This is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady-state and time-dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.
q SPE Member Cowight 1SS5, Sooiefyof PetroleumEnginwrs, Inc. mk paper w preparedfor presentationat the 13rh SPE Symposiumon Re-@ir smulation ho~in $011Antonio,~, lr.s.A-12-15 F-**. mwwwmti~hwansPEP--Mmmmw-ti Mwmation CQntdnedin an abatraci wbmtlted by the author(s).Content@ of the paper. * presented.hcw not been rdbwd by tho eooietyof Potrdeum En@nnm and am wbjeof to wrrocfion by the nutltcw($). The rnawisl, M praentod, dow MM nwcanarily refleof any poaifionof the Sooietyof PetroleumEnglnwm, haofScem,or members.Papainprwentod at SPE rndfw are wblmf to publkdion I'@W by~Mal~m~of IW -ti~m EWiH.~Mnmq@~man~d time*~tio.ll~m ayrtc41mcoPild. meat4MmotahoUfd oonwtoon@ow9õ f where and by whomthe paper iS presented.Write Librarian,SPE, P.O. Sox SS3S3S,Rbhudson, TX 75W3-3SW, U.S.A. Telex, 163246 SPEUT.ABSTRACT Black-oil models neglect difisive mechanisms such as molecular difision or mechanical dispersion; this omission produces a propensity to developing shocks. This paper aims to carry out an exhaustive identification of the kind of shocks that can occur when black-oil models are applied to problems in whtch the bubble-point varies and to establish the conditions under which they are generated. In addition to shocks of Buckley-Leverett type, two other classes of shocks and a bifurcation mechanism, are identified. Except for shocks of Buckley-Leverett type, all other shocks may occur in the presence of capillary forces. The paper contributes to clarifi several aspects of black-oil models and to understand pathologies that occur in their numerical implementation.
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