Steadily translating parabolic dissolution fingers

TL;DR

This paper analyzes the steady shapes and growth velocities of dissolution fingers in porous rocks, revealing parabolic forms and flow characteristics using mathematical methods under the thin-front approximation.

Contribution

It introduces a mathematical framework for steady, parabolic dissolution fingers, detailing their shape, flow, and growth velocity based on the Péclet number regime.

Findings

Fingers have a parabolic shape characterized by steady translation.

Flow inside the fingers is uniform and pressure fields are derived.

Growth velocity depends inversely on curvature at low Péclet numbers and is constant at high Péclet numbers.

Abstract

Dissolution fingers (or wormholes) are formed during the dissolution of a porous rock as a result of nonlinear feedbacks between the flow, transport and chemical reactions at pore surfaces. We analyze the shapes and growth velocities of such fingers within the thin-front approximation, in which the reaction is assumed to take place instantaneously with the reactants fully consumed at the dissolution front. We concentrate on the case when the main flow is driven by the constant pressure gradient far from the finger, and the permeability contrast between the inside and the outside of the finger is finite. Using Ivantsov ansatz and conformal transformations we find the family of steadily translating fingers characterized by a parabolic shape. We derive the reactant concentration field and the pressure field inside and outside of the fingers and show that the flow within them is uniform. The advancement velocity of the finger is shown to be inversely proportional to its radius of curvature in the small P\'{e}clet number limit and constant for large P\'{e}clet numbers.