solution, kg m21 s21.
number of ions per molecule.
density of dry air, kg m23.
density of liquid water, kg m23.
density of fresh liquid water, 1000 kg m23.
density of soil solid grains, kg m23.
density of solid salt, 2165 kg m23.
density of water vapor, kg m23.
density of water vapor at the reference point in the atmosphere, kg m23.
density of water vapor in the NSL, kg m23.
saturated water vapor density, kg m23.
tortuosity factor.
osmotic coefficient, mol J21.
a fitting parameter in the relative liquid water permeability.
solute concentration in liquid phase, kg kg21.
initial solute concentration in liquid phase, kg kg21.
solubility, kg kg21.
molecular diffusivity of water vapor in the air, m2 s21.
ionic diffusion coefficient, m2 s21.
mechanical dispersion coefficient, m2 s21.
evaporation rate, m s21.
surface heat flux density into the soil, W m22.
sensible heat flux density positive upward, W m22.
latent heat of vaporization, J kg21.
mass of solid salt per unit volume, kg m23.
mass of solid salt precipitated in the NSL, kg m23.
molecular weight of dry air, 0.02897 kg mol21.
molecular weight of sodium chloride, 0.0585 kg mol21.
molecular weight of water, 0.018 kg mol21.
ideal gas constant, 8.314 J mol21 K21.
net radiation, W m22.
liquid water saturation.
residual liquid water saturation.
evaporation (positive) or condensation per unit volume of soil, s21.
bulk soil temperature, K.
initial bulk soil temperature, K.
temperature in the NSL, K.
temperature at the reference point in the atmosphere, K.
dry air temperature, K.
a fitting parameter in the relative liquid water permeability.
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ar
as
ax
b1
b2
b3
bk
br
bs
bx
ca
cl
cs
cv
f
fc
g
hr
hrRef
ks
kr
m
n1
n2
pa
pv
px
ql
qv
ra
rH
rHSide
rs
rsc
t
w
z
Dt
Dz
DzNSL
Acknowledgments
This work was supported by the
National Basic Research Program of
China (973 Program, 2012CB417005)
and National Centre for Groundwater
Research and Training (NCGRT). The
authors express their gratitude to the
editors and the anonymous reviewers
for their valuable suggestions. The
data related to this research are
available to all interested researchers
upon request.
ZHANG ET AL.
10.1002/2013WR015127
a fitting parameter in the salt resistance formulation, s m21.
a fitting parameter in the surface resistance formulation, s m21.
a fitting parameter in the osmotic coefficient.
a fitting parameter in the bulk thermoconductivity.
a fitting parameter in the bulk thermoconductivity.
a fitting parameter in the bulk thermoconductivity.
a fitting parameter in the relative liquid water permeability.
a fitting parameter in the salt resistance formulation, s m21.
a fitting parameter in the surface resistance formulation, s m21.
a fitting parameter in the osmotic coefficient.
specific heat capacity of the dry air, 1013 J kg21 K21.
specific heat capacity of freshwater, 4200 J kg21 K21.
specific heat capacity of soil solid grains, J kg21 K21.
specific heat capacity of vapor, 2000 J kg21 K21.
precipitation (positive) and dissolution rate of salt per unit volume, kg m23 s21.
mass fraction of clay in the soil.
gravitational acceleration, 9.8 m s22.
relative humidity in the soil.
relative humidity at the reference point.
intrinsic permeability, m2.
relative permeability.
a parameter representing the density of a virtual medium that stores the precipitated salt, m3
kg21.
fitting parameter in the water retention curve.
fitting parameter in the water retention curve.
atmospheric pressure, 100,325 Pa.
saturated water vapor pressure, Pa.
a fitting parameter in the osmotic coefficient.
liquid water flux, m s21.
water vapor flux, m s21.
aerodynamic resistance to water vapor transfer, s m21.
aerodynamic resistance to the heat transfer, s m21.
aerodynamic resistance on the side of the soil column, s m21.
surface resistance, s m21.
resistance due to salt precipitation as efflorescence, s m21.
time, s.
a fitting parameter in the water retention curve representing the mass fraction of one soil.
vertical space coordinate positive upward, m.
time step, s.
nodal spacing below the NSL, m.
the depth of the NSL, m.
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