Capillary number definition

The Bond and capillary numbers definition might differ in different sources according to the problem under consideration, but the basic concept (relative importance of gravity vs. viscosity and capillarity vs. viscosity) is always the same. Sometimes, the ratio for capillary number is inverted. 

Ransohoff and Radke found that for C <10 lamella generation occurred only by the leave-behind mechanism. The lamellae moderately increased the resistance to flow, raising it by about a factor of five (40). (Here p is the viscosity of the nonwetting phase, U is the total superficial velocity, R is the bead radius, K is the absolute permeability, k is the relative permeability, L is length, and Y is the interSacial tension.) The use of this definition of the capillary number, instead of the usual C = jU/y, with J the viscosity of the wetting phase, was justified in the heoretical analysis ( ). 

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 ... 

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. 

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... 

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... 

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. 

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. 

By the definition of the capillary number, we therefore find that 9 oc Uq slow velocities imply small dynamic contact angles. This expression has been termed Tanner s law. 

Now, the surface tension actively assists in reducing the film resistance to mass transfer, while viscous stresses thicken the films. Substituting Eq. (6.7) and using the definition of the capillary number, we obtain... 

Based on the definition of the capillary number, the two-phase flow patterns in microchannels can be influenced by interfacial forces (fluid viscosity (rf), flow velocity (v), and geometric features (G). The interfacial forces can be modulated by adding surfactants, surface coating, and patterning techniques but also by electrical, thermal, or optical modulations of the interfacial... 

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 ... 

In the definition of the capillary number, the slug velocity is slightly higher than the superficial flow velocity. In analogy with the theoretical predictions of Charles [63] for the pipeline flow of slugs (capsules), the slug flow and average flow velocity can be related by... 

It is of some interest to read the definition of the capillary number from another perspective. Consider an interface where a capillary wave may develop. Let 5 be an estimate of its lenght. The actual value could eventually be d, the cell gap in a Rayleigh-Benard (Marangoni) experiment. Then a quantity, the (mechanical) time constant of such a disturbance upon the interface (the interface m.ay be likened to a stretched membrane) can be defined through the relation... 

The flow in fluid-fluid microstructured channels is characterized using dimensionless numbers. The most important dimensionless number for characterization of all types of flows is the Re number that relates inertial force to viscous force. Due to low flow velocities and characteristic dimension in the micrometer range, Re is often less than 1 meaning that viscous force is dominant over inertial force. The capillary number Ca is the ratio of viscous to interfacial forces. The range of Ca in a typical microchannel is lO " to 10 . Multiplying both numbers. Re and Ca, results in the Weber number We, which represents the ratio between inertial and interfacial forces. The importance of gravity vhth respect to interfacial forces is characterized by the Bond number Bo. The definitions of the dimensionless numbers are summarized in Table 2.2. 

There exist a number of membranes which are permeable to some ions and not to others, and these give rise to definite potential differences when the membrane is interposed between two solutions of the electrolyte. These membranes must be considered capillary in structure and selective permeability attributed to selective ionic adsorption, or in some cases to restriction imposed by the size of the capillaries. 

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