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.

Abstract

Obtaining soil parameters through laboratory tests and solving the governing equations that describe soil settlement can be time-consuming, making immediate on-site predictions of soil settlement challenging. In-situ testing provides a more efficient approach to obtain soil parameters than laboratory tests. Data from the Piezocone penetration test (CPTu) can be used for on-the-spot interpretation of soil mechanical parameters, which can then be incorporated into the governing equations for soil settlement calculation. Physics Informed Neural Networks (PINNs) algorithm uses automatic differentiation method to directly embed partial differential equations (PDEs) into a deep learning neural network and provides solution for these PDEs in a cost-effective manner compared to traditional numerical methods. In this paper, a framework integrating data from CPTu and PINNs to predict soil settlement is proposed and evaluated through comparison with numerical simulations from Finite Element Methods (FEMs). Results show that the framework gave a reasonably good agreement with the FEMs benchmark while substantially reduced the computation time. This method allows for immediate on-site prediction of soil settlement during site investigations, thus better guiding surveying and construction activities.