Iterating on Damping when Solving the Wave Equation - ALRDC

8 th Annual Sucker Rod Pumping
Workshop
Renaissance Hotel
Oklahoma City, Oklahoma
September 25 - 28, 2012
<strong>Iterating</strong> on Damping when Solving the
Wave Equation
Victoria Pons, Ph. D.
Weatherford

Introduction and Motivations (1/3)
• In sucker rod pumping unit, a surface dynamometer
system is used to record the position and load data
at the surface.
• Most companies today use some type of software
analysis to compute the downhole data.
• The most accurate and popular of these methods is
to compute the downhole data from the surface data
by solving the one dimensional damped wave
equation.
• When calculating downhole data using the wave
equation, it is essential to handle damping properly.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 2

Introduction and Motivations (2/3)
• In the modified Everitt-Jennings method, finite
differences are used to solve the wave equation.
• Position and load including stress is computed at
each increment down the wellbore.
• The modified Everitt-Jennings method also
includes an iteration on the net stroke and
damping factor, which automatically selects a
damping factor for each stroke.
• There has been some controversy about the use
of separate upstroke and downstroke damping
factors.
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2012 Sucker Rod Pumping Workshop 3

Introduction and Motivations (3/3)
• The conditions are very different between the
upstroke and downstroke, mainly due to the direction
of the fluid flowing with or against the rods.
• A new method for splitting and iterating on the
upstroke and downstroke damping factor separately,
was presented at the 2012 South Western Petroleum
Short Course in Lubbock TX, April 18-19 th 2012.
• The dual iteration on the damping factors combines a
two step process for damping factor modification
along with a bisection-like algorithm for the accuracy
of the damping factor selection.
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2012 Sucker Rod Pumping Workshop 4

1-Dimensional Damped Wave Equation
• The condensed one-dimensional wave equation
reads:
where the acoustic velocity is given by:
• The above equation only considers friction forces of a
viscous nature.
• Through the use of finite differences, the rod string is
divided into finite difference nodes of length L i (ft),
density ρ i (lbm/ft 3 ) and area A i (in 2 ).
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2012 Sucker Rod Pumping Workshop 5

MODIFIED EVERITT-JENNINGS (1/4)
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.
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MODIFIED EVERITT-JENNINGS (2/4)
The hydraulic horsepower (hp) obtained as follows:
where
Q, production rate in B/D
, fluid specific gravity
F l
, fluid level in feet.
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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2012 Sucker Rod Pumping Workshop 7

MODIFIED EVERITT-JENNINGS (3/4)
• The damping factor relies on the net stroke and
the hydraulic horsepower. Both of which are
unknown or approximated at the beginning of the
calculation.
• The hydraulic horsepower is a factor of the fluid
level, which reflects the amount of fluid in the
wellbore annulus. Therefore unless calculated for
that particular stroke, the fluid level may be
highly inaccurate.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 8

MODIFIED EVERITT-JENNINGS (4/4)
• The original iteration on the net stroke and the
damping factor was therefore modified to
incorporate a fluid level calculation and
initialization step of the variables needed for the
iteration.
• The iteration on damping was further modified to
treat either one damping factor or separate
upstroke and downstroke damping factors.
• Both iterations on damping are enhanced with a
fluid load line calculation (FLLC) complete with a
concavity test.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 9

FLLC & Concavity Testing
• A set of theoretical fluid load lines called
F0_up_actual and F0_down_actual are calculated by
finding the right hand corners of the downhole card.
• Another set of fluid load lines, called F0_up_calc and
F0_down_calc, or measured fluid load lines are
calculated using statistics.
• When comparing the position of the above fluid load
lines, it becomes possible to determine how little or
how much damping is needed.
• The FLLC and Concavity test are made part of the
convergence criteria for both iterations.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 10

Iteration on Damping
• For more details on the iteration on dual damping, cf.
Modified Everitt-Jennings Algorithm with Dual Iteration
on the Damping Factors, Victoria Pons, 2012 South
Western Petroleum Short Course, Lubbock TX, April
18-19 th 2012.
• The iteration on single damping factor is also
enhanced with a FLLC and concavity test.
• The convergence criteria differ from the iteration on
dual damping since both criteria have to be satisfied at
the same time.
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Iteration on Single damping Factor
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Iteration on Dual Damping Factors
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Results
• Results for the comparison of the iteration on single
and dual damping factors and the Delta program are
presented.
• The Delta program uses the Gibbs’ method as coded in
the LOWIS software.
• All of the downhole cards presented as results have
been shifted to the zero line to offer a better
comparison between the methods.
• As a precaution, the damping factors are corrected so
that they do not exceed the value 1 or go below 0.
Therefore a limit value for each end of the range is set
in place with D max = 0.99 and D min = 0.01.
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Example 1
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Example 2
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Example 3
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Example 4
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Example 5
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Example 6
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Example 7
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Example 8
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Conclusions
• The addition of the FLLC and concavity test greatly
improve the results and speed of convergence of the
algorithm.
• The great advantage of the automatic iteration on the
net stroke and the damping factor is the fact that the
damping factor is adjusted automatically without human
intervention.
• For most examples, the results from the iteration on the
single damping factor and from the iteration on dual
damping factors are similar. Both iterative methods
prove consistently better than the results obtained
through the Delta program.
Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 23

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Sept. 25 - 28, 2012
2012 Sucker Rod Pumping Workshop 24

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Sept. 25 - 28, 2012
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