AMG Preconditioning for Sedimentary Basin Simulations in Temistm Calculator
F. Willien, I. Chetvchenko, R. Masson, P. Quandalle, L. Agelas, and S. Requena
Institut Français du Pétrole
Visco3d, the basin simulator of TemisTM suite, accounts for porous medium compaction, heat transfer, hydrocarbon formation and migration. These physical phenomena are modelised by partial differential equations [T3D] like the mass conservation of fluids (water and oil) coupled with Darcy’s law and a compaction law. These equations are discretised using a cell-centered Finite Volume method in space and an implicit Euler scheme in time. On this non linear system, we apply the Newton’s method to linearise and solve the equations.
In this paper, we will focus on solving efficiently ill conditioned linear systems coming from the Newton algorithm in the TemisTM calculator. In particular, the preconditioning step on parallel computer have to be very efficient in spite of becoming dramatically time consuming when the number of processors is high. We present here, the AMG+ILU(0) combinative preconditioning; this approach has been successfully applied to reservoir simulation [GMQR04]. This new type of preconditioner takes advantage of the elliptic part of the system with an Algebraic Multi Grid preconditioner and the hyperbolic part is treated with the ILU(0) factorisation. This kind of preconditioner was first introduced in Visco3d simulator by R.Scheichl and al [MQRS] based on the PETSc library [PETSc] and BoomerAMG implementation in the hypre library [HY]. Following this previous work, we extend the combinative preconditioner to multicomponents hydrocarbon flow. The preconditioning strategy is three fold. The first step is a decoupling stage to concentrate the elliptic part of the system on the pressure block to require a good global preconditioning due to high permeability jump. The second step consists in the AMG factorisation of the pressure block defined previously. And finally the recoupling of the pressure with the other unknowns is achieved with a Block Gauss Seidel algorithm combined to a zero fill-in incomplete LU factorisation for all unknowns.
To evaluate efficiency and robustness of the preconditioner, we will regard the number of Bicgstab iterations to reduce the residual by a factor of e = 10-6 and the cpu time elapsed on a linux cluster composed of 64 bits AMD opteron processors inter-connected with an InfiniBand network. The performance of ILU(0) depends strongly on the heterogeneities and anisotropies of the problem, this is not the case for combinative AMG+ILU(0) preconditioner which is very robust to capture small sand inclusion with an order of magnitude up to 10-12 . The average number of iterations of combinative is less than 10 and is very stable with regards to the number of processors. Contrary to the block ILU(0) preconditioner, the number of iterations is growing with the high heterogeneity , the number of cells and processors. In almost of case tests, we observe considerable reduction of cpu time up to a factor 3-4 with respect to ILU(0).
[GMQR04] J.M. Gratien, J.F.Magras, P.Quandalle, O.Ricois, Introducing a New Generation of Reservoir Simulation Software, ECMOR IX proceedings, Cannes, 2004.
[MQRS]R.Masson, P. Quandalle, S.Requena, R.Scheichl, Parallel Preconditioning for Sedimentary Basin Simulations, Lecture Notes in Computer Science 2907,Proceedings 4th Int. Conf. on Large Scale Scientific Computations, 2003, Sozopol, Bulgaria, Springer, 2004.
[PETSc] S.Balay,W.Gropp,L.C.McInnes, B.Smith, PETSc Users Manual, Technical Report ANL-95/11- Revision 2.1.0, Argonne, IL(2001).
[HY] V.E.Henson, U.M. Yang, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Applied Numerical Mathematics 41, 2002, pp155-177
AAPG Search and Discover Article #90066©2007 AAPG Hedberg Conference, The Hague, The Netherlands