Shocks and Bifurcations in Black-Oil Models
- Ismael Herrera (Instituto de Geofisica, UNAM)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 1996
- Document Type
- Journal Paper
- 51 - 58
- 1996. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements, 5.4 Enhanced Recovery, 4.6 Natural Gas
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Here, it is shown that the ‘traditional' approach to variable bubble-point problems is not consistent, because it does not incorporate important kinds of shocks that occur in black-oil models. This paper contains an exhaustive analysis of shocks that black-oil models may develop and it is shown that a considerable variety of them may indeed occur; a bifurcatiion mechanism is also exhibited. Except for the class of shocks described by Buckley-Leverett theory, all other kinds may be developed even if capillary forces are present. Modelers must be aware of these pathologies in order to handle them properly, This paper also supplies a basis for the developement of consistent black-oil models, applicable to variable bubble-point reservoirs, The analysis and results here presented, also contribute to a better understanding of limitations imposed by the simplifying assumptions of black-oil models.
When applying black-oil models to variable bubble-point problems, it is frequently assumed that the bubble-point may vary inside the undersaturated region [1-4]. However, such assumption is incorrect because it contradicts the basic postulates on which black-oil models are built. Indeed, such postulates do not include molecular diffusion, nor mechanical dispersion, and it has been shown that a consequence of such omission "is the bubble-point conservation law", according to which: when a gas-phase is not present, oil-particles conserve their gas content (dissolved gas:oil ratio), However, such restrictions are not respected in the ‘traditional' approach to modeling variable bubble-point reservoirs [1-4], as it is explained in Section 4 of this article.
To overcome this inconsistency, when bubble-point is variable, it is necessary to introduce shocks in which the bubble-point -i.e., the solution gas:oil ratio: Rs-, is discontinuous. However, this is not done; in the ‘traditional' approach [1-4], since the classical Buckely-Leverett theory was developed [5-9], modelers are prepared to deal with discontinuities of the saturation, but jumps of Rs are not included.
In addition, the bubble-point conservation law imposes very severe restrictions to the manners in which the dissolved gas:oil ratio of and oil-particle can vary, when a gas-phase is absent; physically, it means that when a gas-phase is not present, two oilparticles cannot exchange dissolved gas, even if they are very close. This property in turn, produces a propensity of black-oil models to develop shocks, which becomes apparent in problems with variable bubble-point. The classical Buckely-Leverett theory [5-9] describes an important class of shocks that occur in blackoil -or beta- models. However, this is not by any means the only kind of shocks that black-oil models may develop. On the contrary -as will be shown in the present paper-, a considerable variety of shocks may occur in such models. In addition, there are processes that produce bifurcations of the shocks.
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