Effect of capillary number

Effects of Capillary Number, Capillary Pressure, and the Porous Medium. Since the mechanisms of leave-behind, snap-off, lamella division and coalescence have been observed in several types of porous media, it may be supposed that they all play roles in the various combinations of oil-bearing rocks and types of dispersion-based mobility control (35,37,39-41). However, the relative importance of these mechanisms depends on the porous medium and other physico-chemical conditions. Hence, it is important to understand quantitatively how the various mechanisms depend on capillary number, capillary pressure, interfacial properties, and other parameters. 

Delshad, M., Delshad, M., Bhuyan, D., Pope, G.A., Lake, L.W., 1986. Effect of capillary number on the residual saturation of a three-phase micellar solution. Paper SPE14911 presented at the SPE/DOE fifth Symposium on Enhanced Oil Recovery, Tulsa, 20-23 April. 

Eulcher, R.A., Ertekin, T., Stahl, C.D., 1985. Effect of capillary number and its constituents on two-phase relative permeability curves. JPT (February), 249-260. 

The effects of capillary number are also studied. For this purpose, the computations are performed for the capillary numbers ranging between 0.00625 and 0.2. Figure 3 shows the mixing patterns at the exit of the channel. This figure indicates that the quality of mixing increases as the capillary number increases. 

Fig. 1. The effect of capillary number, N, on the microscopic oil displacement efficiency in porous media of various size distribution (Ref. 1).

Effect of capillary number on trapping of residual saturation at Bond numbers greater than 0.00667. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

In a later investigation, Kraynik and Hansen [62] demonstrated that the shear rate and liquid film viscosity greatly affect the rheological properties of foams. They studied the effect on foam properties and structure with variation of capillary number, Ca, which is the ratio of viscous to surface tension forces in the liquid films, and is given by -... 

The effect of inertia has been investigated by Edvinsson and Irandoust (1996) by using finite element analysis of Taylor flow in a cylindrical capillary. The effects of Ca, Re, and Fr numbers over a wide range were studied, and the simulations revealed that the film thickness was also dependent on Re and Fr numbers. By increasing the Re number the film thickness and the velocity difference between the two phases also increased, while the effect of Fr number (Froude number) was more obvious at higher Ca numbers, with results depending on the flow orientation (downward or upward flow). 

Effect of bond number and capillary number on trapping ... 

The amount of trapped nonwetting phase can be correlated with a linear combination of capillary number and Bond number which accounts for the combined effect of gravity and viscous forces on trapping. 

At a very low rate of penetration (curves 1 and 2), the concentration Cm does not differ appreciably from Cq due to the small Peclet number (Pe = v// D) and predominance of diffusion. At very high values of K, the concentration close to the meniscus tends to zero. This also takes place in the case of forced penetration under the action of an external pressure [19], when Pe > > 1. In this case, surfactant molecules cannot reach the meniscus and influence the contact angle. The effect of capillary radius is similar the smaller the values of r, the more pronounced is the influence of surfactant adsorption. An increase in Z)s2 values leads to some decrease in Cm due to the enhanced diffusion along the nonwetted capillary surface. 

Wenxiang, W., Demin, W., and Haifeng, J. 2007. Effect of the Visco-elasticity of Displacing Fluids on the Relationship of Capillary Number and Displacement Efficiency in Weak Dil-Wet Cores. Paper SPE 109228 presented at the Asia Pacific Pil and Gas Conference and Exhibition, Jakarta, 30 Cctober-1 November. DPI 10.2118/109228-MS. 

Interpretation of IC-AFM images is complicated by the fact that the tip-sample force is a nonlinear function of tip-sample separation. The tip-surface interactions in IC-AI have been modeled extensively and have been recently reviewed [109, 143]. Two important conclusions have come from the modeling. First, the nonlinear interaction of the dynamic tip with the surface can lead to two stable oscillation states one that follows a net attractive path and the other that follows a net repulsive path [147, 148]. A hint of this is seen in the phase versus frequency plot (see Fig. 3.32) where the cantilever initially oscillates along an adhesive path and then abruptly transitions to the repulsive path. Simulated amplitude and phase (z-sweep) curves can reproduce those determined experimentally. These have been interpreted in terms of force based interaction models that include the effect of capillary forces and adhesive forces when they are known or can be estimated. The transition between the bistable states depends on a number of factors including the cantilever Q, Ao, and r p, and the drive frequency as well as the surface properties [149]. In general high Q cantilevers or small Ao favor the net attractive path. 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

The effect of wall heat flux on the length of the heating and evaporation regions, vapor velocity, temperature and pressure in the outlet cross-section is shown in Figs. 8.13, 8.14, and 8.15. These data illustrate some important features of capillary flow at large Euler numbers. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

For the study of flow stability in a heated capillary tube it is expedient to present the parameters P and q as a function of the Peclet number defined as Pe = (uLd) /ocl. We notice that the Peclet number in capillary flow, which results from liquid evaporation, is an unknown parameter, and is determined by solving the stationary problem (Yarin et al. 2002). Employing the Peclet number as a generalized parameter of the problem allows one to estimate the effect of physical properties of phases, micro-channel geometry, as well as wall heat flux, on the characteristics of the flow, in particular, its stability. 

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