Cluster evolution in steady-state two-phase flow in porous media

We report numerical studies of the cluster development of two-phase flow in a steady-state environment of porous media. This is done by including biperiodic boundary conditions in a two-dimensional flow simulator. Initial transients of wetting and nonwetting phases that evolve before steady state has occurred, undergo a crossover where every initial pattern is broken up. For flow dominated by capillary effects with capillary numbers in order of ${10}^{-5}$, we find that around a critical saturation of nonwetting fluid the nonwetting clusters of size $s$ have a power-law distribution $n_s\sim s^{-\tau }$ with the exponent $\tau =1.92\pm 0.04$ for large clusters. This is a lower value than the result for ordinary percolation. We also present scaling relation and time evolution of the structure and global pressure.

Figure 1

Schematic figure that shows the interpretation the biperiodic boundary conditions mean to our simulations. In order to model a situation deep inside a natural reservoir in steady state, we must consider the system size as a small part of a greater system. Statistically, it is the same fluid configurations entering the element as are leaving it. This is illustrated in the upper figure. The way we consider our system is shown on the lower figure where the two-dimensional system is mapped onto a three-dimensional torus.

Figure 2

(Color online) Four different stages of a low $\mathit{Ca}$ flow configuration with viscosity match $M=1$ $\mathit{Ca}=3.2\times {10}^{-5}$ implies a capillary-dominant regime. The different snapshots of the flow patterns are taken at $t=5000\mathrm{s}$, $t=10000\mathrm{s}$, $t=15000\mathrm{s}$, and $t=25000\mathrm{s}$. The system size is $128\times 128$ with a $S_{\mathrm{nw}}=0.5$. The largest connected cluster is colored red in the online version of the article. The two first snapshots correspond to the first regime of the pressure in Fig. 3. In the third snapshot some of the initial configuration is still left and the pressure is building up according to Fig. 3. The last one is in the steady state regime where every initial structure has vanished.

Figure 3

Dynamic evolution of global pressure in a $128\times 128$ system at $\mathit{Ca}=3.2\times {10}^{-5}$ and $M=1$. In the initial phase of the flow evolution there is a small buildup in pressure, but when the wetting front breaks through the initial belt of nonwetting fluid the increase in pressure is violent. At the last, steady state, regime the fluctuations are approximately five times larger than in the first regime.

Figure 4

Distribution of wetting clusters near the nonwetting front at different stages of evolution. The earliest stage (▵) has a slope of 1.71. The cutoff here is due to lack of larger clusters that will evolve at later stages. As time increases (+), the slope is now 1.91, while at steady state (∎) the power-law distribution is completely overrun by the cutoff function.

Figure 5

(Color online) Four different snapshots of configurations with different $\mathit{Ca}$ taken at approximately the same time. In the upper left figure $\mathit{Ca}=3.2\times {10}^{-5}$. For the next three figures $\mathit{Ca}$ is much higher, $\mathit{Ca}=1,10$, and 100. The largest connected cluster is colored red in the online version of the article. The nonwetting front is now compact with high degree of fragmentation. The viscosity ratio of the two fluids is $M=1$, but the effective viscosity may be altered due to decreased permeability caused by fragmentation of the clusters.

Figure 6

Cumulative distribution $N(s^{\prime }>s)$ of nonwetting clusters at steady-state flow regime for $128\times 512$ system at $\mathit{Ca}=3.2\times {10}^{-5}$. The results are from a simulation with $S_{\mathrm{nw}}=0.5$ and the clear cutoff at high $s$ indicates that the critical saturation $S_c>0.5$.

Figure 7

Log-log plot of distribution of nonwetting clusters at $S_{\mathrm{nw}}=0.69$. The upper figure shows the distribution $N\left(s\right)$ for a system of size $1024\times 1024$. The distribution $N(s,L)$ exhibits a power law with a slope of $1.93\pm 0.05$. In the inset, a data collapse of different system sizes is shown with a size dependency of $\chi =1.8$. The lower figure shows the cumulative distribution $N(s^{\prime }>s,L)$ for different system sizes $L\times L$ going from $L=256$ to $L=1024$. The distributions have slopes around $0.91\pm 0.03$. This combined with the data in the upper figure indicates a value of the critical exponent $\tau =1.92\pm 0.04$.

Figure 8

First moment of the largest cluster plotted as a function of saturations $S_{\mathrm{nw}}$ for $L=512$. The three curves are hence for $\mathit{Ca}=3.2\times {10}^{-3}$, $\mathit{Ca}=3.2\times {10}^{-4}$, and $\mathit{Ca}=3.2\times {10}^{-5}$. The percolation thresholds are shifted towards higher values of $S_{\mathrm{nw}}$ as Ca increases. The transitions are located at approximately $S_c=0.675$, $S_c=0.7$, and $S_c=0.725$ and hence collapsed.

Figure 9

Log-log plot of the second moment for different $\mathit{Ca}$ for $L=512$. The data are collapsed using the values for $S_c$ obtained previously. There is only a narrow range where the power law is valid due to finite-size effects. For $S_{\mathrm{nw}}<S_c$ far away from $S_c$ we see a cutoff and for $\mid S_{\mathrm{nw}}-S_c\mid ⪡1∕L^{1∕\nu }$ the function approaches a constant. However, the obtained value $\tilde{\gamma }=1.95\pm 0.06$ is reasonable.

Figure 10

Strength of largest nonwetting cluster for $\mathit{Ca}\to \infty$. The data indicate a threshold at $S_{\mathrm{nw}}=0.91$ which is lower than unity. We can explain this by a lower cutoff of small clusters which will be of significance when operating with no surface tension. The data are from system size $L=512$ and taken from the mean of ten different runs.

Figure 11

Log-log plot of the ratio $M_3∕M_2$ for $L=512$. The solid line has a slope of $-2.14$ and gives $1∕\sigma =2.14\pm 0.08$ using Eq. (33). This gives $\sigma =0.47\pm 0.08$.