Magmatic Intrusions From a Hydraulic Fracture Modeling Perspective

The emplacement of magmatic intrusions in the earth’s crust has been investigated for decades. The driving mechanism is the density difference between the fluid and the rock. In the absence of heterogeneities, this difference creates a constant buoyancy force. This buoyancy governs the internal fluid pressure in excess of the background stress (magmatic overpressure) and creates a self-sustained hydraulic fracture (HF).

From early on, HF was investigated under a 2D plane-strain assumption, revealing a head-tail structure [1, 2, 3]. In this configuration, the propagating head has a constant volume, and viscous fluid flow in the tail dominates the ascent rate. Garagash and Germanovich (2022) [4] extended this approach to a late-time toughness-dominated 3D solution, confirming the head-tail structure and emphasizing a finger-like in-plane shape of such three-dimensional cracks.

Using PyFrac, a planar 3D solver for HF propagation, we compare 3D solutions to the 2D approximations. Considering a homogeneous medium and a continuous point source release of fluid, a family of solutions emerges, ranging from the solution of Garagash and Germanovich (2022) [4] to a zero-toughness limit [5]. These findings serve as a basis to derive the behaviour of buoyant hydraulic fractures produced by a finite volume release.

A recent body of work studied this problem, focusing on the limiting volume necessary for buoyant propagation as well as their ascent rate (see i. e. [6]). Using scaling analysis and numerical simulations, we clarify the entire parametric space. Similarly to the ongoing release case, a family of solutions exists as a function of two dimensionless parameters: A dimensionless viscosity (same as in the continuous release case) and a volume ratio (or, alternatively, a dimensionless buoyancy).

The knowledge of the entire parametric space of 3D finite volume buoyant cracks should help to interpret field emplacement data in a different light, design relevant experiments, study the effects of heterogeneities, and possibly build more computationally efficient, simplified models.

References:

[1]  D. A. Spence and D. L. Turcotte. Magma-driven propagation of cracks. J. Geophys. Res. Solid Earth, 90(B1):575–580, 1985.

[2]  J. R. Lister and R. C. Kerr. Fluid-mechanical models of crack propagation and their application to magma transport in dykes. J. Geophys. Res. Solid Earth, 96(B6):10049–10077, 1991.

[3]  S. M. Roper and J. R. Lister. Buoyancy-driven crack propagation from an over-pressured source. J. Fluid Mech., 536:79–98, 2005.

[4]  D. I. Garagash and L. N. Germanovich. Notes on propagation of 3d buoy- ant fluid-driven cracks. https://arxiv.org/abs/2208.14629arXiv:2208.14629, August 31 2022.

[5]  J. R. Lister. Buoyancy-driven fluid fracture: similarity solutions for the horizontal and vertical propagation of fluid-filled cracks. J. Fluid Mech., 217:213–239, 1990.

[6]  T. Davis, E. Rivalta, D. Smittarello, and R. F. Katz. Ascent rates of 3-D fractures driven by a finite batch of buoyant fluid. J. Fluid Mech., 954:A12, 2023.