Deflated ICCG applied to problems with extreme contrasts in the coefficients

C. Vuik , A. Segal, and J.A. Meijerink

Delft University of Technology
Faculty of Information Technology and Systems
Department of Applied Mathematical Analysis
P.O. Box 5031
2600 GA Delft
The Netherlands


Abstract

Knowledge of the fluid pressure history in the subsurface is important for an oil company to predict the presence of oil and natural gas in reservoirs and a key factor in safety and environmental aspects of drilling a well. A mathematical model for the prediction of fluid pressures in a geological time scale is based on conservation of mass and Darcy's law. The resulting time-dependent three-dimensional non-linear diffusion equation is linearized and integrated in time by the Euler backward method. For the space discretization the finite element method is applied. As a consequence in each time-step a linear system of equations has to be solved.

The matrix itself is sparse, but due to fill-in a direct method requires too much memory to fit in core. Therefore only iterative methods are acceptable candidates for the solution of the linear systems of equations. Since the coefficient matrix of this system is symmetric and positive definite, a preconditioned Conjugate Gradient method (ICCG) seems to be a suitable iterative method. Unfortunately the earth's crust consists of layers with large contrasts in permeability. Hence a large difference of the extreme eigenvalues is common in the system of equations to be solved. This leads to slow convergence of ICCG and conventional termination criteria are no longer reliable.

We prove that the number of small eigenvalues of the IC preconditioned matrix is equal to the number of high-permeability domains, which are not connected to a Dirichlet boundary. The bad effect of these eigenvalues on the convergence of ICCG can be annihilated by using the corresponding 'small' eigenvectors as projection vectors in Deflated ICCG. We describe an approximation of these 'small' eigenvectors and prove that the span of these vectors approximates the 'small' eigenspace of the IC preconditioned matrix. This implies that the convergence behavior of the resulting DICCG is independent of the size of the jump in the coefficients.

We investigate the effect of perturbations of the projection vectors. It appears that large perturbations in the low-permeability parts have only a limited influence on the convergence properties of DICCG. This implies that small components of the projection vectors in these parts can be neglected to save work and memory requirements. Small perturbations of the projection vectors in high-permeability parts have only a small effect on DICCG.

We observe that the resulting method is robust for elliptic problems with highly discontinuous coefficients, and a robust stopping criterion is available, which is not the case for the standard ICCG method. For high accuracies the DICCG method converges considerably faster than the ICCG method. However, for practical accuracies the gain is enormous. This means that in the context of non-linear problems or time-dependent problems DICCG is far superior above ICCG.