Session H56: Flow Driven Pattern Formation in Wet Granular Medium II

Multiscale Dynamics of Colloidal Particle Transport in Porous Media

Colloidal particle transport in porous media is commonly encountered in various industrial activities. In particular, there is significant interest to improve oil recovery or aquifer remediation by subsurface injection of surface-active particles. The transport of particles in such systems is complex due to the interplay of particle adsorption, surface erosion, and pore-space advection. Here, we developed a model transparent porous medium that we can image in 3D using confocal microscopy. Through this medium, we observe, for the first time, the interchange between adsorption and erosion events. With these observations, along with a statistical nonlinear model and a continuum model, we demonstrate how different conditions dictate which transport mechanism is dominant. These findings ultimately show how the pore-scale transport dynamics can be controlled to achieve macroscopic aims in oil recovery or aquifer remediation.   

Laboratory rivers

Alluvial rivers transport sediment, and build their own bed out of it. The flow entrains superficial grains of sediment and deposit them downstream, thus deforming the channel that confines it. This fluid-structure coupling generates ripples, dunes, bars and meanders through various instabilities. It also selects a river's size and slope.

To entrain a sediment grain, the flow-induced shear stress must overcome its weight. This threshold, typical of granular materials, sets the characteristic size of alluvial rivers. Beyond this threshold, however, a river needs to balance the cross-stream fluxes of sediment to maintain its bed. Unfortunately, these fluxes are barely accessible to field measurements.

Creating small rivers in laboratory experiments is an old idea, but only now can we track thousands of individual grains, as they travel downstream, to reveal the statistics of sediment transport, and their consequences on a river's shape.

In a laminar flume, we find that the roughness of the bed causes the traveling particles to roam across its surface. This random walk induces a Fickian flux which tends to homogenize the transport of sediment. Meanwhile, the bed assumes a convex shape which brings the traveling grains near its center. As a result, the sediment flux distributes itself in this self-organized potential well according to Maxwell-Boltzman statistics.

The same mechanism allows laboratory rivers to adjust their cross-section and their width to the sediment discharge: they widen and shallow to accommodate a larger input. Beyond a critical sediment discharge, however, a river destabilizes into a braid of intertwined channels. We suggest that a new instability, driven by bedload diffusion, might explain this transition.

Finally, we look for the expression of these dynamics in large sedimentary structures deposited by rivers: alluvial fans.   

Fluid-Fluid Displacement Patterns in Microfluidic Analogues of Rough Fractures with Controlled Roughness

We report results of immiscible fluid-fluid displacement experiments on 3D-printed artificial fractures with controlled roughness. Our fracture analogue consists of a microfluidic cell with an irregular post pattern and a “free gap” between the top of the posts and the flat surface. We inject a low-viscosity nonwetting fluid to displace a more viscous wetting fluid and observe a transition from porous media flow to flow between parallel plates as a function of free-gap height. When the free gap is smaller than or comparable to the post height, the drainage pattern is similar to that in a 2D porous micromodel. In contrast, when the free gap is significantly larger than the post height, drainage in the gap dominates, and the displacement pattern is controlled by the capillary number. At high Ca, a film of the defending phase is left on the rough surface behind the front; this film then redistributes and forms discrete wetting clusters. At low Ca, the invading fluid occupies the full height of the free gap, with little wetting phase trapped even under unfavorable viscosity ratio. Our observations point to the mechanistic interplay of roughness and capillarity on trapping of the wetting phase in a rough fracture, a process that significantly affects its multiphase-flow properties.   

Erosion of unconsolidated beds by turbidity currents

Turbidity currents are gravity flows of fluids with suspended, denser sediment, which remains aloft due to turbulence generated by the current motion itself. To remain active, turbidity currents must have an ability to entrain material from their base to counteract the sedimentation of particles from the current to the base. A number of decades ago, Bagnold, Engelund, and Fredsøe proposed a physical picture for erosion as a function of the overall velocity of the turbidity current (bed stress). Recently, it has been argued that the high-velocity form of this law is critical in determining the overall mechanics of turbidity currents, particularly their predeliction to erode or deposit sediment in different locations. I re-examine the Bagnold-Engelund-Fredsøe picture, and determines the corresponding erosion law in a way that is consistent with turbidity current mechanics, and has a high-velocity plateau that determines the qualitative features of turbidity current deposition and erosion. This overall picture implies that the mixing of sediment from the boundary layer near the underlying bed with the overall current-suspended sediment will be the rate-limiting step in erosion.
  

Characteristic lengths of stream networks in porous terrain

The conditions for sediment transport of particles in streams has previously been found to give a relationship between the stream's slope and flux. Using this relationship, we establish a lower bound on the size of features in a stream network. Tributaries that violate this lower bound do not receive groundwater and dry up. This technique also gives bounds on a network's drainage density, which is its total length divided by the area it drains; this is the inverse of a characteristic length. These bounds obey scaling laws with respect to physical constants that describe the hydrological environment of the network, such as grain size, porosity, and water table depth, and we observe the effects of these scaling laws in a real-world network.   

Channelling and branching in frangible porous media

Fluid flowing through a fragile porous medium can cause channels to form due to erosion. We write down an effective theory for the process in terms of a multiphase flow model that couples a Darcy equation with a dynamical permeability to a dynamical equation for the rate of erosion. Solving the problem numerically shows that the appearance of tree-like branches as well as channels depending on the boundary conditions, and the rate of flow change at the boundary. Adding deposition and evaporation to the model leads to patterns similar to vascularization in tissues and suggests ways to engineer soft materials to self organize into biomimetic patterns.
  

Fingering Instability in Granular Suspension Injected into Thin Fractures

We discuss observation of a fingering instability when granular beads suspended in a liquid, with the same mass density, are injected into a fracture in the form of a Hele-Shaw cell filled with a similar wetting fluid. When the suspension is injected from a point into the cell, with the same density fluid, we find that the suspension spreads uniformly with a smooth circular front, with an expansion rate proportional to the injection rate. In contrast, when the density of the injected fluid and the interstitial fluid are even slightly different, we find that fingers form when the granular suspension is injected, and as the front spreads above a critical radius. We demonstrate that only the density difference is important, and the patterns form with both higher and lower density interstitial fluid. We will discuss the evolution of the fingers as a function of injection rate and the volume fraction of the beads, and contrast with the Saffman-Taylor instability observed when a less viscous fluid is injected into a more viscous fluid.   

Self-organized compaction fronts in cyclically-sheared sinking grains

Suspension that are sheared cyclically at small amplitude can enter an absorbing steady state where the particles return to their initial positions after each cycle. However, much less is known about the transient dynamics leading up to this reversible steady state. We perform simulations of sheared non-Brownian suspensions where the particles are also sinking under gravity at low speed. We observe a rising front between two regions of different particle area fractions that propagates at constant velocity. Although the suspension is relatively dilute, the front has a sigmoidal shape that is similar to the profile of a dynamic jamming front [1]. Surprisingly, the width of the front is independent of the effective diffusion constant for colliding particles, and depends only on the area fractions of the two regions. To understand our findings, we look for front-like solutions in a set of coupled nonlinear PDEs describing our simulations in the continuum limit. This approach should elucidate the potential for compaction front dynamics in related systems.
[1] Waitukaitis, Roth, Vitelli, and Jaeger, Euro. Phys. Lett. 102 (2013).   

Pattern formation in partially wet granular materials

From sand dunes to Faraday heaping, driven granular matter, i.e., large agglomeration of macroscopic particles, is rich pattern forming system. When a granular material is partially wet (e.g., wet sand on the beach), a different pattern forming scenario arises due to the cohesive particle-particle interactions: Kink-wave fronts were found to be the dominating pattern. Here, we focus on the formation of density-wave fronts in a driven wet granular layer undergoing intermittent gas-liquid-like transition. Fronts, which are curved into a spiral shape, propagate coherently along the circular rim of the container with leading edges. They are stable beyond a critical distance from the container center. Based on the measurement of the critical distance and the rotation frequency, we propose a model for the pattern formation by considering the competition between the time scale for the collapse of cohesive particles and that of the energy injection resisting this process. Possible connections of this pattern to galaxy formation as well as to traffic jam will be discussed.   

DEM modeling of coupled multiphase flow and granular mechanics: wettability control on fracture patterns

As one of the factors that influences multiphase flow in porous media, wettability has been studied for decades, yet many fundamental questions remain. In a recent experimental study, the impact of wettability on the fluid-fluid displacement pattern in a deformable granular pack was investigated. The experiments show the emergence of fracture of the granular pack under certain conditions of injection rate and confining stress. They also show that changes in wettability lead to striking differences in the fracture network morphology.
Here we use discrete element modeling (DEM) to provide insight into the mechanisms underpinning these experimental results. We first develop a 2D model with single-phase flow, and validate it against results from fluid-driven deformation of a confined monoloayer of hydrogel particles [MacMinn et al., PRX 2015]. To study two-phase flow, we couple a dynamic pore-scale network description of fluid-fluid displacement with a DEM model of the mechanical deformation of the skeleton of solid grains. This modeling approach allows us to capture the effects of varying wettability on the fracture pattern, thereby offering a grain-scale mechanistic understanding of the fracturing process.