Experimental and Numerical Study on the Coupled Processes of Salt Migration and Moisture Movement Under Evaporation in the Vadose Zone

Abstract

In arid and semi-arid regions, soil salinization has emerged as an escalating environmental challenge. Soil salinity not only alters the soil structure but also influences water movement and distribution. The coupled processes of water movement, heat transfer, and solute transport in the vadose zone interact dynamically, warranting an in-depth investigation into coupled processes of matter and energy. This study developed a numerical model of coupled water-vapor–heat–salt transport in the vadose zone, validated through evaporation experiments and compared with a conventional model excluding osmotic potential. It is found that salt presence reduces evaporation rates while enhancing soil moisture movement. Liquid water movement is primarily governed by matric and osmotic potential gradient, whereas water vapor movement is dominated by temperature gradients. Matric potential influences water vapor movement only at the soil surface, and the impact of salt on water vapor movement diminishes with increasing water content. Notably, matric potential significantly affects water vapor movement only when soil water vapor relative humidity is below unity. The proposed model effectively describes multi-field coupling transport and clarifies the role of osmotic potential in regulating liquid and vapor water dynamics.

1. Introduction

Soil salinization has emerged as a global environmental and socioeconomic challenge, severely restricting agricultural production and engineering construction [1,2]. This issue is particularly acute in arid and semi-arid regions, where imbalances in salt influx-outflux driven by evaporation and groundwater processes exacerbate soil salinization [3,4,5]. This salinization is a serious environmental hazard. Consequently, understanding the coupled process of salt migration and moisture movement under evaporation in the vadose zone has become a critical research focus.

The vadose zone serves as a vital component of the Earth’s critical zone, facilitating energy and matter exchange in the shallow surface layer. It acts as a key interface connecting groundwater, the Earth’s surface, and the atmosphere, playing a pivotal role in mediating hydrological processes across these interconnected systems [6,7]. In arid and semi-arid regions, soil moisture exists predominantly as liquid water and water vapor, necessitating predictions of moisture movement that account for both matric potential and temperature effects [8,9,10]. Solutes in groundwater further influence flow dynamics via osmotic potential, creating a complex hydrodynamical system where moisture movement, energy conversion, and solute transport are inherently coupled [11,12,13,14]. Studying coupled water-vapor–heat–salt transport is therefore essential for ecological restoration, land salinization mitigation, soil water conservation, and agricultural management.

Early seminal research by Philip and De Vries [15] established the theoretical foundation for temperature and matric-driven soil moisture transport, proposing the PdV model for coupled water-vapor-heat transport. This model describes vapor movement via Fick’s law, linking temperature, relative humidity, saturated vapor density, and matric potential. In their model, soil moisture movement comprises liquid water flow and water vapor transport, controlled by matric potential and temperature. Most scholars employed their theory and conducted further research. Milly [16] formulated a matrix head-based model to simulate the moisture and heat transport in hysteretic, heterogeneous porous media. Saito et al. [17] employed the PdV model to simulate coupled transport of liquid water, water vapor, and heat in the unsaturated zone using the HYDRUS code [18] and achieved a good result. These studies demonstrate the applicability of the PdV model in simulating moisture transport in the vadose zone. However, the model has not fully accounted for the interrelationships between physical quantities, such as the linkage between water vapor relative humidity and water content.

Solute impacts on soil water distribution and movement are equally significant, affecting liquid water via osmotic potential and vapor via changes in relative humidity. Building on the PdV model, Nassar and Horton [19] developed a multi-field coupling model incorporating osmotic potential, describing the influence of solutes on soil liquid water and water vapor by establishing the relationship between osmotic potential, soil water potential, and soil relative humidity.

Subsequently, numerous scholars delved into the coupled transport of water, vapor, heat, and solutes. Noborio, McInnes, and Heilman [11] developed a two-dimensional model of water, heat, and solute transport, employing theoretical and experimental simulations to investigate furrow irrigated soils. Suarez and Simunek [20] established a solute transport model for the vadose zone, accounting for both equilibrium and dynamic chemical conditions. Kumahor et al. [21] experimentally explored the water and solute transport processes in sandy vadose zone soils. Healy et al. [22] developed the VS2DRTI software to simulate the coupled transport process of water, heat, and solute in porous media of the vadose zone. Wen et al. [23] proposed a coupled transport model of water, vapor, heat, and salt in the evaporating vadose zone, incorporating the impacts of salt precipitation and film water. Despite these advancements, the existing research predominantly emphasizes model accuracy, leaving the underlying mechanisms of solute-induced alterations in liquid water and water vapor movement inadequately explored.

In summary, while numerous studies have addressed water-solute transport in the vadose zone, the interactive mechanisms among water, vapor, heat, and solute movement remain understudied. This study presents a coupled water-vapor–heat–salt (WVHS) transport model that considers osmotic potential effects on water movement, vapor humidity, film water thickness, surface tension, and temperature-matric potential interactions. A coupled water-vapor–heat (WVH) transport model is also established and compared to the WVHS model to emphasize the influence of solutes on moisture movement.

The primary objectives of this study are to (1) develop a numerical model of coupled transport of water, vapor, heat, and salt, which explicitly captures the influence of osmotic potential on moisture; (2) elucidate how osmotic potential impacts the spatial distribution and temporal variation of soil water content; (3) identify the dominant factors governing the movement of liquid water and water vapor at different soil depths; and (4) explore the relationship between water vapor fluxes and soil moisture relative humidity.

2. Materials and Methods

2.1. Experimental Setup and Approach

Laboratory evaporation experiments were conducted using a 100 cm in height, 20 cm in diameter open sand column (Figure 1). Volumetric water content, temperature, and electrical conductivity were monitored with ECH2O 5TE sensors (METER Group, Inc., Pullman, WA, USA), while atmospheric parameters (temperature, relative humidity, vapor pressure, and pressure) were measured using a VP4 sensor (METER Group, Inc., WA, USA). Data were logged by a CR1000X data logger (Campbell Scientific, Inc., Logan, UT, USA). The 5TE sensors were set at the soil surface, with an additional six sensors placed at 10 cm intervals starting from a soil depth of 5 cm. VP4 sensors were suspended above the soil and positioned as close as possible to the soil surface. The column was thermally insulated to minimize ambient heat exchange, and a water table control system maintained a constant water table depth. The water table control system was connected to the sand column to maintain a stable water level throughout the experiment.

Silica sand (Xiamen ISO Standard Sand Corp., Xiamen, China) with particle diameters of 0.08–1.25 mm (mean = 0.28 mm) and porosity of 0.304 was used. The sand was dried, packed to a bulk density of 1680 kg/m³, and initially saturated from bottom to top with a 2 g/L NaCl solution, followed by drainage to maintain a water table 55 cm below the surface. The ambient temperature in the laboratory was maintained at a constant 25 °C. Evaporation was induced using an infrared lamp (275 W), with water content, temperature, and electrical conductivity monitored every 15 min at multiple depths for 150 h until a near steady state was reached. To better characterize the distribution of soil salinity, soil samples were collected from various depths after the experiment and used to conduct extraction tests. Three soil samples (each with a mass of 50 g) were taken at each sampling location and soaked in 250 mL of deionized water.

2.2. Numerical Model

2.2.1. Water Vapor Transport

When groundwater contains solutes, water vapor pressure is influenced by both matric and osmotic potential [24], expressed thermodynamically as follows:

$${\displaystyle \frac{RT}{Mg}}\ln H_r={\displaystyle \frac{RT}{Mg}}\ln H_{rl}+{\displaystyle \frac{RT}{Mg}}\ln H_{ro}$$

(1)

where R is the universal gas constant (=8.314 J mol⁻¹K⁻¹); T is the temperature (K); M is the molar mass of water (=0.018015 kg mol⁻¹); g is the gravitational acceleration (=9.81 m s⁻²); H_r is the total relative humidity of water vapor (unitless); H_rl is the relative humidity of water vapor due to matric potential (unitless); H_ro is the relative humidity of water vapor due to osmotic potential (unitless). The above relation can be simplified to the following:

$$H_r=H_{rl}{H}_{ro}$$

(2)

where H_rl can be expressed as follows [15]:

$$H_{rl}=\exp \left({\displaystyle \frac{Mg{h}_l}{RT}}\right)$$

(3)

where h_l is the matric pressure head (m). H_ro can be expressed as follows [25]:

$$H_{ro}=\exp \left(-Mv\varphi C\right)$$

(4)

where v is the number of ions per molecule for ionizing solutes (v_(NaCl) = 2, unitless); ϕ is the osmotic coefficient (unitless); and C is the solute concentration (mol kg⁻¹).

When the liquid water and water vapor in soil reach equilibrium, the vapor density can be expressed as follows:

$${\rho }_v={\rho }_{sv}{H}_r={\rho }_{sv}{H}_{rl}{H}_{ro}$$

(5)

where ρ_sv is the saturated water vapor density (kg m⁻³). Based on Fick’s law, the water vapor flux density is expressed as follows:

$$q_v=-{\displaystyle \frac{D}{{\rho }_l}}\left({\displaystyle \frac{\partial {\rho }_v}{\partial z}}\right)$$

(6)

where q_v is the water vapor flux (m s⁻¹); ρ_l is the water density (kg m⁻³); and D is the vapor diffusivity in soil (m² s⁻¹), defined as follows [17]:

$$D=\epsilon D_a{\theta }_a$$

(7)

where θ_a is the air-filled porosity (m³ m⁻³) and ε is the tortuosity factor (unitless), which can be expressed as follows [26]:

$$\epsilon ={\displaystyle \frac{{\theta }_a^{7/3}}{{\theta }_s^2}}$$

(8)

where θ_s is the volumetric saturated water content (m³ m⁻³), while D_a is the water vapor diffusivity in air (m² s⁻¹), expressed as follows [27]:

$$D_a=2.12\times {10}^{-5}{\left({\displaystyle \frac{T}{273.15}}\right)}^2$$

(9)

Substituting Equations (3)–(5) into Equation (6), the water vapor flux density driven by matric potential, solute concentration, and temperature can be expressed as follows:

$$\begin{array}{l}{q}_v=-{\displaystyle \frac{D}{{\rho }_l}}\left({\displaystyle \frac{\partial {\rho }_v}{\partial z}}\right)=-{\displaystyle \frac{D}{{\rho }_l}}\left({\displaystyle \frac{\partial \left({\rho }_{sv}{H}_{rl}{H}_{ro}\right)}{\partial z}}\right)\\ =-{\displaystyle \frac{D}{{\rho }_l}}\left[{\rho }_{sv}{H}_{ro}{\displaystyle \frac{\partial H_{rl}}{\partial h_l}}{\displaystyle \frac{\partial h_l}{\partial z}}+{\rho }_{sv}{H}_{rl}{\displaystyle \frac{\partial H_{ro}}{\partial C}}{\displaystyle \frac{\partial C}{\partial z}}+H_{rl}{H}_{ro}{\displaystyle \frac{\mathrm{d}{\rho }_{sv}}{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}\right]\\ =-{\displaystyle \frac{D}{{\rho }_l}}\left[{\rho }_{sv}{H}_r{\displaystyle \frac{Mg}{RT}}{\displaystyle \frac{\partial h_l}{\partial z}}-{\rho }_{sv}{H}_{r}Mv\varphi {\displaystyle \frac{\partial C}{\partial z}}+H_r{\displaystyle \frac{\mathrm{d}{\rho }_{sv}}{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}\right]\end{array}$$

(10)

The calculated water vapor flux induced by temperature is always underestimated compared with the actual value; thus, an enhancement factor ƞ was proposed to correct the calculation result. The enhancement factor can be expressed as follows [28]:

$$\eta =a+3{\displaystyle \frac{{\theta }_l}{{\theta }_s}}-\left(a-1\right)exp\left\{-{\left[\left(1+{\displaystyle \frac{2.6}{\sqrt{f_c}}}\right){\displaystyle \frac{{\theta }_l}{{\theta }_s}}\right]}^4\right\}$$

(11)

where a is an empirical constant, and f_c is the mass fraction of clay in the soil (unitless). Thus, the water vapor flux density is expressed as follows:

$$q_v=-{\displaystyle \frac{D}{{\rho }_l}}\left[{\rho }_{sv}{H}_r{\displaystyle \frac{Mg}{RT}}{\displaystyle \frac{\partial h_l}{\partial z}}-{\rho }_{sv}{H}_{r}Mv\varphi {\displaystyle \frac{\partial C}{\partial z}}+H_r\eta {\displaystyle \frac{\mathrm{d}{\rho }_{sv}}{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}\right]$$

(12)

The terms within the right-hand bracket denote the water vapor fluxes induced by matric potential, solute concentration, and temperature, respectively. The following vapor hydraulic conductivities are defined as follows:

$$K_{hv}={\displaystyle \frac{D}{{\rho }_l}}{\rho }_{sv}{H}_r{\displaystyle \frac{Mg}{RT}}$$

(13)

$$K_{Cv}={\displaystyle \frac{D}{{\rho }_l}}{\rho }_{sv}{H}_{r}Mv\varphi$$

(14)

$$K_{Tv}={\displaystyle \frac{D}{{\rho }_l}}{H}_r\eta {\displaystyle \frac{\mathrm{d}{\rho }_{sv}}{\mathrm{d}T}}$$

(15)

where K_hv is the isothermal vapor hydraulic conductivity (m s⁻¹); K_Cv is the osmotic vapor hydraulic conductivity (m² kg s⁻¹ mol⁻¹); K_Tv is the thermal vapor hydraulic conductivity (m² s⁻¹ K⁻¹). Then, Equation (12) can be simplified as follows:

$$q_v=q_{hv}+q_{Cv}+q_{Tv}=-K_{hv}{\displaystyle \frac{\partial h_l}{\partial z}}+K_{Cv}{\displaystyle \frac{\partial C}{\partial z}}-K_{Tv}{\displaystyle \frac{\partial T}{\partial z}}$$

(16)

where q_hv is the isothermal vapor flux (m s⁻¹); q_Cv is the osmotic vapor flux (m s⁻¹); and q_Tv is the thermal vapor flux (m s⁻¹).

2.2.2. Liquid Water Transport

When solutes are present in the soil solution, osmotic potential becomes a critical component of the total water potential. The total water head, comprising matric pressure head, osmotic pressure head, and elevation head, can be expressed as:

$$h=h_l+h_o+z$$

(17)

where h is the total water head; h_o is the osmotic pressure head; and z is the elevation head. Liquid water density can be described based on a modified Darcy law:

$$q_l=-K_{hl}\left({\displaystyle \frac{\partial h}{\partial z}}\right)=-K_{hl}\left({\displaystyle \frac{\partial h_l}{\partial z}}+{\displaystyle \frac{\partial h_o}{\partial z}}+1\right)$$

(18)

where q_l is the liquid water flux (m s⁻¹), and K_hl is the isothermal liquid water hydraulic conductivity (m s⁻¹). The osmotic pressure head is related to solute concentration and temperature through the Van’t Hoff relation as follows [19]:

$$h_o={\displaystyle \frac{-v\varphi CTR}{g}}$$

(19)

As h_o is a function of the temperature and solute concentration, and h_l is a function of the temperature and matric pressure, Equation (18) can be derived based on the chain rule as follows:

$$q_l=-K_{hl}\left({\displaystyle \frac{\partial h_l}{\partial z}}+{\displaystyle \frac{\mathrm{d}{h}_l}{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}+{\displaystyle \frac{\mathrm{d}{h}_o}{\mathrm{d}C}}{\displaystyle \frac{\partial C}{\partial z}}+{\displaystyle \frac{\mathrm{d}{h}_o}{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}+1\right)$$

(20)

The influence of temperature on matric potential fundamentally arises from its effect on the surface tension of soil water. Building on this relationship, substituting Equation (19) into Equation (20) yields the following expression for the flux density of liquid water:

$$q_l=-K_{hl}\left[{\displaystyle \frac{\partial h_l}{\partial z}}+{\displaystyle \frac{h_l}{{\gamma }_0}}{\displaystyle \frac{\mathrm{d}\gamma }{\mathrm{d}T}}{\displaystyle \frac{\partial T}{\partial z}}-{\displaystyle \frac{v\varphi TR}{g}}{\displaystyle \frac{\partial C}{\partial z}}-{\displaystyle \frac{v\varphi CR}{g}}{\displaystyle \frac{\partial T}{\partial z}}+1\right]$$

(21)

where γ is the surface tension of soil water (g s⁻¹) and γ₀ is the surface tension of soil water at 25 °C (g s⁻¹). A coefficient σ (σ < 1) should be multiplied by the third and fourth terms within the bracket of Equation (21). This coefficient is adopted to quantify the efficiency of osmotic pressure in reducing the hydraulic pressure of soil water [19]. Based on the theory of film water, the coefficient can be calculated as follows:

$$\sigma =\left(l_s-l_w\right)/\left({\displaystyle \frac{\delta }{2}}-l_w\right)$$

(22)

where l_s is the hydrated radius of sodium chloride (=3.58 × 10⁻¹⁰ m); l_w is the radius of a water molecule (=2 × 10⁻¹⁰ m); and δ is the film water thickness, which can be calculated as follows [23]:

$$\delta =\sqrt[3]{{\displaystyle \frac{A_{svl}}{3\pi {\rho }_{l}g{h}_l}}}$$

(23)

where A_svl is the Hamaker constant. Then, Equation (21) can be rewritten as follows:

$$q_l=-K_{hl}\left[{\displaystyle \frac{\partial h_l}{\partial z}}+\left({\displaystyle \frac{h_l}{{\gamma }_0}}{\displaystyle \frac{\mathrm{d}\gamma }{\mathrm{d}T}}-\sigma {\displaystyle \frac{v\varphi CR}{g}}\right){\displaystyle \frac{\partial T}{\partial z}}-\sigma {\displaystyle \frac{v\varphi TR}{g}}{\displaystyle \frac{\partial C}{\partial z}}+1\right]$$

(24)

In the bracket of Equation (24), the first term represents water flow driven by matric potential, the second term denotes flow induced by temperature and the third term signifies flow caused by solute concentration. The liquid water hydraulic conductivities are defined as:

$$K_{Cl}=K_{hl}\sigma {\displaystyle \frac{v\varphi TR}{g}}$$

(25)

$$K_{Tl}=K_{hl}\left({\displaystyle \frac{h_l}{{\gamma }_0}}{\displaystyle \frac{\mathrm{d}\gamma }{\mathrm{d}T}}-\sigma {\displaystyle \frac{v\varphi CR}{g}}\right)$$

(26)

where K_Cl is the osmotic liquid water hydraulic conductivity (m² kg s⁻¹ mol⁻¹) and K_Tl is the thermal liquid water hydraulic conductivity (m² s⁻¹ K⁻¹). Then, Equation (24) can be expressed as follows:

$$q_l=q_{hl}+q_{Cl}+q_{Tl}=-K_{hl}{\displaystyle \frac{\partial h_l}{\partial z}}+K_{Cl}{\displaystyle \frac{\partial C}{\partial z}}-K_{Tl}{\displaystyle \frac{\partial T}{\partial z}}-K_{hl}$$

(27)

where q_hl is the isothermal liquid water flux (m s⁻¹); q_Cl is the osmotic liquid water flux (m s⁻¹); and q_Tl is the thermal liquid water flux (m s⁻¹).

2.2.3. Moisture Transport Governing Equation

The governing equation for moisture movement under non-isothermal conditions with solute in the vadose zone is based on the mass conservation law and can be expressed as follows:

$${\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{\partial \left({\theta }_l+{\theta }_v\right)}{\partial t}}=-{\displaystyle \frac{\partial q_l}{\partial z}}-{\displaystyle \frac{\partial q_v}{\partial z}}$$

(28)

where θ_v can be calculated as follows:

$${\theta }_v={\displaystyle \frac{H_r{\rho }_{sv}{\theta }_a}{{\rho }_l}}$$

(29)

The following moisture hydraulic conductivities are defined as follows:

$$K_h=K_{hl}+K_{hv}$$

(30)

$$K_C=K_{Cl}+K_{Cv}$$

(31)

$$K_T=K_{Tl}+K_{Tv}$$

(32)

where K_h is the isothermal moisture hydraulic conductivity (m s⁻¹); K_C is the osmotic moisture hydraulic conductivity (m² kg s⁻¹ mol⁻¹); and K_T is the thermal moisture hydraulic conductivity (m² s⁻¹ K⁻¹).

Substituting Equations (16) and (27) into Equation (28) yields the governing equation for moisture transport in the presence of solutes:

$${\displaystyle \frac{\partial {\theta }_l}{\partial t}}+{\displaystyle \frac{\partial {\theta }_v}{\partial t}}={\displaystyle \frac{\partial }{\partial z}}\left(K_h{\displaystyle \frac{\partial h_l}{\partial z}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(K_T{\displaystyle \frac{\partial T}{\partial z}}\right)-{\displaystyle \frac{\partial }{\partial z}}\left(K_C{\displaystyle \frac{\partial C}{\partial z}}\right)+{\displaystyle \frac{\partial K_{hl}}{\partial z}}$$

(33)

2.2.4. Heat Transport Governing Equation

The governing equation for heat transport is expressed by the following energy conservation equation:

$${\displaystyle \frac{\partial S_h}{\partial t}}=-{\displaystyle \frac{\partial q_h}{\partial z}}-Q$$

(34)

where S_h is the heat storage in soil (J m⁻³); q_h is the total heat flux density (J m⁻² s⁻¹); and Q is the energy source-sink term (J m⁻³ s⁻¹). The soil heat storage is expressed as follows:

$$S_h=C_{s}T{\theta }_n+C_{v}T{\theta }_v+C_{w}T{\theta }_l+L_0{\theta }_v$$

(35)

where C_n, C_w, C_v are the volumetric heat capacities of solid, liquid, vapor phase (J m⁻³ K⁻¹); T is the temperature (K); θ_n is volumetric fraction of solid phase (m³ m⁻³); and L₀ is the volumetric latent heat of vaporization of liquid water (J m⁻³), as defined by Monteith and Unsworth [29]:

$$L_0=L_L{\rho }_l$$

(36)

where L_L is the latent heat of vaporization of liquid water (J kg⁻¹).

The total heat flux density comprises sensible heat from convection of liquid water and water vapor, as well as latent heat from water vapor flow, which can be expressed as follows [17]:

$$q_h=-\lambda {\displaystyle \frac{\partial T}{\partial z}}+C_v\left(T-T_0\right)q_v+C_w\left(T-T_0\right)q_l+L_0{q}_v$$

(37)

where λ is the apparent soil thermal conductivity, which can be expressed as follows [17]:

$$\lambda ={\lambda }_0+\beta C_w\left|q_l\right|$$

(38)

where β is the thermal dispersity (m), which is only considered when the liquid water flux is very large, and λ₀ is the thermal conductivity, which accounts for the tortuosity of the soil porous medium and can be calculated following Chung and Horton [30]:

$${\lambda }_0=b_1+b_2{\theta }_L+b_3{\left({\theta }_L\right)}^{0.5}$$

(39)

where b₁, b₂, b₃ are the empirical regression parameters (W m⁻¹ K⁻¹).

Substituting Equations (35) and (37) into Equation (34), the governing equation for heat transport is expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial T}{\partial t}}\left[C_s{\theta }_n+C_v{\theta }_v+C_w{\theta }_l\right]+{\displaystyle \frac{\partial \left({\theta }_v{L}_0\right)}{\partial t}}\\ ={\displaystyle \frac{\partial }{\partial z}}\left(\lambda {\displaystyle \frac{\partial T}{\partial z}}\right)-{\displaystyle \frac{\partial }{\partial z}}\left[C_v{q}_{v}T+C_w{q}_{l}T+L_0{q}_v\right]\end{array}$$

(40)

2.2.5. Solute Transport Governing Equation

In the absence of precipitation, the governing equation for solute transport is derived from the mass conservation law and expressed as follows:

$${\displaystyle \frac{\partial \left({\theta }_{l}C\right)}{\partial t}}=-{\displaystyle \frac{\partial q_s}{\partial z}}$$

(41)

where q_s is the solute flux (mol m kg⁻¹ s⁻¹) and can be expressed by the advection dispersion equation (ADE) as follows [23]:

$$q_s=-{\theta }_l(D_i+D_m){\displaystyle \frac{\partial C}{\partial z}}+q_{l}C$$

(42)

where D_i is the molecular diffusion coefficient (m² s⁻¹) and D_m is the mechanical dispersion coefficient (m² s⁻¹), which can be calculated as follows [31]:

$$D_m=\left|{\displaystyle \frac{{\alpha }_L{q}_l}{{\theta }_l}}\right|$$

(43)

where α_L is the longitudinal dispersity (m).

Substituting Equation (42) into Equation (41), the governing equation for solute transport is expressed as follows:

$${\displaystyle \frac{\partial \left({\theta }_{l}C\right)}{\partial t}}={\displaystyle \frac{\partial }{\partial z}}\left[{\theta }_l(D_i+D_m){\displaystyle \frac{\partial C}{\partial z}}-q_{l}C\right]$$

(44)

2.2.6. Hydraulic Properties

Prior to the main experiments, pressure heads corresponding to different soil water content values were measured using a matric potential sensor (Model MPS-6, Decagon Devices Inc., Pullman, WA, USA) within the soil column. Based on the data, the soil water characteristic curve (SWCC) of silica sand, as depicted in Figure 2, is fitted using the Van Genuchten (VG) model [32]:

$$\theta \left(h\right)={\theta }_r+{\displaystyle \frac{{\theta }_s-{\theta }_r}{{\left[1+{\left(\alpha \left|h_l\right|\right)}^n\right]}^m}}$$

(45)

where θ_r is the volumetric residual water content; α (cm⁻¹), m (unitless), and n (unitless) are empirical parameters.

The isothermal liquid water hydraulic conductivity K_hl is expressed as follows [33]:

$$K_{hl}=K_s{S}_e^l{\left[1-{\left(1-S_e^{1/m}\right)}^m\right]}^2$$

(46)

where K_s is the saturated hydraulic conductivity (m s⁻¹); S_e is the effective liquid saturation (unitless); and l is the pore connectivity coefficient (unitless). In this experiment, the saturated hydraulic conductivity of the sand was determined using the constant head permeability test. The parameters used in the calculation are given in

Table 1. Model parameters.

Parameter	Symbol	Unit	Value	Source
VG model parameter	α	1/m	2.368	Fitted
VG model parameter	n	-	5.42	Fitted
VG model parameter	l	-	0.5	Fitted
Volumetric saturated water content	θs	m3 m−3	0.304	Measured
Volumetric residual water content	θa	m3 m−3	0.027	Fitted
Volumetric heat capacities of solid phase	Cs	J m−3 K−1	1.92 × 10−6	a
Volumetric heat capacities of liquid water	Cw	J m−3 K−1	4.12 × 10−6	a
Volumetric heat capacities of water vapor	Cv	J m−3 K−1	1.87 × 10−3	a
Empirical parameter related to thermal conductivity	b1	W m−1 K−1	0.228	a
Empirical parameter related to thermal conductivity	b2	W m−1 K−1	−2.406	a
Empirical parameter related to thermal conductivity	b3	W m−1 K−1	4.909	a
Saturated hydraulic conductivity	Ks	m s−1	1.9 × 10−5	Measured
Gravitational acceleration	g	m s−2	9.81	-
Molecular weight of liquid water	M	kg mol−1	0.018015	-
Gas constant	R	J mol−1 K−1	8.314	-
Surface tension of soil water at 25 °C	γ0	N m−1	0.07189	a
Molecular diffusion coefficient of Na+	Di	m2 s−1	5 × 10−11	b
Longitudinal dispersity	ld	m	0.01	b
Number of ions	v	-	2
Osmotic coefficient	ϕ	-	0.932	c
Hydrated radius of NaCl	ls	m	3.58 × 10−10	c
Radius of water molecule	lw	m	2 × 10−10	c
Hamaker constant	Asvl	J	6 × 10−20	b
Empirical parameter related to enhancement factor	a	-	2.3	d
Aerodynamic resistance	ra	s m−1	364.1	e

a, b, c, d, e, and f represent the reference sources, respectively, from Simunek et al. [34], Wen, Lai and You [23], Robinson and Stokes [35], Han et al. [36], and Novak [37].

.

2.2.7. Initial and Boundary Conditions

The initial conditions are established based on measured data. The initial water content, temperature, and salt concentration at various positions were measured using sensors.

The top boundary for moisture movement is specified as a Neumann boundary condition, where the water flow at the soil surface is equal to the evaporation rate, expressed as follows:

$$E={\displaystyle \frac{H_r{\rho }_{sv}-H_{ra}{\rho }_{sa}}{\left(r_a+r_s\right){\rho }_l}}$$

(47)

where E is the evaporation rate at the soil surface (m s⁻¹); H_ra is the relative humidity in the air (unitless); ρ_sa is the saturated relative humidity in the air (kg m³); r_a is the aerodynamic resistance to the water vapor flow (s m⁻¹); and r_s is the soil surface resistance to water vapor flow (s m⁻¹), which can be expressed as follows:

$$r_s=10\times \exp \left[35.63\left({\theta }_{rw}-{\theta }_{top}\right)\right]$$

(48)

where θ_rw (m³ m⁻³) is an empirical parameter, which is often taken to be 0.15, and θ_top (m³ m⁻³) is the water content at the soil surface (top 1 cm layer).

As the water table is fixed, the bottom boundary of moisture movement is set as the Dirichlet boundary condition.

A time-dependent temperature boundary condition was specified for both the top and bottom boundaries, based on measured temperature data. For solute transport, the top and bottom boundaries are defined as Neumann boundary conditions.

The relative root mean square error (RRMSE) serves as a key metric for evaluating simulation results, and its calculation can be expressed as follows:

$$RRMSE={\displaystyle \frac{\sqrt{{{\displaystyle \sum }}_{i=1}^N{\left(M_i-S_i\right)}^2/N}}{Max\left(M_1, M_2,\cdots \cdots , M_n\right)-Min\left(M_1, M_2,\cdots \cdots , M_n\right)}}$$

(49)

where N is the total number of measured samples; M_i is the measured value; and S_i is the simulated value. The RRMSE value is a dimensionless parameter used to evaluate the prediction accuracy of model simulations. The smaller the RRMSE value, the better the simulation fits the measured data. When the RRMSE value is 0, it indicates that the simulated values perfectly match the measured values.

3. Results and Discussion

The governing equations of the WVHS model were solved using the general form PDE interfaces in COMSOL Multiphysics 6.1, a commercial software for multi-physics field coupling calculations based on the finite element method. Additionally, a WVH model was established and solved in this study for comparison with the WVHS model. The governing equations of the WVH model are detailed in Li et al. [38]. The soil profile was discretized into 550 finite elements, and the simulation duration was set to 150 h.

3.1. Experiment and Model Results

Figure 3 illustrates the measured and simulated water content for the WVHS and WVH models at depths of 5 cm, 15 cm, 25 cm, and 35 cm. Both models demonstrate good simulation performance for water content, though the WVHS model exhibits superior accuracy in predicting water content variations. The RRMSE quantifies model prediction accuracy and enables comparison between different models. For the WVHS model, the RRMSE values of water content at the respective depths are 0.797 (5 cm), 0.271 (15 cm), 0.347 (25 cm), and 2.557 (35 cm). Corresponding values for the WVH model are 1.087 (5 cm), 0.242 (15 cm), 0.916 (25 cm), and 9.52 (35 cm). When comparing RRMSE values, the WVHS model outperforms the WVH model at all depths except 15 cm. Notably, Figure 3 shows that the WVH model consistently predicts lower water content than the WVHS model.

In Figure 3, both models simulate lower water content than the measured values at a depth of 5 cm. This discrepancy may be attributed to the use of the VG model for describing the relationship between water content and matric potential. While the VG model exhibits good applicability across a broad range of water content variations, it still introduces certain errors under low water content conditions.

Figure 4 depicts the measured and simulated soil temperatures at depths of 5 cm, 15 cm, 25 cm, 35 cm, 45 cm, and 55 cm. In this study, the simulated values from the two models are nearly identical, with differences only observable in the second decimal place—an error margin negligible for temperature measurements. Thus, Figure 4 presents only the temperature simulation results of the WVHS model. The RRMSE values for the WVHS model at respective depths are 0.018 (5 cm), 0.022 (15 cm), 0.031 (25 cm), 0.032 (35 cm), 0.034 (45 cm), and 0.034 (55 cm)—all of which are below 0.1, indicating excellent temperature simulation performance.

Figure 5 depicts the profiles of measured soil electrical conductivity at different time. Throughout the experimental period, soil electrical conductivity at approximately 25 cm below the soil surface remained nearly constant. Above this depth, conductivity decreased over time, whereas below it, conductivity increased progressively. The reduction in soil electrical conductivity within the upper 25 cm of the soil profile does not necessarily indicate a decrease in soil salinity. Notably, soil water content in this region was low and decreased significantly during the experiment. Since soil electrical conductivity is highly influenced by water content, this resulted in a corresponding decrease in conductivity within the specified zone.

The electrical conductivity measurements of the extraction solution are presented in Figure 6. As shown in Figure 6, the electrical conductivity of the extraction solution near the soil surface was significantly higher than that at other depths, indicating that salt accumulated in this region due to evaporation during the experiment. This process rendered the near surface soil a high salinity zone.

3.2. Spatiotemporal Variation of Soil Water Content Distribution

Figure 7 depicts the soil water content profiles from the WVHS and WVH models. Both models exhibit a similar trend in water content variation, showing a decreasing pattern throughout the experimental period. The closer to the soil surface, the faster and greater the decline in water content. In Figure 7, when the depth reaches a certain threshold, the water content variation diminishes and forms a water content unchanged region. For the WVHS model, this region spans approximately 25–35 cm, whereas it ranges from 40 to 47.5 cm for the WVH model. This phenomenon occurs because when the water table is at a depth of 55 cm, capillary rise in the vadose zone establishes hydraulic connectivity between the water table and the soil surface, allowing water to replenish evaporation-induced soil water loss. As a result, evaporation-driven water content decline is confined to a specific depth from the soil surface, with the WVHS model predicting a shallower depth than the WVH model.

To highlight the discrepancies between model results, we next present the soil water content variation profiles in Figure 8. Here, the soil water content variation (θ_var) is defined as the change in water content relative to its initial value, calculated as follows:

$${\theta }_{var}=\theta \left(z,t\right)-\theta \left(z,0\right)$$

(50)

In Figure 8, four distinct regions of water content variation exist between the soil surface and water table: (1) water content reduction region, (2) water content unchanged region, (3) water content increase region, and (4) the water content reduction region.

In Region 1, the distribution of soil water content variation is similar between the two models. However, the values calculated by the WVH model are consistently higher than those from the WVHS model at the same time and position. Additionally, the spatial extent of Region 1 in the WVH model is broader than that in the WVHS model, indicating that salt presence weakens water content variation. Firstly, salt accumulation near the soil surface reduces the evaporation rate, thereby decreasing the hydraulic gradients required to sustain evaporation. Secondly, as water is lost from the soil through evaporation, salt accumulates at the surface, intensifying upward fluxes of liquid water and water vapor— a phenomenon explicitly described in Equations (16) and (21). These combined mechanisms explain why soil water content variation in the WVHS model is less pronounced than in the WVH model.

Regions 2 and 3 are more clearly distinguishable in the WVHS model than in the WVH model. This can be attributed to the salt-enhanced effect on liquid water transport. Specifically, soil water content increases in Region 3 throughout the experimental period—a phenomenon consistent with previous studies [36,39]. The water content increase is primarily driven by water table recharge, with water vapor condensation serving as an additional contributing mechanism.

The WVHS model can be considered a variant of the WVH model that accounts for salt interaction. Consequently, simulated water content values from the WVHS model are higher than those from the WVH model, suggesting that salt presence enhances moisture transport and dampens soil water content variation.

3.3. Temporal Variation Characteristics of Soil Hydraulic Parameters

This section analyzes six soil hydraulic parameters: isothermal liquid water hydraulic conductivity (K_hl), isothermal vapor hydraulic conductivity (K_hv), thermal liquid water hydraulic conductivity (K_Tl), thermal vapor hydraulic conductivity (K_Tv), osmotic liquid water hydraulic conductivity (K_Cl), and osmotic vapor hydraulic conductivity (K_Cv). These parameters are critical for characterizing soil liquid water and water vapor transport, reflecting the influences of matric potential, temperature, and salt concentration on water movement.

Figure 9 depicts the spatiotemporal variation of soil hydraulic parameters. Figure 9a,e illustrate isothermal liquid water hydraulic conductivity (K_hl) and osmotic liquid water hydraulic conductivity (K_Cl), which are governed by matric potential and salt concentration, respectively. The two parameters exhibit similar variation patterns: increasing with depth while showing minimal temporal change. Their values approach zero at the soil surface, start to increase at 35 cm depth, and peak at the water table depth. As described in Equations (46) and (25), both parameters are linked to isothermal liquid water hydraulic conductivity (K_hl), which correlates positively with effective liquid saturation (S_e). Thus, soil water content emerges as a critical factor influencing these parameters: higher water content corresponds to greater parameter values. This relationship is rational because liquid water movement depends not only on the gradient of driving forces but also on soil water connectivity. Higher moisture content enhances liquid water connectivity in the soil, facilitating easier water transport.

Figure 9c shows the thermal liquid water hydraulic conductivity (K_Tl), which is governed by temperature. This parameter exhibits a distinct variation pattern compared to the isothermal liquid water hydraulic conductivity (K_hl) and osmotic liquid water hydraulic conductivity (K_Cl). Specifically, thermal liquid water hydraulic conductivity (K_Tl) exhibits significant values only within 20 cm of the soil surface (low water content zone), increasing first and then decreasing with depth. As evident from Equation (26), this parameter is primarily influenced by temperature, which affects it by altering the surface tension of liquid water. The surface tension of liquid water undergoes significant changes only under conditions of low water content and a steep temperature gradient—conditions met only in the shallow low water content zone near the soil surface in this study. This explains why the distribution and variation of the thermal liquid water hydraulic conductivity (K_Tl) in Figure 9c are reasonable.

Figure 9b,d,f illustrate the isothermal vapor hydraulic conductivity (K_hv), thermal vapor hydraulic conductivity (K_Tv), and osmotic vapor hydraulic conductivity (K_Cv), which are governed by matric potential, temperature, and salt concentration, respectively. All three parameters exhibit similar variation patterns: decreasing with depth and increasing over time. These parameters primarily characterize the intensity of water vapor transport—the closer to the soil surface, the more active the water vapor movement, resulting in the highest vapor hydraulic conductivity values in the shallow surface zone. Thus, the parameter values increase progressively toward the soil surface.

Additionally, Figure 9 reveals that the three vapor hydraulic conductivities follow an order: thermal vapor hydraulic conductivity (K_Tv) > osmotic vapor hydraulic conductivity (K_Cv) > isothermal vapor hydraulic conductivity (K_hv). K_Tv and K_Cv exhibit values of the same order of magnitude, with the former approximately fourfold greater than the latter, whereas K_hv values are three to four orders of magnitude lower than K_Tv and K_Cv. Significantly, the soil salt concentration remained consistently lower than the temperature gradient throughout the experiment, and the matric potential gradient failed to exceed the temperature gradient by three to four orders of magnitude except in the shallow surface zone. This confirms that temperature is the primary driving factor for soil water vapor transport.

3.4. Temporal Variation of Soil Water Flux

The variation of liquid water and water vapor fluxes in the soil directly reflects the distribution and dynamics of soil moisture, making the analysis of their dynamic behavior crucial for understanding soil moisture variation mechanisms. As described in the numerical model of Section 2, the primary driving forces for liquid and vapor water movement in the vadose zone are matric potential, osmotic potential, and temperature. In this section, liquid and vapor water fluxes under these three driving forces are calculated, with a focus on analyzing the variation of six specific fluxes at defined soil depths. These include the isothermal vapor flux (q_hv), osmotic vapor flux (q_Cv), thermal vapor flux (q_Tv), isothermal liquid water flux (q_hl), osmotic liquid water flux (q_Cl), and thermal liquid water flux (q_Tl) at 0 cm, 15 cm, 25 cm, and 35 cm.

Figure 10 illustrates the variation of moisture fluxes at depths of 0 cm, 15 cm, 25 cm, and 35 cm in the WVHS and WVH models. Comparing Figure 10b–d with Figure 10f–h reveals that liquid water fluxes (q_hl) in the WVHS model are larger than those in the WVH model. This discrepancy arises because salt presence decreases evaporation rates and enhances water upward movement. Notably, the WVHS model exhibits significant osmotic liquid water fluxes (q_Cl), which contribute to higher total liquid water fluxes compared to the WVH model. This finding validates the explanation presented in Section 3.2.

In Figure 10a,e, the isothermal liquid water flux (q_hl) peaks at the experiment’s onset, then gradually decreases over time before approaching a stable value. Initially, rising soil temperature drives surface water evaporation, causing a rapid decline in soil moisture that reduces matric potential and generates a steep matric potential gradient. This gradient accelerates upward liquid water movement, resulting in a high isothermal liquid water flux (q_hl). When surface water content reaches the residual water content, the surface can no longer supply evaporation directly, forcing evaporation to rely on water recharge from deeper layers. Consequently, the isothermal liquid water flux (q_hl) decreases and stabilizes as evaporation rates equilibrate.

In Figure 10b–d,f–h, the isothermal liquid water flux (q_hl) increases with depth and time at each position before stabilizing at a constant value. This behavior is attributed to water content influencing its hydraulic conductivity (K_hl), which increases with moisture. Thus, in zones with higher water content, explaining its depth-dependent increase. The temporal trend at 15 cm, 25 cm, and 35 cm—initial increase followed by stabilization—can be explained as follows: Early evaporation induced surface water loss creates a strong matric potential gradient, driving upward liquid water flow to sustain surface evaporation. As evaporation and water content distributions stabilize, water table recharge balances evaporative loss, leading to steady upward flow from the water table. This equilibrium is reflected in Figure 10 by the stabilization of isothermal liquid water flux (q_hl) values across all layers.

In Figure 10, the osmotic liquid water flux (q_Cl) is negligible at the soil surface but exhibits more pronounced values at depths of 15 cm, 25 cm, and 35 cm, with variations similar to those of the isothermal liquid water flux (q_hl). As demonstrated in Figure 9e, the osmotic liquid water hydraulic conductivity (K_Cl) increases with depth, directly accounting for the depth-dependent increase in osmotic liquid water flux (q_Cl). The congruent variation patterns of isothermal liquid water flux (q_hl) and osmotic liquid water flux (q_Cl) indicate that osmotic potential influences liquid water movement in conjunction with matric potential—an interplay driven by the fact that salt transport inherently accompanies liquid water transport.

Figure 10 demonstrates that the thermal vapor flux (q_Tv) dominates soil water vapor transport. The thermal vapor flux (q_Tv) consistently moves downward throughout the experiment, a behavior that stems from its governance by the temperature gradient, which remains downward directed during the experiment. As shown in Figure 9d, the thermal vapor hydraulic conductivity (K_Tv) decreases with depth, paralleling the decline in temperature gradient with depth. This concurrent decrease in thermal vapor hydraulic conductivity (K_Tv) and temperature gradient results in the depth-dependent decline of thermal vapor flux (q_Tv). Figure 10 illustrates this trend, with the highest thermal vapor flux (q_Tv) values observed at the soil surface and progressively lower values at 15 cm, 25 cm, and 35 cm.

In Figure 10, the isothermal vapor flux (q_hv) exhibits significant values only at the soil surface, with near-zero values at all other depths. This arises because the isothermal vapor hydraulic conductivity (K_hv) is inherently low (Figure 9b), allowing significant fluxes only where the matric potential gradient is steep—a condition met exclusively at the soil surface. According to the soil moisture characteristic curve, when water content approaches the residual water content, minor moisture changes induce substantial matric potential fluctuations. This generates a steep matric potential gradient near the soil surface, where water content remains close to the residual water content, thus driving the observed isothermal vapor flux (q_hv).

In Figure 10, neither the thermal liquid water flux (q_Tl) nor the osmotic vapor flux (q_Cv) exhibited significant values during the experiment. This is consistent with previous studies, which reported no significant thermal liquid water flux (q_Tl) in simulations [8,17]. Temperature influences water movement primarily by altering the surface tension of soil liquid water, but surface tension changes remain negligible unless subjected to extreme temperature gradients. Additionally, when soil water content is high, the effect of surface tension can be nearly dismissed. Consequently, the influence of temperature on liquid water transport is generally negligible.

The influence of salt on water vapor is primarily mediated by its effect on the water vapor relative humidity (H_r) in soil air. As defined by Equation (2), water vapor relative humidity in soil air is the product of the matric-potential-controlled water vapor relative humidity (H_rl) and the osmotic-potential-controlled water vapor relative humidity (H_ro). Thus, the minimal variation in osmotic-potential-controlled water vapor relative humidity (H_ro) accounts for the absence of significant osmotic vapor flux (q_Cv) in the simulation results.

Figure 11 presents the profiles of osmotic-potential-controlled water vapor relative humidity (H_ro). The results show that H_ro remains consistently above 0.9984, approaching unity, indicating that the product of H_ro and H_rl induces negligible changes in the overall water vapor relative humidity (H_r). This suggests a weak influence of salt on water vapor movement in soil pores. Notably, Figure 11 demonstrates that H_ro decreases with decreasing distance from the soil surface. This trend correlates with the lower water content near the surface, suggesting that the influence of osmotic potential on H_r escalates as water content declines. These results validate that osmotic potential significantly impacts soil water vapor transport exclusively under low water content conditions.

Figure 12 depicts the profiles of water vapor relative humidity (H_r). Below the 1.5 cm depth marked by the horizontal red dashed line, soil water vapor approaches saturation, whereas the upper layer shows H_r values dropping to as low as 0.1. This creates a sharp humidity gradient at the 1.5 cm depth, and as described by Equation (10), such a gradient drives the generation of isothermal vapor flux (q_hv).

Figure 13 illustrates the profiles of the isothermal vapor flux (q_hv). The results indicate that significant isothermal vapor flux (q_hv) occurs above the 1.5 cm depth demarcated by the horizontal red dashed line—a region coinciding with the low water vapor relative humidity zone in Figure 12. The theoretical consistency between these two variables—validated by their congruent spatial distributions—affirms the model’s accuracy and the reliability of moisture flux calculations from the WVHS model for the vadose zone.

4. Conclusions

To investigate the impact of salt on soil moisture transport, a numerical model of coupled water, vapor, heat, and salt (WVHS) transport was developed by modifying existing frameworks. Evaporation experiments were performed to validate the model’s accuracy, and simulation results showed good agreement with experimental data. The WVHS model was further compared with the traditional WVH model to analyze how salt influences soil moisture distribution and dynamics. The key conclusions are summarized as follows:

(1)

The presence of salt decreases evaporation rates and enhances upward water movement. Comparative analysis of experimental data with the WVHS and WVH models reveals that the WVH model consistently underestimates soil water content. Specifically, the WVHS model exhibits smaller variations in water content at the same depth compared to the WVH model, indicating that salt in the vadose zone enhances upward soil moisture transport.

(2)

Calculations of soil moisture fluxes and hydraulic conductivities show that liquid water fluxes increase with depth, while water vapor fluxes decrease with depth. Throughout the soil profile, liquid water fluxes are dominated by isothermal liquid water fluxes (q_hl) and osmotic liquid water fluxes (q_Cl). At the soil surface, water vapor fluxes consist of thermal vapor fluxes (q_Tv) and isothermal vapor flux (q_hv), whereas thermal vapor fluxes (q_Tv) represent the primary vapor fluxes at other depths. These findings indicate that liquid water movement is governed by matric and osmotic potentials, while water vapor transport is primarily controlled by temperature. The influence of matric potential on water vapor movement occurs exclusively at the soil surface.

(3)

The influence of osmotic potential on water vapor transport diminishes as water content increases. Osmotic potential affects water vapor relative humidity only when soil water content approaches the residual water content. When water vapor relative humidity is below unity, soil water vapor movement is significantly governed by matric potential.

The WVHS model has been validated through laboratory experiments, and future large-scale simulations will be conducted to expand its application scope. Under conditions of high temperature and low water content, salt precipitation occurs in soil, which affects soil moisture transport. Future research will investigate the impact of salt precipitation on soil moisture movement.