Page 222 - Fundamentals of Enhanced Oil and Gas Recovery

210 Mohammad Ali Ahmadi

α

Figure 7.2 An inclined linear reservoir bed.

Darcy’s equation for oil and water flow rate with respect to dip angle is given as

kk ro A @P o
q o 52 1 P o g sin α (7.11)
μ @x
o

kk rw A @P w
q w 52 1 P w g sin α (7.12)
μ w @x
Replacing water pressure by P w 5 P o 2 P cow so that

kk rw A @P w
q w 52 1 P w g sin α (7.13)
μ @x
w
After rearranging, the equations may be written as
μ
o @P o
2 q o 5 1 P o g sin α (7.14)
kk ro A @x
μ
w @P o @P cow 1 P w g sin α
2 q w 5 2 (7.15)
kk rw A @x @x
Subtracting the first equation from the second one, we get

1 μ μ @P cow
2 q w w 2 q o o 52 1 P w g sin α (7.16)
kA k rw k ro @x
Substituting for

q t 5 q w 1 q o (7.17)
and

q w
f w 5 (7.18)
q t
Also, solving for a fraction of water flowing, we obtain the expression for a frac-
tion of water flowing:
1 1 kk ro A=q t μ @P cow =@x 2 Δρg sin α

f w 5 o (7.19)
1 1 k ro μ =μ k rw
w o