An enhanced pseudo-3D model for hydraulic fracturing accounting for viscous height growth, non-local elasticity, and lateral toughness Dontsov, E. V.; Peirce, Anthony
The goal of this paper is to develop an enhanced pseudo-3D (P3D) model for hydraulic fracturing (HF), whose predictions are more accurate compared to that of the original P3D model, but which requires significantly less computational resources than a fully planar HF simulator. We show that the lack of viscous resistance in the height growth and the local approximation in the computation of elastic interactions, which precludes the incorporation of lateral toughness, are the primary weaknesses of the original P3D model that considers symmetric stress barriers and no leak-off. To account for the viscous resistance, an apparent fracture toughness is introduced. The apparent toughness is calibrated using a one-dimensional HF model resulting in an approximate expression that captures all regimes of propagation. To incorporate non-local elastic interactions, the fracture opening in every vertical cross-section is approximated by a plane-strain solution, and then the 2D elasticity interaction integral is evaluated. To increase the computational efficiency, this 2D integral is further approximated by two one-dimensional integrals. The use of non-local elasticity allows us to include the asymptotic solution in the tip element, and, in particular, to include the effect of lateral fracture toughness. To further increase the accuracy of the P3D model, the flat fracture tip is replaced by its curved counterpart. This also permits us to capture radial behaviour at early times before the fracture has reached the stress barriers. To evaluate the accuracy of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar HF simulator, which is able to capture viscous, toughness, and intermediate propagation regimes. It is shown that the enhanced P3D model is able to approximate the propagation of hydraulic fractures accurately for various regimes of propagation, as well as for different fracture aspect ratios.
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