A Topological Derivative-Based Hydro-Mechanical Fracture Model

Abstract

The present work aims to introduce a hydraulic fracture model based on the concept of the topological derivative. This well stimulating technique also known as fracking was proposed by [1] in 1949 and has been widely used in oil and gas industry [2]. Overall, the fracking technique consists in significantly increase the production surface area of a porous medium from a pre-existing fault in such a way that the gas trapped in the rock formation might be collected in the surface. Furthermore, the fracking process has been the subject of many recent works due to its environmental impacts and economical aspects. In this context, we present an extension of the hydro-mechanical fracturing model proposed by [3] in pursuit of a scenario closer to reality by taking into account the initial stress state (in situ stress) and the fluid inertia during the propagation process. The present model results from adapting the Francfort-Marigo damage model [4] to the context of hydraulic fracture together with Biot's theory [5]. We consider a two dimensional idealization in which the fracturing process is activated by a non constant pressure field distributed over the whole domain. A shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation term is minimized with respect to a set of ball-shaped inclusions by using the topological derivative concept. Then the associated topological derivative is used to construct a topology optimization algorithm designed to simulate the nucleation and propagation process. Finally, some numerical examples depict the role of the in situ stress in the fracture propagation, specific crack path growth (allowing kinking and bifurcations) and also the applicability of the methodology here proposed.

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