[Paper Review] Notes on Propagation of 3D Buoyant Fluid-Driven Cracks

Dmitry Garagash, L. N. Germanovich|arXiv (Cornell University)|Aug 31, 2022

Hydraulic Fracturing and Reservoir Analysis5 citations

TL;DR

This paper develops a 3D mechanical model for buoyant, fluid-driven cracks (dikes) that self-consistently determine fracture breadth through a toughness-dominated hydrostatic head, while the viscous tail governs propagation dynamics. The key contribution is a closed-form solution linking dike breadth to rock toughness and buoyancy, enabling accurate prediction of 3D dike geometry and propagation from finite fluid volumes.

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.

Motivation & Objective

• To resolve the 3D geometry of buoyant, fluid-driven cracks where fracture breadth is not prescribed but determined by fracture mechanics.
• To model the transition from a hydrostatic, toughness-dominated head to a viscous, propagation-dominated tail in 3D dikes.
• To provide a closed-form solution for dike breadth and propagation dynamics that accounts for finite fluid volume injection.
• To validate the model against 3D laboratory experiments using gelatin dikes with constant injection or fixed volume release.
• To demonstrate that dike propagation slows but does not arrest under fixed-volume conditions unless the head absorbs all fluid.

Proposed method

• Formulates a 3D model for buoyant fluid-driven cracks with variable breadth and length, governed by elasticity, fluid pressure, and fracture toughness.
• Applies a PKN-like approximation for the viscous tail, assuming constant breadth and negligible elastic interaction.
• Develops a 3D analog of Weertman’s hydrostatic pulse for the fracture head, solving for the stationary breadth via global energy balance.
• Matches the toughness-dominated head solution with the viscosity-dominated tail solution to form a complete, self-consistent 3D dike solution.
• Uses asymptotic analysis to derive explicit solutions for the head (hydrostatic, high-opening) and tail (thin, viscous-flow-dominated) regions.
• Validates the model by comparing predicted dike breadth and opening profiles with experimental data from gelatin dikes.

Experimental results

Research questions

• RQ1 How does the fracture breadth of a 3D buoyant dike self-organize during propagation, and what controls its final value?
• RQ2 What is the role of the hydrostatic head in determining the terminal breadth of a finger-like dike, and how does it differ from 2D plane-strain models?
• RQ3 How does the viscous tail govern dike propagation dynamics when the fluid volume is finite?
• RQ4 To what extent can 2D PKN models approximate 3D dike behavior when informed by the 3D head solution?
• RQ5 Under what conditions does a dike arrest, and how does fluid volume partitioning between head and tail affect propagation?

Key findings

• The fracture breadth of a 3D buoyant dike is determined by the toughness-dominated hydrostatic head, with a closed-form solution: $ b_* = 0.396 (ar{K}/ ho g)^{2/3} $.
• The 3D hydrostatic head solution is a 3D analog of Weertman’s 2D pulse, with a normalized volume of 0.515, closely matching the 2D case.
• The viscous tail solution scales as $ ilde{ ho} ilde{t}^{1/3} $, with dike length growing slowly when fluid volume exceeds head capacity.
• For fixed-volume injection, dike propagation slows but does not arrest unless the entire fluid volume is consumed in the head, preventing tail formation.
• The 2D PKN model provides a good approximation to 3D behavior if the 3D-derived breadth $ b_* $ is used as input.
• Experimental gelatin dikes show good agreement with the theoretical breadth prediction, though boundary effects can reduce accuracy near tank walls.