ikoku ch4 eng 140 201 deli
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
2/45
142 Deliverability Testingof Gas Wells
4.2 HORIZONTAL FLOW EQUA TIONS
4.2.1 Basic Differential Equations
The basic differential form of Darcy's law for horizontal viscous radial gas low is
1.1271 (27Trh)k dp
qgr= 1000f.L~ (4.1)
where
qgr = gas flow rate at radius r, res bbl/day
r = radial distance, ft
h = zone thickness, ft
f.L
=
gas viscosity, cp
k = formation permeability, md
p = pressure, psi
1.1271 = conversion constant from Darcy units to field units
The equivalent of qgr in terms of gas flow at standard conditions is qbr, in
standard cubic feet per day and is related to qgr by
pTbZb
qbr
= 5.615qgr
PbTz (4.2)
Combining Eqs. 4.1 and 4.2 yields
3.9764 x 10-2khTbzb dr
pdp=-
qbrPbTzf.L r (4.3)
4.2.2 Steady-State Flow
For steady-state low, qbr has the same value at all radii. A constant pressure is
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
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144 Deliyerability Testingof Gas Wells
t
oil
qbT
Slope =
o:7""ii3ih
I n - -
w
Fig. 4.1 Graphof'" vs. n(r/rw) steady tate).
Define the volume average eal gaspseudopressure:
- - f:~ f1 dV
- f:~\f121Thdr
\f1
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- ;(re2 - rw2)h~
J'wav (4.13)
i
nd since re2 - rw2
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
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146 Deliyerability Testingof Gas Wells
Or
1422qT r
1\1 - 1\1 = -kh
In -;z (4.20)
1422qT
(
r
e
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\Ie I\Iw - kh rw (4.21)
Or
1422qT r e
l\Ie -
I\Iw
= kh-in
-;z (4.22)
and
- 1422qT
( reI
)
\1-1\1 =In---+s
w
kh rw 2 (4.23)
Or
-
1422qT
(
re
1)
\I-l\Iw = -kh In-;z - 2 (4.24)
These equations, presented n terms of the real gaspseudopressure unction,
will result in more exact engineering. They require a
1\1
-
p chart or an equation
for converting pressures o pseudopressureand vice versa. It is at times permissi-
ble to use approximate techniques, especially when only one-time calculations are
needed. Two such approximations of Eq. 4.5 are given below.
1. Pressure-squaredmethod. The first modification of the exact solution is to
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
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8/19/2019 Ikoku Ch4 Eng 140 201 Deli
18/45
158 Delil erability Testing
of
Gas Wells
•
Based on a large number of empirical observations, Rawlins and Schellhardt
postulated that the relationship between flow rate and pressure can be expressed
s
where
or
q
= C P/ - Pw/t
(4.66)
q
= gas flow rate at base conditions
P
=
average reservoir pressure obtained by shut-in
of
the well to com-
plete stabilization, psi a .
Pw/
= flowing wellbore pressure at sandface, psi a
C
=
the performance coefficent which describes the position
of
the
stabilized deliverability curve. The value
of C
depends on the
units of
q.
n
=
an exponent that describes the inverse
of
the slope
of
the stabil
ized deliverability curve
n = tan 8
The straight line shown in Fig. 4.7 is an approximation
of
the true behavior.
Ideally the curve should be slightly concave and have unit slope
(8
=
45°)
at low
flow rates and somewhat greater slope at high flow rates. The change
of
slope
results from increased turbulence in the region of the well bore plus changes in
other skin factors that are rate dependent
s
flow rate increases. The practice
is
to
use a straight-line deliverability curve in most instances. In general, the exponent
n
will range between about 0.5 and 1.0.
Points on the back-pressure curve should be plotted on equal scale log-log
graph paper. As straight a line as possible should be drawn through the points.
he
value of the exponent
n
in the back-pressure
or
deliverability equation
(Eq. 4.66) may be determined from the slope of the straight line
or
by substituting
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-
8/19/2019 Ikoku Ch4 Eng 140 201 Deli
19/45
-
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