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Novel Hybridizable Discontinous Galerkin method paves the way to tackle realistic 3D problems in seismic imaging

For several decades, the Oil and Gas industry has been committed to produce more and more hydrocarbons in response to the growing world demand for energy. Always seeking deeper and farther, exploration and development has become economically challenging as a result of increased geological and above ground complexity, stronger environmental constraints and pressure on costs.

The progress of multi component seismic data acquisition and the fast evolution of new technologies in the rock physics labs provide the opportunity to develop new families of algorithms which include more complex physics. 

Modern imaging techniques such as Full Wave Form Inversion (FWI) rely on intensive usage of full 3D physical Wave Equation engines. 

FWI can be formulated in different domains: time domain or frequency domain. In the case of frequency domain, the difficulties related to frequency domain inversion lie in the solution of huge linear systems, which cannot be achieved today when consider-ing realistic 3D elastic media, even with the progress of high-performance computing.

We consider Discontinuous Galerkin (DG) methods formulated on fully unstructured meshes, which are more convenient than finite difference methods on Cartesian grids to handle the topography of the subsurface. Moreover, DG methods are more adapted than Continuous Galerkin  (CGFEM)  methods to deal with hp-adaptivity. Nevertheless, the main drawback of classical DG methods is a larger number of degrees of freedom as compared to CGFEM methods (see Figs 1.a and 1.b).

In the framework of the HPC4E project and in collaboration with Inria teams Hiepacs, Magique 3D and Nachos, we are developing a new class of DG method: the Hybridizable DG (HDG) methods. The principle of HDG consists in introducing a Lagrange multiplier representing the trace of the numerical solution on each face of the mesh. Hence, it reduces the number of unknowns of the global linear systems (see Fig. 1.c) and the volume solution is recovered thanks to a local computation on each element.

Fig. 1.a : Degrees of freedom associated to CGFEM

Fig 1.b : Degrees of freedom associated to DG

Fig 1.c : Degrees of freedom associated to HDG

Fig. 1.a : Degrees of freedom associated to CGFEM Fig 1.b : Degrees of freedom associated to DG Fig 1.c : Degrees of freedom associated to HDG

 

First results are very encouraging and demonstrate the value of this method by reduc-ing by a factor up to 3 the memory consumption (see Fig. 2.a)  and by 4 the CPU time (see Fig. 2.b)  of the direct solver.

The HDG method for seismic imaging has been implemented as a proof of concept (see Figs. 2.a and 2.b) and is now being implemented into the FWI algorithm. Coupled with the progresses of linear solvers such as Mumps or Maphys and using advanced methods of High Performance Computing, we expect to be able to tackle realistic 3D problems in the next years.

Fig. 2.a : Memory consumption of classical DG and hybridizable DG Fig. 2.b : Speedup of hybridizable DG vs classical DG
Fig. 2.a : Memory consumption of classical DG and hybridizable DG Fig. 2.b : Speedup of hybridizable DG vs classical DG
Fig 3.a : Configuration of the anisotropic media Fig 3.b : Real part of the horizontal component of the displacement
Fig 3.a : Configuration of the anisotropic media Fig 3.b : Real part of the horizontal component of the displacement

 

Henri Calandra
Expert Numerical Methods and High Performance  Computing - TOTAL

 

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