During soil freezing, both heat transfer and water migration coexist in the freezing process which occurs in a coupled manner. When a temperature gradient forms in the soil, the temperature gradient drives the heat to flow from the higher temperature zone towards the lower one and the pore water to migrate from the unfrozen zone to the freezing fringe, and then to the frozen front. The water migration from warmer unfrozen zone can influence the heat conduction process due to the effect of convection and latent heat of phase change, while the heat conduction may induce phase change and in turn change the hydraulic conductivity of the soil.
Because of the importance of the soil freezing process, many classical models have been developed, such as finite difference method (FDM), finite element method (FEM) and finite volume method (FVM), to describe the coupled heat-fluid transport phenomenon of the soil freezing process when ice lensing does not occur [1, 2, 3, 4, 5, 6]. However, the traditional numerical methods are based on the discretization of macroscopic continuum equations. This scheme makes traditional numerical methods face great challenges with solving flow with complex interface or flow with complex boundary. Besides, the traditional numerical methods have difficulties dealing with the microscale force such as interactive force in the pre-melted film. Recently, numerical models based on LBM (Lattice Boltzmann method) for simulating the heat and mass transfer phenomena with phase transformation in frozen soil during freezing process are presented [7, 8]. However, the freezing fringe, the balance between surface tension and gravity, and the pre-melted film in frozen front were not considered. Here, in this works, based on the Shen-Chen model which gives a Lattice Boltzmann approach for multiphase fluid flows, and allows to calculate the temporal and spatial evolution of the density distribution functions fi for an arbitrary number of components [9, 10, 11, 12], we further developed a new Lattice Boltzmann model for the simulation of soil freezing. The suction of the pre-melted film in frozen front and freezing fringe was regarded as a kind of suction force, two kinds of resistance in unfrozen zone and freezing fringe were regarded as a kind of body force, and the balance between the surface tension and gravity was regarded as a kind of block force.
The aim of this study was to develop a comprehensive three-phase Lattice Boltzmann model capable of describing the heat and mass transfer during unsaturated soil freezing. The model is capable of predicting the water content distribution along the depth of soil and the water content variations with temperature in unfrozen zone, freezing fringe and frozen zone.
2 Basic models
2.1 Physical model of soil freezing
The physical model is schematically illustrated in Figure 1. During soil freezing, as temperature gradient forms in soil, heat moves gradually from high temperature to the lower one, and moisture migrates from unfrozen zone to the frozen zone. The heat and mass transfer caused by the temperature gradient can be divided into three parts (frozen zone, freezing fringe and unfrozen zone). In frozen zone, due to the intermolecular interactions, a certain amount of unfrozen water exists in the pre-melted film between the surfaces of soil grains and ice lens [13, 14], and the pre-melted film forms the channel for the migration of liquid water towards the solidification front in soils to supply the growth of ice lenses . In freezing fringe, the temperature is lower than freezing point Tm, and higher than the ice entering point Tie. A certain amount of unfrozen water exists between the surfaces of soil grains and ice grains due to the Gibbs-Thomson effect [16, 17], and the formation of the ice grains rises the resistance for the migration of liquid water. In unfrozen zone, water exits among the soil pore space where the adhesion force and water-air interface tension keep balance with gravity.
2.2 Soil structure
In nature, soils are porous media with stochastic particle size distribution which is an important soil characteristic, and this characteristic has significant effects on the heat and mass transfer during soil freezing process [18, 19, 20]. In this paper, random generated volume fraction for each lattice was used to describe this characteristic.
In theoretical and experimental work on fluid flow in porous media, it is typically attempted to find functional correlations between the particle size distribution and some other macroscopic properties of the porous medium. Among the most important of such properties are the porosity φ and the specific surface area S, which give the ratios of the total void volume and the total interstitial surface area to the bulk volume, respectively .
The value of porosity is equal to the probability that a given point in volume V is not overlapped by any of the obstacles, this can be calculated by:
where K is the number of obstacles, fv = V0/V is the average volume fraction of obstacles, V0 is the average volume of obstacles.
The specific surface area is S = A/V (A is the value of the total surface area of all obstacles), which is given by the total number of obstacles times the surface area of a single obstacle times the probability that a given point on the surface of an obstacle is not overlapped by any other obstacles, i.e.,
where A0 is the average surface area of obstacles.
It is observed that both these two macroscopic properties are related to the average volume fraction of obstacles fv. Similarly, we get the macroscopic properties (porosity and specific surface area) from the average volume fraction of obstacles fvl and the average number of obstacles in a single lattice N, where fvl = Nfv. Consequently, it is reasonable to use the randomly generated volume fraction to indicate the stochastic particle size distribution.
2.3 Lattice model
The D2Q9 lattice model was used in this simulation, as shown in Figure 2. This model is very common, especially for solving fluid flow problems. It has high velocity vectors, with the central particle speed being zero, and the streaming velocity for the D2Q9 model takes the value:
The associated lattice weights wi are:
The sound speed (cs) is equal to
2.4 Enthalpy conservation
The conservation of enthalpy for the three phases is:
where ki and Ti respectively refer to the thermal conductivity and temperature of phase i, Ci is the specific heat, ρs, ρl and ρg are the local density of solid, liquid and gas, respectively.
2.4.1 Initial conditions
The initial conditions are as follows:
2.4.2 Boundary conditions
The boundary conditions are as follows:
Periodic temperature boundary conditions are applied on the side boundaries, as shown in Figure 3.
2.4.3 Heat conduction with phase change in porous media
During soil freezing process, the thermal diffusion satisfies the usual Lattice Boltzmann equation:
where gi is the temperature distribution function in the i-th velocity direction, and τh is the relaxation time related to the thermal diffusion as α = (τh – 0.5). The equilibrium distribution function for can be calculated as usual as: (X, t) = wig.
2.5 Mass and momentum conservation
2.5.1 Mass conservation
In a lattice with a random obstacle volume fraction fv, the mass conservation for each of the two fluids and the solid fraction can be written in the following equations:
where Γm is the melting rate, ug and ul are the local fluid velocities of liquid and gas, respectively.
2.5.2 Momentum conservation
Momentum equations for the fluid mixture are seen as a single fluid:
where ρ = ∑δρσ is the total density of the mixture. The total momentum of the fluid mixture is:
2.5.3 Initial conditions
The initial conditions are as follows:
2.5.4 Boundary conditions
Bounce back boundary conditions are applied on both top (down) boundaries and ice-fluid interfaces in order to maintain mass conservation. The periodic boundary conditions are applied on the side boundaries in order to maintain the continuity of flow, as shown in Figure 4.
2.6 Multiphase fluid model
To our best of knowledge, the Shen-Chen model is the most widely used multiphase LB model for the simulation of multiphase fluid [9, 10, 11, 12]. In Shen-Chen model, distribution function of all components in soil freezing satisfies the usual Lattice Boltzmann equation:
where is the density distribution function of the σ-th component in the i-th velocity direction, and τσ is the relaxation time related to the kinematic viscosity for the component σ as νσ = The equilibrium distribution function for can be calculated as usual as:
where the density and momentum of the σ fluid component, can be calculated through:
ueq in equation (15) is determined by the relation:
where u′ is the common velocity of the fluid mixture, and it is calculated as:
and Fσ = + + is the sum of the interactive force between fluid particles, (responsible of cohesion force), between solid boundary and fluid particles, (responsible of adhesion forces) and external body forces, , such as gravity and buoyancy forces.
The interactive force acting on the particles of species σ located in position x can be calculated as:
where σ and σ are respectively the first and second components of the fluid mixture and Gcoh is the parameter that can be used to tune the surface tension between the two modeled fluids. Analogously, the cohesion force between particles of the fluid σ and the solid boundary, can be evaluated as:
where s is a flag variable that is equal to 1 if the lattice node i belongs to the solid boundary and it is equal to 0 if si points a fluid lattice node.
Eventually, the action of a constant body force that mimics the effect of a buoyancy force can be added as:
where g is the gravity and Δρ the difference in density between the two components.
3 Developed models
3.1 Melting problem in soil freezing
For bulk water freezing, the problem of half space conduction melting with thermal diffusivity which is called Neumann-Stefan problem, has been solved analytically in 1860 by Neumann. Here, the enthalpy-based method by Jiaung et al. [22, 23], which has been successfully used by Huo and Rao  for solid–liquid phase change phenomenon of phase change material under constant heat flux, was modified for the LB approaches of melt-solid moving boundary in porous media.
In enthalpy-based method, the melting term is introduced as a source (crystallization) or sink (melting) term in the collision step. In summary, at the time-step n and iteration kn, the macroscopic temperature is calculated by:
where Tn,kn ≡ Tkn(t = n). The local enthalpy is obtained by:
with the liquid fraction fl of the previous iteration. Finally, the enthalpy is used to linearly interpolate the melt fraction
where Ens and Enl are the enthalpies of the solid and liquid at the melting temperature Tm.
However, in freezing fringe, according to Gibbs-Thomson equation, the melting temperature in porous media is related to the pore throat, that is to say, the particle size distribution plays an important role in the pore water melting temperature . According to Saruya et al. , the difference between the warmest temperature Tf at which ice can first form and the normal bulk melting temperature Tm0 ≈ 273.15 K can be obtained by ΔTf = Tm0 – Tm = 2γslTm/(ρLfRp), where, Rp is the characteristic radius of a pore throat, ρLf ≈ 3.1 × 108 J/m3 is the latent heat of fusion per unit volume and γsl ≈ 29 × 10–3 J/m2 is the ice-liquid surface energy, ΔTf which describes the difference between the warmest temperature Tf at which ice can first form and the normal bulk melting temperature Tm0 ≈ 273.15 K. So, the actual water melting temperature in porous media is:
The characteristic radius of pore throat is Rp = αpR, where, R is the particle radius, and αp = 0.7 is a correlation coefficient. The average characteristic radius of a single lattice in the simulation can be calculated as:
In frozen zone, due to the intermolecular interactions between the particles, liquid, and ice (e.g., van der Waals forces), interfacial melting between ice surface and soil particle surface below bulk melting point Tm can be found, and the thickness of this pre-melted film upon approaching the melting point Tm follows in a logarithmic growth law :
where T0 ≈ Tm - 17 K. The constant a(0) ≈ 0.84 nm corresponds to the decay length of the non-ordering (average) density.
As shown in Figure 5, the average thickness of pre-melted film in a single lattice is also related to the average obstacle volume fraction fv, and the relationship is:
In the developed model, the bulk melting temperature Tm in equation (24) was replaced by the variable melting temperature (Tmi). Consequently, the formation of pre-melted is possible.
The collision is calculated by:
where τh is the relaxation time related to the thermal diffusion for the component σ as kσ =
Then, a sub-loop is introduced into every time step until the temperature and the melt fraction field converge to within a set tolerance. Finally, the macroscopic temperature can be calculated as:
3.2 Shen-Chen model for multiphase fluid in soil freezing
During soil freezing, water in the freezing fringe and the unfrozen zone moves towards the frozen front by the suction in pre-melting film and freezing fringe. At the same time, the resistance along the path in the unfrozen zone and the freezing fringe blocks the movement, as shown in Figure 1. Similar to the adhesion force between fluid particles and solid boundaries in the Shen-Chen model, the suction in frozen front can be thought as a kind of interactive force between ice and water particles According to Darcy law, we can infer that the resistance along the path in unfrozen region can be described by a body force which is related to the distance from freezing fringe. Consequently, the sum of the interactive force for mixture fluid flow in soil freezing is:
3.2.1 Adhesion force
The interactive force (adhesion force ) in soil is related to the soil particle distribution, so the adhesion force can be calculated as:
where f is a flag variable that is equal to the soil particle volume fraction of the lattice node i.
3.2.2 Suction force
According to the kinetics of ice growth in porous media , The suction force is caused by the pore water pressure difference between frozen zone, freezing fringe and unfrozen zone. First, due to the pore water pressure decrease between ice grain surfaces and soil particle surfaces, water moves from unfrozen zone towards the freezing fringe. Then, because of the deep decrease of water pressure in pre-melted film in frozen front, water moves again from the freezing fringe towards the frozen front. We can found that the suction only forms in the freezing fringe and pre-melted film near the ice particles. So, it is reasonable to consider the suction as an adhesion force between the ice particles and fluid particles, which can be calculated by:
where l is a flag variable that equals to fl for the particles of fluid σ with phase change to happen near the phase change boundary, and it is equal to 0 for the particles of fluid σ without phase change to happen. Gfre is the parameter that describes the magnitude of suction force which is related to the pore water pressure drop in freezing fringe and pre-melted film.
According to the generalized Clapeyron equation:
where Pcl is the pore water pressure in the freezing fringe and pre-melted film at temperature T, P is the soil particle pressure in warm region, Tm is the bulk water freezing point, Lf is the latent of ice melting, ρ is the ice density. If there is a supply of ground water at a pressure PR, the pore water pressure difference between the ground water and pore water near ice particles is:
As we know, the fluid should move from the higher pressure to the lower one. So, in this study the parameter Gfre is related to the maximum pressure difference that locates in the frozen front where the temperature is equal to the ice entering temperature Tie:
Consequently, the suction force can be described as:
where Ac is the cross-sectional area of flow.
Therefore, Gfre is a parameter that relates to the surface tension at ice-water interface and the pore structure of soil particles.
3.2.3 Resistance along the path
In soil freezing, due to the suction, water moves from unfrozen zone to the freezing fringe, then to the frozen front. According to the kinetics of ice growth in porous media , there are two kinds of resistance along the path way, one is the hydraulic resistance due to the flow in the porous medium, and the other is the resistance due to the flow in the thin films around the particles.
The hydraulic resistance is:
where kp is the permeability from Carman-Kozeny equation:
The resistance in the thin films is:
where μp is the viscosity of water in thin films.
Finally, the whole resistance along the path is:
3.2.4 Block force
Before soil freezing, as shown in Figure 6 (a), there is no pressure gradient between pore water and ground water, and to keep the pore water stand still in the space among soil grains, the adhesion force and surface tension at the water-air interface need to keep balance with gravity (Ft = ρghA). In soil freezing, as shown in Figure 6 (b), pressure gradient starts to form in the pore water, the tension force is gradually replaced by the pressure gradient to keep the balance with gravity, and the water flow begins after the pressure gradient overcomes the gravity. So, this varying force can be thought as a block force instead of the body force and it can be calculated as:
Based on the Shen-Chen model, this paper presented a new Lattice Boltzmann model for the simulation of soil freezing. In this model, the suction of the pre-melted film in freezing fringe was regarded as a kind of suction force, the adjustment coefficient of the suction force was a parameter that related to the particle size, water-air surface tension, and ice entering temperature. Two kinds of resistance were regarded as a kind of body force related to the water films between the ice grains and soil grains, and a block force instead of gravity was introduced to keep balance with gravity before soil freezing.
This study is supported by Natural Science Foundation of China (Grant NO. 51776049), Special Foundation for Major Program of Civil Aviation Administration of China (Grant No. MB20140066) and National Materials Service Safety Science Center open fund.
φ porosity of porous medium (dimensionless)
S specific surface area (m2/m3)
Tm bulk water melting temperature (K)
Tie ice entering temperature in porous medium (K)
K number of obstacles
fv average volume fraction of a single obstacles (dimensionless)
V0 average volume of obstacles (m3)
A0 average surface area of a single obstacle (m2)
fvl average volume fraction of obstacles in a single lattice (dimensionless)
N average number of obstacles in a single lattice
cs sound speed (1/s)
e discrete velocity (dimensionless)
w lattice weight (dimensionless)
k thermal diffusion (W/m K)
T temperature (K)
C specific heat (J/kg K)
ρ density (kg/ m3)
Tcold fixed temperature on up boundary (K)
Thot fixed temperature on down boundary (K)
τ relaxation time (s)
α thermal diffusion (W/m K)
g temperature distribution function (dimensionless)
fl liquid fraction (dimensionless)
Ens enthalpy of the solid (J/kg)
Enl enthalpy of the liquid (J/kg)
Rp characteristic radius of a pore throat (m)
R particle radius (m)
αpa correlation coefficient (dimensionless)
γsl ice-liquid surface tension (N/m)
Lf latent heat of ice (J/kg K)
L film thickness (m)
u velocity of fluid (m/s)
Fcoh cohesion force (N)
Fadh adhesion force (N)
Ffre suction force (N)
Fblo resistance along path way (N)
Fb body force (N)
ν kinematic viscosity (kg/m s)
f density distribution function (dimensionless)
P pressure (Pa)
PR ground water pressure (Pa)
Pcl pore water pressure by Clapeyron equation (Pa)
Gcoh parameter control the surface tension between the two modeled fluids (dimensionless)
Gads parameter control the adhesion force between fluid and soil particles (dimensionless)
Gfre parameter control the suction force between ice and fluids (dimensionless)
μ bulk water viscosity (kg/m s)
μp viscosity of water in thin films (kg/m s)
kp permeability (dimensionless)
σ fluid component
iindex of speed direction of lattice
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Published Online: 2018-04-02