The arrest and recession dynamics of a deflating radial hydraulic fracture in a permeable elastic medium

Abstract

In this paper we describe the post injection deflation dynamics of a radially symmetric hydraulic fracture in a permeable elastic medium. Depending on the parameters of the problem, the fracture may arrest almost immediately after injection has ceased, or continue to propagate despite the fluid loss to the porous medium. After arrest the fracture continues to deflate while the stress intensity factor decreases to zero, after which it recedes until the fracture finally collapses. In order to establish rigorous numerical solutions to explore this deflation dynamics, we make use of recent research (Peirce and Detournay, 2021) that derived vertex asymptotes for the arrest and recession phases of the deflation process as well as multiscale asymptotic solutions that connect these vertex asymptotes: during arrest, through the arrest–recession transition, and during recession. If only vertex asymptotes are used to capture the arrest and recession, the solution exhibits jump discontinuities through this arrest–recession transition point. We describe how the multiscale asymptotes can be used to obtain a smooth solution valid through the arrest–recession transition. Significantly, after the arrest–recession transition, the jump-induced transients decay and the solutions using vertex asymptotes converge to those that use multiscale asymptotes. Thus unless it is important to obtain a smooth solution through the arrest–recession transition, a practical, and more efficient, approach would be to use an algorithm based solely on vertex asymptotes. We also provide numerical confirmation of the emergence, beyond the arrest–recession transition time, of the dominant balance used in the asymptotic analysis to establish the linear recession asymptote wˆ∼rˆ. We present a novel scaling analysis to establish the characteristic power laws for the arrest time, arrest radius, arrest aperture, and the deflation time, in terms of two new dimensionless parameters, which make it possible to unify the power laws between the zero and finite toughness cases. These power laws show close agreement with those obtained by regression of the numerical results over a range of the dimensionless parameters. Numerical solutions are provided to illustrate the solution landscape in parameter space and the effect that each of the dimensionless parameters has on: the duration of the period of propagation after injection has ceased, the duration of the arrest period, and the time from the initiation of recession to collapse.