Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells

8/10/2019 Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells

1/7

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


2/7

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.


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]

509


3/7

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



4/7

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)


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


5/7

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



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)


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


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.


Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells

Documents

Transcript of Theoretical Stability Analysis of Flowing Oil Wells and Gas-Lift Wells