Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a time-dependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion is not valid in this problem. After diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability. For the IC preconditioner the same behavior is observed. To annihilate the effect of the extreme eigenvalues a deflated CG method is used. The convergence rate improves considerably and the termination criterion becomes again reliable. Finally a cheap approximation of the eigenvectors is proposed.
Vuik et al. REF show that for a class of layered problems, two-sided scaling can reduce the number of extreme eigenvalues, making it equal to the number of layers of high permeability.
An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients
Journal of Computational Physics
Mathematics