Competition is the underlying mechanism controlling viscous fingering and wormhole growth

TL;DR

This paper models viscous fingering and wormhole growth as competition-driven phenomena, deriving empirical solutions and analytical models that explain pattern formation and match observed power-law decay in field data.

Contribution

It introduces a unified framework based on competition for water, providing empirical and analytical solutions for finger and wormhole growth patterns.

Findings

Patterns follow a power-law decay with depth, exponent -1.

Empirical solutions quantify flow rates in multi-finger systems.

Analytical models predict deterministic pattern formation.

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 re-active 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 Damk ohler values) follow a power law decay of the number of fingers/wormholes with depth with an exponent of -1 consistent with field observations