Nonlinear Darcy flow dynamics during ganglia stranding and mobilization in heterogeneous porous domains

We study the steady-state displacement of nonwetting liquid ganglia during immiscible two-phase flows in realistic, stochastically reconstructed porous domains, focusing primarily on the nonlinear Darcian regime that arises when capillary to viscous (or gravity) forces become comparable at the pore scale. During this process, the ganglia undergo a continuous cycle of dynamic coalescence and fragmentation, resulting in two populations (a mobile and a stranded one) with distinct structural and rheological features, that continuously exchange mass between them under “stationary” flow conditions. We use a lattice Boltzmann model for the explicit solution of flow and interfacial dynamics at the pore scale driven by a constant body force field (i.e., gravity), and a periodic clustering algorithm for the identification and classification of mobile and stranded ganglia. Our simulation results reveal that an increase in the applied Bond number (Bo) leads to a gradual mobilization of the initially stranded ganglia population, resulting in a power-law scaling with an exponent being a strong function of the nonwetting-phase saturation. The linear Darcian scaling for the nonwetting phase is progressively restored at high Bo, while the wetting phase appears to maintain a linear Darcian scaling over the entire range of Bo values. We show that the mobilization process is characterized by a critical Bo, which is independent of saturation, above which new flow paths are created, in a similar fashion as a yield-stress fluid flows in a porous medium. Our results also offer a unique insight on the distinct structural characteristics of the mobile and stranded populations (e.g., ganglia size and length), as well as on their velocity and orientation with respect to their size.

Figure 1

A schematic of a ganglion breakup (fragmentation) event during immiscible two-phase flow within a disordered two-dimensional (2D) porous domain, as it is visualized at the pore scale. This event generates a stranded ganglion in the upstream region and a mobile one that flows further downstream driven by the surrounding w-phase. The ganglia (nw-phase) are shown in yellow (brighter color in gray scale), the w-phase in gray, and the solid sites in black.

Figure 2

(a) A typical reconstructed 2D porous domain used in this study. The black and white colors denote solid (impermeable to flow) and void (permeable) space, respectively. (b) Scaled velocity magnitude, $\left|u\right|/{\widehat{u}}_t$, during creeping single-phase flow within the domain. Lighter colors correspond to higher velocities, while darker blue colors correspond to solid sites or flow stagnation regions. The calculated intrinsic permeability of the domain is $k=9.9\delta x^2$. Reported lengths are in lattice space units $\left[\delta x\right]$.

Figure 3

Snapshots of ganglia distributions over the entire computational domain at steady-state flow conditions for $S_{nw}=0.3$ and three different values of $\mathrm{Bo}$: (a) $\mathrm{Bo}=0.01$, (b) $\mathrm{Bo}=0.1$, and (c) $\mathrm{Bo}=0.2$. The mobile ganglia are shown in yellow, the stranded ganglia are shown in red, and the solid sites in darker blue color. The complete videos for these simulations are provided in the Supplemental Material [49]. Reported lengths are in lattice space units, $\delta x$, while the simulation time is denoted above each snapshot in $\delta t$ units.

Figure 4

(a) Superficial (Darcy) two-phase velocity, ${\widehat{u}}_t$, scaling with applied $\mathrm{Bo}$ for various values of the nw-phase saturation, $S_{nw}$. The dot-dashed line corresponds to our reference single-phase flow scaling, namely, for $S_{nw}=0$. (b) Mobile nw-phase saturation, $S_{nw,mb}$, as a function of applied $\mathrm{Bo}$. Note that at high $\mathrm{Bo}$ values all curves reach asymptotically a plateau, $S_{nw,mb}=S_{nw}-S_{nw,r}$, which corresponds to a fixed residual nw-phase saturation value of $S_{nw,r}\approx 0.05$.

Figure 5

Superficial (Darcy) velocity scaling with $\mathrm{Bo}$ (a) for the w-phase and (b) for the nw-phase (ganglia). (c) The rescaled nw-phase velocity with respect to $\left|\text{Bo}-{\mathrm{Bo}}_c\right|$.

Figure 6

Mean presence maps $p(x,y)$ of ganglia saturation (nw-phase) over all recorded time steps at steady-state conditions for $S_{nw}=0.3$ and (a) $\text{Bo}=0.06$, (b) $\text{Bo}=0.08$, and (c) $\text{Bo}=0.1$. All lengths are in lattice space units, $\delta x$.

Figure 7

Mean correlation maps of nw-phase flux, $C(x,y)={\langle P(x,y,t)\times u_{nw}(x,y,t)\rangle }_t$, over all recorded time steps at steady-state conditions for $S_{nw}=0.3$ and (a) $\text{Bo}=0.06$, (b) $\text{Bo}=0.08$, and (c) $\text{Bo}=0.1$. All lengths are in lattice space units, $\delta x$.

Figure 8

Probability density functions of the (a) total, (b) mobile, and (c) stranded ganglia populations for $S_{nw}=0.30$ and different values of $\mathrm{Bo}$. Also shown is the initial ganglia size distribution at $t=0$, and the curves corresponding to the power-law scalings with an exponent of $-0.5$ (dashed line) and $-2$ (dotted line).

Figure 9

Probability density maps of ganglia velocity in the flow direction, $u_{x,g}/u_0$, and size, $s_g/{\lambda }_s^2$, over two orders of magnitude of $\mathrm{Bo}$ for $S_{nw}=0.4$. The dashed red line denotes the applied threshold criterion between mobile and stranded ganglia and corresponds to a relative permeability of 0.15.

Figure 10

Probability density maps of ganglia flow direction with respect to the applied body force, $\theta$, and size, $s_g/{\lambda }_s^2$, over two orders of magnitude of $\mathrm{Bo}$ for $S_{nw}=0.4$. The scattered points in the left side of these maps should be attributed to very small stranded ganglia with sizes determined by the specific structural properties of our reconstructed domains. These constitute (at least partly) the reported residual saturation, $S_{nw,r}$.

Figure 11

(a) Mean ganglion length ($l_{\max }$, crosses) and width ($l_{\min }$, circles) as a function of size, $s_g/{\lambda }_s^2$, for different $\mathrm{Bo}$ values. The dashed line represents a power law of $s_g^{0.5}$ and the dotted line represents $s_g^{0.72}$. (b) Product of mean ganglion length and width as a function of size. The dashed line represents $s_g^{1.22}$.

Figure 12

Probability density functions (PDFs) of the rescaled (a) longitudinal, $u_x$, and (b) transverse velocity, $u_y$, components, and (c) orientation, $\theta ={tan}^{-1}(u_y/u_x)$, of the w-phase velocity vector ${\vec{u}}_w$ for various values of the nw-phase saturation in the limit where all ganglia are immobile (i.e., $\mathrm{Bo}\le {\mathrm{Bo}}_c$).