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Theoretical Stability Analysis of
Flowing
Oil Wells
and
Gas-Lift Wells
E.F.
Blick
SPE,
U.
of Oklahoma
P.N. Enga
SPE
U.
of Oklahoma
P C
Lin, U. of Oklahoma
Summary. The unsteady equations
of
motion for flow out
of
naturally flowing and gas-lift wells are derived and then solved by
the Laplace transform method. This analysis produces a characteristic equation with coefficients that allow determination
of
the
stability
of
a particular well. .
Introduction
Many oil wells, naturally flowing
or
otherwise, reach a stage in
their flowing life when liquid rates are low. Such wells may be
candidates for flow instabilities, commonly called heading. Heading
can be defined as a flow regime characterized by regular and perhaps
irregular cyclic changes in pressure at any point in the tubing string.
Numerous studies
l
-
17
of
heading have been reported since the
pioneering work of Donahue I in 1930. Among them, the first com
prehensive discussion
of
the phenomenon
of
heading was given by
Gilbert
2
in his pioneering paper.
In this present study, a mathematical model is developed to
describe well and reservoir variables that are affected by pressure
fluctuations in the well/reservoir system. These variables include
tubing inertance, tubing capacitance, wellbore storage, and flow
perturbation from the reservoir. In the model, a series
of
differ
ential equations that express the pressure-dependent variables are
Laplace transformed and combined by Cramer s rule to obtain a
characteristic equation with coefficients
K I , K 2,
and
K 3. By
using
Routh s cri teria, the model predicts that a well is stable when
K I ,
K
2
, and K3 are all positive or all negative. However, when one
or
two
of
the values
of
K o K 2,
and
K 3
have a sign that is
different, the model predicts that the well is unstable.
Model
for
Unsteady
Flow
In Wells
A model for unsteady flow in gas-lift wells is developed in this
section. The model can be modified to describe the unsteady flow
in a naturally flowing oil well by changing a few terms.
It is assumed that all the physical flow variables experience only
small disturbances from steady state. These are represented by
Pw =Pw o+Pwj,
1)
P=Po+P', . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
q=qo +q' , (3)
etc. From Appendix A, Eq. A-26, the relationship between the
bottornhole-flowing-pressure (BHFP) perturbation caused by a fluc
tuating flow out
of
the reservoir is
r dqR
(Bllln
re/rw)
Pwj(t)=-J
{l-exp[-ab(t-r)]dr}.
o
dt 0.OO708kh
(4)
The increased flow out
of
the annulus,
qA,
caused by the annulus
capacitance effect is (from Eq. A
-11)
qA = -
C
s
( d:; ).
................................
5)
Now at Natl. Hydrocarbon Corp.
Copyright 1988 Society of Petroleum Engineers
508
The increased flow from the tubing,
qT,
caused by tubing
capacitance effect is
d ~ p
qT=C
T
.
6)
dt
The total flow-rate perturbation, q can be expressed by
q'=qR+qA +qT
(7)
The change in pressure drop, ~ P : in tubing section below the valve
caused by inertance effect, gas/liquid ratio change, F
gLl
and flow
rate change, ~ q : can be expressed as
(
a ~ p I ) a ~ p I )
aq
~ p = (F
g
)\+ q'+M
I
-
. . . . . . . . . .
8)
aF
g
0
aq
0
at
Similarly, the change in pressure drop, t1pi, in the tubing section
above the valve can be expressed as
(
a ~ p
) (
a ~ p
) aq
~ p i (FgL)z+ q'+M
2
- .
9)
aF
g
0
aq
0
at
The difference in the BHFP and tubing-head pressure
is
expressed
as
P w J - P t J = ~ p i + ~ p i . (10)
The change in the tubing-head flowing pressure, Prj can be ex
pressed in terms
of
change
in
the gas/liquid ratio,
FgL2,
flow rate,
q, and choke diameter,
d
as follows:
PtJ=(
apt ) (FgLH+( apt )
q,+(
apt ) d' .
(11)
aF
g
0 aq 0 ad 0
In Appendix B the above set
of
equations is solved by the Laplace
transform method.
IS
This solution shows that a well will be stable
if K I , K 2,
and
K 3
have like signs. Conversely,
if
there is a
difference in sign between K I, K 2, and K 3, the well is unstable
(it will
head up ).
For a gas-lift well, assume a straight-line inflow performance,
[(
aPt ) a ~ P I ) a ~ p 2 ) ] J
)
2
=
+ +
- C
s
aq 0
aq
0 aq 0 ab
[(
at1pl) a ~ p 2 ) ]
J(M
I
+M
2
)-C
T
+ , 13)
aq
0
aq
0
SPE Production Engineering, November 1988
8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
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TABLE 1 NATURALLY FLOWING WELL PROPERTIES
Casing
10,
in.
Casing weight, Ibmlft
J
(assume straight line), bbl/(psi-O)
k, md
t >
/1-
cp
c,
psi- '
F
wv
, psi/ft
r
w
, in.
D,ft
r
I r
w
(F
gdo'
Mcf/bbl
qo'
B/O
Po
psi
Pc' psi
7
26
0.4
17
0.30
30
10 -5
0.35
5
4,000
800
0.1
300
1,800
200
TABLE 2 COMPUTED FLOW PROPERTIES OF
NATURALLY FLOWING WELL
Tubing size, in.
Tubing weight, Ibmlft
Pwfo psi
Plf, psi
Choke size, do, in. *
(op
ffloq)
0
bbl/psi-O
(aAplaq) 0 , bbl/psi-O* *
C
T,
ft
3
/psi
C
s
, ft
3
/psi
M, psi-sec
2
/ft
3
K,
,
seconds
2
K
2
, seconds
K3
Case 1
2 l8
6.5
1,050
85
24.9/64
0.283
-0.35
0.055
0.354
958
442
-233
0.97
Case 2
1.9
2.75
1,050
220
15.1/64
0.73
-0.05
0.020
0.407
2,238
1,231
6,011
1.27
Computed from 3 Ptto =1435(F
gdO.
546
/(d
o
)
,.
69
1
Qo
psig.
Computed from Gilbert s2 charts.
and
For a naturally flowing oil well,
aPt )
(ail
p
,)] J)
(ailp)
K
2
= - + C
s
+JM-C
T
- ,
aq
0
aq
0 ab
aq
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)
Stability Example-Naturally Flowing il Well. Assume a
naturally flowing oil well with the properties listed in Table 1. Two
different tubing sizes are used: Case 1 uses a
2 -in.
[7.3-cm] tubing;
Case 2 uses a 1.9-in. [4.8-cm] tubing. From the equations developed
in this paper and the well properties of Table 1, the values given
in Table 2 can be computed (see Appendix C for example calcu
lations). Thus, it is seen that Case 1 (2 -in. [7.3-cm] tubing) was
unstable because
K
2 was negative and K, and
K
3 were positive.
When a smaller tubing (1.9 in. [4.8 cm]) was used (Case 2), the
well was stable (all K values were positive) with no heading. Field
experience has shown that it is not uncommon to stabilize a well
by replacing a larger tubing with a smaller one.
SPE Production Engineering, November 1988
TABLE 3 GAS UFT WELL PROPERTIES
D,ft
Static well pressure, psi
J,
bbl/(psi-O)
(F
gL) 0 Mcflbbl
Tubing size, in.
Gas-injection pressure, psi
Ap across valve, psi
8,000
2,000
0.5
1.0
3.5
600
100
TABLE
4 COMPUTED
FLOW PROPERTIES
OF GAS LIFT WELL
qo'
B/O
Pw, psi
Optimum FgL' Mcf/bbl
Pff, psi
Valve depth, ft
Choke size, in. *
CT, ft
3
/psi
M, , psi-sec
2
1ft
3
M
2
, psi-sec
2
1ft3
(apfflaq)o
bbllpsi-O* *
(at:.p,/aq+aAp2Iaq)o bbl/psi-O**
C
s
,
ft
3
/psi
K,
,
seconds
2
K
2 ,
seconds
K3
Case 1
200
2,300
6.3
450
1,730
27.5/64
2.38x10-
4
481
23.2
2.25
-1.68
1.65
962
1,675
1.28
Computed 3 from Ptto =1435(F
.dO.
546
/(d
o
)
,.
89
1
Qo
psig.
Computed
from
Gilbert s
2
charts.
Case 2
400
2,100
4.5
341
3,530
41.8/64
3.47 x
10-
4
293
49.8
0.85
-1.24
1.93
750
-1,302
0.8
Stability
Example-Continuous-Gas-Lift
Well_ Assume a
continuous-gas-lift well with the properties given in Table
3.
Two
different flow rates are used, 200 and 400 BID [32 and 64 m
3
/d].
The data in Table 4 can be computed for these cases. The increase
from 200 (stable flow) to 400 BID [32 to 64 m
3
/d] (Case 2) neces
sitated opening the choke size to
41.8164
in. [0.65 cm]. This caused
the tubing-head pressure to drop from 450 to 341 psi [3.1 to 2.35
MPa]. The required optimum FgL dropped from 6.3 to 4.5
Mcf/bbl [1.1 to 0.8x10
3
m
3
/m
3
].
These changes caused
(ap 1 laq)o
to drop, leaving a value too small to offset the negative
sum of
(ailp,/aq+ailp2Iaq)o'
Hence, the coefficient K2 was
negative for Case 2 (400 BID [64 m
3
/d] , which means that Case
2 was unstable. Thus, one cannot produce this well at 400 BID
[64
m
3
/d] without flow-oscillation (heading) problems.
Conclusions
A mathematical model has been proposed for unsteady flow in
naturally flowing oil wells and continuous-gas-lift wells. This model
produces a characteristic equation that allows determination of the
stability
of
the well.
f K
K
2
,
and
K3 of
the characteristic
equation are oflike sign, the well is stable (small flow perturbations
from steady state damp out with time). f any of the coefficients
have a different sign, the system is unstable (small flow perturbations
increase with time).
It
was found that the sign
of (aptflaq+
ailplaq)o
strongly influenced the sign of K2 and hence the stability
of
the well.
f
(apliaq+ailplaq)o
is
negative, a strong probability
exists that the well will be unstable.
Nomenclature
a = defined by Eq. A-27,
hours-
A = annulus area,
ft2
[m
2
]
AI = tubing area,
ft2
[m
2
]
b = defined by Eq. A-24
B =
reservoir volume factor
c = compressibility, psi - [kPa - , ]
C
s
=
wellbore storage constant, ft3/psi [m
3
/kPa]
C
T
= tubing capacitance, ft3/psi [m
3
/kPa]
d = choke diameter, in
X;4
in., in. [cm]
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8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
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d = fluctuating choke diameter, in Y ;4 in in. [cm]
D = well depth,
ft
[m]
E
=
Young's modulus for steel, psi
[kPa]
FgL
=
gaslliquid ratio, Mcf/bbl
[m
3
/m
3
]
F;L = fluctuating gas/liquid ratio, Mcf/bbl
[m
3
/m
3
]
Fwv = specific weight of liquid, psi1ft [kPa/m]
gc = unit conversion factor, 32.17 Ibm-ftllbf-sec
2
[1
kg m/N s2]
h
= height
of
fluid
in
annulus,
ft [m]
h
J
=
formation thickness,
ft
[m]
J = productivity index, bbllD-psi
[m
3
Id kPa]
k =
permeability,
md
K
1,K
2,K3
=
characteristic equation coefficients
Kbe
= effective bulk modulus, psi [kPa]
KbL
= bulk modulus of liquid, psi [kPa]
Kbt = bulk modulus of tubing, psi [kPa]
510
L
=
length of tubing,
ft
[m]
M =
tubing inertance, (psi-sec
2
)/ft3 [(kPa'
s2)/m
3
]
Mg
=
molecular weight
of
gas,
g/gmol
P
= pressure, psi [kPa]
= average pressure, psi [kPa]
Pc =
annular casinghead pressure, psi [kPa]
Pg
= gas pressure
in
annulus, psi [kPa]
Po
=
steady-state reservoir pressure, psi [kPa]
PtJ
= flowing tubing-head pressure, psi [kPa]
Pw
=
BHP, psi [kPa]
PwJ =
BHFP, psi [kPa]
Pwj = fluctuating BHP, psi [kPa]
PwJo = steady-state BHP, psi [kPa]
IIp
= pressure drop in tubing, psi [kPa]
IIp =
fluctuating pressure drop
in
tubing, psi [kPa]
q
=
volumetric flow rate,
BID [m
3
Is]
q =
perturbation flow rate out of wellhead,
BID
[m
3
/s]
qA
= perturbation
flow
rate out of annulus into
tubing, BID
[m
3
/s]
qo = steady-state flow rate out of well, BID [m
3
Is]
qR = perturbation flow rate out of reservoir into
tubing, BID [m
3
/s]
qT
= perturbation flow rate out
of
tubing because
of
compressible effects,
BID
[m
3
Is]
re =
reservoir radius,
ft [m]
reD
= reservoir diameter, dimensionless
rw
=
wellbore radius,
ft
[m]
R =
universal gas constant,
(ft-Ibt)/(lbm mol-OR)
[(m kN)/(kmol
K)]
s =
Laplace transform variable
t =
time, seconds
tD
= dimensionless time
T = temperature,
OF
[0C]
v =
velocity,
ft/sec [m/s]
V = volume,
ft
3
[m
3
]
V = average gas volume in tube, ft3 [m
3
]
Vg
= gas volume,
ft3 [m
3
]
Vgs
=
gas volume at surface,
ft3
[m3]
V
L
=
liquid volume,
ft3
[m
3
]
V
t
=
tubing volume,
ft3 [m
3
]
w =
mass
flow
rate,
BID [m
3
/d]
z =
gas compressibility factor
Y = specific heat ratio
p = viscosity, cp [Pa' s]
p = density, Ibm/ft
3
[kg/m3]
Ii
g = average gas density,
Ibm/ft
3
[kg/m3]
PL =
liquid density, Ibm/ft
3
[kg/m
3
]
T =
dummy time, seconds
TD
= dimensionless dummy time
p
=
porosity
Subscripts
o =
steady-state variable
I
=
variable evaluated
in
tubing section below
gas-lift valve
2
=
variable evaluated in tubing section above
gas-lift valve
References
1. Donahue, F.P.: Classificationof Flowing Wells With Respect to Ve
locity, Pet. Dev. and Tech. (1930) 86, 226.
2. Gilbert, W.E.: Flowing and Gas-Lift Well Performance, Drill.
Prod. Prac. (1954) 126-57.
3. Ros, N.C.J.: Simultaneous Flow
of
Gas and Liquid as Encountered
in Well Testing, JPT(Oct. 1961) 1037-40; Trans., AIME, 222.
4. Fancher, G.H. Jr. and Brown, K.E.: Prediction
of
Pressure Gradients
for Multiphase Flow in Tubing, SPEJ (March 1963) 59-62; Trans.,
AIME,228.
5. Duns, H. Jr. and Ros, N.C.J.: Vertic al Flow of Gas and Liquid
Mixtures in
Wells,
Proc., Sixth World Pet. Cong., Frankfurt (1963)
451.
6. Hagedorn, A.R. and Brown, K.E. : Experimental Study of Pressure
Gradients Occurring During Continuous Two Phase Flow in Small
Diameter Vertical Conduits,
JPT ApriI1965)
475-78;
Trans.,
AIME,
234.
7. Marshall, R.S.:
The
Later Stages
of
an Oil Well, Including a Dis
cussion of Heading Phenomena, undergraduate entry, AIME Student
Paper Contest, Mid-Continent Area, Stillwater, OK (April, 1967).
8.
Zarrinal,
F.,
Brown, K.E., and Shozo, T.: Tubing Size Determi
nat ion, technical report, API Project No. 89, U.
of
Tulsa, Tulsa, OK
(July 1967).
9. Simmons, W.E. : Optimizing Continuous-Flow Gas-Lift Wel ls, Pet.
Eng. (Aug.-Sept. 1972).
10. Grupping,
A.
W. et al.: Computer Program Helps Analyze Unsteady
Flowing Oilwells, Oil Gas
J.
(Sept. 8, 1980).
11. Grupping, A.W. et al.: Computer Program Helps Analyze Unsteady
Flowing
Wells,
Oil Gas J. (Sept. 1980) 55-59.
12. Grupping, A.W., M.H. Boersma, and Bos,
C.F.M.:
Computer
Program Helps Predict Effect of Bean Changes on Unsteady Flowing
Oil
Wells,
Oil Gas J. (June 15, 1981).
13. Nind, T.E.W.: Principles a/Oil Well Production, McGraw Hill Book
Co. Inc., New York City (1981) 159-65.
14. Tiemann, W.D. and DeMoss, E.E.: Gas-Li ft Increases High-Volume
Production from the Claymore
Field,
JPT (April 1982) 696-702.
15. Grupping, A.W., Luca, C.W.F., and Vermeulen, F.D.: Heading
Action Analyzed for Stabilization, Oil Gas
J.
(July 23,
1984)
47-51.
16. Grupping,
A.W.,
Luca, C.W.F., and Vermeulen,
F.D.: These
Methods Can Eliminate
or
Control Annulus Heading, Oil Gas
J.
(July 30, 1984) 186-92.
17. Torre, A.J. et al.: Casing Heading in Flowing Oil
Wells,
SPEPE
(Nov. 1987) 297-304; Trans., AIME, 283.
18. Hale, F.J.:
Introduction to Control System Analysis and Design,
Prentice
Hall Inc., Englewood Cliffs, NJ (1973) 83-90.
19. Merritt,
H.E.:
Hydraulic Control Systems, John Wiley Sons Inc.
(1967) 16-17.
20. Lee, J.: Well Testing, SPE Textbook Series, Richardson, TX (1982)
2,106,109-11.
Appendix
A UnsteadyState
Flow
Variables
Fig. A-I
is
a diagram
of
a continuous-flow gas-lift system. As the
well flows, gas
is
injected into the annulus at a constant mass flow
rate, w through a surface injection choke. This
gas
enters the tubing
through a valve in the tubing wall.
Tubing inertance, tubing capacitance, and annulus capacitance
are unsteady-flow parameters affected by pressure variation in the
weli/reservoir system.
Tubing Inertance, M.
Tubing inertance,
M,
characterizes the
pressure drop caused by fluid acceleration along a pipe or tubing.
Consider the fluid in the control volume in Fig. A-2. Because
the net force
on
the fluid will tend
to
accelerate the fluid, the fol
lowing force balance can be written:
dv
(p+llp)A
t
-p A
t
-T7f'DL=pA
t
.
(A-l)
dt
Because q=Atv, Eq. A-I can be simplified to
T7f DL
dq
IIp= M
.
(A-2)
At
dt
SPE Production Engineering, November 1988
8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
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d -
choke
diameter
liquid
+
gas
=
0
0
0
0
0
0
0
0
- - t - - , - - - - I o ~ O
000
r - - - - - - ~ - injected
gas
gas gas
o
valve
Fig. A 1 Continuous flow gas-lift system.
r i ------------------,
I
t
I
p + ~ p
.:
v
I
I
: p
___________ -4. --_ t-_-_ _-
___________________ J
I
L
Fig. A 2 Control volume for pipe flow.
The first term on the right
in
Eq.
A 2 is
the pressure drop caused
by friction; the second term is the pressure drop owing
to
ac
celeration.
The tubing inertance is defined as
pL
M . A-3)
At
For continuous-flow gas-lift system, the
fluid
density in the tubing
of length L
2
above the point of annulus gas injection valve po
sition), P2 is different from that below, PI. Thus, there are two
inertance terms, MI and M
2
, for the gas-lift model:
inertance for the tubing portion below the gas-injection point, and
inertance for the tubing portion above the gas-injection point.
Tubing Capacitance, CT. It has been shown 19 that the effective
bulk modulus,
K
be
, of
a tube containing gas and liquid can be ex
pressed as
I I V I -1
Kbe
= Kbt
+
KbL
+
V: K
bg
, . . . . . . . . . . . . . . . . . A-4)
SPE Production Engineering, November 1988
D
gas
pg
Vg
A=cross sectlonal
qA= flow out of annulus
Fig. A 3 L1quld flow out of the annulus.
0.018
0.016
0.014
0.012
0.010
b
0.008
0.006
0.004
r L s o
0.002
0.000
100 200
300
400 500 600 700
800
900
Fig. A 4 b vs.
tD
where K
bt
, K
bL
, and K
bg
are the bulk modulus
of
the tube, liquid,
and gas, repectively. Because the bulk modulus is defined by
dV
then
ddV VI
ddp
dt
but -ddVldt=qt, flow out of tubing owing to elasticity of gas,
liquid, and tubing wall. Hence Eq. 6),
d.1.p
qT=C
T
- -
dt
511
8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
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where
C
T
=VI(_I_+_I_+ Vg _1_ ................... A-5)
Kbl KbL
VI
K
bg
Wellbore Storage Const ant,
C
s
. The wellbore storage effect can
be derived with the aid of Fig. A-3. The volumetric flow rate of
liquid out of the annulus into the tubing is
dh
q A A
.
A-6)
dt
The pressure at the bottom of the annulus, neglecting gas hydro
static pressure, is,
PWf=Pc+Fwvh . A-7)
Solving for h from Eq. A-7 and substituting into Eq. A-6 yields
A dpwf dpc
qA A-S)
Fwv dt dt
Assuming that the gas volume changes adiabatically and that
P
g
and
Vg
are the average annular gas pressure and volume, respec
tively,
PgVg=K=constant,
A-9)
if
the casinghead pressure is approximately equal to the average
gas pressure in the annulus. Hence,
dpc K dVg Pc
-=- - -=-qA
A-lO)
dt
V
g
2 dt Vg
With K=PgVg' dVgldt= -qA, and
Vg=A(D-h),
Eq. A-lO can be
substituted into Eq. A-S to obtain
dpwf
qA
=-Cs-
A-II)
dt
where
Cs=Akwv+
: ~ h ) r
1
A-12)
Reservoir Flow Fluctuations, qR' The diffusivity equation for
radial flow in a porous medium is
20
;j2p 1 ap t >JLe ap
= A-13)
ar2 r
ar
k at
The generalized solution of Eq. A-13
is
20
O.OO70Skh
f
(po
-Pwf)
P= (tD,reD), A-I4)
qBJL
where
reD=re1rw, A-15)
O.OOO264kt
tD= , A-16)
t >JLcr
w
2
and t
is
in hours.
5 2
By rearranging Eq. A-I4
qBJL
Po-Pwf= (tD,reD)' A-17)
O.OOO70Skh
f
Pwfo
is the steady-state BHFP and Pw/ is the fluctuating value,
then
Pwf=Pwfo +Pw/ A-IS)
Substituting Eq. A-IS into Eq. A-I7 yields
The quasisteady-state solution
is
qo
(Po-Pwfo)=-, A-20)
J
where
J
0.OO70Skh
f
BJL
In
re1rw
A-2I)
Subtracting Eq. A-20 from Eq. A-I9 yields
qRBJL (tD,reD)
Pw/ = - . . A-22)
O.OO70Skh
f
For a finite radial-flow system with a fixed constant pressure at
the exterior boundary,
r
e' and
consta 1t
flow rate at the wellbore,
r
w
, a tabulated solution for (tD,reD)
is
available.
20
However, we
have discovered by regression analysis that an approximation to
the exact tabulated solution
20
is
re
(tD,reD)=ln-[I-exp( -brD)]' A-23)
rw
For values of b see Fig. A-4.
The regression analysis showed that b can be approximated by
0.S92
b=
A-24)
tDo.792reDo.217
If qR
is
a function of time, then Eq. 22 can be replaced by a
Faltung-type integral:
Pw/(t)
= J
dqR
[
BJL l (t-r,reD)dr.. A-25)
dr
O.OO70Skh
f
Substituting Eqs. A-16 and A-23 into Eq.
25
yields
[ IdqR (BJL In re1rw)
Pw/(t) = - J - {I-exp[ab(t-r)]}dr,
. dr 0.OO70Skh
f
A-26)
where
O.OOO264k
a=----
A-27)
t >JLcr
w
2
SPE Production Engineering, November 1988
8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
6/7
Appendix
B-Laplace Transform olution
to Flow
Equatlon
Eqs. 4 through II can be Laplace-transformed to obtain
qR(S)
-Pw/(s) = , B-I)
1
+s/ab)
qA(S)= -sCAPw/(s),
............................. B-2)
qT(S)=SCTtlp'(S), ............................... B-3)
q'(S)=qR(S)+qA(S)+qt(S), ........................
B-4)
................................... (B-5)
(
OtlP2) (OtlP2)
tlP2'(S)= (FgLh'(s)+ q'(s)+M
2
sq'(s),
oF
gL
0 oq 0
................................... (B-6)
P; f(s)=Pt/(S) + lpi '(s) + lP2 '(s),
.................. B-7)
and
................................. B-8)
where
S
is the Laplace transform variable.
Eqs.
B 1
through B-8 are a set
of
eight algebraic equations with
eight unknowns:
Ptf(s),
Pw/(s),
tlPI
'(s),
tlP2
'(s),
qR (s), qt (s),
q
(s),
and q'(s). This set
of
equations can be solved by a number
of
methods, including Cramer s rule, to obtain
................................. B-9)
B-IO)
and
................................ B-ll)
The denominator in each
of
the terms above
(K
I
S2+K
2
S+K
3
)
is called the characteristic function. When the characteristic function
is set equal to zero, the resultant equation
is
called the character
istic equation:
K
I
S2
+K
2
S+ K
3
=0.............................
B-12)
SPE Production Engineering, November 1988
Control theory
18
has shown that for systems
of
Laplace trans
form equations like Eqs. B-9 through B-ll to be stable (the small
fluctuations
P t/, Pw/,
and
q'
will approach zero for unit impulse
inputs on the choke diameter and/or gas/ liquid ratio), the two roots
of
the characteristic equation (Eq. B-12) must both be negative (if
both are real) or have negative real parts
if
they are complex con
jugate.
t is
possible to show by Routh s criteria
18 or
by simply solving
the characteristic equation by means of the quadratic equation that
a necessary and sufficient condition to have all negative real roots
(or all negative real parts,
if
roots are complex conjugate) is that
the coefficients
of
the characteristic equation be all positive
or
all
negative. That is, the well is stable
if
or
Therefore, a well will become unstable (head up)
if
a single root
or both roots are positive or have positive real parts.
Appendix
C-Example
Calculations of
Stability Constants
The following are calculations for Case 1 for the naturally flowing
well data
of
Table 1.
Areas. The tubing area
is
The annulus area is
A
=1(/4(6.276
2
-2.875
2
)/144=0.169
ft2.
BHFP (steady state).
Pfwo
is
Pfwo
=po
-q /J=
1
800-300/0.4=
1,050 psi.
Tubing-Head
Pressure, Ptfo, and Pressure Drof Ap.
From Gilbert s2 Fig. 25, for
q=2oo
B/D [32 m /d]
andpfwo=
1,050 psi [7.2 MPa], equivalent depth
=4,200
ft [1280 m]. Actual
depth=4,OOO ft [1219 m]. Equivalent depth
of
tubing-head
pressure=4,2oo-4,000=200
ft [61 m]. For 200 ft [61 m],
Ptfo=50
psi [345 kPa].
Similarly, from Gilbert s2 Fig. 26, for
q=4oo BID
[64 m
3
/d]
and Pfwo = 1,050 psi [7.2 MPa], equivalent depth=4,7oo ft [1433
m]. Actual depth=4,OOO ft [1219 m]. Equivalent depth
of
tubing
head pressure
=4,700-4,000
=700 ft [213 m].
For
700 ft [213 m],
Ptfo=120 psi [827 kPa].
Hence, for
q=3oo
B/D [48 m
3
/d],
Ptfo=50+(120-50)(3OO-
200)/(400-200)=85
psi [586 kPa].
tlP=Pfwo -Ptfo = 1,050-85 =965 psi.
oPtfo Ptfo 85
= = =0.2833.
oq q 300
oJ1p/oq.
At q=2oo BID [32 m
3
/d], tlP=ffw
o
-Ptfo=
1,050-50=
1,000 psi. At q=400 BID [64 m /d],
tlP=Pfwo-Ptfo=
1,050-120=930 psi.
Otlp 930 - 1 000
= = 0.35.
oq
400-200
Tubing Capacitance,
CT'
Kbt =tE/d=(0.23
in.)(30 x 10
6
psi)/(2.876 in.)=2.4x 10
6
psi.
KbL
= 10
5
psi for petroleum fluid.
513
8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells
7/7
ji = Pwo +Ptfo)/2+
15
=(1 ,050+85)/2+
15
=582.5 psia.
K
bg
=-yji = 1.25(582.5)=728 psi.
- -
VglV
L
= VgslVL> VglV
gs
)
VgslVL
=(F
g
L>(1 ,(00)/5.61 = 178F
gL
= 178(0.1)= 17.8.
~
zRT
Ps
Vgs
zsRTs P
Assume constant gas temperatures and constant values
of
z.
= Ps _5__
=0.026.
Vgs P h(l,050+85)+15
VglV
L
=(17.8)(0.026)=0.46.
VglV,= VgI Vg + V
L
)=(1 + VLlVg)-1 =(1 + 110.46)-1 =0.315.
(
1 1 Vg 1 )
CT=V,
K
b
, kbL V, K
bg
=(4,000)(0.032)(112.4 X
10
6
+
1110
5
+0.315/728)
=0.055
ft
3
/psi.
Wellbore Storage Constant,
C
s
.
h= Pwfo -Pc)IFwv=(1 ,050-200)/(0.35) =2,429 ft
D-h=4,000-2,429=1,571
ft
Cs=A/[Fwv+Pcl(D-h)]
=(0.169)/(0.35
+200/1 ,571)=0.354 ft3/psi.
Inertance, M.
p g=jiIR T=(582.5)(144)/(1 ,545/22.5) 520)=2.35 Ibm/ft3.
R =RIM
g
.
M=D[(VglV,(pg)+(I- VglV,)pLlIA,gc
=4,000[ 0.315) 2.35) +(1-0.315)50.4]/0.032(144)(32.17)
Jlab.
0.OOO264k
a=
pp
w
2
514
(0.000264)(17)
=287 hours
I.
0.3) 30) 10 -5) 5 /12)2
For a typical heading cycle period of t=1 hour,
tD =at=(287)(I)=287.
For reD =800, from Fig. A-4, b=0.002.
Jlab= O.4 B/D) 5.61 ft3 Ibbl)/(0.002)(287)(lIhour)(24 hourslD)
=0.163 ft3/psi.
Kl Term.
KI =M(Cs- CT+Jlab) =958(0.354-0.55 +0.163)
=442 seconds
2
[
0Ptfo
Ot.l
p
) Ot.lp]
K
2
=
(Jlab+Cs)-C
T
-
+JM
oq oq oq
=[(0.283 -0.35)(0.163 +0.354)-0.055( -0.35)](15,388)
+(0.4)(958)115,388=
-233
seconds.
(
1 bbl
) 24
hOUrS)(3,600 seconds)
Note that 15,388= .
5.61 ft3 D hour
K3 Term.
OP f
o
Ot.lp)
K3= J+l=(0.28-0.35)(0.4)+1=0.97.
oq oq
SI Metric onversion Factors
bbl x 1.589 873
E-Ol
bbl/(psi-D) x 2.305 916
E-02
cp x 1.0*
E-03
ft
x
3.048*
E-Ol
ft3 x
2.831 685
E-02
in x
2.54*
E+oo
Ibm x 4.535 924
E-Ol
psi x 6.894 757
E+oo
scf/bbl x 1.801
175 E-Ol
Conversion factor is
exact
m
3
m
3
/(kPa d)
Pa s
m
m
3
cm
kg
kPa
std
m
3
/m
3
SPEPE
Original SPE manuscript received for review March 13, 1986. Paper accepted for publi
cation July 6 1987. Revised manuscript received Oct. 29, 1987. Paper (SPE 15022) first
presented at the 1986 SPE Permian Basin Oil Gas Recovery Conference held in Midland,
March 13-14.
SPE Production Engineering, November 1988
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