Capillary number equation

This expression provides a direct relationship between the minimum and maximum interfacial tensions and the cap angle for any given capillary number. Equation (7-294) is plotted in Fig. 7-20. If we are not careful to think about what we have done, the result (7-294) may seem rather strange as it does not seem to contain any indication of the drop size. This clearly cannot be correct. The fact is that the left-hand side of (7-294) still contains one unknown, namely the translational velocity of the drop U that appears in the capillary number, and this does depend on the drop size. Hence, to complete the calculation, we need to obtain one additional result, namely the drag on the bubble and, hence, by means of a balance with buoyancy, the translational velocity as a function of 6C. 

To calculate the pressure drop using the above equation, the film thickness is crucial. It can be estimated using Bretherton s or Aussillon and Quere s correlations depending on the capillary number (Equations 7.2 or 7.3) as demonstrated in Example 7.6. 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

Viscous and inertial forces are related to surface tension by the dimensionless Capillary and Weber numbers. Capillary number (Co), as shown in Equation 4.3, describes the relative importance ofviscosity and surface tension, where p represents the viscosity, u is the velocity and a is the surface tension. 

As defined by Radke and Ransohoff (Equation 7), the "snap-off" capillary number, C, contains the effective grain radius, R the permeability, K anS the relative permeability of the nonwetting phase, k ( ). In field applications, the values of all of these parameters are set by the reservoir. Also contained in C are the total superficial velocity, U, and the distance between injection and production wells, L. Within narrow limits, L can be changed by... 

Here J denotes the Leverett J function, and is the porosity.) Khatib and co-workers used Equation 9 in their demonstration of the dependence of coalescence and gas mobility on the limiting capillary number (41). 

Much is known about the surface tensions between surfactant/ water mixtures and air at 1 atm. However (except for thermodynamic equations), hardly anything is known about tensions between aqueous solutions saturated with CO2 at 10 MPa and their conjugate C02 rich phases. Although interfacial tension measurements at such pressures are very uncommon, values of the capillary number and their dependence on surfactant and hydrocarbon structures cannot be determined without such data. 

If Cq is known as a function of the capillary number and the surfactant properties, the functional form of the frequency and bubble volume can be approximated from the linear results. However, a model for Cq in constricted angular tubes does not exist. If one assumes that snap off occurs as soon as the thread becomes axisymmetric, then the base state thread radius is approximately the half width of the channel at the point snap off occurs. The experimental observations of Arriola and Ni along with the theoretical predictions of Ransohoff and Radke indicate that snap off takes place very near the constriction neck. Therefore, the radius of the bubbles formed should be slightly larger than the half width of the constriction neck. In fact, approximating Cq by the constriction half width, one observes from equations 14 and 15, that the snap off frequency and bubble volume are independent of the liquid flow rate once the critical liquid flow rate has been exceeded. Ni measured the dependence of snap off on the bubble velocity, the velocity of... 

The capillary pressure will have an effect only at high capillary numbers when the curvatures of the front and rear ends of the Taylor bubble are not symmetrical. At low velocity and in narrow channels, the frictional pressure drop is viscosity-dominant and can be calculated using the Hagen-Poisseuille equation... 

Capillary Number in Oil Mobilization. The capillary number is a dimensionless ratio of viscous to capillary forces it provides a measure of how strongly trapped residual oil is within a given porous medium (5). Various definitions have been used for capillary number, but the following equation is common ... 

In this equation, ( ) is the porosity in fraction, and u is the Darcy velocity of the displacing fluid. The velocity used by Abrams (1975) is v/[( )(Soi - Sm)]. He also modified the capillary number by multiplying the viscosity ratio (liw/lio)°-" ... 

Note that in this definition, the porosity term is included. If the velocity is used, the preceding equation becomes Eq. 7.82. It is expected that for a group of rocks with different porosities, if the porosities are included, the calculated capillary numbers should be closer to their average. However, the data that is shown in Table 7.7 do not consistently support this expectation. The ratio of the average to the standard deviation decreases for the data from Chatzis and Morrow (1984) if the porosity is included, but it increases for data from Taber et al. (1973). From these two data sets, it seems as though the capillary numbers that do and do not include porosity are equally good. 

In the case of discontinuous oil, L may be equal to Db, which is the diameter of a single oil blob. The capillary number required to mobilize the single oil blob is calculated using the preceding equation with L = Db. In the case of continuous oil whose size could be several times of Db, and L would be several times of Db, then the capillary number required to mobihze the continuous oil would be several times lower than that required to mobilize a single oil blob. In other words, the critical capillary number required to mobilize discontinuous oil is higher than that to mobilize continuous oil. This is another justification that chemical flood should be conducted early in the secondary recovery mode instead of in the tertiary recovery mode. 

Let us use the simple equation, Eq. 7.84, to calculate the capillary number in a typical waterflood case. Assume that injection velocity is 1 ft/day, which is 3.528 X 10 m/s, the water viscosity is 1 mPa s, and the interfacial tension is 30 mN/m. The corresponding capillary number is then... 

In a simulation model, we need to input a capillary desaturation curve model. Stegemeier (1977) presented a theoretical equation to calculate CDC based on the capillary number originally proposed by Brownell and Katz (1947). This equation requires several petrophysical quantities. Thus, it would probably be even more difficult to calculate a CDC using the Stegemeier equation than to obtain a CDC in the laboratory. In the laboratory, if several points of residual saturation versus capillary number are measured, we can use those measured points to fit a theoretical model. In UTCHEM, a form of Eq. 7.121 is used ... 

Nc)max mean at critical capillary number and maximum desaturation capillary number (Nc) is capillary number and Tp is the parameter used to fit the laboratory measurements. The definition of capillary number used in the preceding equation must be the same as that used in the simulation model. One example of CDC using Eq. 7.121 is shown by the curves in Figure 7.35, and some of the CDC parameters are presented in Table 7.9. The data points in Figures 7.35 and 7.36 are calculated using Eq. 7.124, to be discussed later. 

This equation shows that the fraction is a function of the relative permeability ratio, k /kio- It increases as the ratio is increased. We would be interested to know how the ratio changes as the IFT is decreased or the capillary number is increased. 

Various attempts were made to determine the individual mass transfer coefficients in Equation (30) in nonreactive systems. But, because the concentration profiles in the liquid surface film and in the slugs are strongly affected by fast chemical reactions, caution must be exercised in extending the results to reactive systems. The thickness of the liquid film was shown to be a function of the capillary number Ca. This relationship was found to be true for channels with circular cross-sections [114]... 

The above equation eorrectly approximates the volumes of the droplets at low values of the Capillary numbers for a range of geometries of the T-junctions, with the exact value of the eonstant a of order one depending on the aspect ratios of the widths and height of the mierochannels forming the T-junction device. 

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