Bubble train displacement

In the present paper, pore level descriptions of bubble and bubble train displacement in simple constricted geometries are used in developing mobility expressions for foam flow in porous media. Such expressions provide a basis for understanding many of the previous core flood observations and for evaluating the importance of foam texture and interfacial mobility. Inclusion of the effects of pore constrictions represents an extension of the earlier efforts of Hirasaki and Lawson (1). 

Extensive mobility control applications of foams are limited by inadequate knowledge of foam displacement in porous media, plus uncertainties in the control of foam injection. Because of the importance of in situ foam texture (bubble size, bubble size distribution, bubble train length, etc.), conventional fractional flow approaches where the phase mobilities are represented in terms of phase saturations are not sufficient. As yet, an adequate description of foam displacement mechanisms and behavior is lacking, as well as a basis for understanding the various, often contradictory, macroscopic core flood observations. 

Once in the medium, bubbles can be displaced through pore constrictions only by the concerted action of long, continuous bubble trains. As illustrated in Figure 1(a), bubble 1 will not move through constriction E until the bubble train behind it catches up and pushes it through the constriction. The latter is possible since the pressure drop across the long continuous train is much larger than across the individual bubble. 

Figure 1. Illustration of foam displacement (a) For bubble 1 to be displaced through constriction E, the bubble train behind it must first advance through constriction C, form a continuous train, and then push bubble 1 through constriction E. (b) The displacement of bubble 2 first requires the advancement of bubble 4 through E, bubble 5 through C, etc., to form a continuous train. Once this train pushes bubble 2 through F, the train momentarily breaks with 3 trapped at F and 5 trapped at E.

The above discussion is concerned with single bubble displacement. To obtain results analogous to Equations 6, 8, and 9 for bubble trains, it is necessary to account for changes in curvature at the bubble ends (in the Plateau border regions) due to the compression between adjacent bubbles. Referring to Figure 5(b), the contact radius Rc can be related to the capillary pressure Pc = Pi - Pi = P - P by... 

As a final point in this section, we should mention that as the bubble trains advance through the constricted channels, the capillary resistance will assume its maximum value (the mobilization pressure) only when the lamellae in the train assume their most unfavorable positions with respect to displacement. At other times, the capillary resistance will be below this maximum value with the result that the actual work required to maintain foam flow at a given rate will be below that which would be required if the mobilization pressure was operative at all times. This is easily understood if one pushes a bubble through a single constriction in a tube and notes that the pressure in the train builds up to the mobilization pressure as the drainage surface advances into the constriction and then rapidly falls as the front bubble experiences a Haines jump. To account for such effects in the present model, the Km term in Equation 63 would vary with time as the bubble train moved through the constricted channels. 

When gas alone is injected into a porous medium where foam had been flowing, the liquid saturation can be reduced below the irreducible liquid saturation. It would be expected that the liquid phase becomes discontinuous at this point and the further reduction of the liquid saturation occurs as a result of liquid flowing from the core as bubble train lamellae. This does not occur in conventional gas-liquid displacement, and the lower limit of the liquid saturation corresponds to the irreducible value. 

Although the current permeability model properly reflects many of the important features of foam displacement, the authors acknowledge its limitations in several respects. First, the open pore, constricted tube, network model is an oversimplification of true 3-D porous structures. Even though communication was allowed between adjacent pore channels, the dissipation associated with transverse motions was not considered. Further, the actual local displacement events are highly transient with the bubble trains moving in channels considerably more complex than those used here. Also, the foam texture has been taken as fixed the important effects of gas and liquid rates, displacement history, pore structure, and foam stability on in situ foam texture were not considered. Finally, the use of the permeability model for quantitative predictions would require the apriori specification of fc, the fraction of Da channels containing flowing foam, which at present is not possible. Obviously, such limitations and factors must be addressed in future studies if a more complete description of foam flow and displacement is to be realized. 

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