Notes on Propagation of 3D Buoyant Fluid-Driven Cracks

TL;DR

This paper develops a 3D model for buoyant, magma-driven fractures in the crust, analyzing how fluid dynamics and rock toughness influence fracture propagation and geometry, with implications for understanding magma emplacement.

Contribution

It introduces a novel analytical model for buoyant, finger-like fractures that accounts for the interplay between viscous fluid flow and rock toughness in three dimensions.

Findings

Fracture breadth remains nearly stationary during propagation.

The model provides a closed-form solution relating fluid pressure, toughness, and fracture geometry.

The analysis applies to low viscosity magma diking scenarios.

Abstract

Magma-driven fractures are the main mechanism for magma emplacement in the crust. A fundamental question is how the released fluid controls the propagation dynamics and fracture geometry (depth and breadth) in three dimensions. Analog experiments in gelatin have shown that fracture breadth remains nearly stationary when the process in the fracture head (where breadth is controlled) is dominated by solid toughness, whereas viscous fluid dissipation is dominant in the fracture tail. We model propagation of the resulting buoyant, finger-like fracture of stationary breadth with a slowly varying opening along the crack length. The elastic response to fluid loading in a horizontal cross-section is local and can be treated similarly to the classical Perkins-Kern-Nordgren (PKN) model of hydraulic fracturing. The propagation condition for a finger-like crack is based on balancing the global energy release rate due to a unit crack extension with the rock fracture toughness. It allows us to relate the net fluid pressure at the tip to the fracture breadth and rock toughness. Unlike laterally propagating PKN fracture, where breadth is known a priori, the final breadth of a finger-like vertically ascending fracture is a result of processes in the fracture head. Because the head is much more open than the tail, viscous pressure drop in the head can be neglected leading to a 3D analog of Weertman hydrostatic pulse. This requires relaxing the local elasticity assumption of the PKN model in the fracture head. As a result, we resolve the breadth, and then match the viscosity-dominated tail with the three dimensions, toughness-dominated head to obtain a complete closed-form solution. We then analyze the buoyancy-driven fracture propagation in conditions of either continuous injection or finite volume release for sets of parameters representative of low viscosity magma diking.