8th Annual Sucker Rod Pumping
Workshop
Renaissance Hotel
Oklahoma City, Oklahoma
September 25 - 28, 2012
Wave Equation:
Derivation and Analysis
Victoria M. Pons, Ph. D.
Weatherford
Jeffrey J. DaCunha, Ph. D.
Pioneer Natural Resources
Reciprocating Rod Lift
• The most widely used mean of artificial lift is sucker rod
pumping.
• In reciprocating rod lift the work done by the generator at
the surface is translated downhole through the polished
rod and the rod string into work at the pump.
• The work at the surface of the pumping unit is measured
by a surface dynamometer, capable of recording the
position and load of the rod string.
• Energy is irreversibly and continuously lost from the
system due to Friction and Elasticity.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
2
Irreversible Energy losses
• Elasticity: Due to the load of the fluid and the load of
the rod string below, in the case of a vertical well, the
rod string can be compared to an ideal slender bar. It
will elongate and contract as stress waves move
through it.
• Viscous Friction: Fluid is constantly opposing the
movement of the rods. The well fluids impart a
viscous force at the outer surface of the rods
resulting in continuous energy loss.
• Mechanical Friction: Occurs when tubing is in contact
with rods and rod couplings, relevant only in the case
of deviated wells.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
3
Surface and Downhole Data
• Because of elasticity and friction, the work done at the
surface is not directly translated downhole.
• To know how much actual work is done downhole, a
downhole dynamometer can be used. Drawback: very
costly.
• A more efficient solution is to calculate the position and
load at the pump using the surface position and load.
• The position and load can be illustrated as a function of
two variables and graphed to give a surface and downhole
card.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
4
Conventional pumping unit
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2012 Sucker Rod Pumping Workshop
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The 1D Damped Wave Equation
• Calculating downhole conditions is difficult because
of the sucker rod’s elasticity.
• This takes the form of elastic force or stress waves
traveling along the string at the speed of sound.
• The rod string is physically equivalent to an ideal
slender bar, therefore the propagation of stress
waves is a one dimensional phenomenon.
• The wave equation describes the motion and stress
wave propagation phenomena in the rod string.
• In the one dimensional damped wave equation, the
damping term stands for the irreversible energy
losses that occur along the rod string.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
6
Forces acting on a Rod Element
Forces:
•Buoyant weight of the rod
element W,
•Tension force representing the
upward pull on the rod element
FX,
•Tension force representing the
pull from below on the rod
element FX+ΔX,
•The damping force opposing the
movement, FD, resulting from
fluid friction on the rod element’s
surface.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Newton’s Second Law
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2012 Sucker Rod Pumping Workshop
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Breakdown of Forces (1/2)
• Using the stresses present in the rod sections and
Hooke’s law the tension forces can be rewritten as:
• The acceleration can be written as:
the mass as
Where ρ is the density and g the gravity constant.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Breakdown of Forces (2/2)
Since the friction force considered is of viscous
nature only, it is proportional to the velocity of the
rod element:
Where c is the damping coefficient.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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The 1D Damped Wave Equation (1/2)
• The conservation of energy for the rod element reads:
• The acoustic velocity in the rod string is given by
• The damping factor is defined as
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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The 1D Damped Wave Equation (2/2)
• Therefore the condensed form of the above equation
reads:
Acceleration
Elasticity
Damping
• Cf. Sucker-Rod Pumping Manual, by Gábor Takács.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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True Loads vs. Effective Loads
• The difference between true loads and effective loads
is that when using true loads the buoyant force is
added to the load values.
• The Gibbs method uses true loads, meaning that the
resulting downhole card is translated vertically
downward by the value of the buoyant force.
• The modified Everitt-Jennings method uses effective
loads, meaning the resulting downhole card rests on
the zero load line.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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The Gibbs Method
• The Gibbs Method fits a function to the measured
surface position data and surface load data using
harmonic analysis.
• From this function, the wave equation is
implemented.
• Advantages include a smoother data set on
which to apply the wave equation, unlike taking
hundreds of numerical derivatives (finite
differences) which can actually add noise to the
data.
• The damping term is set in the field once and the
downhole card is then computed.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Polished Rod Position
80
70
60
PRP, in
50
40
30
20
10
0
0
2
4
6
Time, sec
8
10
12
© Brex, LLC 2012
Polished Rod Load
20000
18000
16000
14000
PRL, lbs
12000
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4
6
Time, sec
8
10
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© Brex, LLC 2012
20000
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Measured
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1 terms
8000
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0
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© Brex, LLC 2012
20000
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Measured
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2 terms
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0
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© Brex, LLC 2012
20000
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Measured
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3 terms
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0
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© Brex, LLC 2012
20000
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4 terms
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© Brex, LLC 2012
20000
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Measured
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5 terms
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0
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20000
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Measured
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7 terms
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0
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20000
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Measured
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9 terms
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0
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20000
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Measured
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11 terms
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0
0
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Measured
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15 terms
8000
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0
0
2
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8
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20000
18000
16000
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Measured
10000
20 terms
8000
6000
4000
2000
0
0
2
4
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8
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© Brex, LLC 2012
Polished Rod Position
80
70
60
PRP, in
50
40
30
20
10
0
0
2
4
6
Time, sec
8
10
12
© Brex, LLC 2012
Polished Rod Load
20000
18000
16000
14000
PRL, lbs
12000
10000
8000
6000
4000
2000
0
0
2
4
6
Time, sec
8
10
12
© Brex, LLC 2012
Surface Dynagraph
20000
18000
16000
14000
PRL, lbs
12000
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8000
6000
4000
2000
0
0
10
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30
40
PRP, in
50
60
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80
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Surface Dynagraph - 1 term
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
60
70
80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 2 terms
20000
15000
Load, lbs
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5000
0
0
10
20
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40
50
60
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80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 3 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
60
70
80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 4 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
60
70
80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 5 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
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50
60
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-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 7 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
60
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80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 11 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
60
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80
-5000
-10000
Stroke, in
© Brex, LLC 2012
Surface Dynagraph - 15 terms
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
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-5000
-10000
Stroke, in
© Brex, LLC 2012
Dynagraphs
20000
15000
Load, lbs
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5000
0
0
10
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50
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-5000
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Stroke, in
© Brex, LLC 2012
Dynagraphs - Zero Damping
20000
15000
Load, lbs
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5000
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0
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Stroke, in
© Brex, LLC 2012
Dynagraphs - Too Much Damping
20000
15000
Load, lbs
10000
5000
0
0
10
20
30
40
50
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-5000
-10000
Stroke, in
© Brex, LLC 2012
Dynagraphs – Way Too Much Damping
20000
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Load, lbs
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Stroke, in
© Brex, LLC 2012
The Everitt-Jennings Method
• T.A. Everitt and J.W. Jennings used finite differences to
solve the wave equation in 1990, cf. An Improved Finite
Difference Calculation of Downhole Dynamometer
Cards for Sucker-Rod Pumps, SPE 18189, SPE Annual
Technical Conference and Exhibition, Houston Oct. 2-5.
• The Everitt-Jennings method incorporates an iteration
on the net stroke and damping factor.
• Weatherford developed the MEJ method in 2008.
• With the MEJ, it is possible to compute position, load
and stress at any level down the taper.
• It permits the use to manage a large group of wells with
the automatic selection of the damping factors.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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The Everitt-Jennings Method
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2012 Sucker Rod Pumping Workshop
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Finite Differences
Approximates the solutions to differential
equations by replacing derivative expressions with
finite difference quotients.
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Everitt-Jennings Algorithm (1/2)
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Everitt-Jennings Algorithm (2/2)
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2012 Sucker Rod Pumping Workshop
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Hydraulic horsepower
The hydraulic horsepower (hp) obtained as follows:
where
Q, production rate in B/D
, fluid specific gravity
Fl, fluid level in feet.
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2012 Sucker Rod Pumping Workshop
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Production Rate
The pump production rate is given by:
Where
SPM, pumping speed in strokes/minute
S, net stroke in inches
D, pump diameter in inches.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Damping Factor
The damping factor can be computed through the
equation:
Where
HPR, polished rod horsepower in hp
HH, hydraulic horsepower in hp
g, gravity constant
τ, period of a stroke in seconds
S, net stroke in inches.
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2012 Sucker Rod Pumping Workshop
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Iteration on Single Damping factor
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Iteration on Dual Damping factors
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2012 Sucker Rod Pumping Workshop
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Deviated Wells (1/3)
• In the case of
deviated
wells,
mechanical
friction
becomes an
non
negligeable
force.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Deviated Wells (2/3)
• The dynamic behavior of the rod string is different for
deviated wells than for vertical wells.
• In vertical wells, the rod string is assumed to not
move laterally.
• The only friction to consider is the friction of viscous
nature, since mechanical friction is not consequential
enough to be considered.
• In deviated wells however, mechanical friction
becomes non-negligible since there is extensive
contact between the rods, the rod couplings and the
tubing.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
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Deviated Wells (3/3)
• Also, since the well is deviated, some sections of the
rod string can be bent between two couplings in the
middle of a “dog leg” turn, which introduces the
concept of curvature of the rod string.
• While analyzing the behavior of the rod string, it is
therefore essential to capture the behavior of the
longitudinal stress waves as well as the lateral stress
waves of the rod element.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
54
Rod Pumping Book by Sam Gibbs
• ROD PUMPING
Modern Methods of
Design, Diagnosis,
and Surveillance
• Available with Ronda
Brewer.
• Visit www.samgibbs.net
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
55
Copyright
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the Southwestern Petroleum Short Course (SWPSC), rights to:
– Display the presentation at the Workshop.
– Place it on the www.alrdc.com web site, with access to the site to be as
directed by the Workshop Steering Committee.
– Place it on a CD for distribution and/or sale as directed by the Workshop
Steering Committee.
Other use of this presentation is prohibited without the expressed
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author(s) may publish this material in other journals or magazines if
they refer to the Sucker Rod Pumping Workshop where it was first
presented.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
56
Disclaimer
The following disclaimer shall be included as the last page of a Technical Presentation or
Continuing Education Course. A similar disclaimer is included on the front page of the Sucker Rod
Pumping Web Site.
The Artificial Lift Research and Development Council and its officers and trustees, and the Sucker
Rod Pumping Workshop Steering Committee members, and their supporting organizations and
companies (here-in-after referred to as the Sponsoring Organizations), and the author(s) of this
Technical Presentation or Continuing Education Training Course and their company(ies), provide
this presentation and/or training material at the Sucker Rod Pumping Workshop "as is" without any
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Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop
57