Fractures as Interface Conditions for Biot’s equations in the Frequency-space Domain

  • Favino, Marco (University of Lausanne)

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Biot’s equations describe the hydro-mechanical coupling of solid displacement and fluid pressure in porous media. In grophysical applications, in order to simulate seismic wave propagation in porous media, Biot’s equations are simulated in the frequency-space domain [3]. This allows to evaluate attenuation and velocity dispersion of seismic waves at a given frequency. However, fractured media are complicated to simulate. Fractures are heterogeneities character- ized by material properties which also differ by orders-of-magnitude from the ones of than the embedding background and are characterized by one dimension, the aperture, which is orders-of- magnitude smaller than the characteristic size of the background. This renders the generation of meshes a tedious and time-consuming task, which is difficult to automate. In order to alleviate the problems related to mesh generation, one common approach is to employ the so-called hybrid-dimensional models, in which fractures are considered objects of a lower dimension than the embedding background. There are two main approaches to derive these models: one is to consider the fractures as immersed objects [4], the other is to replace the fractures with interface conditions [1, 2]. While the immersed approach is actually well-suited for flow problems, since fractures are usually more permeable than the background, it does not allow to properly account for the mechanical deformations in the fractures, which are usually more compliant than the background. In this talk, we present a derivation of a hybrid-dimensional model for fractured media, where fractures are replaced with interface conditions. The resulting equations involves the standard fields usually employed in Biot’s equations, i.e., the solid displacement and the fluid pressure and do not require any additional variable. We discussed the analytical properties of the resulting differential problem and, for a standard finite element discretization, we study the accuracy of the method by means of numerical experiments.