Wrobel, M., Mishuris, G. (2015). Hydraulic fracture revisited: Particle velocity based simulation. International Journal of Engineering Science, 94, 23-58.We develop a new effective mathematical formulation and resulting universal computational algorithm capable of tackling various HF models in the framework of a unified approach. The scheme is not limited to any particular elasticity operator or crack propagation regime. Its basic assumptions are: (i) proper choice of independent and dependent variables (with the direct utilization of a new one ? the reduced particle velocity), (ii) tracing the fracture front by use of the Stefan condition (speed equation), which can be integrated in closed form and provides an explicit relation between the crack propagation speed and the coefficients in the asymptotic expansion of the crack opening, (iii) proper regularization techniques, (iv) improved temporal approximation, (v) modular algorithm architecture. The application of the new dependent variable, the reduced particle velocity, instead of the usual fluid flow rate, facilitates the computation of the crack propagation speed from the local relation based on the speed equation. This way, we avoid numerical evaluation of the undetermined limit of the product of fracture aperture and pressure gradient at the crack tip (or alternatively the limit resulting from ratio of the fluid flow rate and the crack opening), which always poses a considerable computational challenge. As a result, the position of the crack front is accurately determined from an explicit formula derived from the speed equation. This approach leads to a robust numerical scheme. Its performance is demonstrated using classical examples of 1D models for hydraulic fracturing: PKN and KGD models under various fracture propagation regimes. Solution accuracy is verified against dedicated analytical benchmarks and other solutions available in the literature. The scheme can be directly extended to more general 2D and 3D cases.publishersversionPeer reviewe
The problem of hydraulic fracture propagation is considered by using its recently suggested modified formulation in terms of the particle velocity, the opening in the proper degree, appropriate spatial coordinates and e-regularization. We show that the formulation may serve for significant increasing the efficiency of numerical tracing the fracture propagation. Its advantages are illustrated by re-visiting the Nordgren problem. It is shown that the modified formulation facilitates (i) possibility to have various stiffness of differential equations resulting after spatial discretization, (ii) obtaining highly accurate and stable numerical results with moderate computational effort, and (iii) sensitivity analysis. The exposition is extensively illustrated by numerical examples.?Peer reviewe
A novel hydraulic fracture (HF) formulation is introduced which accounts for the hydraulically induced shear stress at the crack faces. It utilizes a general form of the elasticity operator alongside a revised fracture propagation condition based on the critical value of the energy release rate. It is shown that the revised formulation describes the underlying physics of HF in a more accurate way and is in agreement with the asymptotic behaviour of the linear elastic fracture mechanics. A number of numerical simulations by means of the universal HF algorithm previously developed in Wrobel & Mishuris (2015) are performed in order to: i) compare the modified HF formulation with its classic counterpart and ii) investigate the peculiarities of the former. Computational advantages of the revised HF model are demonstrated. Asymptotic estimations of the main solution elements are provided for the cases of small and large toughness. The modified formulation opens new ways to analyse the physical phenomenon of HF and also improves the reliability and efficiency of its numerical simulations.
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