The near-tip behavior of a hydraulic fracture determines the local dynamics of the fracture front, and therefore affects the global fracture geometry. Several physical mechanisms may compete to determine the near-tip behavior. In this paper, we consider the simultaneous interplay of fracture toughness, fluid viscosity, and leak-off, which together cause the solution to vary at multiple scales in the near-tip region. In order to avoid having a mesh size that is able to resolve the finest length scale, an Implicit Level Set Algorithm (ILSA), which uses a suitable asymptotic solution for the tip element to locate a fracture front, is employed. The latter asymptotic solution comes from the analysis of a semi-infinite fracture propagating steadily under plane strain elastic conditions. Equipped with an accurate closed-form approximation for this asymptotic solution, which resolves the effects of the fracture toughness, fluid viscosity, and leak-off at all length scales, we analyze the problem of the simultaneous propagation of multiple parallel hydraulic fractures.


Hydraulic fracturing is a process, in which a pressurized fluid is injected into a rock formation to induce crack propagation. This technology is used primarily to stimulate oil and gas wells [1], but, in addition, is used for waste disposal [2], rock mining [3], as well as for CO2 sequestration and geothermal energy extraction [4]. To increase the efficiency of operation in petroleum applications, multiple hydraulic fractures from different perforations are often generated simultaneously from one well-bore. In this situation, outer fractures induce an additional compressive stresses on inner fractures and cause non-uniform fracture growth. This phenomenon is called stress shadowing and has been addressed in numerous studies [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] to name a few. It can significantly affect the fracture geometry and the associated production rate. For this reason, it is important to develop numerical models that are able to predict simultaneous growth of multiple hydraulic fractures and that can be used to design more efficient hydraulic fracture stimulations.

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