### Abstract

The simulation of sedimentary basins aims at reconstructing its historical evolution in order to
provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of
this simulation is the numerical solution of a complex system of time dependent, three-dimensional
partial differential equations (PDE) of mixed parabolic-hyperbolic type. A discretisation (Finite
Volumes + Implicit Euler) and linearisation (Newton) of this system leads to very ill-conditioned,
strongly non-symmetric and large systems of linear equations with three unknowns per mesh element,
i.e. pressure, geostatic load, and hydrocarbon saturation.
The preconditioning which we will present for these systems consists in three stages. First of all
the equations for pressure and saturation are locally decoupled on each element. This decoupling aims
not only at reducing the coupling, but also at concentrating in the "pressure block" the elliptic
part of the system which is then in the second stage preconditioned by efficient methods like AMG.
The third step finally consists in "recoupling" the equations
(e.g. Block Gauss-Seidel, combinative techniques,...).

In almost all our numerical tests on real test problems from case studies we observed a
considerable reduction of the CPU-time for the linear solver, up to a factor 4.3 with respect to
ILU(0) preconditioning (which is used at the moment in TEMIS3D). The performance
of the preconditioner shows no degradation with respect to the number of elements, the size of
the time step, high migration ratios, or strong heterogeneities and anisotropies in the porous media.