Fingering patterns in hierarchical porous media

Porous media with hierarchical structures are commonly encountered in both natural and synthetic materials, e.g., fractured rock formations, porous electrodes, and fibrous materials, which generally consist of two or more distinguishable levels of pore structure with different characteristic lengths. The multiphase flow behaviors in hierarchical porous media have remained elusive. In this study, we investigate the influences of hierarchical structures in porous media on the dynamics of immiscible fingering during fluid-fluid displacement. Divided by the breakthrough, such a displacement process includes pre- and postbreakthrough stages during which the fingering evolution is dominated by viscous and capillary effects, respectively. Through conducting a series of numerical simulations, we found that the immiscible fingering can be suppressed due to the existence of secondary porous structures. To characterize the fingering dynamics in hierarchical porous media, a phase diagram, which describes the switch among the three fingering modes (the suppressing, crossover, and dendrite mode), is constructed by introducing a scaling parameter, i.e., the ratio of timescales considering the combined effect of characteristic pore sizes and wettability. The findings presented in this work provide a basis for further research on the application of hierarchical porous media for controlling immiscible fingerings.

Figure 1

Sketch of the numerical model of a two-level hierarchical porous medium. (a) Representative area of the numerical model with fluid displacement, in which the boundary conditions are set as velocity boundary at the left side, outflow boundary at the right side, and both bottom and top sides are set as periodic boundaries; (b) first-order porous structure with $R_1$ the radius of first-order obstacles and $d_1={\beta }_1{R}_1$ the first-order throat sizes; (c) second-order porous structure with $d_2={\beta }_2{R}_1$ the second-order throat sizes.

Figure 2

Fingering patterns with three different modes: (a) suppressing mode $\left({\beta }_1=0.83,{\beta }_2=0.075,\theta ={15}^{\circ }\right)$ with breakthrough time of approximately 4 sec; (b) crossover mode $\left({\beta }_1=1.11,{\beta }_2=0.14,\theta ={45}^{\circ }\right)$ with breakthrough time of 2 sec; (c) dendrite mode $\left({\beta }_1=0.83,{\beta }_2=0.14,\theta ={120}^{\circ }\right)$ with breakthrough time of 2 sec.

Figure 3

Fingering indices as a function of injection time, including (a) displacement efficiency $F_e$, (b) effective interface length $F_l$, (c) fractal dimension $F_d$, and (d) second-order invasion ratio $F_i$, and the cases corresponding to the dendrite, crossover, and suppressing modes in Fig. 2, and the corresponding breakthrough time is marked by a short dash line.

Figure 4

Phase diagram of fingering patterns in hierarchical porous media at different contact angles measured within the invading phase $(\theta ={15}^{\circ }$, 30°, 45°, 60°, 90°, and 120°) and (a) second-order throat sizes $({\beta }_2=0.075$, 0.12, 0.14, 0.16, 0.18, and 0.20) with fixed first-order throat size of $0.83R_1$; (b) first-order throat sizes $({\beta }_1=0.54$, 0.83, and 1.11) with fixed second-order throat size of $0.14R_1$. Among those, “” denotes suppressing mode and “” dendrite mode, while the rest are crossover mode. All results are taken at $t=20\mathrm{s}$.

Figure 5

(a) Two-sphere model for calculating the capillary pressure as invading fluid moves through the throat where $d$ is the throat size, α is the filling angle, and θ is the contact angle; (b) Filling angle $\alpha$ vs normalized capillary pressure, i.e., the ratio of capillary pressure $P_c$ in Eq. (8) to $\gamma /\phantom{\gamma d_2}{d}_2$.

Figure 6

Cloud chart and contour lines of hierarchical number Hi as a function of contact angle and (a) second-order relative throat size ${\beta }_2$ and (b) first-order relative throat size ${\beta }_1$; and scatter “” denotes suppressing mode, “” crossover mode, and “” dendrite mode, which correspond to those in Fig. 2.

Figure 7

The correlation between hierarchical number Hi and (a) displacement efficiency $F_e$, (b) effective interface length $F_l$, (c) fractal dimension $F_d$, and (d) second-order invasion ratio $F_i$. All results are taken at $t=20\mathrm{s}$.