Linear elastic fracture mechanics (LEFM) combined with lubrication theory (linear hydraulic fracture mechanics-LHFM for short) have successfully predicted hydraulic fracture growth for planar geometry in model materials such as Poly(methyl methacrylate) (PMMA) and glass (

No abstract

Summary The rheology of rocks transitions from a cataclastic brittle behaviour to plastic flow with increasing pressure and temperature. This brittle-plastic transition is empirically observed to occur when the material strength becomes lower than the confining stress, which is termed Goetze’s criterion. Such a criterion works well for most silicates but is not universal for all materials. We aim to determine the microphysical controls and stress-strain behaviour of rocks in the brittle-plastic transition. We use a micro-mechanical approach due to Horii and Nemat-Nasser, and consider representative volume elements containing sliding wing-cracks and plastic zones. We find solutions for frictional slip, plastic deformation and crack opening at constant confining pressure, and obtain stress-strain evolution. We show that the brittle-plastic transition depends on the confining stress, fracture toughness and plastic yield stress but also critically on the friction coefficient on preexisting defects. Materials with low friction are expected to be more brittle, and experience transition to fully plastic flow at higher pressure than anticipated from Goetze’s criterion. The overall success of Goetze’s criterion for the brittle-plastic transition in rocks is likely arising from the low toughness, high strength, and medium friction coefficient character of most rock forming minerals.

One investigates the post-shut-in growth of a plane-strain hydraulic fracture in an impermeable medium while accounting for the possible presence of a fluid lag. After the stop of fluid injection, the fracture may present three distinct propagation patterns: an immediate arrest, a temporary arrest with delayed propagation, and a continuous fracture growth. These three patterns are all followed by a final fracture arrest yet the fracture behaviour prior to that results from the interplay between the dimensionless toughness $$\mathcal {K}_m$$ K m , the shut-in time $$t_s/t_{om}$$ t s / t om , and the propagation time $$t/t_s$$ t / t s . $$\mathcal {K}_m$$ K m characterizes the energy dissipation ratio between fracture surface creation and viscous fluid flow under constant rate injection. $$t_s$$ t s and $$t_{om}$$ t om represent respectively the timescale of shut-in and the coalescence of the fluid and fracture fronts. The immediate arrest occurs when the fracture toughness dominates the fracture growth at the stop of injection ($$\mathcal {K}_m \gtrapprox 4.3$$ K m ⪆ 4.3 ). It may also occur upon an early shut-in at low dimensionless toughness associated with an overshoot of fracture extension and a significant fluid lag. For intermediate values of $$\mathcal {K}_m$$ K m and $$t_s/t_{om}$$ t s / t om , the fracture may experience a temporary arrest followed by a restart of fracture propagation. The period of the temporary arrest becomes shorter with higher dimensionless toughness and later shut-in until it drops to zero. The fracture behaviour after shut-in then transitions from temporary arrest to continuous propagation. These propagation patterns result in different evolution of fracture dimensions which possibly explains the various emplacement scaling relations reported in magmatic dikes.