Breakout --- Wave Equation.ppt

8/10/2019 Breakout --- Wave Equation.ppt

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8thAnnual 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


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Sept. 25 - 28, 20122012 Sucker Rod Pumping Workshop 2

Reciprocating Rod Lift

The most widely used mean of artificial lift is sucker rodpumping.

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, adownhole 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 becauseof 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.



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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 elementFX,

Tension force representing the

pull from below on the rodelement FX+X,

The damping force opposingthe movement, FD, resulting

from fluid friction on the rod

elements surface.


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Newtons Second Law



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Breakdown of Forces (1/2)

Using the stresses present in the rod sections andHookes law the tension forces can be rewritten as:

The acceleration can be written as:

the mass as

Where is the density and gthe 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 Gbor Takcs.



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True Loads vs. Effective Loads

The difference between true loads and effective loadsis 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 verticallydownward 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 thedata.

The damping term is set in the field once and the

downhole card is then computed.



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0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

PRP,

in

Time, sec

Polished Rod Position

Brex, LLC 2012


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0

2000

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0 2 4 6 8 10 12

PRL,

lbs

Time, sec

Polished Rod Load

Brex, LLC 2012


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0

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12000

14000

16000

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0 2 4 6 8 10 12

Measured

1 terms

Brex, LLC 2012


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0

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0 2 4 6 8 10 12

Measured

2 terms

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0

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Measured

4 terms

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0

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0 2 4 6 8 10 12

Measured

5 terms

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0

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0 2 4 6 8 10 12

Measured

7 terms

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0

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20000

0 2 4 6 8 10 12

Measured

9 terms

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0

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0 2 4 6 8 10 12

Measured

11 terms

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0

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Measured

15 terms

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0

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0 2 4 6 8 10 12

Measured

20 terms

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0

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0 2 4 6 8 10 12

PRP,in

Time, sec

Polished Rod Position

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0

2000

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8000

10000

12000

14000

16000

18000

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0 2 4 6 8 10 12

PRL,

lb

s

Time, sec

Polished Rod Load

Brex, LLC 2012


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0

2000

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16000

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PRL,

lb

s

PRP, in

Surface Dynagraph

Brex, LLC 2012


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-10000

-5000

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5000

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0 10 20 30 40 50 60 70 80

Load,

lbs

Stroke, in

Surface Dynagraph - 1 term

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-10000

-5000

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5000

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Load,

lbs

Stroke, in

Surface Dynagraph - 2 terms

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-5000

0

5000

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Load,

lbs

Stroke, in


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-5000

0

5000

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0 10 20 30 40 50 60 70 80

Load,

lbs

Stroke, in


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Load,

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Stroke, in


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-10000

-5000

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Load,

lbs

Stroke, in


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-5000

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Load,

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Stroke, in


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Load,

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Stroke, in

Dynagraphs

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-10000

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Load,

lbs

Stroke, in

Dynagraphs - Zero Damping

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Load,

lbs

Stroke, in

Dynagraphs - Too Much Damping

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42/57

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, loadand 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.



43/57

The Everitt-Jennings Method



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Finite Differences

Approximates the solutions to differentialequations 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.



48/57

Production Rate

The pump production rate is given by:

WhereSPM, pumping speed in strokes/minute

S, net stroke in inches

D, pump diameter in inches.



49/57

Damping Factor

The damping factor can be computed through theequation:

Where

HPR, polished rod horsepower in hp

HH, hydraulic horsepower in hp

g, gravity constant

, period of a stroke in seconds

S, net stroke in inches.



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Iteration on Single Damping factor



51/57


52/57

Deviated Wells (1/3)

In the case ofdeviated

wells,

mechanical

friction

becomes an

non

negligeable

force.

Sept. 25 - 28, 2012 2012 Sucker Rod Pumping Workshop 52


53/57

Deviated Wells (2/3)

The dynamic behavior of the rod string is different fordeviated 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 viscousnature, since mechanical friction is not consequential

enough to be considered.

In deviated wells however, mechanical friction

becomes non-negligible since there is extensivecontact between the rods, the rod couplings and the

tubing.



54/57


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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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Display the presentation at the Workshop.

Place it on the www.alrdc.com web site, with access to the site to be asdirected by the Workshop Steering Committee.

Place it on a CD for distribution and/or sale as directed by the WorkshopSteering Committee.

Other use of this presentation is prohibited without the expressed

written permission of the author(s). The owner company(ies) and/orauthor(s) may publish this material in other journals or magazines ifthey refer to the Sucker Rod Pumping Workshop where it was firstpresented.


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The Artificial Lift Research and Development Council and its officers and trustees, and the SuckerRod Pumping Workshop Steering Committee members, and their supporting organizations andcompanies (here-in-after referred to as the Sponsoring Organizations), and the author(s) of thisTechnical Presentation or Continuing Education Training Course and their company(ies), providethis presentation and/or training material at the Sucker Rod Pumping Workshop "as is" without anywarranty of any kind, express or implied, as to the accuracy of the information or the products orservices referred to by any presenter (in so far as such warranties may be excluded under anyrelevant law) and these members and their companies will not be liable for unlawful actions and anylosses or damage that may result from use of any presentation as a consequence of anyinaccuracies in, or any omission from, the information which therein may be contained.

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Breakout --- Wave Equation.ppt

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