Lattice model for spontaneous imbibition in porous media: The role of effective tension and universality class

Recently, anomalous scaling properties of front broadening during spontaneous imbibition of water in Vycor glass, a nanoporous medium, were reported: the mean height and the width of the propagating front increase with time $t$ both proportional to $t^{1/2}$. Here, we propose a simple lattice imbibition model and elucidate quantitatively how the correlation range of the hydrostatic pressure and the disorder strength of the pore radii affect the scaling properties of the imbibition front. We introduce an effective tension of liquid across neighboring pores, which depends on the aspect ratio of each pore, and show that it leads to a dynamical crossover: both the mean height and the roughness grow faster in the presence of tension in the intermediate-time regime but eventually saturate in the long-time regime. The universality class of the long-time behavior is discussed by examining the associated scaling exponents and their relation to directed percolation.

Figure 1

Lattice imbibition model. (a) Two-dimensional network of $\mathcal{N}=N\times L$ pores with $N=6$ and $L=4$. Pores are indexed $n=0,1,2,...,\mathcal{N}-1$ from the left bottom to the right top. Solid (red) and dashed (blue) lines indicate filled and empty pores, respectively. The coordinates $({\overline{x}}_n,{\overline{y}}_n)$ of the center of a pore $n$ are given by ${\overline{x}}_n=n-N\lfloor \frac{n}{N}\rfloor +\frac{1}{2}$ and ${\overline{y}}_n=\lfloor \frac{n}{N}\rfloor +\frac{1}{2}$ and with $\lfloor x\rfloor$ the integer not larger than $x$. The interface $\mathcal{I}$ consists of the pores $n=7,9,10,11,12,13,14$. (b) $p_{10}^{\left(\mathrm{L}\right)}$ is the Laplace pressure at the free end of the meniscus in pore $n=10$ and $p_{(5,1)}$ is the hydrostatic pressure at $(5,1)$. (c) Six independent vertical columns from (a). For the column at $x=\frac{5}{2}$, the pore $n=8$ is the highest filled pore, the upper end of which is at $y=2$, and the pressure in the column is given by $p_{(5/2,y)}^{\left(0\right)}=(y/2)p_8^{\left(\mathrm{L}\right)}$ for $0\le y<2$ and 0 for $y\ge 2$ according to Eq. (3). (d) The effective tension $T_{s,s^{\prime }}=I$ in case of $f_{s_{\ell }}=1,f_{s_r}=0$, and $f_{s_r^{\prime }}=1$.

Figure 2

Variation of the hydrostatic pressure with height $\left(y\right)$ obtained from the pore-network model  [11, 24]. The pressure linearly decreases with height $y$ in a specific time. The dashed line is a linear fit to the data. Insets: Pressure variation in terms of $x$ at two different fixed heights. The pressure does not vary considerably in the $x$ direction.

Figure 3

Time evolution of the mean height $H$ and the roughness $W$ for $N=32,R=N/2,I=0$, and different values of $\Delta$. (a) Plots of $H$ versus time $t$. Inset: The same plots in logarithmic scales showing that $H\sim t^{1/2}$. The bottom dashed line has slope $\frac{1}{2}$. (b) Plots of $W$ versus $t$. Inset: The same plots in logarithmic scales showing that $W\sim t^{1/2}$. The smaller $\Delta$ is, the later the regime showing the square-root scaling appears. The top dashed line has slope $\frac{1}{2}$. (c) Plots of the relative roughness $w=\frac{W}{H}$ versus $t$. Inset: The ratio $\frac{w}{w_{\mathrm{th}}\left(\Delta \right)}$ becomes independent of $\Delta$ in the long-time limit as predicted in Eq. (13).

Figure 4

Liquid configuration at time $t={10}^4$ in the lattice imbibition model for $N=32,\Delta =0.1,R=N/2$, and $I=0$.

Figure 5

Roughness $W$ as a function of time $t$ for $N=32,I=0,\Delta =0.1$, and different ranges $R$ of pressure correlation. The roughness increases with time as $W\sim t^{\beta }$ with $\beta$ increasing with $R$. The top dashed line and the bottom dotted line have slopes 0.45 and 0.3, respectively. Inset: The exponent $\beta$ as a function of $R/N$ for different $N$'s.

Figure 6

Time evolution of the liquid configuration with (a) $I=0$ and (b) $I=0.08$ for $N=32,\Delta =0.4,R=N/2$. The insets show magnified views at $t=2\times {10}^4$ (a) for $280\le y\le 380$ and (b) for $130\le y\le 230$. The effective tension makes the cluster of filled pores more compact than without tension and leads the front to stop in the long-time limit.

Figure 7

Time evolution of the mean height $H$ and the roughness $W$ under the effective tension for $N=64,R=N/2,\Delta =0.4$, and different values of $I$. (a) Plots of $H$ versus time $t$. $H$ grows faster with $I>0$ than with $I=0$ in the intermediate-time regime and saturates in the long-time limit. Inset: The estimated exponent $B$ in the relation $H\sim t^B$ increases from 0.5 to 0.7 before the saturation of $H$. (b) Plots of $W$ versus $t$. The roughness is not larger with $I>0$ than that with $I=0$ in the whole time regime. As the mean height does, the roughness increases faster with $I>0$ than with $I=0$ in the intermediate-time regime and saturates in the long-time limit. Compared with the mean height, the time-dependent behavior of $W$ is complicated. Inset: The estimated exponent $\beta$ in the relation $W\sim t^{\beta }$ finds its maximum value around 0.7.

Figure 8

Collapse of the scaled data $H/t_1^B=H{I}^{B{\eta }_1}$ as functions of $t/t_1=t{I}^{{\eta }_1}$ with ${\eta }_1=2.0$ and $B=0.53$ for $\Delta =0.4$ and different values of system size $N$ and the effective tension $I$ [see Eq. (16)]. Two lines with slopes $B=0.53$ and $B_{*}=0.70$, respectively, as in Eqs. (9) and (15) are also drawn.

Figure 9

Scaling behavior of the saturation height $H_{\mathrm{sat}}$ with respect to the system size $N$ and the effective tension strength $I$. (a) Collapse of the plots of $H_{\mathrm{sat}}{I}^{{\nu }_H}$ versus $N$ for different $I$'s and given $\Delta$. The upper collapsed data are for $\Delta =0.1$ and the lower for $\Delta =0.4$. (b) Collapse of $H_{\mathrm{sat}}{N}^{-{\alpha }_H}$ versus $I$ for different $N$'s and given $\Delta$. The upper collapsed data are for $\Delta =0.4$ and the lower for $\Delta =0.1$. The data collapse in both plots are found with ${\nu }_H=0.47$ and ${\alpha }_H=1.0$ for $\Delta =0.1$ and ${\nu }_H=0.36$ and ${\alpha }_H=0.4$ for $\Delta =0.4$.

Figure 10

Collapse of the scaled data $H{I}^{{\nu }_H}{N}^{-{\alpha }_H}$ as functions of the scaling variable $t/t_H=t{I}^{{\eta }_H}{N}^{-z_H}$ for $\Delta =0.4$ and different values of $N$ and $I$ [see Eq. (18)]. The values of the scaling exponents are ${\nu }_H=0.36,{\alpha }_H=0.4,{\eta }_H=1.0$, and $z_H=0.6$. The bottom dashed line has slope $B_{*}=0.70$.

Figure 11

Collapse of the scaled data of $W{I}^{\theta }{N}^{-\zeta }$ as functions of $t/t_2=t{I}^{{\eta }_2}$ for $t/t_2\gg 1$ with ${\eta }_2=1.2$, $\theta =0.43,$ and $\zeta =0.2$ for $\Delta =0.4$ and different values of system size $N$ and the effective tension $I$ [see Eq. (21)]. The top dashed line has slope ${\beta }_{*}=0.67$ as in Eq. (10).

Figure 12

Scaling behavior of the saturation roughness $W_{\mathrm{sat}}$ with respect to the system size $N$ and the effective tension strength $I$. (a) Collapse of the plots of $W_{\mathrm{sat}}{I}^{{\nu }_W}$ versus $N$ for different $I$'s and given $\Delta$. The upper collapsed data are for $\Delta =0.4$ and the lower for $\Delta =0.1$. (b) Collapse of $W_{\mathrm{sat}}{N}^{-{\alpha }_W}$ versus $I$ for different $N$'s and given $\Delta$. The upper collapsed data are for $\Delta =0.4$ and the lower for $\Delta =0.1$. The data collapse in both plots are found with ${\nu }_W=0.45$ and ${\alpha }_W=1.1$ for $\Delta =0.1$ and ${\nu }_W=0.33$ and ${\alpha }_W=0.5$ for $\Delta =0.4$.

Figure 13

Collapse of the scaled data $W{I}^{{\nu }_W}{N}^{-{\alpha }_W}$ as functions of the scaling variable $t/t_W=t{I}^{{\eta }_W}{N}^{-z_W}$ for $\Delta =0.4$ and different values of $N$ and $I$ [see Eq. (23)]. The scaling exponents are ${\nu }_W=0.33,{\alpha }_W=0.5,{\eta }_W=1.1$, and $z_W=0.45$. The top dashed line has slope ${\beta }_{*}=0.67$.