### 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 non-linear
partial differential equations (PDE) of mixed parabolic-hyperbolic type in 3D.
A discretisation and linearisation of this system leads to very large,
ill-conditioned, non-symmetric systems of linear
equations with three unknowns per mesh cell, i.e. pressure, geostatic
load, and oil saturation.
This article describes the parallel version of a preconditioner for these
systems, presented in its sequential form in [7]. It consists
of three steps: in the first step a local decoupling of the pressure and
saturation unknowns aims at concentrating in the ``pressure block'' the
elliptic part of the system which is then, in the second step,
preconditioned by AMG. The third step finally consists in recoupling
the equations. Each step is efficiently parallelised using
a partitioning of the domain into vertical layers along the $y$-axis and a
distributed memory model within the PETSc library (Argonne National
Laboratory, IL). The main new ingredient in the parallel version is a
parallel AMG preconditioner for the pressure block, for which we use the
BoomerAMG implementation in the Hypre library [4].

Numerical results on real case studies, exhibit (i) a
significant reduction of CPU times, up to a factor 5 with respect
to a block Jacobi preconditioner with an ILU(0) factorisation of each
block, (ii) robustness with respect to heterogeneities,
anisotropies and high migration ratios, and (iii) a speedup of up
to 4 on 8 processors.