We study the dynamic deflation of a hydraulic fracture subject to fluid withdrawal through a narrow conduit located at the centre of the fracture. Recent work revealed a self-similar dipole-flow regime, when the influence of material toughness is negligibly small. The focus of the current work is on the influence of material toughness, which leads to an additional self-similar regime of fracture deflation with fixed frontal locations in the toughness-dominated regime. The two limiting regimes can be distinguished by a dimensionless material toughness $\Pi _k$, defined based on a comparison with the influence of the viscous thin film flow within the fracture: $\Pi _k \to 0$ indicates the dipole-flow regime, while $\Pi _k \to \infty$ indicates the fixed-length regime. For intermediate $\Pi _k$, the fracture’s front continues to propagate during an initial period of deflation before it remains pinned at a fixed location thereafter. A regime diagram is then derived, with key scaling behaviours for the frontal dynamics, pressure and volume evolution summarised in a table for the self-similar stage. A comparison is also attempted between theoretical predictions and available experimental observations of viscous backflows from transparent solid gelatins.

We study the dynamics of fracture deflation following hydraulic fracturing of an infinite elastic solid, with fluid removal from a narrow conduit at the centre. This process involves coupled lubricating flow and elastic deformation, now subject to appropriate descriptions of fluid removal through the conduit towards the ambient, driven by elastic stresses and extraction/suction. When the influence of material toughness is negligible, the dynamics is found to be governed by two dimensionless parameters that describe the relative influence of elasticity-driven backflow ($\Pi _c$) and ambient-pressure-driven backflow ($\Pi _e$), respectively. We also found that the fracture’s thickness eventually approaches zero at the centre, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment $M$. The fracture’s front continues to elongate according to $x_f \propto t^{1/9}$, while the total fluid volume within the fracture decreases according to $V \propto t^{-1/9}$. The model and solutions might find use in practical problems to estimate the rate of backflow and effective permeability of a fractured reservoir once pressure is released.

We present a parallel preconditioning method for the iterative solution of thetime-harmonic elastic wave equation which makes use of higher-order spectral elements toreduce pollution error. In particular, the method leverages perfectly matched layerboundary conditions to efficiently approximate the Schur complement matrices of a blockLDLTfactorization. Both sequential and parallel versions of the algorithm are discussed andresults for large-scale problems from exploration geophysics are presented.

This paper first studies the transition matrix formulation for the analysis of responses of an elastic halfspace with a buried tunnel subjected to obliquely incident waves. The basis functions are constructed using the moving P-, SV-, and SH-wave source potentials and to represent the scattered and refracted wave fields in series forms. The associated T-matrix expression of elastic inclusion is derived using Betti's third identity. Second, this study proposes a technique for calculating the integral representation of basis functions in the wave-number domain using the method of steepest descent. Finally, typical numerical results obtained under incident plane waves are presented for verification.