Page 324 - Geology and Geochemistry of Oil and Gas

WETTABILITY AND CAPILLARITY 285
equal to

1 1
p cwb ¼ s þ cos y (A.32)
r 1 r 2
where s is the interfacial tension between the oil and water, r 1 and r 2 are the radii of
capillary in two perpendicular directions, and y is the contact angle.
The above equation can be derived from Eq. A.5:
1 1

p p ¼ Dp ¼ s þ (A.33)
1
2
R 1 R 2
(where Dp is the pressure difference across the oil–water interface, and R 1 and R 2 are
the curvatures of interface in two perpendicular directions), by substituting r 1 = cos y
for R 1 and r 2 = cos y for R 2 .

A.4. BUOYANCY
Where the diameter of the gas bubble is less than that of the migration channel,
the gas babble/globule is able to float upward in the channel. In this case, the
capillary pressure plays little or no part in the resistance to gas flow as the gas
globule does not need to displace the water from the pore channel to migrate upward
(a bubble is a small globule).
Fig. A.14 shows an example of gas bubble floating upward due to buoyancy force.
For a Reynolds number, N R p0.4, the drag coefficient, C D ¼ 24/N R for a bubble of
spherical form. Thus, for a laminar or viscous flow, the drag force, D, on the gas
bubble is equal to

2
2
2
D ¼ AC D rðn Þ=2 ¼ ð24m=rndÞrðn Þ=2ðpd =4Þ ¼ 3mnpd (A.34)
2
where A is the largest projected area (in ft ), r is the mass per unit volume (in slug/
3
2
ft ), m is the viscosity of the fluid (in lb-s/ft ) (or slug/ft s), n is the upward velocity of
the bubble in ft/sec, and d is the diameter of the spherical gas bubble (in ft).
Inasmuch as the buoyant force, B, is acting upward, whereas the drag force, D,
and the weight of the gas bubble, W g , are acting down, D+W g ¼ B:
3
3
ðpd =6Þg ðpd =6Þg ¼ 3mnpd (A.35)
w g
Solving for upward velocity of bubble, n,
2
n ¼ d ðg g Þ=18m w (A.36)
w
g
2
Assuming a temperature of 251C, a pressure of 20 kg/cm , a specific weight
3
3
of methane gas of about 0.015 g/cm , a specific weight of water of 1 g/cm , a
water viscosity at this conditions of approximately 10 2 P, and a migrating gas
bubble having the radius of 0.1 cm, the critical velocity is approximately equal to
0.22 cm/s.