Environmental Engineering Reference
In-Depth Information
Q
loss
=
ˀ
r
to
U
to
(
−
T
wb
)
2
T
f
(4)
where,
r
to
is the outer tubing radius,
U
to
is the overall heat transfer coefficient based
on
r
to
,
T
f
is the fluid temperature (saturation temperature,
T
sat
) inside the injection
well and
T
wb
is the temperature in the well-porous medium boundary. The overall
heat transfer coefficient is given by
1
U
to
=
r
to
r
ti
h
t
+
r
to
ln
(
r
to
/
r
ti
)
r
to
ln
(
r
ins
/
r
to
)
r
to
+
+
k
t
k
ins
r
ins
(
h
c
+
h
r
)
r
to
ln
(
r
co
/
r
ci
)
r
to
ln
(
r
wb
/
r
co
)
+
+
(5)
k
c
k
cem
where
k
t
,
k
ins
,
k
c
and
k
cem
are the thermal conductivity of the tubing, insulation,
casing and cementing, respectively.
r
ins
and
r
co
are the insulation and casing external
radius,
r
ci
is the casing internal radius and
r
wb
is the cementing-porous medium
interface radius.
h
t
is the convective heat transfer coefficient for the water-steam
mixture,
h
c
and
h
r
are the convective and radiation heat transfer coefficients in the
annulus.
The thermic model involves a 2D transient heat diffusion equation in cylindrical
coordinates
k
er
r
∂
k
ez
∂
1
r
∂
∂
T
e
∂
∂
∂
T
e
∂
=
ˁ
e
C
pe
∂
T
e
∂
+
(6)
r
r
z
z
t
where,
T
e
is the porous medium temperature,
k
er
is the conduction coefficient in the
radial direction,
k
ez
is the conduction coefficient in the vertical direction
z
,
ˁ
e
is the
porous medium density,
C
pe
is the porous medium heat capacity and
r
is the radius.
One initial condition and four boundary conditions are necessary to solve Eq. (
6
).
The geothermal gradient was used as initial condition
T
ei
=
T
ei
wh
+
g
T
z
cos
ʸ
(7)
where,
T
ei
is the initial porous medium temperature and
T
ei
wh
is the initial wellhead
temperature. The boundary conditions are given by;
T
e
=
T
surface
ₒ
z
=
0
,
r
≥
r
wb
(8)
T
e
=
T
reservoir
ₒ
z
=
L
,
r
≥
r
wb
(9)
∂
T
e
∂
r
=
0
ₒ
0
≤
z
≤
L
,
r
ₒ∞
(10)
rk
e
∂
T
e
∂
Q
loss
=−
2
ˀ
ₒ
0
≤
z
≤
L
,
r
=
r
wb
(11)
r
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