Capillary number calculation

We next determine the residual saturation, m , by first calculating the dimensionless capillary number ... 

The results of the two-phase, steady state, cocurrent flow experiments are summarized in Table II. The various fluid systems from Table I are listed along with selected injection rates and corresponding capillary numbers. Each velocity shown is the sum of the superficial velocities for the two fluids or, equivalently, the ratio of the superficial velocity of either fluid to its fractional flow. The utility of this quantity is discussed later. The viscosities given and used in calculating Ca are those of the wetting phases in the various systems. Most of the data are for 1 1 injection ratios. The observed flow mechanisms are given in the last column of Table II. 

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... 

The capillary number for flow in a typical oil reservoir undergoing water-flooding can easily be calculated. At a flow rate of about 0.26 m/day (3 X 10 m/s), an oil-water interfacial tension of 30 mN/m, and water viscosity... 

We now repeat the calculation using the Bravo and Fair correlations. The vapor-phase mass transfer coefficient is exactly the same as determined above. We need only compute the interfacial area density and the liquid-phase mass transfer coefficient. We begin by computing the capillary number... 

This section discusses the definitions of capillary number and how to calculate capillary numbers. 

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. 

This section discusses how to select the parameters to calculate capillary number. Initially, capillary number was proposed to correlate the residual saturation of the fluid (oil) displaced by another fluid (water) in the two-phase system. In surfactant-related flooding, there is multiphase flow (water, oil, and microemulsion), especially at the displacing front. If we use up/a to define the relationship between capillary number and residual oil saturation, which phase u and p and which o should be used then To the best of the author s knowledge, this issue has not been discussed in the literature. The following is what we propose. 

Similarly, the water phase also has two capillary numbers (Nc)wo for the oil phase displacing the water phase and (Nc)wm for the microemulsion phase displacing the water phase. When the definition Nc = up/o is nsed to calculate the capillary number, Uo and Po of the oil phase and 0 0 should be used for (Nc)wo and Um and Pm of the microemulsion phase and Owm should be used for (Nc)wm- When the definition Nc = k(Ap/L)/o is used, ko, Apo, and o o should be used for (Nc)wo and k , Apm, and Owm should be used for (Nc) - Two residual oil saturations are calculated S ro (residual water saturation in the water-oil conjugates) calculated using (Nc)wo, and Swm (residual water saturation in the water-microemulsion conjugates) calculated using (Nc) - The final residual water saturation should be saturation-weighted, as in... 

The reader may ask is (Nc)ow for the water phase displacing the oil phase the same as (Nc) o for the oil phase displacing the water phase, for example In principle, they are different from the preceding discussion. However, the formulas to calculate capillary number are empirical. In most practical cases. 

Nc)ow and (Nc)ow are not differentiated, simply calculating the single form of Nc. This is more obvious when the definition Nc = k(Ap/L)/a is used to calculate capillary number then we use the same absolute permeability (k), the pressure drop (Ap) along the core with the length L, and the interfacial tension... 

A similar discussion can be applied to (Nc)om and (Nc)mo, and (Nc)wm and (Nc)mw hi practice, a further approximation may be made. For instance, in measuring three-phase relative permeabilities, Delshad et al. (1987) used Nc = k(Ap/L)/o, where o was the average of the two IFTs Omo (IFT between the microemulsion and oil phases) and Omw (IFT between the microemulsion and water phases). In this case, only a single form of capillary number was calculated for the three-phase flow. 

According to current practice, we generally use average parameters to calculate the capillary number for the whole system, regardless of a two-phase or three-phase system. When the definition Nc = up/a is used to calculate the capillary number, the velocity (u) and viscosity (p) are those of the injection fluid. When the definition Nc = k(Ap/L)/o is used, the absolute permeability (k) and the total pressure drop (Ap) along the distance L are used. In either case, we use Omo for a type ll(-) system, Omw for a type ll(-i-) system, and an average IFT for a type III system—for example, the arithmetic average of 0 0 and Omw. as Delshad et al. (1987) did. 

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... 

To further reduce waterflood residual oil saturation, the capillary number must be higher than the preceding calculated value. In general, the capillary number must be higher than a critical capillary number, (Nc)c, for a residual phase to start to mobilize. Practically, this (Nc)c is much higher than the capillary number at normal waterflooding conditions. Another parameter is maximum desaturation capillary number, (Nc)max, above which the residual saturation would not be further reduced in practical conditions even if the capillary number is increased. Lake (1989) used the term total desaturation capillary number for (Nc)max. In practical conditions, total desaturation (i.e., zero residual saturation) may not occur due to some films or blobs trapped in pores. 

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 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. 

The time scale U deserves a comment. Although such time scale is useful for some calculations, however, in the case under consideration it is much shorter than the time scale, which governs the process under consideration. Indeed, the capillary number for the spreading... 

Calculation of capillary number from Equation 9.36 gives ... 

It is important to note that the definition of Ca is based on bubble velocity (m ). The bubble velocity in vertical capillaries was found to depend on the two-phase superficial velocity u and the capillary number (Ca ), which is calculated with the two-phase superficial velocity. On the basis of the experimental results obtained with different capillary diameters and liquids, Liu etcd. [16] proposed the following relationship ... 

Using Equation 7.4, the bubble velocity can be estimated. In Figure 7.7 the bubble velocity and the superficial gas velocity is plotted as the function of the two-phase superficial velocity u. It is evident that the bubble velocity is much higher than the calculated gas velocity and exceeds even the two-phase superficial velocity. Figure 7.7b shows the estimated thickness of the liquid wall film based on Equations 7.2 and 7.3. For low capillary number... 

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. 

Many are dimensionless and are often referred to collectively as either the "capillary number" or the "critical displacement ratio" by workers dealing with surface phenomena and oil recovery. Although the experimental data are still rather limited as far as capillary number results are concerned for different types of rocks, the consensus by the various workers is good. Different authors have examined the pore dimensions and geometry in both synthetic and real systems to calculate the critical value of the... 

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