15.30 David Santillán (Universidad Politécnica de Madrid)
Phase-field model of fracture propagation in elasticity and poroelastic media.
The simulation of the propagation of fluid-driven fractures in elastic and poroelastic media is an emerging computational challenge. Moreover, it plays an important role in natural and engineering processes, such as
the transport of magma in the lithosphere, the geologic sequestration of carbon dioxide or the enhanced recovery of oil and gas.
Recently, phase-field models have been proposed as a promising framework to describe and simulate brittle fluid-driven fracturing. These approaches have a great flexibility to capture complex fracture patterns, such as branching, joining or kriging. However, the performance of these models has not been thoroughly assessed. In this meeting we describe the performance of phase-field models to simulate brittle fluid-driven fractures in elastic and poroelastic media. We present a new approach that incorporates a careful formulation for the fluid flow in the fracture, and validate the model by comparing the numerical results against analytical solutions of fracture propagation. The model results show very good agreement under the toughness-, toughness with leak-off- and viscous-dominated regimes. We also present some validation with laboratory experiments, as well as some engineering applications.
Recently, phase-field models have been proposed as a promising framework to describe and simulate brittle fluid-driven fracturing. These approaches have a great flexibility to capture complex fracture patterns, such as branching, joining or kriging. However, the performance of these models has not been thoroughly assessed. In this meeting we describe the performance of phase-field models to simulate brittle fluid-driven fractures in elastic and poroelastic media. We present a new approach that incorporates a careful formulation for the fluid flow in the fracture, and validate the model by comparing the numerical results against analytical solutions of fracture propagation. The model results show very good agreement under the toughness-, toughness with leak-off- and viscous-dominated regimes. We also present some validation with laboratory experiments, as well as some engineering applications.
16:30 Andrea Davini (Università di Roma I 'La Sapienza')
Convergence of the solutions of the discounted HJ equation
We consider a continuous coercive Hamiltonian on the cotangent bundle of the compact connected manifold \(M\) which is convex in the momentum. We prove that the viscosity solutions \(u_\lambda:M\to\mathbb{R}\) of the critical Hamilton-Jacobi equation with discount factor \(\lambda>0\) converge uniformly, as \(\lambda\) goes to 0, to a specific solution \(u_0:M\to\mathbb{R}\) of the limit equation. We characterize \(u_0\) in terms of Peierls barrier and projected Mather measures. As a corollary, we infer that the ergodic approximation, as introduced by Lions, Papanicolaou and Varadhan in 1987 in their seminal paper on periodic homogenization of Hamilton-Jacobi equations, selects a specific corrector in the limit. The talk is based on a joint work with A. Fathi, R. Iturriaga and M. Zavidovique.
MTM2017-85934-C3.