Flow-Driven Branching in a Frangible Porous Medium

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Flow-Driven Branching in a Frangible Porous Medium

Nicholas J. Derr, David C. Fronk, Christoph A. Weber, Amala Mahadevan, Chris H. Rycroft, and L. Mahadevan

Phys. Rev. Lett. 125, 158002 – Published 6 October 2020

Abstract

Channel formation and branching is widely seen in physical systems where movement of fluid through a porous structure causes the spatiotemporal evolution of the medium. We provide a simple theoretical framework that embodies this feedback mechanism in a multiphase model for flow through a frangible porous medium with a dynamic permeability. Numerical simulations of the model show the emergence of branched networks whose topology is determined by the geometry of external flow forcing. This allows us to delineate the conditions under which splitting and/or coalescing branched network formation is favored, with potential implications for both understanding and controlling branching in soft frangible media.

Received 21 June 2020
Accepted 18 August 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.158002

Physics Subject Headings (PhySH)

Applications of soft matter Flows in porous media Porous media

Fluid Dynamics

Authors & Affiliations

Nicholas J. Derr ¹, David C. Fronk ², Christoph A. Weber³, Amala Mahadevan ⁴, Chris H. Rycroft^1,5, and L. Mahadevan ^1,2,6,*

¹John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
³Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany
⁴Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02450, USA
⁵Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁶Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*Corresponding author. Lmahadev@g.harvard.edu

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Vol. 125, Iss. 15 — 9 October 2020

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Images

Figure 1
Schematic of model fields, length scales, and erosion criteria. (a) Branched patterns in porous media can emerge on macroscopic lengths $L$ due to interactions at the pore size $l$ . In this simulated pattern, eroded regions of low solid fraction $ϕ (x, t)$ are blue. At a given point on the macroscale (black dot), $ϕ$ is the fraction of a pore-scale integration volume (inset) occupied by rigid grain microstructure (red circles). Fluid-mediated forces on grains induce stresses over a macroscopic region (shaded green circle) characterized by the communication length $ξ$ . The response fraction $φ$ is the spatial average of $ϕ$ throughout this region. (b) The erosion threshold function $ψ (φ) = c_{0} \tanh [ω (φ - φ_{*})] + c_{1} \in [0, 1]$ represents resistance to grain dislodgement at response fraction $φ$ .
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Figure 2
Homogeneous model of erosion in the presence of an externally controlled flux $Q (t)$ defined by (7). (a) Phase trajectories through forcing-response space with initial condition $(Φ, G^{2}) = (0.8, 1.4)$ are shown for varying $\dot{Q}$ . The erosion threshold $ψ (Φ)$ has $ω = 8$ , $φ_{*} = 0.7$ . Constant-flux trajectories reach the threshold quickly, stopping erosion in finite time. For $\dot{Q} \neq 0$ , sustained erosion takes place at long times along the slow manifold $G_{s}^{2} (Φ)$ , plotted here as a translucent, thick line. (b) The erosion rate along the manifold, $f_{s} = G_{s}^{2} - ψ$ , is plotted for three thresholds with $φ_{*} = 0.7$ and varying sharpness $ω$ , subject to $\dot{Q} = 0.1$ . Bounds on the rate $f_{0} > f_{s} > f_{1}$ are plotted as black lines.
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Figure 3
Erosion patterns as functions of boundary conditions obtained by solving (4) and (5). The solid fraction $ϕ (x, t)$ is shown at $t = 50$ . The integrated flux through the system is ramped from zero to a final magnitude $F = {\hat{q}}_{in} w_{in}$ over a duration $T = 10$ . Flow enters on the bottom wall and exits through the top wall (a)–(d) or via evaporation in the bulk (e). If the regions of inflow and outflow are of similar size as in (a) and (b), flow is concentrated in straight channels. If they are of different sizes as in (c)–(e), one inflow channel branches into many at the outflow. The boundary widths satisfy $w_{in} = 0.1$ or 10; $w_{out} = 0.1$ , 10 or 20. Simulation parameters: grid size $1000^{2}$ , $ϕ_{0} = 0.8$ , $φ_{*} = 0.7$ , $σ_{ϕ} = 0.02$ , $ζ = 0.1$ , $ξ = 0.1$ , $ω = 6.5$ , $F = 0.8$ . (See SM movie 1 [22] for visualizations of the dynamics of arborization that correspond to this figure.)
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Figure 4
Erosion patterns as functions of flux dynamics and threshold shape. The solid fraction $ϕ (x, t)$ is plotted with the Fig. 3 color scheme at $t = 50$ , subject to varied ramp time $T$ and sharpness $ω$ . Increasing $T$ or $ω$ promotes confinement of erosion to a footprint which is smaller for more slowly increasing fluxes and larger for sharper thresholds. Boundary fluxes: $w_{in} = 0.5$ , $w_{out} = 10$ , $\hat{s} = 0$ , $F = 0.5$ . Sharpness and ramp duration: $ω = {1, 8, 15}$ , $T = {0, 3, 10}$ . Other parameters: grid size $1024^{2}$ , $ϕ_{0} = 0.8$ , $φ_{*} = 0.7$ , $σ_{ϕ} = 0.02$ , $ζ = 0.08$ , $ξ = 0.05$ . (See SM movie 2 [22] for visualizations of the dynamics of arborization that correspond to this figure).
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