### Abstract or Additional Information

In less than a decade hydraulic fractures (HF) have become the focus of considerable public attention as new technology to develop multiple HF from deep horizontal wells has enabled the extraction of hydrocarbons from impermeable shale deposits. This process, commonly referred to as ‘fracking’, has caused great public concern due to hydrocarbons from HF operations potentially polluting underground water. There is thus considerable interest in developing new analytic and computational models of propagating hydraulic fractures.

One of the most challenging problems encountered when modeling a propagating hydraulic fracture is locating the unknown fracture free boundary. In this talk I will discuss the asymptotic analysis of the governing equations near the free boundary, which characheterizes the possible singular behavior of the field variables near the free boundary. I then show how these asymptotic solutions can be used to develop an implicit level set algorithm (ILSA) that is able to resolve the free boundary problem on a relatively coarse mesh. I will then demonstrate recent work [1] in which I have used the ILSA methodlogy to model multi-scale behavior in planar hydraulic fractures propagating in three dimensional elastic media. I will also describe how we have recently used this methodology to develop fully coupled eXtended Finite Element (XFEM) models of propagating hydraulic fractures [2,3]. I will illustrate these models with comparisons to analytic solutions, laboratory experiments, and how they have been used to provide reference solutions to construct more accurate reduced models.

- A Peirce, Modeling multi-scale processes in hydraulic fracture propagation using the implicit level set algorithm, Comp. Meth. in Appl. Mech. & Eng., 283, p 881-908, 2015.
- E. Gordeliy & A. P. Peirce, Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems, Comp. Meth. in Appl. Mech. & Eng., 283, p474-502, 2015.
- E. Gordeliy & A. P. Peirce, Implicit level set schemes for modeling hydraulic fractures using the XFEM, Comp. Meth. in Appl. Mech. and Eng., 266, p125-143, 2013.

Further relevant papers can be found at: http://www.math.ubc.ca/~peirce