Stability of Displacement Fronts in Porous Media Growth of Large Elliptical Fingers
- Kwoon C. Mui (Carnegie-Mellon U.) | Clarence A. Miller (Carnegie-Mellon U.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- April 1985
- Document Type
- Journal Paper
- 255 - 267
- 1985. Society of Petroleum Engineers
- 5.2.1 Phase Behavior and PVT Measurements, 4.1.5 Processing Equipment, 5.8.5 Oil Sand, Oil Shale, Bitumen, 4.1.2 Separation and Treating, 5.4 Enhanced Recovery, 5.1.1 Exploration, Development, Structural Geology, 5.7.2 Recovery Factors
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A theoretical analysis is presented for growth of a single, large elliptical finger in a porous medium. The objective is to obtain an understanding of how large fingers develop once a displacement front has become unstable. The results provide information beyond that available from linear analysis, which has been used in most previous work and applies only to infinitesimal perturbations.
Separate analyses are given for finger growth during instability of (1) the front accompanying displacement of one immiscible fluid by another, (2) the combustion front in a dry, forward combustion process, (3) a condensation front, and (4) the combustion front in dry reverse combustion as employed in the linking step of underground coal gasification. In Cases 1 and 2, the fastest: growing finger is found so that growth rate and finger size can be estimated. In contrast, the linear anal- sea for these cases predict no fastest-growing disturbance. Thus, important additional information is provided by the elliptical finger model.
In Case 3, the condensation front, a fastest-growing disturbance is found as well. Results show that effects on stability of the mobility ratio and of the volume decrease accompanying condensation depend only on the shape of the finger (ratio of major and minor axes) while heat transfer effects also depend on the absolute size of the finger. Since the stabilizing effect of heat transfer is least for large fingers, such fingers are favored.
For Case 4, we found that the channel in underground coal gasification always advances at the maximum speed permitted by the reaction kinetics. Actual results confirm permitted by the reaction kinetics. Actual results confirm the basic validity of those obtained by Dunn and Krantz using linear analysis, but the elliptical finger model provides a clearer understanding of the physical phenomena provides a clearer understanding of the physical phenomena controlling finger size and speed.
Sweep efficiency is a vital consideration in every secondary or tertiary oil recovery process. Reservoir heterogeneity adversely affects all existing processes. But even in an ideal, homogeneous porous medium, sweep efficiency may be poor when there is an inherent instability of the front between displacing and displaced fluids. The role of stability analysis is to predict when such inherent instability occurs and, if it does, to provide some information on the size of the unstable perturbations or "fingers" that develop and their rates of growth.
Most previous theoretical work on displacement front stability has employed linear stability analysis, which is applicable only when perturbation amplitude is infinitesimal. With this method it is possible to predict whether any small perturbations in front shape become larger with time (i.e., whether the front is unstable). It is also possible to calculate the growth rates of various unstable perturbations. If a particular perturbation grows faster than all others, it ultimately dominates and its configuration and growth rate are presumed to characterize the overall instability. The size and growth rate of the fastest-growing disturbance obtained from linear theory can be used, strictly speaking, only when perturbation amplitude is infinitesinial. However, in the absence of additional information, they often are assumed to be approximately correct for large amplitudes as well. It is, of course, only when large fingers develop that sweep efficiency is appreciably affected. Thus, an analysis directly applicable to large fingers is of considerable interest.
Some previous efforts have been made to determine the growth rate of large fingers. Saffinan and Taylor analyzed growth of a single large finger for a two-dimensional (2D), simple displacement of a fluid with low mobility (k/; ) by a fluid with high mobility. They used conformal mapping techniques to calculate finger shape and the (constant) rate of finger advance. McLean and Saffinan recently extended the analysis to include an effective interfacial tension (IFT) at the finger surface, a concept previously employed by Chuoke et al in linear analysis. previously employed by Chuoke et al in linear analysis. Outmans and Nayfeh followed the alternate approach of including higher-order (nonlinear) terms in the equations describing development of an unstable perturbation of a flat front during a simple displacement perturbation of a flat front during a simple displacement process. Such studies show, for instance, that an initially process. Such studies show, for instance, that an initially symmetric perturbation such as a sine wave becomes asymmetric as the perturbation develops, a result that is consistent with experimental observations but not predicted by linear analysis. predicted by linear analysis. Experiments on growth of large fingers have been of two types. Some workers have studied displacement of one fluid by another in a Hele-Shaw cell (i.e., between two closely spaced parallel plates). The equation relating the average fluid velocity to the applied pressure gradient in such a cell has the same basic form as Darcy's law.
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