Section: Research Program

High-order numerical methods for modeling wave propagation in porous media: development and implementation

We aim at achieving the characterization of conducting porous media which are media favoring the conversion of seismic waves into electromagnetic waves.. This project is identified as a ”New scientific challenge” which is a set of research projects funded by the E2S project of UPPA. The shape and form of porous media can vary depending on the size of the pore and the structure of the solid skeleton. Porous media are found in the nature (sandstone, volcanic rocks, etc) or can be manufactured (concrete, polyurethane foam, etc) as depicted in [68]. Instead of modeling such media as strongly heterogeneous, homogenization is used to describe the material on a macroscopic scale. Biot’s theory describes the solid skeleton according to linear elasticity and adds to this the Navier-Stokes equation for a viscous fluid and Darcy’s law governing the motion of the fluid  [63], [61]. For simplified linear elasticity, there are one equation of motion and one constitutive law, with the unknowns being the displacement field in the solid and the solid stress. In poro-elasticity, the added unknowns are the fluid displacement relative to the solid and the fluid pressure. There are two equations of motion, coupled with two constitutive laws. By plane wave analysis, one obtains three types of waves: S wave, fast P wave and slow P wave (Biot’s wave). While the first two types are similar to those existing in elastic solid, the existence of a third wave with drastically smaller speed adds to the complications already encountered in elasticity. This is obviously even more challenging for conducting poroelastic media where the three poroelastic waves are coupled with an electric field. In this case, it is not realistic to use a unique scheme for all the waves. Standard finite element methods coupled with time schemes have indeed difficulties to deliver accurate solutions because there is a need of adapting the mesh size to the smallest wave velocity and the time discretization to the largest wave velocity. It is then tricky to numerically reproduce the Biot’s wave while approximating correctly the regular elastic waves P and S. Moreover, there is a challenging question about the boundary condition to be used for limiting the computational domain. We have launched a Ph.D project (Rose-Cloé Meyer) aiming at developing a new piece of software for the simulation of time-harmonic waves in conducting porous media. This project is developed in collaboration with Steve Pride from the Lawrence Berkeley National Laboratory who has elaborated the corresponding physical theory [83], [87], [73], [74]. Next, once a new numerical method is developed, it is validated by comparing the numerical solution to an analytical one. This is a key step to us for assessing the accuracy of our simulations. Nevertheless, analytical solutions are not available for realistic media such as poroelastic or viscoelastic media represented by heterogeneous parameters. Engineers still argue that simulations may be inaccurate and could lead to wrong conclusions. Fortunately, it is possible to produce experimentally quite complex configurations where multi-physics measurements are used to monitor the wave propagation. There is thus a possibility of moving further on the validation of the numerical methods by comparing simulations and experiments. What is very exciting is that experiments are used to validate numerical methods which have the objective of simulating new phenomena that are not possible to reproduce in a lab. We have launched two Ph.D thesis (Chengyi Shen and Victor Martins Gomes) in collaboration with Daniel Brito (LFCR-UPPA) on the comparison of simulations with experiments. This topic is connected to another project that we have with Total on the use of waves for characterizing carbonates.