Page 103 - Introduction to Petroleum Engineering

88 MULTIPHASE FLOW

20

10 Primary
Capillary pressure (psi) 0 Secondary Secondary
drainage

imbibition

–10 drainage

–20
0.00 0.20 0.40 0.60 0.80 1.00
Water saturation

FIgURE 5.4 Hysteresis of capillary pressure.

5.3 RELATIVE PERMEABILITY

Darcy’s equation for calculating volumetric flow rate q through a linear, horizontal,
water‐saturated permeable material is proportional to pressure gradient:

∆
kA p
q =−0 001127 (5.5)
.
w
µ ∆ x
w
The variables are expressed in the following units: volumetric flow rate q (bbl/day),
permeability k (md), cross‐sectional area A (ft ), pressure p (psi), fluid viscosity μ
2
(cp), and length Δx (ft). We represent flow of water through a material that is not
entirely saturated with water by introducing a new term called water relative perme-
ability, as shown as follows:
kk ( ) A p∆
S
.
q =−0 001127 rw w (5.6)
w
µ w ∆ x
Water relative permeability k (S ) is a dimensionless variable that depends on water
rw
w
saturation. If the permeable material also contains some oil, then we can write a sim-
ilar expression for the oil flow rate using an oil relative permeability k (S ) that
o
ro
depends on oil‐phase saturation:
kk ( ) A p∆
S
.
q =−0 001127 ro o (5.7)
o
µ o ∆ x