Two-Fluid Cocurrent Flowing Porous Media

So far we have assumed that all the pore space in the porous medium was occupied by the same fluid, such as air or water or oil. However, there are very important problems in which two immiscible fluids are present in the same pore space, e.g., the simultaneous flow of oil and gas or of oil and water, which occurs in petroleum reservoirs, or the air-blowing of a filter cake to drive out a valuable filtrate. i 

If in the experimental apparatus shown in Fig. 12.2 we fill all the pores with water and then force air through the system, the fraction of water in the outlet stream behaves as shown in Fig. 12.5. Initially only water will flow out of 

Why does this water not flow out It is held in place by surface forces. This can be made plausible by comparing the surface of Igal of fluid in a cylindrical container with 1 gal of fluid in a typical sandstone. An ordinary 1-gal paint can has a surface area of 1.5 ft, A gallon of fluid contained in the pores of a sandstone made of spherical grains of diameter 0.01 in and porosity 0.3 has 216 ft of surface area. One may think of the fluid in the pores of such a sandstone as being spread out as a film of average thickness 0,007 in = 0.18 mm, 

If only one fluid is present, then this is ail solid-fluid surface and the interaction is simply one of adhesion at the surface. If two fluids are present, then in addition to two kinds of solid-fluid interface there will be a fluid-fluid interface. The pressure difference which can exist across such a fluid-fluid interface is of the form  

From this physical description we concluded that a particle of fluid stops moving when the displacing force (which equals the pressure gradient times the length of the droplet times its cross-sectional area perpendicular to the flow) is balanced by the surface force (which equals the surfajce tension divided by the radius of the drop times its cross-sectional area). Equating these, we find that the fluid particle should stop moving when 

---