Transcript important note

9th Annual Sucker Rod Pumping
Workshop
Renaissance Hotel
Oklahoma City, Oklahoma
September 17 - 20, 2013
Wave Equation:
Derivation and Analysis
Victoria M. Pons, Ph. D.
Weatherford
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.
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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.
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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.
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Conventional pumping unit
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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.
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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,
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• The damping force opposing
the movement, FD, resulting
from fluid friction on the rod
2013 Sucker Rod
Pumping Workshop
element’s
surface.
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Newton’s Second Law
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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.
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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.
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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
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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.
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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.
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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.
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Polished Rod Position
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PRP, in
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0
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Time, sec
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Polished Rod Load
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Surface Dynagraph - 1 term
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Surface Dynagraph - 2 terms
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Surface Dynagraph - 3 terms
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Surface Dynagraph - 4 terms
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Surface Dynagraph - 5 terms
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Surface Dynagraph - 7 terms
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Surface Dynagraph - 11 terms
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Surface Dynagraph - 15 terms
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Dynagraphs - Zero Damping
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Dynagraphs - Too Much Damping
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Dynagraphs – Way Too Much Damping
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Gibbs Method
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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.
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The Everitt-Jennings Method
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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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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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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.
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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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Iteration on Single Damping factor
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Iteration on Dual Damping factors
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Deviated Wells (1/3)
• In the case of
deviated
wells,
mechanical
friction
becomes an
non
negligeable
force.
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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.
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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.
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Rod Pumping Book by Sam Gibbs
• ROD PUMPING
Modern Methods of
Design, Diagnosis,
and Surveillance
• Available with Ronda
Brewer.
• Visit www.samgibbs.net
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presented.
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Disclaimer
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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
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Sept. 17 - 20, 2013
2013 Sucker Rod Pumping Workshop
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