Page 180 - Fundamentals of Enhanced Oil and Gas Recovery

168 Forough Ameli et al.

Ramey [72] represented that rate of heat transfer at the radius of drill hole is a
transient process and is calculated using the following equation:
ð
_
Q l 5 ½ 2πk hf T h 2 T e ފ (5.98)
f ðtÞ
where k hf is the k of formation (Btu/h ft F), T h is the T at cement-formation inter-

face ( F), T e is the stabilized T ( F), f(t) is the transient dimensionless time function.
If injection time is long enough, the transient function would be applied.
Dimensionless time function:
r ffi
a f t
ffiffiffiffiffi
t ðÞ 5 ln 2 2 0:29 Ramey for t . 1week (5.99)
r hd
2
where α f is the thermal diffusivity (k/ρc) of formation (ft /h) t 5 time (hour), r hd is
the radius of hole drill (ft).
Integration of the model considering the quasisteady states as a series of steady-
state mechanisms would be
_
Q 5 2πr to U to ðT s 2 T h Þ (5.100)
l
where U to is the overall heat transfer coefficient fluid cement/formation interface
(Btu/h ft 2 F) based on outside tube surface area, assuming ΔT 5 T s 2 T h

T s ftðÞ 1 T e k hf r to =U to
T h 5 (5.101)
ftðÞ 1 k hf r to =U to
In short time, f(t) is a function of U to .

5.2.10.1 Overall Heat Transfer Coefficient
To calculate the overall heat transfer coefficient in an annulus, use the following corre-
lation represented by Willhite [73]:

21

r to r to ln r to =r ti 1 r to ln r co =r ci r to ln r hd =r co
U to 5 1 1 1 1 (5.102)
r ti h f k htub h nc 1h r k hcas k hcem
where h f is the film htc between flowing fluid and inside tube, h nc is the natural con-
vection htc in annulus based on r to and T to 2 T ci , h r is the radiation htc based on r to
and T to 2 T ci , k hcas is the thermal conductivity of casing based on average casing T,
k hcem is the thermal conductivity of cement based on average cement T,P; k htub is the
thermal conductivity of tubing; h is calculated in Btu/h ft 2 F, and k is calculated in

Btu/h ft F.