Competition Is the Underlying Mechanism Controlling Viscous Fingering and Wormhole Growth

Abstract

Viscous fingering and wormhole growth are complex nonlinear unstable phenomena. We view both as the result of competition for water in which the capacity of an instability to grow depends on its ability to carry water. We derive empirical solutions to quantify the finger/wormhole flow rate in single‐, two‐, and multiple‐finger systems. We use these solutions to show that fingering and wormhole patterns are a deterministic result of competition. For wormhole growth, controlled by dissolution, we solve reactive transport analytically within each wormhole to compute dissolution at the wormhole walls and tip. The generated patterns (both for viscous fingering and wormhole growth under moderate Damköhler number values) follow a power law decay of the number of fingers/wormholes with depth with an exponent of −1 consistent with field observations.

A Three‐Dimensional Study of Wormhole Formation in a Porous Medium: Wormhole Length Scaling and a Rankine Ovoid Model

“…The power‐law distribution of wormhole lengths shows its scale‐invariant property: scaling the wormhole length by a factor only result in a proportionally scaled power‐law distribution of the lengths. Similar power‐law scaling has been observed in the lengths of 2‐D patterns, such as viscous fingers and dissolution channels in a fracture (Budek et al., 2015; Cabeza et al., 2020; C. E. Cohen et al., 2008; Roy et al., 1999; Szymczak & Ladd, 2006, 2009), with their exponents m around 1. In our study, the wormholes, forming in a 3‐D porous medium, have lengths following a power‐law distribution with the exponent m around 1.4, which is slightly higher than the exponents reported for the 2‐D patterns (Budek & Szymczak, 2012; Cabeza et al., 2020; C. E. Cohen et al., 2008; Szymczak & Ladd, 2006).…”

“…The power-law distribution of wormhole lengths shows its scale-invariant property: scaling the wormhole length by a factor only result in a proportionally scaled powerlaw distribution of the lengths. Similar power-law scaling has been observed in the lengths of 2-D patterns, such as viscous fingers and dissolution channels in a fracture (Budek et al, 2015;Cabeza et al, 2020;C. E. Cohen et al, 2008;Roy et al, 1999;Szymczak & Ladd, 2006, 2009, with their exponents m around 1.…”

“…E. Cohen et al, 2008;Roy et al, 1999;Szymczak & Ladd, 2006, 2009, with their exponents m around 1. In our study, the wormholes, forming in a 3-D porous medium, have lengths following a power-law distribution with the exponent m around 1.4, which is slightly higher than the exponents reported for the 2-D patterns (Budek & Szymczak, 2012;Cabeza et al, 2020;C. E. Cohen et al, 2008;Szymczak & Ladd, 2006).…”

“…Among the wormholes, the longer ones, having a shorter distance to the outlet and thus causing a higher pressure gradient in the adjacent porous matrix, are likely to conduct more flow and develop faster. Collectively, we see wormholes develop by competing for flow (Cabeza et al, 2020;Hoefner & Fogler, 1988;Szymczak & Ladd, 2006).…”

Wormholing in Anisotropic Media: Pore‐Scale Effect on Large‐Scale Patterns

“…An analogy between viscous fingering and wormholing can be made by replacing the continuous porosity field by a sharp transition: the wormholes approximated as fully dissolved and the matrix assumed completely undissolved. This analogy assumes that: (a) the pressure field in the undissolved rock satisfies the Laplace equation; (b) the hydraulic resistance of the wormholes is negligible in comparison to the matrix; and (c) wormhole growth speeds are proportional to the pressure gradient ahead of their tips (Cabeza et al., 2020).…”