## Section: New Results

### Improved numerical approaches for seismic wave equation at a fluid-solid interface in the oil industry

Participants : Ronan Madec, Dimitri Komatitsch, Julien Diaz, Pierre Thore.

We have studied in [29] how to implement plane wave sources with any incidence angle for the spectral-element time-domain method. We have also implemented absorbing boundary conditions following the ideas of [47] , which is crucial to ensure that no significant spurious waves propagate back into the main domain. We have validated the method and checked its accuracy for the numerical modeling of seismic wave propagation, in particular in order to compute the response of a free surface with topography under the incidence of a plane wave with different angles by comparing the results obtained to the Method of Fundamental Solutions (MFS), which is a new method to solve the wave equation in the frequency domain.

In the oil industry, many important oil reservoirs are located offshore and it is therefore of interest to be able to simulate seismic wave propagation in deep offshore geological media and it is crucial to reduce the cost of the calculations in the thick but homogeneous and therefore simple water layer. Explicit numerical methods used to model wave propagation must satisfy a stability condition called the Courant-Friedrichs-Lewy condition (CFL) that depends on the maximum velocity of the medium and on the ratio between the time step and the size of a grid cell. In the case of a deep offshore model, the velocity of P waves in the thick homogeneous water layer located upon the ocean bottom is generally slower than in the solid part of the model located underneath the ocean flow and therefore the local CFL is different within each medium. But to ensure numerical stability, one has to take a small global time step which is imposed by the solid part of the medium, and therefore waste some calculation time in the simpler fluid part of the model. An idea that can be used to overcome this is to implement substepping in time to reduce the computation time by taking a local time step and still honor the CFL condition by increasing the time step in the layer of slower velocity (the water layer). In [33] we therefore implemented in a spectral-element method an idea developed in [61] that ensures the conservation of energy along the interface.