9 th annual sucker rod pumping workshop renaissance hotel oklahoma city, oklahoma september 17 - 20,...

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.

Sept. 17 - 20, 2013 2013 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.

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.


Conventional pumping unit


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.


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.


Newton’s Second Law


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.


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.


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 .


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.


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.


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.


Gibbs Method


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.


The Everitt-Jennings Method


Finite Differences

Approximates the solutions to differential equations by replacing derivative expressions with

finite difference quotients.


Everitt-Jennings Algorithm (1/2)


Everitt-Jennings Algorithm (2/2)


Hydraulic horsepower

The hydraulic horsepower (hp) obtained as follows:

where

Q, production rate in B/D

, fluid specific gravity

Fl, fluid level in feet.


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.


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.


Iteration on Single Damping factor


Iteration on Dual Damping factors


Deviated Wells (1/3)

• In the case of deviated wells, mechanical friction becomes an non negligeable force.



• 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.



• 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.


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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9 th annual sucker rod pumping workshop renaissance hotel oklahoma city, oklahoma september 17 - 20,...

Documents

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reciprocating rod

rod sections

rod couplings

polished rod

suckerrod pumping manual

rod elements surface

rod element forces