Three-dimensional buoyant hydraulic fractures: constant release from a point source

Abstract

Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.

Ascent rates of 3-D fractures driven by a finite batch of buoyant fluid

“…These scalings have been confirmed by Möri & Lecampion (2022) using 3-D numerical simulations computed with open-source codes (Salimzadeh et al 2020;. Möri & Lecampion (2022) show that for a 3-D, constant injection-rate, ascending fracture, the shape of the tip-line depends on a dimensionless ratio that includes viscosity and fracture toughness. At early times, close to the injection point, viscous resistance limits tip-line propagation, resulting in a V-shaped tip line.…”

“…In this study we are interested in how the ascent speed decays with time. In the case of buoyant ascent fed by a constant flux, the ascent speed in the viscosity-dominated regime is v ∝ t −1/6 , whereas later, after transitioning to the toughness regime, the velocity is constant (Lister 1990b;Germanovich et al 2014;Möri & Lecampion 2022). These limiting regimes correspond with the asymptotic solutions for an aperture near the tip-line (Detournay 2004).…”

“…We expect that if the time scale of injection is much lower than the time scale of propagation, then once the crack has exceeded the critical radius of Davis et al (2020), it will begin to elongate in the direction of the stress gradient. In these cases, the upwards ascent speed should lie somewhere between the two limiting regimes: that predicted by (2.5) for a batch of finite volume and the ascent rate determined for a constant rate of volume addition, (2.4) (Möri & Lecampion 2022). Even when the injection rate is low, once the entire fluid batch has been injected into the fracture and as t increases, the general trend of the deceleration rate should follow a t −2/3 asymptote.…”

“…Three-dimensional (3-D) fractures, with finite extent in the y direction, pose additional challenges to analysis. Workers have begun to investigate the case of constant rate of volume injection in 3-D (Möri & Lecampion 2022), proving approximate analytical solutions for the height as a function of time, the width and the aperture of the fracture. In contrast, there has been no analysis of the ascent speed of a 3-D fracture containing a fixed volume of fluid.…”

“…This expression for A c is derived by requiring that the fracture walls close shut at the lower tip (K I (z = −c) = 0) and that the upper tip is critically stressed (K I (z = +c) = K c ), where c is the fracture's half-length and z the vertical distance from its centre (Pollard & Townsend 2018). Working independently, Dahm (2000), Salimzadeh et al (2020), Davis, Rivalta & Dahm (2020), Smittarello et al (2021) and Möri & Lecampion (2022) each extended the 2-D analytical model of Secor & Pollard (1975) to quantify the critical volume of fluid for ascent of a 3-D fracture. The solutions derived in these studies have the same scaling with parameters but differ by around 10 % because of different numerical constants.…”