Page 224 - Fundamentals of Enhanced Oil and Gas Recovery

212 Mohammad Ali Ahmadi

ð
kk rw @ AP w Þ
q w 5 (7.22)
μ @r
w
where A is the flow area, k is the reservoir permeability, k ro and k rw are relative perme-
ability to oil and water, respectively, P o and P w are pressure of oil and water, respec-
tively, q o and q w are flow rate for oil and water, respectively, r is the radius of the
wellbore, μ and μ are oil and water viscosities.
w o
Recall that capillary pressure is defined as
(7.23)
P c 5 P o P w
where P c is the capillary pressure.
Hence, we have P w 5 P o 2 P c
Substituting P w into Eq. (7.22), we have

½
ð
kk rw @ AP o 2 P c ފ
q w 5 (7.24)
μ @r
w
In terms of pressure gradient, Eqs. (7.22) and (7.24) become
ð
@ AP o Þ μ o
5 q o (7.25)
@r kk ro
½
@ AP o 2 P c ފ μ w
ð
5 q w (7.26)
@r kk rw
Subtracting Eq.(7.26) from Eq. (7.25),wehave
½
ðފ
@ AP c μ w μ o
2 5 q w 2 q o (7.27)
@r kk rw kk ro
The total liquid flow rate (q t ) is defined as
(7.28)
q t 5 q o 1 q w
whereas the fractional flow f w ðÞ can be defined in terms of both oil and water. For
water, the expression applies

q w
f w 5 (7.29)
q t
Substituting Eq. (7.27) into Eq. (7.29) yields

ðފ=@r kk ro =q t μ
½
1 2 @ AP c o
f w 5 (7.30)
1 1 k ro μ =k rw μ o
w