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9 th Annual Sucker Rod Pumping Workshop Renaissance Hotel Oklahoma City, Oklahoma September , 2013 Wave Equation: Derivation and Analysis Victoria M. Pons, Ph. D. Weatherford

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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 , Sucker Rod Pumping Workshop2

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Sept , Sucker Rod Pumping Workshop3 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. Sept , Sucker Rod Pumping Workshop4

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Conventional pumping unit Sept , Sucker Rod Pumping Workshop5

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The 1D Damped Wave Equation Calculating downhole conditions is difficult because of the sucker rods 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 , Sucker Rod Pumping Workshop6

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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 F X, Tension force representing the pull from below on the rod element F X+ΔX, The damping force opposing the movement, F D, resulting from fluid friction on the rod elements surface. Sept , Sucker Rod Pumping Workshop7

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Newtons Second Law Sept , Sucker Rod Pumping Workshop8

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Breakdown of Forces (1/2) Using the stresses present in the rod sections and Hookes 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 , Sucker Rod Pumping Workshop9

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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 , Sucker Rod Pumping Workshop10

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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 , Sucker Rod Pumping Workshop11

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The 1D Damped Wave Equation (2/2) Therefore the condensed form of the above equation reads: Acceleration ElasticityDamping Cf. Sucker-Rod Pumping Manual, by Gábor Takács. Sept , Sucker Rod Pumping Workshop12

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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 , Sucker Rod Pumping Workshop13

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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 , Sucker Rod Pumping Workshop14

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© Brex, LLC 2012

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Gibbs Method Sept , Sucker Rod Pumping Workshop38

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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 The Everitt-Jennings method incorporates an iteration on the net stroke and damping factor. Weatherford developed the MEJ method in 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 , Sucker Rod Pumping Workshop39

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The Everitt-Jennings Method Sept , Sucker Rod Pumping Workshop40

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Finite Differences Approximates the solutions to differential equations by replacing derivative expressions with finite difference quotients. Sept , Sucker Rod Pumping Workshop41

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Everitt-Jennings Algorithm (1/2) Sept , Sucker Rod Pumping Workshop42

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Everitt-Jennings Algorithm (2/2) Sept , Sucker Rod Pumping Workshop43

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Hydraulic horsepower The hydraulic horsepower (hp) obtained as follows: where Q, production rate in B/D, fluid specific gravity F l, fluid level in feet. Sept , Sucker Rod Pumping Workshop44

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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 , Sucker Rod Pumping Workshop45

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Damping Factor The damping factor can be computed through the equation: Where H PR, polished rod horsepower in hp H H, hydraulic horsepower in hp g, gravity constant τ, period of a stroke in seconds S, net stroke in inches. Sept , Sucker Rod Pumping Workshop46

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Iteration on Single Damping factor Sept , Sucker Rod Pumping Workshop47

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Iteration on Dual Damping factors Sept , Sucker Rod Pumping Workshop48

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Deviated Wells (1/3) In the case of deviated wells, mechanical friction becomes an non negligeable force. Sept , Sucker Rod Pumping Workshop49

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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 , Sucker Rod Pumping Workshop50

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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 , Sucker Rod Pumping Workshop51

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Rod Pumping Book by Sam Gibbs ROD PUMPING Modern Methods of Design, Diagnosis, and Surveillance Available with Ronda Brewer. Visit Sept , Sucker Rod Pumping Workshop52

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Sept , Sucker Rod Pumping Workshop53 Copyright Rights to this presentation are owned by the company(ies) and/or author(s) listed on the title page. By submitting this presentation to the Sucker Rod Pumping Workshop, they grant to the Workshop, the Artificial Lift Research and Development Council (ALRDC), and the Southwestern Petroleum Short Course (SWPSC), rights to: –Display the presentation at the Workshop. –Place it on the 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 written permission of the author(s). The owner company(ies) and/or 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.

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Sept , Sucker Rod Pumping Workshop54 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.

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