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• W1988887213 abstract "Member SPE-AIME Abstract A new approach to quasi-Newton methods for nonlinear equations called quasi-Newton successive substitution (QNSS) is proposed. The use of QNSS does not require the computation nor the inversion of the Jacobian matrix at every iteration. Hence for large sparse systems which arise from the discretization of the reservoir flow equations, substantial savings can be achieved. The new method is applied to the solution of the pressure equation associated with a compositional model. An example shows the drastic improvement in convergence which can be achieved with QNSS compared to a previously published technique. The handling of gas solubility in the water phase using three-phase oil-gas-water flash calculations is also described. The QNSS method is very general and can be applied to any existing simulators. Introduction Broyden introduced a class of methods called quasi-Newton (QN) methods for solving nonlinear simultaneous equations where only approximate Jacobians are used. Although QN methods are quite popular in other fields, there is no published popular in other fields, there is no published report of their application to reservoir simulation. The main reason of this lack of application is due to the large sparse matrices which arise from the discretization of the reservoir flow equations, making the available forms of QN methods very cumbersome to use. Approximate Jacobians have been used in the past to solve the equations associated with past to solve the equations associated with compositional models, but these approaches do not have the desirable characteristics of QN methods. In these approaches, certain derivatives are neglected in order to form an approximate Jacobian with respect to pressure. After each iteration of the pressure variable, the global composition of each block is calculated, and flash calculations are performed to determine the phase compositions and performed to determine the phase compositions and densities. The storage and work per iteration in these methods are smaller than those of Newton's method, but the former techniques require more iterations. A new form of QN methods called QNSS is proposed in this paper which is very efficient to use, especially with large sparse matrices. The technique is applied to a compositional model with approximate pressure Jacobian. The compositional model handles pressure Jacobian. The compositional model handles three-phase oil-gas-water flash calculations and is an extension of a previous model which includes only two-phase oil-gas flash calculations. The results obtained are very encouraging and suggest the feasibility of using QNSS in reservoir simulators, the Jacobians of which are complicated and expensive to compute. QUASI-NEWTON METHODS Only the essential equations related to QN methods are reported in the following. The reader is invited to refer to the review papers of Dennis and More and Sargent for an in-depth treatment of the subject. Some Quasi-Newton Formulae Consider a system of n nonlinear equations (1) and let (2) (3) where the superscript (k) denotes the iteration Level." @default.
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• W1988887213 date "1983-11-15" @default.
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• W1988887213 title "A New Approach to Quasi-Newton Methods With Application to Compositional Modeling" @default.
• W1988887213 doi "https://doi.org/10.2118/12242-ms" @default.
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