Page 188 - Fundamentals of Enhanced Oil and Gas Recovery

176 Forough Ameli et al.

Figure 5.13 Heat loss to o/u.

2 ð x 2t 2 2 ð N 2t 2
erfc xðÞ 5 1 2 erf xðÞ 5 1 2 p e dt 5 p e dt (5.117)
π π
ffiffiffi
ffiffiffi
0 x
Heat conduction to semiinfinite media is calculated as

@T k h T s 2 T R Þ
ð
qj 52 k h 5 p (5.118)
z50 ffiffiffiffiffiffiffiffi
@z παt
z50
A h
ð
_

Q loss 5 2 qt 2 t ÞdA h (5.119)
ð
0

where t is the process time (injection), t is the time for the heated interface to
advance to a specific location, t 2 t is the time that o/u has been in contact with the

heated zone.
Ordinary differential equation (ODE) equation in terms of A h is as follows:
ð t
ð
_ hdA h k h T s 2 T R Þ dA h
Q 5 M R T s 2 T R Þ 1 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt (5.120)
ð
in
dt 0 πα t 2 t Þ dt

ð

αt
M 2
t D 5 4 2 (5.121)
M R h
where t D is the dimensionless time, M is the volumetric heat capacity of o/u, M R is
the heat capacity of reservoir, α is the thermal diffusivity of o/u.
Heated area as a function of time is
_ m s H s M R h
A h 5 Gðt D Þ (5.122)
4 T s 2 T R ÞαM 2
ð
r ffiffiffiffiffi
p
ffiffiffiffiffi t D
Gt D 5 e erfc t D 1 2 2 1 (5.123)
t D
ðÞ
π