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Two-phase multicomponent filtration: instabilities, autowaves and retrograde phenomena

Published online by Cambridge University Press:  26 April 2006

V. S. Mitlin
V. S. Mitlin
Affiliation:
Institute of Earth Physics of the Academy of Sciences of the USSR, Bol. Gruzinskaya ul. 10, 123810 Moscow, USSR
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Abstract

The different modifications of the models of two-phase multicomponent filtration (Collins 1961; Nikolaevsky et al. 1968) enable one to study the dynamics of filtration flows, taking into account phase transitions. The equations of multicomponent filtration are quite complicated and only in a few individual cases do they allow for an exact solution. The most frequently used of these appears to be the solution of the stationary problem of the flow of a multicomponent mixture toward a well or a system of wells (Khristianovich 1941). In the present paper we show that at certain values of pressure, temperature and composition of the multicomponent mixture a stationary solution of the problem may not exist. The absence of a stationary solution is related to the possibility of a spatially homogeneous solution losing its stability under a perturbation (Mitlin 1986a, 1987b). We obtain an analytical criterion for instability. As an illustration, we present the results of the numerical solution of the planar linear problem of the evolution of a multicomponent system whose pressure and composition are perturbed with respect to their constant values, which are equal at both ends. We have done a numerical analysis of the plane-radial problem of the operation of a gas–condensate well with different mass fluxes, applying the conditions of a real deposit. There are several ranges of flux where the flow becomes pulsating. It is shown that the time within which the stationary solution sets in is a non-monotonic function of flux and on approaching the stability limit diverges in inverse proportion to the undercriticality of debit. We have analysed the connection between the observed instabilities and the thermodynamics of two-phase multicomponent mixtures. It is shown that the instabilities are associated with the system entering the region of retrograde condensation. We discuss the relation of retrograde phenomena to the effect of negative volume of heavy components and, as a consequence, to the negative compressibility of an individual volume of a two-phase mixture moving in a porous medium. It is shown that the observed autowave modes are relaxation oscillations in a distributed system. By using the method of perturbations in the interphase equilibrium time, we have analysed the loss of stability in a more general – non-equilibrium – model. We show that the instabilities are generated according to the Landau–Hopf scenario and calculate the period of auto-oscillations. The one-mode approximation of the theory leads to the Van der Pol equation. In conclusion we present an experimental confirmation of the theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 220 , November 1990 , pp. 369 - 395
Copyright
© 1990 Cambridge University Press

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