Successful capturing the mechanical behaviors of rock is the first and important step for hydraulic fracturing simulation. For heterogonous materials like rock, the numerical fracturing models must correctly represent the energy dissipation during the fracture propagation. However, finite element analysis using strain softening model becomes highly affected by the mesh size and alignment causing non-physical predictions of damage or softening process zone. Therefore, the energy dissipation becomes unstable since that it is highly affected by the different size of element used in the simulations, which is so called mesh size dependency. In this paper, nonlocal formulation damage mechanics is employed as the rock constitutive model in hydraulic fracturing simulation to capture the energy. Nonlocal formulation abandons the classical assumption that the strength degradation at certain point can only results from the strain-stress states at the point itself, but the states over the whole domain or at least a certain representative volume defined by a characteristic length. Also, the simulation involves a numerical algorithm that couples the fracture fluid flow with the fracture deformation in a moving boundary during the fracture propagation using lubrication theory.


For quasi-brittle materials, the cohesive strength and fracture energy are basic parameters to describe the behaviors of fracture propagation [Park et al., 2008]. It is observed that there is an intermediate region between uncracked and cracked parts defined as the fracture process zone. The material softening localized in the fracture process zone consumes energy. The relationship between the fracture energy and size of the localization zone has been verified by the experiments. Especially, due to the micro cracks and voids, relatively larger fracture process zone is found in quasi-brittle materials, and results in the difference between the strength measured in laboratory-size sample and the strength of actual structures. This phenomenon is associated with the size effect [Bazant and Planas, 1998]. In addition, in finite element implementation, the size of localization zone is related to element size. In other words, in traditional strain softening model, the fracture energy depends on the size of element. This effect is called mesh size dependency or mesh size. To capture the fracture energy, [Klein and Gao, 1998; Gao and Ji, 2003] introduced the fracture localization zone in virtual internal bond (VIB) model to simulate the fracture process zone of materials. The fracture localization zone is consistent with fracture band model developed by [Bazant and Cedolin, 1979]. As an intrinsic length of materials, the size of fracture localization zone is calculated in conjunction with the J integral.

The constitutive relation, in traditional strain softening model, is calibrated to match the experiment data for all element size. However, the actual softening curve and fracture energy should integrally consider both the elemental strain-stress curve and the size of fracture process zone (size of element in FEM implements). The quantitative analysis will be highly effected since that the model fails to capture energy dissipation during the fracture propagation. The present work will employ the nonlocal damage model to capture the fracture energy with an intrinsic characteristic length, in which the size of element does not have to be the size of fracture localization zone that is more convenient for curved fracture propagation.

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