Hydraulic fracturing is a subsurface stimulation technology that has been deployed at a massive scale in North America, and made it possible to produce hydrocarbons from low-permeability rocks, like oil shales and gas shales, which traditionally had been considered uneconomical. Despite this large-scale deployment, our understanding of the physics and controlling parameters in hydraulic fracturing is still very limited. One of the fundamental challenges is the ability to describe the interaction of rock tensile failure of an advancing fracture with the network of intersecting pre-existing fractures and cracks in the rock. Such interaction results in extension and coalition of initial cracks, and is wildly different from the well-understood process of the propagation by fluid injection of a smooth penny-shaped crack in a homogeneous porous medium.
From a traditional computational modeling standpoint, capturing fracture generation, growth and coalescence requires that each possible location of a fracture be added to the geometry and meshing process. As a result, this entails that the coupled flow-geomechanics equations be solved on an evolving unstructured mesh that dynamically adapts to each fracture zone—a computationally demanding, and poorly-scalable, approach.
Here, we present an approach to incorporate discrete fractures in the model, implicitly, through the Extended Finite Element Method (XFEM) . In this framework, the discontinuity is modeled by enriching the basis functions so that they can reproduce the discontinuous deformation along and across the surface of discontinuity. Use of XFEM for modeling hydraulic fracking was proposed a long time ago for two-dimensional problems , but its applicability for real-case three dimensional cases has not been addressed.
In our work, the geometry of each fracture is incorporated implicitly through a level-set technique . A multi-grid level-set discretization is proposed and implemented to accurately regenerate the complexity of the fracture’s geometry without significant computational effort. We then use this representation to regenerate the enrichment functions. Here, we propose a new implementation for XFEM that is fully vectorized and hence very efficient to perform three-dimensional studies. We validate our XFEM model first by comparing with simulations using contact elements. We then present representative simulations with multiple fractures to show the capability of the framework. We demonstrate that this framework is well suited for optimization and inverse analysis, which often requires running a large ensemble of simulations.
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 Réthoré, J., Borst, R. de & Abellan, M.-A. (2006). A two-scale approach for fluid flow in fractured porous media. International Journal for Numerical Methods in Engineering, 71(7):780–800.
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