Multiphase flow and granular mechanics

In this perspective we provide a brief overview of the state of knowledge and recent progress in the area of multiphase flow through deformable granular media. We show, with many examples, that the interplay between viscous, capillary, and frictional forces at the pore scale determines the mode of fluid invasion. We pay particular attention to the central role of wettability on the morphology of granular-pack deformation and failure. Beyond their intrinsic interest as processes that give rise to spectacular pattern formation, these coupled phenomena in granular media can control continental-scale fluxes like methane venting from the seafloor and geohazards like earthquakes and landslides. We conclude this perspective by pointing to fundamental knowledge gaps and exciting avenues of research.

This article appears in the following collection:

Figure 1

Visual examples of the powerful interplay between multiphase fluids and the mechanics of granular media. (a) Particle self-assembly at the nanoscale (from Wang et al. [35]). (b) Sand castle in moist sand [192]. (c) Detail of craquelure [art credit: Mona Lisa (La Gioconda) by Leonardo da Vinci]. (d) Desiccation cracks on the soil surface (from Weinberger [43]). (e) Labyrinth patterns formed as a result of air invasion into a frictional suspension (from Sandnes et al. [40]). (f) Venting of methane bubbles from the ocean seafloor (from Skarke et al. [21]).

Figure 2

Lock-exchange flow of (a) a miscible and (b) an immiscible fluid pair in a glass-bead pack. (c) Fluid-fluid interface pinning in a microfluidic chip. The interface de-pins and moves away from the vertical position only where the local pressure difference between the two fluids is greater than the threshold capillary entry pressure. Adapted from Zhao et al. [44].

Figure 3

Lenormand et al. [57] studied drainage in porous media and found that the fluid-fluid front can advance through (a) invasion percolation, (b) stable displacement, or (c) viscous fingering, depending on Ca and $M$. The character of displacement is synthesized in the (d) phase diagram of Lenormand et al. [57]. Adapted from Lenormand [58].

Figure 4

Water displacing viscous silicone oil in wettability-controlled quasi-two-dimensional porous medium. Water was injected with different Ca under wettability ranging from strong drainage to strong imbibition. The displacement front was shown to advance through invasion percolation, cooperative filling, and corner flow at low Ca. At high Ca water advanced through viscous fingers, either leaving a film of oil or moving through films of water on the solid surfaces. Reprinted from Zhao et al. [91].

Figure 5

Pore-scale displacement events that govern quasistatic fluid-fluid flow are (a) burst (i.e., Haines jump), (b) touch, (c) overlap (i.e., coalescence), and (d) corner flow (i.e., coating of posts with wetting fluid). These pore-scale events naturally augment the dynamic pore-network model (“moving-capacitor” model). The moving-capacitor model utilizes the analogy between (e) immiscible fluid-fluid displacement in porous media and (f) electrical current, where local fluid-fluid interfaces are represented through capacitors and event capillary entry pressures inform the voltage drop corresponding to dielectric breakdown in a capacitor. Adapted from Primkulov et al. [72] and Primkulov et al. [94].

Figure 6

Numerical simulation of fluid-fluid displacement under different Ca and wettability using quasistatic [72] and dynamic moving-capacitor [73] models. The simulations cover the majority of the Ca-$M$ parameter space along with the dominant flow regimes demonstrated in experiments (Fig. 4). Adapted from Primkulov et al. [73].

Figure 7

Schematic diagram of the two modes of fluid–fluid displacement (gas-water) in a deformable granular medium. (a) The fluid-fluid interface before displacement. (b) Fluid invasion when the medium behaves rigidly. (c) Invasion by conduit opening; the exerted fluid pressure is sufficient to overcome confinement, cohesion and friction at grain contacts. Reprinted from Jain and Juanes [111].

Figure 8

3D PLIF reconstructions of the fluid invasion pattern (in white) superimposed on the conduit opening (in red) for experiments in which a fluid (silicon oil) is injected at the bottom of a pack of glass beads to displace a more viscous fluid (glycerol). Reprinted from Dalbe and Juanes [136].

Figure 9

Schematic of the experimental setup for frictional flows: (a) air is injected into a Hele-Shaw cell loaded with polydisperse glass beads that have settled in a water-glycerol solution; (b) the invading air-fluid interface accumulates a front of close-packed grains in the gap between the plates. (c) Phase diagram of frictional-flow morphologies in the space of the injection rate (increasing to the right) and the packing density (decreasing to the top). Reprinted from Sandnes et al. [41].

Figure 10

(a) Experimental setup of hydrocapillary fracturing experiments, where a thin bed of water-saturated glass beads is confined in a cylindrical acrylic cell, subject to a weight placed on a disk that rests on top of the beads. Air is injected into the center of the cell at a fixed flow rate. (b) Phase diagram of drainage in granular media, showing three invasion regimes: viscous fingering (VF), capillary fingering (CF), and fracturing (FR). The tendency to fracture is characterized by the fracturing number $N_{\text{f}}$: drainage is dominated by fracturing in systems with $N_{\text{f}}\gg 1$. At lower $N_{\text{f}}$ values, the type of fingering depends on the modified capillary number, ${\text{Ca}}^{*}$. Adapted from Holtzman et al. [112].

Figure 11

Comparison of fracture networks that develop for different wettability conditions (strong drainage, weak imbibition, and strong imbibition), under the same modified capillary number and confining stress. Shown are the contours of the evolving fracture patterns at different times during injection (see color map). Adapted from Trojer et al. [160].

Figure 12

Jamming transition analysis for the same injection rate and initial packing fraction (${\varphi }_0=0.77$) and four different wetting conditions ranging from weak imbibition to drainage: $\theta ={75}^{\circ }, {90}^{\circ }, {120}^{\circ }$, and ${140}^{\circ }$. (a) Mean particle stress $P$ as a function of packing density $\varphi$ in the compacting granular layer. Inset: Determination of the critical packing fraction at jamming, ${\varphi }_c$, for $\theta ={75}^{\circ }$. (b) Interface morphology at the jamming transition identified from panel (a) (black line), compared with that at breakthrough—when the invading fluid first reaches the outer boundary (red line). The comparison confirms that the jamming transition determines the onset of fracturing and that this transition occurs earlier in imbibition ($\theta ={75}^{\circ }$) than in drainage ($\theta ={140}^{\circ }$). Adapted from Meng et al. [161].

Figure 13

Visual phase diagram of the invading fluid morphology at breakthrough corresponding to different substrate wettabilities (contact angle $\theta$) and initial packing densities ${\varphi }_0$. Four distinct morphological regimes are identified: (I) cavity expansion and fracturing, (II) frictional fingers, (III) capillary invasion, and (IV) capillary compaction. Reprinted from Meng et al. [161].