Page 321 - Geology and Geochemistry of Oil and Gas

282 APPENDIX A

Fig. A.10. Changes in contact angle as a result of movement of water–oil interface. y is the contact angle
at static position, y a is the contact angle when oil is displaced by water (advancing angle), and y b is the
contact angle when water is displaced by oil (receding angle).

Fig. A.11. Flow through a two-branch capillary and trapping of oil in a small-diameter capillary.

and
2
n ¼ q=A ¼ d Dp =32 mL (A.18)
t
3
where q is the volumetric rate of flow (cm /sec), d is the diameter of capillary (cm), Dp t
2
is the total pressure drop (dyn/cm), A is the cross-sectional area (cm ), m is the viscosity
(cP), L is the flow path length (cm), and n is the velocity (cm/s).
The total pressure drop, Dp , is equal to
t
Dp ¼ Dp þ P c . (A.19)
t i
where P c is the capillary pressure (see Eq. A.1) and Dp is the applied pressure (dyn/
i
2
cm ). Solving for n in each capillary, by combining Eqs. A.1, A.18, and A.19 gives
2
n 1 ¼ d =32 m L 1 ðDp þ 4s cos y=d 1 Þ (A.20)
1 1 i
and
2
n 2 ¼ d =32 m L 2 ðDp þ 4s cos y=d 2 Þ (A.21)
i
2
2
Setting L 1 ¼ L 2 and m ¼ m , and dividing Eq. A.20 by Eq. A.21 gives the
1
2
following relationship:
2 2
n 1 =n 2 ¼ ðd Dp þ 4s cos yd 1 Þ=ðd Dp þ 4s cos yd 2 Þ (A.22)
2
i
i
1
Therefore, when Dp P c ,
i
2
n 1 =n 2 d =d 2 (A.23)
1 2
and when Dp P c ,
i