Inexact Adaptive Newton Methods

Abstract

Inexact Adaptive Newton Methods W.I. Bertiger; W.I. Bertiger Chevron Oil Field Research Co. Search for other works by this author on: This Site Google Scholar F.J. Kelsey F.J. Kelsey Chevron Oil Field Research Co. Search for other works by this author on: This Site Google Scholar Paper presented at the SPE Reservoir Simulation Symposium, Dallas, Texas, February 1985. Paper Number: SPE-13501-MS https://doi.org/10.2118/13501-MS Published: February 10 1985 Cite View This Citation Add to Citation Manager Share Icon Share Twitter LinkedIn Get Permissions Search Site Citation Bertiger, W.I., and F.J. Kelsey. "Inexact Adaptive Newton Methods." Paper presented at the SPE Reservoir Simulation Symposium, Dallas, Texas, February 1985. doi: https://doi.org/10.2118/13501-MS Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll ProceedingsSociety of Petroleum Engineers (SPE)SPE Reservoir Simulation Conference Search Advanced Search AbstractThe Inexact Adaptive Newton method (IAN) is a modification of the Adaptive Implicit Method1 (AIM) with improved Newton convergence. Both methods simplify the Jacobian at each time step by zeroing coefficients in regions where saturations are changing slowly. The methods differ in how the diagonal block terms are treated.On test problems with up to 3,000 cells, IAN consistently saves approximately 30% of the CPU time when compared to the fully implicit method. AIM shows similar savings on some problems, but takes as much CPU time as fully implicit on other test problems due to poor Newton convergence.IntroductionUsing the Inexact Adaptive Newton method (IAN), we have observed up to 40% savings in CPU time for black oil simulation over existing methods in Chevron's Black Oil Simulator. The method applies to problems ranging from coning studies to full field studies.Rather than using the Jacobian from a fully implicit method at each time step, IAN and AIM use an approximation to the Jacobian obtained by setting some of the entries to zero to solve the fully implicit nonlinear equations at reduced cost.The difference between IAN and the AIM method proposed by G. Thomas and D. Thurnau1 is that AIM not only zeros elements outside the diagonal block but also adjusts the diagonal block matrix to preserve a material balance property during each Newton iteration. The IAN matrix is closer to the true Jacobian in north but does not preserve material balance. Being closer to the Jacobian results in quicker convergence to the solution of the nonlinear problem at which point material balance is preserved by IAN.The IAN method was implemented in Chevron's Black Oil Simulator (CRS3D). In order to obtain efficiency, two new pieces of software were written.Vectorized Gaussian elimination routine for IAN and AIM matrices. This routine eliminates the unknowns labeled explicit.Memory management for iterative linear equation solvers.We have been successful in testing problems with up to 6,000 cells, although the iterative linear equation solver has had difficulty with IAN matrices in the 6,000 cell case.In order to further verify the performance of IAN versus AIM, a simplified version of the methods was coded in Chevron's fully implicit compositional simulator. Keywords: formulation, equation, permeability, maximum time step, flow in porous media, non-i inear residual, adaption criterion, explicit variable, ian matrix, implicit method Subjects: Reservoir Fluid Dynamics, Reservoir Simulation, Flow in porous media This content is only available via PDF. 1985. Society of Petroleum Engineers You can access this article if you purchase or spend a download.