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 [24] 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:

$$\begin{array}{c}{T}^{n,k}={\displaystyle \sum _{i}{g}_{i}^{n,k}}\end{array}$$(22)

where *T*_{n,kn} ≡ *T*_{kn}(*t* = *n*). The local enthalpy is obtained by:

$$\begin{array}{}{En}^{n,kn}=c{T}^{n,kn}+{L}_{f}{f}_{l}^{n,kn-1}\end{array}$$(23)

with the liquid fraction *f*_{l} of the previous iteration. Finally, the enthalpy is used to linearly interpolate the melt fraction

$$\begin{array}{}{f}_{l}^{n,kn}=\left\{\begin{array}{ll}0& {En}^{n,kn}<{En}_{s}=c{T}_{m0},\\ \frac{{En}^{n,kn}-{En}_{s}}{{En}_{l}-{En}_{s}}& {En}_{s}\le {En}^{n,kn}\le {En}_{s}+{L}_{f},\\ 1& {En}^{n,kn}>{En}_{s}+{L}_{f};\end{array}\right.\end{array}$$(24)

where *En*_{s} and *En*_{l} are the enthalpies of the solid and liquid at the melting temperature *T*_{m}.

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 [25]. According to Saruya *et al*. [25], the difference between the warmest temperature *T*_{f} at which ice can first form and the normal bulk melting temperature *T*_{m0} ≈ 273.15 K can be obtained by *ΔT*_{f} = *T*_{m0} – *T*_{m} = 2*γ*_{sl}T_{m}/(*ρL*_{f}R_{p}), where, *R*_{p} is the characteristic radius of a pore throat, *ρL*_{f} ≈ 3.1 × 10^{8} J/m^{3} is the latent heat of fusion per unit volume and *γ*_{sl} ≈ 29 × 10^{–3} J/m^{2} is the ice-liquid surface energy, *ΔT*_{f} which describes the difference between the warmest temperature *T*_{f} at which ice can first form and the normal bulk melting temperature *T*_{m0} ≈ 273.15 K. So, the actual water melting temperature in porous media is:

$$\begin{array}{}{T}_{m}={T}_{m0}-\frac{2{\gamma}_{sl}{T}_{m0}}{\rho {L}_{f}{R}_{p}}\end{array}$$(25)

The characteristic radius of pore throat is *R*_{p} = *α*_{p}R, where, *R* is the particle radius, and *α*_{p} = 0.7 is a correlation coefficient. The average characteristic radius of a single lattice in the simulation can be calculated as:

$$\begin{array}{}{R}_{p}={\alpha}_{p}{\left(\frac{3{f}_{v}}{4\pi}\right)}^{\frac{1}{3}}\end{array}$$(26)

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 *T*_{m} can be found, and the thickness of this pre-melted film upon approaching the melting point *T*_{m} follows in a logarithmic growth law [26]:

$$\begin{array}{}L\left(T\right)=a\left(0\right)ln\frac{{T}_{m}-{T}_{0}}{{T}_{m}-T}\end{array}$$(27)

where *T*_{0} ≈ *T*_{m} - 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 *f*_{v}, and the relationship is:

Figure 5 Schematic of pre-melted film

$$\begin{array}{}L={\left[\frac{3}{4\pi}(1-{f}_{l}+{f}_{l}{f}_{v})\right]}^{\frac{1}{3}}\end{array}$$(28)

Combine Eq. (27) with (28), we get the relationship between the average obstacle volume fraction *f*_{v} and melting temperature of water pre-melted film:

$$\begin{array}{}{T}_{mi}={T}_{m}-\frac{{T}_{m}-{T}_{0}}{{e}^{\left(\frac{{\left[\frac{3}{4\pi}(1-{f}_{l}+{f}_{l}{f}_{v})\right]}^{\frac{1}{3}}}{a\left(0\right)}\right)}}\end{array}$$(29)

In the developed model, the bulk melting temperature *T*_{m} in equation (24) was replaced by the variable melting temperature (*T*_{mi}). Consequently, the formation of pre-melted is possible.

The collision is calculated by:

$$\begin{array}{}{g}_{i}^{n,k+1}\left(\mathit{X}+{e}_{i}\right)={g}_{i}\left(\mathit{X}\right)-\frac{1}{{\tau}_{h}}\left({g}_{i}\left(\mathit{X}\right)-{g}_{i}^{eq}\left(\mathit{X}\right)\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-{t}_{i}\frac{{L}_{f}}{c}\left({f}_{l}^{n,k}\left(\mathit{X}\right)-{f}_{l}^{n-1}\left(\mathit{X}\right)\right)\end{array}$$(30)

where *τ*_{h} is the relaxation time related to the thermal diffusion for the component *σ* as *k*_{σ} =
$\begin{array}{}{c}_{s}^{2}({\tau}_{h}-0.5).\end{array}$

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:

$$\begin{array}{}T=\sum _{i}{T}_{i}\end{array}$$(31)

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