Modeling multi-scale processes in hydraulic fracture propagation using the implicit level set algorithm

In this paper we describe an implicit level set algorithm (ILSA) (Peirce and Detournay, 2008) suitable for modeling multi-scale behavior in planar hydraulic fractures propagating in three dimensional elastic media. This multi-scale behavior is typically encountered when multiple physical processes compete to determine the location of the fracture free boundary. Instead of having to match the mesh size to the finest active length scale, or having to re-mesh as the dominant length scales change in space and time, the novel ILSA scheme is able to represent the required multi-scale behavior on a relatively coarse rectangular mesh. This is achieved by using the local front velocity to construct, for each point of a set of control points, a mapping that adaptively identifies the dominant length scale at which the appropriate multi-scale universal asymptotic solution needs to be sampled. Finer-scale behavior is captured in a weak sense by integrating the universal asymptotic solution for the fracture width over partially filled tip elements and using these integrals to set the average values of the widths in all tip elements. The ILSA solution shows good agreement with a multi-scale reference solution comprising a radial solution that transitions from viscosity to toughness dominated propagation regimes. The ILSA scheme is also used to model blade-like hydraulic fractures that break through stress barriers located symmetrically with respect to the injection point. For the zero toughness case, the ILSA solution shows close agreement to experimental results. The multi-scale ILSA scheme is also used to provide results when the material toughness KIc is non-zero. In this case different parts of the fracture-free-boundary can be propagating in different regimes. It is hoped that the multi-scale ILSA solutions presented here will form a set of reference results that can be used to benchmark simulators that use a propagation criterion based on only one dissipative process (either toughness or viscosity). The multi-scale ILSA solutions at larger times (for which plane strain conditions develop in vertical cross sections) are compared with and show close agreement to plane strain exact solutions for height-growth and the fracture width in vertical cross sections. This comparison provides some measure of the accuracy of the multi-scale ILSA scheme. The multi-scale ILSA solutions are also used to identify the regimes of applicability of pseudo 3D (P3D) approximate solutions. These ILSA solutions can also be used to design improved P3D models.