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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process

Wenyu Song
• School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Tel./Fax: +86 451 86412078, China
• Other articles by this author:
/ Yaning Zhang
• Corresponding author
• School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Tel./Fax: +86 451 86412078, China
• Department of Bioproducts and Biosystems Engineering, University of Minnesota, St. Paul 55108, Falcon Heights, U.S.A.
• Email
• Other articles by this author:
/ Bingxi Li
• School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Tel./Fax: +86 451 86412078, China
• Email
• Other articles by this author:
/ Fei Xu
• School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Tel./Fax: +86 451 86412078, China
• Other articles by this author:
/ Zhongbin Fu
• School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Tel./Fax: +86 451 86412078, China
• Other articles by this author:
Published Online: 2017-06-16 | DOI: https://doi.org/10.1515/phys-2017-0042

## Abstract

In the current study, a macroscopic lattice Boltzmann model for simulating the heat and moisture transport phenomenon in unsaturated porous media during the freezing process was proposed. The proposed model adopted percolation threshold to reproduce the extra resistance in frozen fringe during the freezing process. The freezing process in Kanagawa sandy loam soil was demonstrated by the proposed model. The numerical result showed good agreement with the experimental result. The proposed model also offered higher computational efficiency and better agreement with the experimental result than the existing numerical models. Lattice Boltzmann method is suitable for simulating complex heat and mass transfer process in porous media at macroscopic scale under proper dimensionless criterion, which makes it a potentially powerful tool for engineering application.

PACS: 02.60.Cb; 05.60.-k; 44.30.+v; 65.20.De

## 1 Introduction

Regions with frozen ground cover approximately 55% to 60% of the exposed land surface in the Northern Hemisphere [1]. In such regions, the application of porous material often meets significant challenges. The most common challenge is that porous material suffers damages caused by frost heave. More specifically, frost heave is caused by heat and moisture transfer process and solidification of water inside the material [25]. Such process occurs at pore scale which makes the experimental investigation hard to be carried out. Therefore, the numerical analysis of heat and moisture transfer phenomenon in the porous material during freezing process is of great importance for developing the strategy to prevent the frost heave damage.

In general, the existing numerical models canbe divided into three categories: empirical and semi-empirical, analytical, and numerical physically-based [6, 7]. In empirical and semi-empirical models, one or several experimentally established relationships or coefficients are adopted to solve the freezing depth [8]. Analytical models are mainly used specifically to solve heat conduction problems under certain assumptions. The most commonly used analytical model is Stefan’s formulation, which solves the position of freezing front of pure substance during the freezing process [9, 10]. Unlike the former two, in numerical physically-based models, the momentum equation and energy equation are both solved. To the best of author’s knowledge, such models provide higher accuracy on simulating freezing process than the other two kinds. The development of numerical physically-based models began in the early 1970s [1113]. In the early stage, due to the limitation of computing capabilities, certain processes (e.g. the condensation process of vapor) had to be neglected [1114]. As research continues, the complexity of freezing process in porous media was gradually revealed [1517]. Consequently, a large variety of improvements by introducing equations with different levels of complexity have been developed to date [18]. Such approaches introduce various parameterizations which make these formulations very flexible, and the porous media community has come to value flexible formulations above predictive ones because somewhere a combination of parameters must exist that makes them correct and appropriate [19]. As a result, comprehensive analysis investigating the rationale behind the freezing process has not been carried out [18]. The nonlinear relationship between temperature, ice and liquid water content at the freezing front makes the numerical model hard to reach convergence. What’s more, heat and mass transfer process in porous media is clearly affected by its microscopic structure. The popular approaches (Finite Difference, Finite Element and Finite Volume) employ the discretization of macroscopic continuum equations. Substantially, this scheme ignores the randomness of the system, which makes it hard to solve flow with complex interface or flow with complex boundary.

Since the late 1980s, a new mesoscopic method based on the discrete kinetic theory, named lattice Boltzmann method (LBM), has become a promising tool for the investigations of a wide range of complex flows, including multiphase flows, porous flows, thermal flows and reactive transport [20]. In fact, LBM shares the same fundamental idea with percolation theory (which is also a great success in porous media research): the complicacy of heat and mass transfer process in porous media is caused by the interaction between simple physical process and complex structure. What’s more, the lattice network in LBM also is a site percolation network. The fundamental physics of porous media is classical Newtonian mechanics, in LBM, it is expressed by discrete kinetic theory. The major advantage of LBM is that it behaves like a solver for the conservation equations in the bulk flow; meanwhile, the kinetic equation offers many benefits of molecular dynamics, including clear physical pictures, easy implementation of complex boundary conditions and fully parallel algorithms [20].

It should be noted that due to the mesoscopic nature of LBM, itis hard to perform simulations at macroscopic scale since such simulations will require massive computational resource [20].To overcome such limitation, a macroscopic lattice Boltzmann model is presented in the current study. The proposed model is validated by Mizoguchi’s experiment. It not only provides the same accuracy as the mesoscopic LBM model but also reduces the need for computing capability as well, which makes it a potentially powerful tool for engineering application.

## 2.1 Reconstructing the structure of porous media

In general, porous media is a heterogeneous assembly of solid skeleton and pores [21]. The stochastic pore structure makes the porosity randomly distributed. Therefore, it is important to obtain the proper distribution of porosity in porous media. In the current study, firstly, a stochastic generation-growth method is adopted to generate the geometry of part of the porous media. The porosity of geometry is then calculated by the porosity distribution for a larger scale model. The reconstruction process is conducted as follows:

1. At a given core distribution probability φd, randomly locate the growing cores of particles. φd is calculated by the number of particles.

2. Define the growing probability φg,i of the existing growing cores. There are 8 possible growing directions, while generating particles of different shapes, the shape is controlledby φg,i.

3. In each cycle, grow one particle along the given directions from each growing core; calculate the porosity ε, stop growing when ε reaches the given porosity ε0, otherwise go back to step (1).

4. Divide the whole area by a grid with a certain resolution, calculate the porosity in each unit, and get the porosity distribution function.

5. Generate the new model by randomly locating the cell with different porosities according to the porosity distribution function obtained in (4).

It is noteworthy that the reconstructed structure is mainly determined by three parameters: ε, φd and φg,i, where ε ensures the consistency of macroscopic porosity with thereal object; φd determines the number of pores, for granular porous material, φd determines the number of solid particles; φg,i controls the shape of pores. More specifically, if φg,i has the equal value in each direction, one can get a structure made of round-shape particles, while the value of φg,i differs in each direction, one can get triangle, square or particles with any shape.

By using this procedure, once the microscopic structure of the porous material is known, one could reproduce a corresponding macroscopic model with the statistical properties based on the microscopic model. As shown in Figure 1, the microscopic model is a porous media of 1000×1000 randomly located particles with a total porosity ε0 = 0.5. Then the microscopic model is divided by a 100×100 grid; each lattice represents the average porosity of an area of 10×10 particles. By using this strategy, the amount of computational grid is reduced to 1/100 of the original one, yet the randomness of structure is well preserved.

Figure 1

Microscopic model and corresponding macroscopic model generated by reconstruction algorithm, values on the color bar represent porosity

## 2.2 Two phase displacement model

To the best knowledge of the authors, since Shan and Chen pseudopotential LBM model (SC model hereafter) was first brought up by Shan and Chen in 1993 [2225], it has been the most widely used multiphase LBM model due to its simplicity and versatility [26]. Therefore, SC model is adopted to simulate the displacement process of water and air in the soil during the freezing process. In SC model, each phase is represented by a set of lattices, respectively. In the proposed model, single relaxation time collision operator (LBGK) equation is adopted for both phases. In the computer implementation, this equation can be expressed by: $fσ,ix,t+Δt=fσ,ix,t−1τσ,νfσ,ix,t−fσ,ieqx,t$(1) $fσ,ix+eiΔt,t+Δt=fσ,ix,t+Δt$(2) Equations 1 and 2 represent collision step and streaming step, respectively. In the equations above, fσ,i(x, t) represents the density distribution function of phase σ in ith direction at lattice site x and time t; feq is the equilibrium distribution function; τν is the dimensionless relaxation time; c = Δx/Δt is the lattice speed which signifies the propagation velocity of disturbance, in which Δx and Δt are lattice spacing and time step, respectively. In the current study, Δx and Δt are chosen as Δx = Δt = 1. Also, a D2Q9 (2 dimensions, 9 velocity directions) lattice is adopted. As shown in Figure 2, the definition of the discrete velocity ei of a D2Q9 lattice is expressed as follows: $ei=0,0i=0cosi−1π/2,sini−1π/2i=1,2,3,42cosi−5π/2+π/4,sini−5π/2+π/4i=5,6,7,8$(3)

Figure 2

Layout of discrete velocity directions of a D2Q9 lattice

In Eq. 1, feq stands for the equilibrium distribution function, which is given by: $fσ,ieq=ωiρσ1+3c2ei⋅ueq+92c4ei⋅ueq2−32c2ueq2$(4) in Eq. 4, the weight factor ωi is assigned with: $ωi=4/9i=01/9i=1,2,3,41/36i=5,6,7,8$(5) The viscosity of each phase in lattice unit is expressed by: $νσ=cs2τσ,ν−0.5Δt$(6) In which cs signifies the lattice sound speed. The macroscopic quantities such as density and momentum of each phase at a lattice node are expressed by: $ρσ=∑ifσ,i$(7) $ρσuσ=∑ifσ,iei$(8)

Then, the mass fraction χ of phase σ is carried out from the density of each phase, which can be expressed by: $χσ=ρσρσ+ρσ¯$(9) To introduce the external force and bulk force (e.g. gravitational force, fluid-fluid interaction force, cryosuction force, etc.), in the proposed model, a velocity shift scheme [22] is employed. The equilibrium speed $\begin{array}{}{\mathbf{u}}_{\sigma }^{eq}\end{array}$ in Eq. 4 is modified by the following relation: $ueq=u′+τσ,νFσρσ.$(10) in which, the common velocity of the fluid mixture is expressed by u, which is calculated by: $u′=∑σρσuστσ,ν∑σρστσ,ν$(11) It is noteworthy that in Eq. 10, Fσ represents the sum of total interactive forces. In the current study, $\begin{array}{}{\mathbf{F}}_{\sigma }={\mathbf{F}}_{\sigma }^{Coh}+{\mathbf{F}}_{\sigma }^{B}+{\mathbf{F}}_{\sigma }^{T},\end{array}$ in which $\begin{array}{}{\mathbf{F}}_{\sigma }^{Coh}\end{array}$ represents the cohesive force between fluid particles from the same phase; $\begin{array}{}{\mathbf{F}}_{\sigma }^{B}\end{array}$ is the sum of bulk forces (e.g. gravitational force, buoyancy force, etc.); $\begin{array}{}{\mathbf{F}}_{\sigma }^{T}\end{array}$ represents the cryosuction force caused by the change of water potential.

In the proposed model, the existence of vapor in porous media is neglected. Therefore, the multiphase system is treated as a binary mixture formed by air and water. For fluid-fluid interaction, only the repulsive force between different phases is taken into consideration, the repulsive force of each phase at one lattice site is calculated by: $FσCohx=−ψσx,tGσCoh∑iωαψσ¯x+eiei$(12) where σ and σ stand for the first and the second phase of the fluid mixture, respectively; $\begin{array}{}{G}_{\sigma }^{Coh}\end{array}$ is a coefficient to adjust the surface tension between two phases; ψσ is effect mass, which is carried out by: $ψσ=ρσ,0(1−exp⁡(−ρσρσ,0))$(13) The gravitational force applied to phase σ is given by: $FσB=ρσg$(14)

## 2.3 LBM model for melt/solidification

During the freezing process, water migrates along the direction of temperature gradient in porous media. The interpretation of melt/solidification process in LBM is quite convenient. In LBM, each phase, component or physics is described by an individual set of lattice. To simulate the melt/solidification process, the third layer of D2Q9 lattice is introduced. Analogous to section 2.2, the LB equation for thermal diffusion process is expressed by: $gix+eiΔt,t+Δt=gix,t−1ταgix,t−gieqx,t$(15) in which, the discrete velocity ei is the same as in Eq. 3. The equilibrium distribution function $\begin{array}{}{g}_{i}^{eq}\end{array}$ is carried out by: $gieq=ωiT1+3c2ei⋅ueq+92c4ei⋅ueq2−32c2ueq2$(16) The weight factor ωi is given by Eq. 5, equilibrium speed ueq is given by Eq. 10. It is noteworthy that Eq. 16 is the general form for both the convection and conduction heat transfer process. For conduction heat transfer process, one can replace ueq with 0, which yields: $gieq=ωiT$(17) The thermal diffusivity α is calculated in analogy with the kinematic viscosity, which is expressed by: $α=cs2τα−0.5Δt$(18) The thermal conductivity λ is expressed by: $λ=αρCp$(19)

Similarly, the macroscopic temperature is calculated by: $∑igi=T$(20) In the current study, to simulate the heat conduction process with melt/solidification, a LB model proposed by Jiaung [27] is adopted. In Jiaung’s model, an extra source term is introduced in Eq. 15, which yields: $gix+eiΔt,t+Δt=gix,t−1ταgix,t−gieqx,t+ωiLC(fl,it+Δt−fl,it)$(21) A subroutine (with iteration) is employed at every time step to solve the source term in Eq. 21, which is executed as follows:

1. For the next time step t + Δt in the kth iteration, Eq. 21 is given by: $gikx+eiΔt,t+Δt=gix,t−1ταgix,t−gieqx,t+ωiLCfl,ik−1t+Δt−fl,it$(22) in which the superscripts k-1 and k represent the previous and current iterations of the current time step, respectively.

2. In the current kth iteration, the macroscopic temperature is calculated by Eq. 20).

3. In the current kth iteration, total enthalpy is calculated by: $Hk=CTk+flk−1L$(23)

4. In the current kth iteration, liquid fraction is carried out by: $flk=0HHl$(24) where Hs and Hl represent the enthalpy at the freezing temperature and melting temperature, respectively.

5. Repeat steps (i)-(iv) until the residuals reach the following convergence criterion: $min(fk−fk−1fk−1,Tk−Tk−1Tk)<10−8$(25)

It is noteworthy that in real physical units, the value of thermal diffusivity is rather small. Therefore, the value of τα in Eq. 18 is very close to 0.5, which will cause numerical instability. Huber et al. [28] proposed a scheme in which the total time of evolution tmax is used as the timescale, then a non-dimensional version of Eq. 21 can be obtained: $∂T⋆∂t⋆=αtmaxl2∇2T⋆−1St∂fl∂t⋆$(26) In which the dimensionless scales T = (TT0)/(T1T0); t = αt/l2; l is the length of the system; T0 and T1 represent the initial temperature and the boundary temperature, respectively; St = C(T0T1)/L is the Stefan number.

## 2.4 Implementation of water potential

As mentioned above, in porous media, water moves from one place to another driven by gravitational force, pressure difference, capillary force, osmosis, etc. The sum of driving forces is described by water potential Ψ. The total water potential ΨTot is given by: $ΨTot=ΨP+ΨZ+ΨS+ΨA+ΨT$(27) in which ΨP, ΨZ, ΨS, ΨA and ΨT represent pressure, gravitational, solute (osmotic), air pressure and temperature potentials, respectively. ΨS and ΨA are neglected for low salinity porous media and standard atmospheric pressure, which yields: $ΨTot=ΨP+ΨZ+ΨT$(28) In the proposed model, Ψp is caused by capillary action, which is determined by $\begin{array}{}{\mathbf{F}}_{\sigma }^{Coh}\text{\hspace{0.17em}and\hspace{0.17em}}{\mathbf{F}}_{\sigma }^{Ads};{\mathit{\Psi }}_{Z}\end{array}$ is represented by $\begin{array}{}{\mathbf{F}}_{\sigma }^{B};{\mathit{\Psi }}_{T}\end{array}$ is represented by $\begin{array}{}{\mathbf{F}}_{\sigma }^{T}.\end{array}$

During the freezing process, generally speaking, the porous media is divided into two regions: frozen region and unfrozen region. In the unfrozen region, the gradient of temperature potential is derived from Clapeyron’s equation, which is: $∇Ψ=LlTlΔVl∇T$(29) Ll and ΔVl represent the latent heat of condensation and specific volume change during vapor condensation, respectively. At the freezing front, the gradient of water potential caused by frozen water is expressed as: $∇Ψ=LsTsΔVs∇T$(30) where Ls and ΔVs are the latent heat of solidification and specific volume change during the freezing process, respectively. In Eq. 30, Ls and ΔVs are known as the property of water, T = 273 K is the freezing temperature of water, which yields: $∇Ψ=13.48×106∇T$(31) Here we give a demonstration of converting the real physical units into lattice units. The real physical unit of temperature potential is Pa, which signifies the cryosuction force during the freezing process. Thus, the temperature potential can be written as: $ΨT,r=Pr=ρrur22$(32) In lattice unit, this suction pressure is expressed by: $P=ρu22$(33) Hence, the suction pressure in lattice unit becomes: $P=ρu2ρrur2Pr$(34) it is known that $uru=cs,rcs$(35) assuming that in a porous media, the speed of sound in real physical unit cs,r = 200 m/s, the density of water in the real physical unit ρr = 1000 kg/m3; the speed of sound in lattice unit cs = $\begin{array}{}1/\sqrt{3},\end{array}$ the density of water in lattice unit ρ = 1 it can be calculated that $P=111.97×107Pr$(36) Therefore, Eq. 31 in lattice unit yields: $∇Ψ=13.48×10611.97×107∇T=0.113∇T$(37) Note that Eq. 37 needs to be corrected according to the type of porous media. The temperature potential is introduced as body force on the freezing front, which can be expressed by: $FσT=GT∇ΨΔx$(38) in which GT is determined by the type of porous media.

## 2.5 Macroscopic LBM model for porous media

The existing studies have revealed that the application of LBM has contributed to improve the understanding of fundamental aspects of fluid flow in porous media [20, 29]. However, the ability of macroscale simulation with LBM is still limited due to its mesoscopic nature. One challenge is that to conduct a macroscalesimulation, LBM will require massive lattices to represent the structure of porous media. To get rid of this limitation, Dardis and McCloskey [30, 31] proposed a method which can overcome this difficulty by parameterizing a porous medium regarding its solid density ns(x), or equivalently its porosity since ns = 1 − ε [32]. By applying this strategy, LBM is elevated from mesoscopic scale to macroscopic scale. This strategy introduces a solid density value at each lattice node and simulates the porous medium by a ‘probabilistic’ bounce-back based on solid density. To implement this model, considering the traditional collision step expressed by Eq. 1 as an intermediate step, which can be expressed by: $fσ,i⋆⋆x,t+Δt=fσ,i⋆x,t−1τσ,νfσ,i⋆x,t−fσ,ieqx,t$(39) Then, the macroscopic porous medium step has the form: $fσ,ix,t+Δt=fσ,i⋆⋆x,t+Δt−nsfσ,i⋆⋆x,t+Δt−fσ,i′⋆⋆x+eiΔt,t+Δt$(40) where i′ is the index of the direction opposite ei.

Note that ns = 0 recovers the traditional free-fluid model while ns = 1 introduces a bounce-back-like condition that makes the medium impermeable. For 0 ≤ ns ≤ 1, we have a ‘probabilistic’ bounce-back condition as shown in Figure 3.

Figure 3

Schematic view of probabilistic bounce-back scheme

It is important to note that according to Eq. 31, during the freezing process, the cryosuction force is several orders of magnitude higher than the sum of other forces. Therefore, in Eq. 28, Ψp and ΨZ can be neglected. Another tricky issue in LBM is, converting units from real physical units to lattice units is somehow complex. Fortunately, Eq. 26 offers the dimensionless form of heat conduction equation with melt/solidification. It can be concluded that according to Eq. 26, two similarity criterions need to be followed, which are:

Fourier number $Fo=αtmaxl2$(41)

Stefan number $St=CΔTL$(42) For the fluid flow with low velocity in porous media, another similarity criterion should also be followed, which is:

Darcy number $Da=Kd2$(43) Therefore, by using the similarity criterions above, complex unit converting procedure is simplified. Another advantage is that in Eq. 41, α can be assigned with any value to ensure the stability of the numerical program.

## 3.1 Introduction of experiment

In 1990, Mizoguchi conducted a freezing experiment on Kanagawa sandy loam soil [33], which has been adopted as a benchmark in various studies [20, 3436]. In the experiment, four identical cylinders were filled with Kanagawa sandy loam soil. The size of the cylinders was 20 cm in height, 8cm in diameter. The initial conditions for the specimens were identical, which were 6.7°C in temperature and 0.33 in volumetric water content. The first specimen was used for measuring initial condition, and the other three were prepared for the freezing test. The cylinders were thermally isolated on side face and bottom face. A metal slab with channels inside was placed on the top surface; the channels were filled with circulating fluid at a constant temperature of −6°C. Several thermal couples were placed at various depths to measure the temperature distribution. As the experiment continued, the specimens were unloaded at 12, 24 and 50 h, respectively. The soil cylinders were sliced into 1 cm thick to measure the water content.

## 3.2.1 Soil model generation

As shown in the middle image of Figure 4, the computational domain is divided by 200 x 500 lattices to maintain the same aspect ratio as the cylinder used in Mizoguchi’s experiment. In our previous research, 200×500 lattices provide sufficient representativeness of the characteristics of soil. The porosity of sandy loam soil varies between 0.4 and 0.5 [37]. It is assumed that the porosity of soil in the experiment is 0.45. The core distribution probability φd = 0.35; the growing probabilities φg,i are random in each direction at growing core. Then, the soil model with chaotic structure is obtained. Next, the model is divided into a 40×100 grid as shown in the right image of Figure 4, each lattice in the grid represents the average porosity of an area of 5×5 particles. By using this strategy, the amount of computation is reduced to 1/25 of the original, yet randomness of the structureis well preserved, which offers 25 times speed-up on the aspect of computation speed.

Figure 4

Boundary conditions in Mizoguchi’s experiment and in numerical simulation

## 3.2.2 Initial conditions and boundary conditions

The initial temperature is assigned with T0 = 6.7 C; the initial volumetric water content is assigned with 0.33, which is executed by assigning a uniform density ratio of ρσ / ρσ = 1/0.37 to the fluid lattices of each phase. GT is assigned with 0.05, $\begin{array}{}{G}_{\sigma }^{Coh}\end{array}$ is given 0.4 to make sure that the replacement between water and air mainly happens near the freezing front. $\begin{array}{}{G}_{\sigma }^{Ads}\end{array}$ is given 0 to exclude capillary force. Buoyancy force is also neglected. On the boundary AB and DC, fluid lattices are given a porosity εAB = εDC = 0, which according to Section 2.5, yields a bounce-back boundary condition. Boundary AD and BC are set to be periodic for fluid lattices. The parameters used in the numerical simulation are listed in Table 1.

Table 1

The parameters used in the simulation

As shown in Figure 4, thermal lattices at boundary AB are applied with Dirichlet boundary condition with a constant temperature of −6°C. The distribution functions g4, g7 and g8 are unknown at boundary AB, as shown in Figure 5. According to Eq. 20, the temperature at boundary AB is expressed by: $TAB=∑i=08gi=g0+g1+g2+g3+g5+g6+g′ω4+ω7+ω8$(44) In which g signifies the sum of residual unknown distribution functions. g is calculated by: $g′=TAB−g0+g1+g2+g3+g5+g6ω4+ω7+ω8$(45) Assuming that the residual unknown distribution functions meet the relation: $gi=ωig′$(46) Then, the unknown distribution functions can be expressed by: $ω4=19g′ω7=136g′ω8=136g′$(47) The adiabatic boundary condition is applied to boundary DC, as shown in Figure 5, which is: $∂T∂yy=y0=0$(48) The temperature at boundary DC is calculated by a three-point finite difference scheme [38], which is expressed by: $TDC=23∂T∂yy=y0+4T(y0+Δy)−T(y0+2Δy)3$(49) With this scheme, the adiabatic boundary condition is converted in Dirichlet boundary condition, which can be calculated with the same scheme as boundary AB.

Figure 5

Illustration of unknown distribution functions at boundary AB and DC

## 3.2.3 Critical porosity during freezing process

It is noteworthy that ice growth in soil fills the pores and obstructs the flow of water, which reduces the effective porosity and eventually leads to the decrease of permeability. In the proposed model, the decrease of effective porosity εe caused by ice growth is described by: $εe=ε−θi$(50) where θi signifies the volume content of ice in each lattice.

What’s more, according to percolation theory, for porous medium, there is a percolation threshold, which means below this threshold, the permeability (effective porosity) of porous medium remains zero regardless of the change of porosity, which can be expressed by: $εe=εε≥εc0ε<εc$(51) In which εc signifies the critical porosity when the porous medium reaches percolation threshold. The value of εc varies from different porous medium. For loam soil, a recent study conducted by Wu et al. [36] suggested that εc ≈ 0.15 would be a reasonable assumption.

## 3.2.4 Thermal properties

The thermal properties for the materials in the simulation are given in Table 2. It should be noted that each lattice is substantially the mixture of water/ice, air, and porous skeleton. The corrected specific heat is calculated by: $C=Cσχσ+Cσ¯χσ¯+CSχS$(52) The corrected density and latent heat is calculated by: $ρ=ρσχσ+ρσ¯χσ¯+ρSχS$(53) $H=Hσχσ$(54) The corrected thermal conductivity is calculated by Maxwell’s equation, which is: $λ=λσ2λσ+λσ¯−2χσ¯λσ−λσ¯2λσ+λσ¯+χσ¯λσ−λσ¯$(55) in which χ is calculated by Eq. 9.

Table 2

Thermal properties of the materials for numerical simulation

## 3.3 Model validation

During the freezing process, the porous media is divided into three regions, as shown in Figure 6. It can be concluded from the sections above that during the freezing process, water migration in both unfrozen region and frozen fringe is driven by the gradient of water potential. It is widely observed that water migrates rapidly near the freezing front. According to Eqs. 29 and 30, the gradient of water potential in the frozen fringe is several orders of magnitude higher than that in the unfrozen region. As a matter of fact, freezing provides a much quicker process than drying due to its high water potential gradient in frozen fringe, which is clearly evident in Figures 7, 8, 9.

Figure 6

Illustration of three regions in freezing process

Figure 7

Comparison between experimental and numerical results of water content distribution after 12 hours freezing

Figure 8

Comparison between experimental and numerical results of water content distribution after 24 hours freezing

Figure 9

Comparison between experimental and numerical results of water content distribution after 50 hours freezing

The comparisons among experimental result, numerical results obtained by mesoscopic LBM model, finite difference model in the previous studies [20, 35], and proposed macroscopic model in the current study at 12, 24 and 50 hours are presented in Figures 7, 8, 9, respectively. It can be observed that the result from finite difference model, which is proposed by Hansson et al. [35], briefly meets the experimental result at 12 and 24 hours. However, it fails to predict the water content distribution at 50 hours. Mesoscopic LBM model offers better agreement with experiment, but the water content around freezing front is over-estimated in the frozen region and under-estimated in the unfrozen region, respectively. In comparison with the former two numerical models, the proposed macroscopic LBM model showed the best agreement with the experimental result. The moisture concentration at the freezing front, a decrease of water content in frozen fringe and slow recovery in deeper position are well described.

## 4 Results and discussion

It is concluded from model validation process that the proposed macroscopic LBM model offered the best agreement with experiment results while the traditional macroscopic FDM model proposed by Hansson failed to predict the water content distribution at 50 hours. In Hansson’s model, moisture migration in soil was treated as a diffusion process of single component. Therefore, the displacement process of water and air was consequently neglected, which is obviously a debatable approach. In mesoscopic LBM model, this displacement process was taken into consideration, which leads to a better agreement with the experimental result. However, such model ignored the evolution process of porosity in frozen fringe, which leads to an over-estimated porosity and caused over-estimated water content in the frozen region. Unlike the former two models, the proposed model not only takes displacement process into account but also solves the evolution of porosity in frozen fringe as well. As described in Eq. 51, the permeability of soil in frozen fringe behaves nonlinearly due to the existing percolation threshold. This scheme introduces an extra resistance into the frozen fringe which is absent in the previous mesoscopic LBM model. Moreover, thermal properties are corrected with porosity ε, while permeability is corrected by effective porosity εe. Thus, this strategy ensures the accuracy in simulating both heat transfer and fluid flow processes.

What calls for special attention is that in the proposed model, cryosuction force which is expressed by Clapeyron’s equation in Eq. 30, is reduced by Eq. 38 with a coefficient of 0.05. Consistent with the current study, Style et al. [39] proposed a theoretical study and also suggested that Clapeyron’s equation should be modified. In Style’s study, the cryosuction force was reduced by applying a coefficient to fluid viscosity, which was selected by μ = 41μ0. However, from authors’ point of view, such approach proposed by Style is contradictory to the existing theory. As the freezing process continues, the average pore size is reduced by the formation of ice. According to theory,

Knudson number $Kn=λl$(56) In which λ stands for the mean free path.

The corrected viscosity is expressed by: $μ=μ01+9.638Kn1.159$(57) It is evident that while pore size reduces, the corrected viscosity should be reduced instead of increased. Therefore, the investigation into the corresponding aspect is of vital necessity.

The volumetric water content (VWC) distribution and volume of fraction (VOF) distribution are shown in Figures 1015. It can be observed from Figures 1012 that water migration is affected by porosity distribution. According to the proposed model, critical porosity is assigned with 0.15, as a consequence, those pores with higher initial porosity will have more time for accumulating water. What’s more, it can be observed from Figures 1315 that VOF distribution is also affected by porosity distribution as well. Water accumulates periodically at the first centimeters of the sample. Vast research has revealed that the formation of ice lens is affected by the type of soil. By using the proposed model, one will be able to obtain more detailed information about ice lens formation and distribution with a reasonable porosity distribution in the future.

Figure 10

VWC distribution after 12 hours freezing

Figure 11

VWC distribution after 24 hours freezing

Figure 12

VWC distribution after 50 hours freezing

Figure 13

VOF distribution after 12 hours freezing

Figure 14

VOF distribution after 24 hours freezing

Figure 15

VOF distribution after 50 hours freezing

Obtaining the proper temperature distribution is another important aspect in numerical simulation. As mentioned previously, water migration in porous media during freezing process is mainly driven by water potential difference caused by the temperature gradient. It can be observed from Figure 16 that experimental and numerical results at 12, 24 and 45 hours of freezing are in good agreement. However, the temperature obtained by numerical simulation in the frozen region at 12 hours freezing is lower than that in the experiment. Considering that in the current study, critical porosity is set to be 0.15, which leads to a possibility that the resistance of flow is relatively high at the beginning stage of simulation, hence there is not enough time for water to migrate, which introduces lower latent heat in such region.

Figure 16

Comparison between experimental and numerical results of temperature distribution after 12, 24 and 45 hours freezing, respectively

## 5 Conclusions

Lattice Boltzmann method has become a promising tool for the investigations of a wide range of complex flows, including multiphase flows, porous flows, thermal flows and reactive transport [20] by courtesy of its mesoscopic nature. However, this mesoscopic nature makes it require high computing capability while dealing with macroscopic simulations. The current research presented a lattice Boltzmann model for solving the water migration process in unsaturated porous media during the freezing process at macroscopic scale. The proposed model was validated with experimental results. The proposed model showed better agreement with experiment in the aspect of minimum water content around freezing front. The proposed model also showed relatively higher computational efficiency than the existing mesoscopic LBM model. It can be concluded from the study that:

1. Lattice Boltzmann method is suitable for simulating the complex heat and mass transfer process in porous media at macroscopic scale under proper dimensionless criterions.

2. Water migration in frozen fringe is affected by the change of permeability of porous media during the freezing process. In such region, the evolution of permeability behaves nonlinearly due to the existence of percolation threshold, which introduces additional resistance. Further investigation into the relationship between effective porosity and the influence factor of critical porosity of porous media at percolation threshold during freezing process is needed.

3. The cryosuction force which calculated by Clapeyron’s equation is over-estimated.What’s more, theoretical analysis shows that such overestimation should not be corrected by introducing increased fluid viscosity. Therefore, further investigation into the interaction between capillary behavior and porosity change in porous media during freezing process is needed.

## Nomenclature

c Lattice speed

cs Speed of sound

C Specific heat

d Characteristic length

e Discrete velocity

f Density distribution function of fluid lattice

fl Liquid fraction

F Force

g Density distribution function of thermal lattice

g Gravitational acceleration

g Sum of unknown distribution functions

H Enthalpy

k Number of iteration

K Permeability

l Characteristic length of model

L Latent heat

ns Solid density

P Pressure

t Time

T Temperature

u Velocity

u Common velocity of the fluid mixture

ΔV Difference of volume

x, y Axial coordinate

x Lattice site

y0 Axial coordinate of boundary

α Thermal diffusivity

χ Mass fraction

Δt Time step

Δx Lattice spacing

ε Porosity

ε0 Given porosity

θi Volume content of ice

λ Thermal conductivity

ν Kinematic viscosity

ρ Density

ρ0 Given density

τ Relaxation time

φd Core distribution probability

φg Growth probability

ψ Effect mass

Ψ Water potential

ω Weight factor

## Superscripts

B Body force

c Critical

Coh Cohesive force

e Effective

eq Equilibrium state

k Number of iteration

T Temperature

## Subscripts

A Air pressure

i Index of speed direction of lattice

l Liquid state

max Scale of total simulation

P Pressure

s Solid state

S Osmotic

Tot Total

T Temperature

Z Position

α Thermal conductivity

σ First phase

σ Second phase

ν Kinematic viscosity

## Acknowledgement

The authors gratefully acknowledge the supports from the Open Foundation for Preliminary Research of National Center for Materials Service Safety and the Major Program of Civil Aviation Administration of China (Grant No. MB20140066).

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Accepted: 2016-12-19

Published Online: 2017-06-16

Citation Information: Open Physics, Volume 15, Issue 1, Pages 379–393, ISSN (Online) 2391-5471,

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