SPE-23429 A Simple Method of Predicting The Performance of Sucker Rod Pumping System
SPE-23429 A Simple Method of Predicting The Performance of Sucker Rod Pumping System
SPE-23429 A Simple Method of Predicting The Performance of Sucker Rod Pumping System
SPE 23429
A Simple Method for Predicting the Performance of a Sucker-Rod
Pumping System
A. Khodabandeh and S. Miska, New Mexico Inst. of Mining & Technology
SPE Members
ABSTRACT
mathematical models for the sucker rod pumping system for the
digital computer started in early 1960's. Gibbs! was the first
who successfully modeled a sucker rod pumping system, and one
of his most important contribution is simulation of the subsurface
elements. He used the finite difference solution of the following
wave equation and its initial and boundary conditions, which
together can be used to predict the performance of the sucker
rod string:
a2u(x,t)
at
a 2 a2u(x,t) - cau(x,t)
--2
ax
(1)
at
For the sucker rod string, other models were developed by using
different solution techniques for the partial differential
equation.2,3 These models all ignore the effect of the fluid
inertia and asSUme that the fluid surrounding the rod is
incompressible. There are models that take into consideration
the dynamic effect of fluid. 4,5 In the later models, the dynamic
effects of fluid were modeled by a system of partial differential
equations, which were solved by the method of characteristics.
INTRODUCTION
Sucker rod pumping systems are the most popular artificial lift
method in the oil industry.
Accurate prediction of the
performance of the system can improve efficiency of a system.
The most important parts to model are the subsurface elements
such as the sucker rod string and downhole pump. The
149
A SIMPLE METHOD FOR PREDICTING THE PERFORMANCE OF A SUCKER ROD PUMPING SYSTEM
in the same direction as the rod. In other words, the sign for
characteristic impedance is negative when the observer moves in
the same direction as the rod and positive when he moves in the
oppOsite direction. By this convention, compression is positive
and tension is negative.
BERGERON TECHNIQUE
As the name indicates, this technique was developed by
Bergeron. 6 In his work, mostly the cases were considered in
which the friction losses were negligible. Later, the friction
losses were included in this model. 9 The Bergeron technique
does not give an expression to describe how force and velocity
are changing with time and space. Instead, the dynamic force
and velocity is measured by an imaginary observer moving up
and down the rod with acoustic velocity. This approach has
already been used successfully by Lubinski10 for predicting
transient pressure surges due to drill string motion and dynamic
loading of drill pipe during tripping operation. Encouraged by
the results ofLubinski's work, the Bergeron technique is adopted
for the modeling of the sucker rod pumping system~ The
following equations, which are equivalent to the Bergeron
graphical presentation, can be used to determine the dynamic
changes for a solid rod (derivation in Appendix A).
Vat
and
(3)
where Sr is the characteristic impedance and can be determined
by
s = apA
a
Jg:
Var
(4)
= !:.
Fbr
=0
(6)
= V*,
Fb t
= KY,
(7)
= V*,
(2)
SPE 23429
(5)
150
SPE 23429
MATHEMATICAL MODEL
and
Fa"" -Fb"_I,1-!
= -S,(Va",,- Vb,,_l,,_l)
(9)
(15)
where
(10)
(16)
(11)
where Fd is the damping force and will be explained later.
At the polished rod, the velocity of the rod at any time is
known, providing one of the lower boundary conditions.
Therefore, the dynamic force on the polished rod at time t is
PRF(t) '" Fal,l-! +S,(PRV(t) - Val,l-!)
(12)
= PRF(t) - Wb
(13)
= -S,(PLV(t) - Vb"_I.I-!)
PLV(t)
=0
(17)
(14)
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A SIMPLE METHOD FOR PREDICTING THE PERFORMANCE OF A SUCKER ROD PUMPING SYSTEM
where PIF is the force in the rod at the plunger. When the
plunger is moving down, the traveling valve is open and the
standing valve is closed, thus
PIF(t) ,. Pj (L,t)Arotl
(19)
>
(20)
<
Pj(L,t)Arotl
(21)
Damping Forces
The primary cause of energy losses along the sucker rod striIig
are the various frictional forces. These forces oppose the
movement of the rod. To account for these forces, an element
of the rod is considered. Using the equation of motion and
writing a force balance on the element, the final results can be
presented as l 2,14
(22)
SPE 23429
L1
,.
L2
(25)
(23)
CONCLUSIONS
where
c ,. 1fap
(24)
2L
F
II
1fap A.:1x V
2Lg P
t
(25)
MODEL VALIDATION
To verify the validity of the model, the computer-generated
dynamometer cards were compared to measured dynamometer
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SPE 23429
NOMENCLATURE
a
c
g
t
u
A
A, C, L,
P, R
BU
E
F
Fa
Fb
Fd
L
Sr
Va
Vb
Vo, V"
P
po,r
PLF
PLV
PRF
PRL
PRV
Wb
Subscripts
n
rod,r
t
APL
BPL
PL
PR
ACKNOWLEDGMENTS
REFERENCES
Greek Symbols
Damping
Fluid
Element
Sucker rod
Bergeron time unit (BU)
Above plunger
Below plunger
Plunger
Polished rod
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A SIMPLE METHOD FOR PREDICTING THE PERFORMANCE OF A SUCKER ROD PUMPING SYSTEM
SPE 23429
APPENDIX A
(A-4)
(A-I)
= a~
FL,t - Fo,t-!
where a is the propagation velocity (acoustic velocity). It should
be noted that the velocity at B' in Figure ll-b is V.. while
velocity of A' and A" is V".
= (PaA)(VL,t - VO,t_I)
(A-6)
APPENDIXB
Now, consider Figure ll-e which shows the situation one second
later. During this time, one end has moved with velocity V" and
the other end with velocity V... Meanwhile, V" has propagated
some distance through the rod with velocity a. Since the rod is
considered after one second, the distance that velocity has been
propagated is also a. The momentum for this element at 1=0 is
(Aap)V.., and at 1=1 is (Aap)V". Since the change in momentum
of a body is equal in magnitude and direction to the sum of the
forces acting on the body, therefore
or
P* = -(paA)(V* - V.)
(A-3)
It can be observed that when V" > V.. the force is negative and it
causes compression. When V" < V.. the force is positive and puts
the rod in tension. By this convention, a negative change in
force causes tension and a positive change causes compression.
To calculate velocity and force at a point along the rod, the
following time unit is defIned. This time unit is called Bergeron
time unit denoted by BU, and it is equal to the ratio of the length
of the element over the sound velocity of the element. If L is
the length of the element and a is the propagation velocity,
during one BU the wave front propagates a distance of L. If the
momenta at times 1 and t+ 1 are (pAa)V.. and (pAa)V", then
154
Graphical Solution
For the construction of the graphical presentation of Figure 2, it
is assumed that the change in velocity at point a, in Figure 1,
occurs at 1=0. At 1=-1, force and velocity are zero at point b.
Line b_Iao presents the situation of an observer that leaves point
b at 1=-1 and moves toward point a with the velocity of
propagation. Since he moves in the opposite direction of the
rod, slope of the line is positive, and its value is equal to S, .
The observer arrives at the point a at 1=0. The abscissa of the
point ao is equal to the velocity at time zero. The meaning of
the line aobl is similar to b_IQo' An observer starts at 1=0 from
point a and moves with the velocity of propagation toward point
b. Since he moves in the same direction as the rod, slope of the
line is negative. At 1 = 1, the observer arrives at the point b and
encounters the condition at point b, which is zero force. Point
bl on Figure 2 represents the force and velocity of the point b at
1=1. By following the same procedure, the remaining portion
of the graph on the Figure 2 can be constructed.
SPE
TABLE 1
Dynamic Force and Velocity
at the Ends of a Rod Without Damping
Fa
Vb
sv*
sV"
2V"
-sV"
2V"
-sV"
sV"
sV"
-SV"
-SV"
sV"
5
6
7
8
2V"
2V"
0
0
TABLE 2
Dynamic Force and Velocity for a Rod with Damping
t
Fa
Vb
Sy*
SY*
(1+24)SY*
(1+24)SY*
(2(A+I)A+I)SY*
(2(A+I)A+I)SY*
A+I)A+I)2SY*
K+S
2SY*
S+K
2Sy*
K+S
(A+I)2SY*
K+S
(A+I) 2Sy*
K+S
155
23429
SPE 2342.9
TABLE 3
Input data obtained from OTC
PUMPING UNIT
A - Center of the well to the saddle bearing
C - Saddle bearing to equalizer bearing
P - Pitman length
K - Saddle bearing to crank shaft
Stroke per min.
ROD
(in):
(in):
(in):
(in):
(SPM):
72.750
72.750
71.870
105.006
11.48
PRESSURE
Well head pressure (psi)
0.0
TABLE 4
Input data for the field case
PUMPING UNIT
A - Center of the well to the saddle bearing
C - Saddle bearing to equalizer bearing
P - Pitman length
K - Saddle bearing to crank shaft
Stroke per min.
(in):
(in):
(in):
(in):
(SPM):
ROD
Sec. L (ft) OD (in)
1
1500.0 .8750
2
3600.0 .7500
Plunger diameter
Fluid level
Specific gravity
PRESSURE
Well head pressure (psi)
30.0
156
150.000
154.000
180.000
236.888
10.05
SPE
23429
IE------.....,..-----~~v
h l 5.'._
laz
I
I
Ohpot
I
I
I
I
Fagure 4- Graphical solution for K
<
Sr'
..................................-,
............
__
~------+-------v
..-
>
Sr'
157
SPE
2:5 429
1000 T!,T
o
o
10
20
30
PoII.hed rod dl.placement. In
40
110
-----~----.-,---'---.-----,,.---~---.
20
60
40
30
A;-
a)
o
==--l
po-I
--iB
1- Po
b)
c)
A'
r~~~.;::::::::::~
B'
~t--_---,-i_._V_o_ _I - Po
Figure 11- a) Elastic rod at equilibrium, b) after change of force, c) one second later.
158