Deflation Accelerated Preconditioned Conjugate Gradient Method in Finite Element Methods in Oil Reservoirs

Oil reservoirs generally contain several layers with highly varying permeabilities. Hence modeling oil and water flows often involves solving partial differential equations with very large contrasts in the coefficients. Since the injection and production wells are very small compared to dimensions of the reservoir, the injection wells and production wells are modelled by the use of delta-functions appearing in the right hand side of the partial differential equation for the pressure. In the presentation we consider a reservoir that consists of several layers with extreme contrasts of the permeability at the interfaces between the adjacent layers. Further, the finite element mesh is refined in the vicinity of the production and injection wells.<br><br>The finite element discretization of the above equation gives a stiffness matrix with extremely varying coefficients and hence the spectrum consists of large eigenvalues and eigenvalues that are almost zero, which gives a very high condition number. Hence a very bad convergence behavior for an iterative solver such as the conjugate<br>gradient method results. A preconditioner, like ILU, removes almost all the small eigenvalues, however, some small eigenvalues due to the large ratio of the coefficients at the interfaces persist. These small eigenvalues are removed by deflation based on a set of vectors that approximate the span of the corresponding eigenvectors.<br><br>Herewith the speed of convergence is successfully enhanced and the computational cost are reduced significantly. By the use of a proper choice for the deflation vectors, we show that the speed of convergence of our method does not depend on either the value of the contrasts in the coefficients or on the number of layers with varying coefficients. Further, the method is scalable in a parallel computing environment.