# Wave Equation: Derivation and Analysis

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Wave Equation: Derivation and Analysis
Title slide Keep the title brief and to one line? The second line (in smaller type style) is for the date of the presentation – depending on your presentation, you may also need to include Location and/or Presenter’s name/s and Job title if needed – as a such a second line may be used for these purposes. Do not include unnecessary information in your presentation title – starting off with clear and simple messages will set the style and focus the audience for what is to follow. 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. 2013 Sucker Rod Pumping Workshop Sept , 2013

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 , 2013 2013 Sucker Rod Pumping Workshop

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

Conventional pumping unit
2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

Forces acting on a Rod Element
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. 2013 Sucker Rod Pumping Workshop Sept , 2013

2013 Sucker Rod Pumping Workshop
Newton’s Second Law 2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

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 . 2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2013 Sucker Rod Pumping Workshop Sept , 2013

2013 Sucker Rod Pumping Workshop
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. 2013 Sucker Rod Pumping Workshop Sept , 2013

2013 Sucker Rod Pumping Workshop
Gibbs Method 2013 Sucker Rod Pumping Workshop Sept , 2013

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. 2012 Sucker Rod Pumping Workshop Sept , 2012

The Everitt-Jennings Method
2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
Finite Differences Approximates the solutions to differential equations by replacing derivative expressions with finite difference quotients. 2012 Sucker Rod Pumping Workshop Sept , 2012

Everitt-Jennings Algorithm (1/2)
2012 Sucker Rod Pumping Workshop Sept , 2012

Everitt-Jennings Algorithm (2/2)
2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
Hydraulic horsepower The hydraulic horsepower (hp) obtained as follows: where Q, production rate in B/D , fluid specific gravity Fl, fluid level in feet. 2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
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. 2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
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. 2012 Sucker Rod Pumping Workshop Sept , 2012

Iteration on Single Damping factor
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Iteration on Dual Damping factors
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2012 Sucker Rod Pumping Workshop
Deviated Wells (1/3) In the case of deviated wells, mechanical friction becomes an non negligeable force. 2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
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. 2012 Sucker Rod Pumping Workshop Sept , 2012

2012 Sucker Rod Pumping Workshop
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. 2012 Sucker Rod Pumping Workshop Sept , 2012

Rod Pumping Book by Sam Gibbs
ROD PUMPING Modern Methods of Design, Diagnosis, and Surveillance Available with Ronda Brewer. Visit 2012 Sucker Rod Pumping Workshop Sept , 2012

2013 Sucker Rod Pumping Workshop
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2013 Sucker Rod Pumping Workshop
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 warranty of any kind, express or implied, as to the accuracy of the information or the products or services referred to by any presenter (in so far as such warranties may be excluded under any relevant law) and these members and their companies will not be liable for unlawful actions and any losses or damage that may result from use of any presentation as a consequence of any inaccuracies in, or any omission from, the information which therein may be contained. The views, opinions, and conclusions expressed in these presentations and/or training materials are those of the author and not necessarily those of the Sponsoring Organizations. The author is solely responsible for the content of the materials. The Sponsoring Organizations cannot and do not warrant the accuracy of these documents beyond the source documents, although we do make every attempt to work from authoritative sources. The Sponsoring Organizations provide these presentations and/or training materials as a service. The Sponsoring Organizations make no representations or warranties, express or implied, with respect to the presentations and/or training materials, or any part thereof, including any warrantees of title, non-infringement of copyright or patent rights of others, merchantability, or fitness or suitability for any purpose. Sept , 2013 2013 Sucker Rod Pumping Workshop

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