An lmproved Finite-Difference
Calculation of Downhole
Dynamometer Cards for
Sucker-Rod Pumps
T.A. Everltt,
*
SPE, Chevron Oil Field Research Co., and J.W. Jennlngs,* *
SPE, Texas A&M U.Summary. This paper presents a finite-difference representation of the wave equation developed for diagnostic analyses of sucker- rod pumping systems. A consistent method of computing the viscous damping term associated with the damped-wave equation is also
lntroductlon
Sucker-rod pumping is the most widely used means of artificial lift. About 85% to 90% of all producing wells in the U.S. are rod- -
pumped. Thus, a reliable method of analyzing these pumping sys- tems is a necessity.
For many years, the surface dynamometer has been used to ana- lyze sucker-rod systems. Interpretation of actual pump conditions from surface dynamometer cards is often difficult, if not impossi- ble. Results obtained from surface cards are strictly qualitative and are dependent on the analyzer's expertise.
The ideal analysis procedure would be to measure the actual pump conditions with a downhole dynamometer. However, this situation is not economically feasible. Therefore, an accurate method of cal- dating downhole pump cards from measured surface cards is need- ed. This paper presents a method for calculating these downhole cards that uses a finite-difference representation of the wave equa- tion. First, a brief description of previous calculation techniques is given.
Previous Methods. Past work involving the analysis of sucker-rod
pumping systems can be divided into two categories. One category involves predicting the performance of new sucker-rod installations by calculating surface load from known surface position and pump load. The other category deals with the diagnosis of existing pump- ing installations by determining actual pump conditions from meas- ured surface conditions. This paper focuses on the latter category. Snyderl was the first to develop a method for calculating down- hole forces and displacements. His technique incorporates the method of characteristics to solve the undamped-wave equation. Snvder assumed that the tension in the rod is the result of two force waves, f (downward wave) and g (upward wgve). The values of f and g, calculated from the surface dynamometer card, would be constant over the entire rod string for the undamped solution. Snyder
corrected for damping using a concentrated damping force to ad- vance the values of f and g down the rod string. These two force waves are then used to compute the downhole pump card. Snyder's method is rigorously valid oniy for a uniform sucker-rod string. Gibbs and Neely2 developed an analytical technique in 1966 for obtaining subsurface conditions. The method uses truncated Fou- rier series approxirnations of the ID, damped-wave equation to de- termine load and displacement. The relative smoothness of the loadltime and displacement/time curves is important in a Fourier analysis; however, the load function approaches a square wave at the pump. The Fourier series solution oscillates at the discontinui- ties of this square wave, restricting the number of terms that can be taken in the series solution and still preserve accuracy. In turn, the smaller number of terms in the series causes the solution to be less accurate. Gibbs and Neely's analytical method has become the primary means of calculating downhole dynamometer cards.
'Now at Shell Ofkhore Inc. "Now retired.
Copyright 1992 Society of Petroleum Engineers
Knapp3 was the first to present a method for computing down- hole dynamometer cards using finite differences. His formulation does not account for variable rod diameter or rod material. Knapp's theory was used in the development of the model presented here.
Model Development
The behavior of the sucker-rod pumping system is complex. This study entails modeling a portion of this system, namely the sucker- rod string from the surface to the pump. The wave equation is ideal for this purpose because the problem at hand involves the propa- gation of waves in a continuous medium.
Wave Equation. The 1D wave equation is a linear hyperbolic dif-
ferential equation that describes the longitudinal vibrations of a long, slender rod. Using this equation with viscous damping, we can ap- proximate the motion of the sucker-rod string. In its simplified form, the wave equation is given by
a2u a2u au
,,2-=- fc-,
. . .
ax2 at2 at (1)
where v = m .
Eq. 1 is for the simplified case of a constant rod diameter. Mul-
tiplying through by (pA/144gC) modifies Eq. 1 to account for vari-
able rod diameter:
a2u PA a 2 ~ PA
au
m-=--
+c- - > ....f.............ax2 i44gr at2 i44gr at
(2)
which is the form of the wave equation used to develop the finite- difference model. Several researchers4-6 gave a detailed deriva- tion of the wave equation, beginning with a force balance on an element of the sucker-rod string.
Generally, solving the wave equation would require two bound- ary conditions and two initial conditions because the equation con* tains second-order derivatives in both time and space. However, the problem solved here does not require initial conditions because oniy periodic (steady-state) solutions are desired. Because the ef- fects of the initial conditions have faded in periodic solutions, only two boundary conditions are required.
The two required boundary conditions are time histories of polished-rod load and displacement. These conditions can be ob- tained directly from a surface dynamometer point plot, a graph of polished-rod load vs. displacement recorded at evenly spaced incre- ments of time. Surface cards are typically recorded as continuous plots, however, and not as point plots at equal time increments. In this case, pumping-unit kinematics must be used to attain a rela- tionship between time and polished-rod displacement. Svinos7 de- veloped a versatile method for performing the kinematic analysis of pumping units. His method was used in this study to obtain sur- face position at evenly spaced time increments. Constant speed was assumed for the prime mover, and inertia effects were neglected.
Timesteps U) 1 2 3 . . . n o U U U U U U U U U U U U U U U U U U U U U U U U U U U U u 1 U U U U U U U U U U U U U U U u u U U U U U U U U U U U U 2 ? u u u u u u u u u u u u u u u u u u u u u u u u u u u ? 3 ? ? u u u u u u u u u u u u u u u u u u u u u u u u u ? ? A
= .
? ? ? u u u u u u u u u u u u u u u u u u u u u u u ? ? ? tn a:
.
? ? ? ? u u u u u u u u u u u u u u u u u u u u u ? ? ? ?z
.
? ? ? ? ? u u u u u u u u u u u u u u u u u u u ? ? ? ? ?.
? ? ? ? ? ? u u u u u u u u u u u u u u u u u ? ? ? ? ? ?.
? ? ? ? ? ? ? u u u u u u u u u u u u u u u ? ? ? ? ? ? ? m ? ? ? ? ? ? ? ? u u u u u u u u u u u u u ? ? ? ? ? ? ? ?u
-
Displacements That Can Be Calculated ? - Displacements That Cannot Be Calculated -- - -- Fig. 1-Dlsplacements not calculated.
Finite-Difference Analogs. Finite differences are used in the model
development to obtain the numerical solution of the ID, dampeú- wave equation. Taylor series approximations are used to generate finitedifference analogs for the derivatives of displacement that ap-
pear in the wave equation. Eq. 3 is the result of substituting these
Taylor series approxirnations into Eq. 2 .
(pAl144gc) + +(pAI144gc)- where a = -
Ar2 2
Eq. 3 is used to transmit the surface position downhole by cal- culating displacements at each node along the rod string until the last node just above the pump is reached. The Appendix gives a complete derivation of Eq. 3.
Note that Eq. 3 requires knowing displacements two nodes be- hind in space, ui, j and ui- 1 , relative to the node being calculat-
ed, u i + l , i . Therefore, to start the solution, the displacements uo,
and u l , must be known for al1 tirnesteps. The displacements at
uo, are known from the surface dynamometer card. The displace-
ments at u l , are calculated with Hooke's law:
Substituting the polished-rod load, FPR, for F and a first-order-
correct forward-difference analog for aulax yields
Thus, the displacements needed to start Eq. 3 are obtained. Note
that FpR is the dynamic polished-rod load (the surface-recorded
load minus the buoyed weight of the rods).
Another interesting aspect of Eq. 3 is that the determination of
ui+ l , , requires knowing the displacements ui, j - i and ui, j+ 1. At
the lower end of the timestep scale (when j=O), u i j j - ~ does not exist; at the upper end of the timestep scale (when j=n), u i , j + i
does not exist. Therefore, the two endpoints at each node cannot
be calculated. Fig. 1 illustrates this phenomenon.
If the calculations are started from the surface'node (i=O) with only one cycle of lo& and positions, the solution at the pump (i=m) will not represent a complete cycle because the endpoints at Nodes 2 through m cannot be calculated. To solve this problem, enough surface points (loads and positions) must be repeated before the calculations begin so that when points are lost as the calculations progress downhole, the pump card will still represent a complete cycle. The number of points that will be lost can be determined
TABLE 1-DATA FOR CASE 1 Rod Data
Diameter, in. 0.75 Material Steel
Length, ft 2,000 Elasticity, psi 30.5~ lo6
Downhole Data
Plunger diameter, in. 2.5 Pump depth, ft 2,000
Pumping speed, strokeslmin 15 Fluid level, ft 2,000
Pump condition Full Fluid specific gravity 1 .O
Pumping-Unit Data
Unit designation C-114-119-54 Stroke length, in. 54
Manufacturer Lufkin Rotation Clockwise
TABLE 2-DATA FOR CASE 2 Rod Data
Diameter Length Elasticity
(in.) (ft) Material (PSi)
1.250 3,000 Fiberglass 8.03 x lo6
0.875 3,000 Steel 30.5 x lo6
Downhole Data
Plunger diameter, in. 1.5 Pump depth, ft 6,000
Pumping speed, strokeslmin 10 Fluid level, ft 6,000
Pump condition Fluid pound Fluid specific gravity 1 .O Pumping-Unit Data
Unit designation C-228-213-100 Stroke length, in. 1 O0
Manufacturer American Rotation Clockwise
10000 7
o 1 t J
O 20 40 60
Position (In.)
Flg. 2-Surface dynamometer card for Case 1.
from the number of nodes (number of lost points=number of nodes X 2 -2). For example, if the rod string has 20 nodes, 38 of the original surface points wiU not be transmitted to the pump. In this case, 38 surface points must be repeated before the calcula- tionsbegin(i.e., Pointn+l=Point 1, Pointn+2=Point2.. .Point n+38=Point 38) to obtain a complete pump cycle.
As mentioned before, Eq. 3 transmits displacements downhole to the node just above the pump. To obtain pump displacement, a different equation must be used because (EA)+ and [pAl
(144g,)] + do not exist at the pump. The equation used is the sim- plified form of Eq. 3 for constant rod diameter and rod material:
Now that pump displacement is known, pump load may be cal- culated. Hooke's law (Eq. 4) is used, substituting a second-order- correct backward difference for aulax:
Thus, equations have been generated for calculating pump dy- namometer cards. In the model development, higher-order Taylor series approximations could have been chosen for the derivatives. It is expected, however, that the additional accuracy would be small and would not compensate for the increased computational speed and supplementary storage requirements.
Stability Criterion. Because the model developed here is an ex- plicit finitedifference method, the stabiiity of the solution is of sig- nificant importante and must be considered at this point. Gibbs*
6000-
-
Diagnostic Model-- --
Predlctive Model 9 m P dP
3
-2000 t I I I 1 O 10 20 30 40 50 Position (In.)Flg. 4-Comparison of numerical diagnostic and predlctlve models for Case 1.
noted the stability criterion of the finitedifference predictive model as
Remember that the predictive and diagnostic models solve two different problems. The predictive model calculates (predicts) sur- face dynamometer cards, whereas the diagnostic model calculates downhole cards from known surface conditions. The diagnostic model solves for displacements ahead in space; the predictive model, on the other hand, solves for displacements ahead in time. Thus, the stability criterion for the diagnostic model is not the same as
Eq. 8. Everitt4 derived the stability criterion for the diagnostic model. For brevity, only the final result is shown here:
A xlvAt 5 l.
...
(9)This criterion is the converse of Eq. 8, the stability criterion of the predictive model.
Model Verltlcatlon
Using the finitedifference equations presented earlier, we devel- opedalgorithms and progra&ed them in FORTRAN on a personal computer. Standards used to test the model were the fínitedifference predi~tive model and the Fourier series (analytical) diagnostic model. Schafer and Jenningss programmed these two models and studied the parameters involved. These programs were used to ve@ the numerical diagnostic model developed here.
Predicave-Model Comparisons. The predictive model calculates
surface load from known surface position and pump load. The proce- dure for comparison is as follows.
9 ID P d P
8
A-
Diagnostic Model----
Predictive Model -20000 1 I I 0 I 20 40 60 80 100 Position (In.)Flg. 5-Comparison of numerlcal diagnostlc and predlctlve models for Case 2.
TABLE 3-DATA FOR CASE 3 Rod Data
Diameter Length Elasticity
(in.) (ft) Material (PS~)
0.875 1,500 Steel 30.5~
loe
O. 750 2,400 Steel 30.5~
lo6
Downhole Data
Plunger diameter, in. 2.25 Pump depth, ft 3,900
Pumping speed, strokeslmin 8 Fluid level, ft 2,850
Fluid specific gravity 0.92
Pumping-Unit Data
Unit designation M-228-256-120 Stroke length, in. 121
Manufacturer Lufkin Rotation Counterclockwise
TABLE 4-DATA FOR CASE 4 Rod Data
Diameter, in. 0.75 Material Steel
Length, ft 3,179 Elasticity, psi 30.5~
lo6
Downhole Data
Plunger diameter, in. 1.5 Pump depth, ft 3,234
Pumping speed, strokeslmin 10 Fluid level, ft 3,234
Fluid specific gravity 0.8
Pumping-Unit Data
Unit designation C-114-143-64 Stroke length, in. 64
Manufacturer Lufkin Rotation Counterclockwise
1. Generate surface cards from known downhole conditions with the predictive model.
2. Calculate pump cards with the numerical diagnostic model by use of the surface cards generated in Step 1.
3. Compare pump cards calculated with the numerical diagnos- tic model to actual pump cards in the predictive model.
Tables 1 and 2 are examples of artificial data sets used in these
comparisons. Figs. 2 and 3 show the surface cards generated with the predictive model for these data sets. Figs. 4 and 5 compare the predictive-model and diagnostic-model pump cards. These ex- arnples demonstrate the diagnostic model's ability to reproduce the actual pump cards from surface information.
Analytical-Model Comparisons. The Fourier series (analytical)
model is probably the most well-known means of calculating down- hole dynamometer cards at this time. Tables 3 and 4 give exam-
ples of actual wells used to compare the numerical and analytical 15000
-
-
a n 1 uB
Ao
1 1 1 1 I O 25 50 75 100 125 Position (In.)Flg. 6-Suriace dynamometer card for Case 3.
diagnostic models. Figs. 6 and 7 show the recorded surface dy-
namometer cards for these wells, and Figs. 8 and 9 compare the
pump cards calculated with the two diagnostic models. These cases illustrate good agreement between the two models.
Numerical vs. Analyticai. Afier good agreement between the nu-
merical and analytical diagnostic models is shown, the question arises as to which model is more accurate. A description of the procedure used to answer this question follows.
1. Generate surface dynamometer cards with the predictive model. 2. Caiculate pump dynamometer cards from these surface cards with the numerical diagnostic model.
3. Calculate pump cards from the surface cards with the analyti- cal diagnostic model.
4. Plot the pump cards calculated with both diagnostic models on the same graph with the actual pump card from the predictive model
.
8000-
c..2
d
a
O A 2000 4,
I I O 20 40 60 80 Position (In.)Flg. 7-Suriace dynamometer card for Case 4.
-
Numeriml Modo17500
-
...
Analytical Model- 2 m 0 I I t 1 1
25 50 75 100 125
Position (In.)
Flg. 8-Comparlson of numerlcal and analytlcal diagnostic models for Case 3.
5. The calculated pump card that matches the pump card from the predictive model more closely is the more accurate model.
Data sets described in Tables 1 and 2 are examples used in per- forming this analysis. The generated surface cards are shown in Figs. 2 and 3 . Figs. 10 and 11 illustrate the results of the compari-
sons. In each case, 10 to 15 Fourier coefficients were used in the analytical model, as suggested by Schafer and Jemings. The an- alytical model tends to round the corners of the pump card. This is explained by the difficulty in approximating a square wave (load at the pump) with a Fourier series.
Damplng Calculatlon
Past methods of sucker-rod anaiysis lacked a consistent means of simulating the damping forces inherent in sucker-rod pumping sys- tems. Nicol6 presented a workable method of calculating damp- ing, but it requires knowledge of fluid viscosity.
The intent of this section is to define a viscous damping coeffi- cient, c, that may be used in Eq. 2 to model these damping forces. The damping coefficient used is similar to that presented by Gibbs9 :
where HH=7.36 X 10 -6qyZ.
Eq. 10 is valid for any rod material, whereas Gibbs' damping coefficient is valid only for steel rods. Ref. 4 gives a complete deri- vation of Eq. 10. Calculation of hydraulic horsepower,
HH,
re- quires knowledge of the fluid level, which may or may not be known, and of pump production rate, which is dependent on the net pump stroke. This net pump stroke may be obtained from a pump dynamometer card; however, the damping coefficient must be known before the pump card can be calculated accurately.
This predicament leads to an iterative procedure for calculating the cor- rect damping coefficient.1. Calculate the damping coefficient with a fraction of the polished-rod stroke as a first guess for the net pump stroke.
2. Calculate the pump dynamometer card and determine the net pump stroke from this pump card.
3 . Recalculate the damping coefficient, calculate a new pump
card, and again determine the net pump stroke.
Experience has shown that the net pump stroke converges with- in an acceptable tolerance in two iterations. The next task is iterat- ing until the damping coefficient converges.
4. Recalculate the damping coefficient with the now-fixed net pump stroke.
5. Calculate the pump dynamometer card with the new damping coefficient.
6 . Determine the pump horsepower by computing the area of the
pump card.
-
Nunmiul Uod.l...-
Anrlytiul Uod.l -1000 1 I 1 O 20 40 60 Position (In.)Flg. 9-Comparlson of numerlcal and analytlcal dlagnostlc models for Case 4.
7 . If the difference between the pump horsepower and the hy-
draulic horsepower is within an acceptable tolerance, the proce- dure is complete. Otherwise, adjust the damping coefficient accordingly and repeat Steps 5 and 6 .
Fig. 12 is a flow chart iliustrating the damping calculation proce- dure. Running many cases has shown that the damping coefftcient converges within at most four to five iterations.
To verify the damping calculation procedure, the predictive model was used to generate surface cards with a damping coefficient of unity. Pump dynamometer cards were calculated, aliowing the nu- merical diagnostic model to compute its own damping coefílcient. P u m ~ cards from both models were then com~ared to verifv the damPing calculation. Fig. 13 is an example of &S comparison bing
data listed in Table 2. For this exam~le. the dam~ine coefficient calculated in three iterations is withh 2% of ac&al.-
Concluslons
1. The finitedifference diagnostic technique is an excellent means of analyzing the behavior of sucker-rod systems.
2. The equations presented are valid for tapered-rod strings, rod strings with sinker bars, and steel and fiberglass rods.
3 . The stability criterion for the finitedifference diagnostic model
is the converse of the stability criterion for the finitedifference predictive model.
4. The viscous damping term associated with the wave equation may' be computed with knowledge of the fluid level.
5. The finte-difference solution better approximates square-wave pump loads than the truncated Fourier series solution (10 to 15 Fou- rier coefficients).
Nomenclature
A =' d cross-sectional area, in.2
c = damping coefficient, seconds-1 E = Young's modulus of elasticity, psi
FpR = polished-rod load, lbf
F,,, = pump load, lbf
g, = units conversion factor, 32.20bm-ft)/(lbf-sec2)
HH = hydraulic horsepower, hp
HpR = polished-rod horsepower, hp
Li = length of individual rod section, fi
q = pump production rate, BID S = polished-rod stroke, ft
S,, = net pump stroke, in.
t = time, seconds
u = rod deformation (displacement), ft
v = velocity of force propagation in rods, ft/sec
x = axial distance along the rod string, fi
Z = fluid level or net lift, ft
y = specific gravity of fluid, fraction SPE Production Engineering,
6000-
-
.
Numerical Dlagnostic. .
.
. . . .
Analytlcal Disgnostlc- - -
Predlctlve-
2000 I I 1 1 1 O 1 O 20 30 40 50 Position (In.)Flg. 10-Accuracy of numerlcal and analytlcal dlagnostlc models for Case 1.
p = rod density, lbmlft3
T = period of pumping cycle, seconds
Subscripís
i = axial distance @sitive downward)
j = time
m = node at pump
Superscripts
+
= element below element of interest-
= element above element of interestAcknowledgments
We thank the Texas Petroleum Research Cornmittee and those responsible for the Hughes Tool Professorship in Petroleum Engi- neering at Texas A&M U. for supporting this study.
I
Calculate HPhyd 1 Calculate cI
1
~alculate ~ u m p ~ a r d l1
I = l , + l1
1üetermlne Net Pump Stroke
-
I
Calculate Pump Card ?
and Pump Horsepower
No
v
1
Flg. 12-Flow chart of damplng calculatlon.1
Referentes
1. Snyder, W.E.: "A Method for Computing Down-Hole Forces and Dis- placements in Oil Wells Pumped With Sucker Rods," paper 851-37-K
presented at the 1963 Spring Meeting of the API Mid-Continent Dis- tnct Div. of Production, Amarillo, March 27-29.
2. Gibbs, S.G. and Neely, A.B.: "Computer Diagnosis of Down-Hole Conditions in Sucker Rod Pumping Wells," JPT (Jan. 1966) 91-98; Trans., AIME, 237.
3. Knapp, R.M.: "A Dynamic Investigation of Sucker-Rod Pumping," MS thesis, U. of Kansas, Topeka (Jan. 1969).
4. Eventt, T.A.: "An Improved Finite Difference Calculation of Down- hole Dynamometer Cards for Sucker Rod Pumps," MS thesis, Texas A&M U., College Station (Dec. 1987).
5. Schafer, D.J. and Jennings, J.W. : "An Investigation of Analytical and Numencal Sucker-Rod Pumping Mathematical Models," paper SPE 16919 presented at the 1987 SPE Annual Technical Conference and Exhibition, Dalias, Sept. 27-30.
6. Nicol, T.H.: "Dynamic Analysis of Sucker Rod Pumping," MS the- sis, U. of Oklahoma, Norman (Dec. 1982).
7. Svinos, J.G.: "Exact Kinematic Analysis of Pumping Units," paper SPE 12201 presented at the 1983 SPE Amual Technical Conference and Exhibition, San Francisco, Oct. 5-8.
8. Gibbs, S.G.: "Predicting the Behavior of Sucker-Rod Pumping Sys- tems," JPT (July 1963) 769-78; Trans., AIME, 228.
9. Gibbs, S.G.: "Method of Determining Sucker-Rod Pump Perform- ance," U.S. Patent No. 3,343,409 (Sept. 26, 1967).
-
Numerical Diagnostic-
Numerical Model, C = 1.02, 3 lterati0nS- - -
Predictive Model, C = 1 .O (Actual)6000
-
4000 m n d o 20008
J o-
2000 1 1 1 I l O 20 40 60 80 100 Position (In.)~
Flg. 13-Calculatlon of damplng coefflclent wlth numerlcal model for Case 2.
6000-
4060-
? O
126 SPE Production Engineering, February 1992
...
Analytical Diagnostic---
Predictiw.
.
. . .
.
.
b3
2000a
A o . ... ......
..
...
-
2000 1 1 1 1 J O 20 40 60 80 1 O0 Position (In.)Flg. 11-Accuracy of numerical and analytlcal dlagnostlc models for Case 2.
Appendlx-Derivatlon of Eq. 3
Equation 2 ,
the ID, damped-wave equation, is the basis used in deriving the finite-difference analogs.
Taylor series approximations are used to generate íinitedifference analogs for the derivatives of displacement that appear in the wave equation.
The analogs for the first and second derivatives of displacement with respect to time are straightforward. Note that the subscript i denotes axial distance (positive downward) and the subscript j
denotes time:
( a u ~ a t ) ~ i , j = ( u ~ , ,+ -ui, j ) ~ ~ t ,
. . .
('4-1). . .
and ( a 2 ~ 1 a t 2 )1
i , j = (ui, ,+ -2ui, +ui,,-
l ) / ~ t 2 . (A-2)The first derivative analog is a first-order-correct forward differ- ence; the second is a second-order-correct central difference.
Translating the second derivative with respect to distance into differences is not quite as straightforward. This derivative may be written as
. . .
where ( a u l a ~ ) l ~ + ~ ~ , = ( u ~ + ~ , -uii ,)/AX .(A-4)
. . .
..
and (aulax)
1
i -,
=cui,,
- U i - l , ,)IAX. (A-5)Now, substituting Eqs. A-4 and A-5 into Eq. A-3 yields
Eq. A-6 can be recognized as a normal second-order-correct cen- tral difference that has been rearranged slightly. The rationale for treating this difference in this manner is to account for variations in rod diameter and rod material. However, Eq. A-6 is not the fi- nal form of the difference analog to be used for the space deriva- tive. Another preferred aspect of the finite-difference model is the ability to allow for a variable Ax.
With tapered-rod strings, the solutions at the rod tapers are often desirabie so that maximum rod stresses may be analyzed. A constant-
Ax scheme would require interpolation at the rod tapers, whereas a variabledx method would permit a different Ax for each rod size, providing exact solutions at the rod tapers. Thus, rewriting Eq. A-6 to account for a variable Ax yields
where &=[(Ax) + +(Ax) -112.
Eq. A-7 is the final form of the finite-difference analog for the second derivative with respect to distance.
The next step is to substitute Eqs. A-1, A-2, and A-7 into Eq. 2 ,
. . .
(A-8)Grouping in similar terms yields
1 E A + PA +*)ui,j+l
=(-)
Ax Ax U'+'"=( 144gcAt2 144gcAtBefore joinlng Shell Offshore Inc. in New Orleans, T.A. Everltt worked at Chevron O11 Fleld Research Co. in La Habra, CA, for 3 years as a research englneer in the Ar- tificial Llft Group. He wae lana-an " m""" "" Everltt Jennlngs treasurer and 1988-
89 Mlcrocomputer puter User Group chairman for the Los Angeles Basln Sec- tion. Everltt holds BS and MS degrees in petroleum englneer- lng from Texas A&M U. Jlm W. Jennlngs is the Hughes Tool Professor of Petroleum Englneering at Texas A&M U., where he speclallzes in artificial llft and well-performance predlc- tions. He prevlously was vice presldent of production research at Gulf. A 1992 Dlstlnguished Lecturer, Jennings has also served on the Forum Serles, Reprint Serles, and Ferguson Medal committees. He holds BS and MS degrees from the Colorado School of Mines and a PhD degree from the U. of Pittsburgh.
PA 1 E A -
- * ] u i , + ( 1 4 4 g c A t 2 )Ui. j - i
-
( X )
U i - 1, j -144gcb
. . .
(A-9)Multiplying through by and factoring out pAl(144g,At2) gives
U i , j+ i
-
pA
( E )
+
-(E)
-jUi,,.
( 2 + c A t ) - - - -
At2 1448,
Dividing through by (EAIAx) + and taking an average for
@A1144gc) gives the final form of the main working equation:
ui+l,j = { [ a ( l + c A t ) ] ~ ~ , j + ~ -[a(2+cAt)-(EAlAx)+
- ( E A I A x ) - ] u ~ , ~ + ~ u ~ , ~ - ~ -(EAIAx)-ui-1, j } l ( E A I A ~ ) + ,
where CY = .
Eq. A-1 1 is identical to Eq. 3, the equation used to transmit the surface position downhole by calculating displacements at each node along the rod string.
SI Metrlc Converslon Factors
ft
x
3.048* E-01 = m in. x 2.54* E+OO = cm lbfx
4.448 222 E+OO = Npsi x 6.894757 E+OO = kPa
*Conversion factor is exact. SPEPE Original SPE manuscript received for review Oct. 2, 1988. Revised manuscript received July 5, 1990. Paper accepted for publication March 15, 1991. Paper (SPE 18189) first presented at the 1988 SPE Annual Technical Conference and Exhibition held in Houston. Oct. 2-5.
