Research ArticleStudy on Equivalent Viscous Damping Coefficient of Sucker RodBased on the Principle of Equal Friction Loss

Qin Li Bo Chen Zhiqiang Huang Haipin Tang Gang Li and Lei He

School of Mechanical Engineering Southwest Petroleum University Chengdu China

Correspondence should be addressed to Bo Chen 714093521qqcom

Received 24 October 2018 Revised 28 February 2019 Accepted 20 March 2019 Published 15 April 2019

Academic Editor Andres Saez

Copyright copy 2019 Qin Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Equivalent viscous damping coefficient is an important parameter of wave equation for sucker rod string In this paper based onthe principle of equal friction loss when the viscous energy consumption and the local damping energy consumption are taken intoaccount effects of equivalent viscous damping coefficients are obtainedThrough deducing energy consumption equation of oil andenergy consumption equation of the coupling theoretical formula for equivalent damping coefficient of sucker rods is receivedResults show that the smaller the K is (K is the ratio of sectional area of tubing to sucker rod) the larger the proportion of dampingcoefficient caused by viscous energy consumption in the equivalent damping coefficient of sucker rod system is When Klt 0095the proportion of damping coefficient caused by viscous energy consumption is more than 90 Reducing the sudden change ofcross-section area at sucker rod coupling has remarkable effect on reducing damping force of the sucker rod system The researchprovides a theoretical basis for the application and design of sucker rod and tubing

1 Instruction

Sucker rod equipment is widely used in artificial lift wells[1ndash3] Its failure modes are various such as pump leakagepump blockage sucker rod facture sucker rod eccentricwear and so on In order to accurately predict and judgethe working condition of oil wells the sucker rod stringwave equation is regularly used to characterize the suckerrod pumping system The wave equation contains viscousdamping coefficient The magnitude of viscous dampingcoefficient has a great influence on the results of predictionand diagnosis [4ndash7]

Since the 1960s many researchers have carried out thestudy of equivalent viscous damping coefficient [8] Nowthere are many methods to determine the equivalent viscousdamping coefficient such as experiential algorithm deduc-tion formula of hypothetical conditions calculated by surfacedynamometer cards etc [9ndash12] SGGibbs considered thatthe nonviscous damping force is negligible Assuming thatthe suspension point motion is a simple harmonic motionthe average velocity of the rod is expressed by the rootmean square value of the instantaneous velocity of the suckerrod and the dissipated work in one cycle of the rod string

is equal to the dissipated work by the equivalent viscousdamping Based on these assumptions the formula of theequivalent viscous damping coefficient is derived [13ndash15]The formula of sucker rod damping force on the conditionof laminar flow (Relt2300) is derived by AMPearvillejbut the formula can only be applied to laminar flow inPoiseuille flow [16] Based on the principle of equivalentmethod the formula of dimensionless equivalent viscousdamping coefficient is derived by Rubio D [17] Zhang Qiused the energy consumption of the upstroke to replace thedownstroke energy consumption By calculating the frictionloss caused by the viscous resistance of the sucker rodstring in a cycle the damping coefficient formula suited forvarious flow patterns is derived based on the principle ofequal friction loss [18] Using the curve method MJBasitionproposed that the intersection of the water powerdampingcoefficient and pump powerdamping coefficient curves isthe viscous damping coefficient of the sucker rod Thus theequivalent viscous damping coefficient is obtained [19] Basedon the Gibbs formula Everitt TA deduced the formulaof viscous damping coefficient by cyclic iteration methodThis method can be applied to any material sucker rod butGibbs damping coefficient formula can only be applied to

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9272751 12 pageshttpsdoiorg10115520199272751

2 Mathematical Problems in Engineering

steel sucker rod [20] On the basis of the sucker rod stringwave equation Sun Renyuan et al proposed the equivalentviscous damping coefficient formula which was related tothe actual pump dynamometer cards by using the numericalintegration method and established the corresponding iter-ative algorithm to calculate the equivalent viscous dampingcoefficient [21] ISteliga and Dong-Yu Wang proposed thatthe sucker rod dynamic model including the influences ofrod-fluid-tubing viscous friction and rod-tubing Coulombfriction is developed And then a method for calculatingvibration damping of sucker rod string based on indicatordiagram is proposed The periodic variation of vibrationdamping coefficient is analyzed with surface dynamometercards and pump dynamometer cards However this methodcan only be used to analyze the dynamometer cards of aspecific oil well [22 23]

All the above studies are aimed at the specific well con-ditions and the viscous damping coefficients are calculatedwhen the size of sucker rod and tubing are unchangedUnfortunately the energy consumption of sucker rod cou-pling and rod guides in the process of pumping is notconsidered The calculation model needs to be optimizedOn the basis of Gibbs formula and Zhang Qi formula thispaper assumes that the fluid is Couette flow in a circularpipe and considers the energy consumption of coupling androd guides Based on the principle of equal friction lossthe equivalent viscous damping coefficient of sucker rod isderived And the influence of sucker rod and tubing sizeon the equivalent viscous damping coefficient is analyzedwhich provides a theoretical basis for sucker rod and tubingmatching application

2 Dynamic Equations of Sucker Rod String

Dynamic differential equation of sucker rod string (Figure 1)with viscous damping coefficient [14] is as follows

12059721199061205971199052 = 1198862 12059721199061205971199092 minus 119888120597119906120597119905 (1)

where 119906 is the displacement of the sucker rod node 119909 isthe distance between the node and the wellhead 119905 is the runduration of sucker rod

The suspension point displacement function and loadfunction expressed by Fourier series are used as boundaryconditions

119863119909=0 = 12059002 + 119899sum0

120590119899 cos 119899120596119905 + 120591119899 sin 119899120596119905 (2)

119906119909=0 = V02 +119899sum0

V119899 cos 119899120596119905 + 120575119899 sin 119899120596119905 (3)

where 120590119899 120591119899 V119899 and 120575119899 are Fourier coefficientCombining (1)-(3) the displacement function of sucker

rod at any depth of x section can be given by

119906 = 12059002119864119860119903 119909 +V02 +

119899sum0

119874119899 cos 119899120596119905 + 119875119899 sin 119899120596119905 (4)

3 The Principle of Equal Frictional Loss

When the wave equation is used to solve the dynamics ofsucker rod string the damping term is included in the equa-tion Damping of sucker rod string system is a combinationof many factors such as viscosity temperature and flowpattern of oil so it is difficult to solve the real dampingof the system In general equivalent damping coefficient isused to replace the real damping The substituted conditionis that the frictional loss done by the viscous damping termin the wave equation is equal to the energy loss in the actualpumping well during a period of motion and the viscousdamping coefficient is called the equivalent viscous dampingcoefficient This described the principle of equal frictionalloss [13]

119882119891 = 119882119889 (5)

where119882119891 is the frictional loss of sucker rod string duringa period of motion and 119882119889 is the energy consumption inactual pumping wells

119882119891 = int1198710int1198790119888120588119903119860119903 (120597119906120597119905 )

2 119889119905119889119909 (6)

Without considering the Coulomb friction of single-stagerod vertical well the actual energy consumption includes theviscous energy consumption in the up stroke anddown strokeand the local viscous energy consumption of the oil passingthrough the sucker rod coupling in the actual operation ofsucker rod

Combining (5) and (6) the equivalent viscous dampingcoefficient is deduced

119888 = 119882119889120588119903119860119903 int1198710 int1198790 (120597119906120597119905)2 119889119905119889119909 (7)

4 Viscous Energy Consumption alongthe Tubing of Oil

Within a running cycle (Figure 3) of the sucker rod stringthe oil moves up with the sucker rod and the direction of netflow is the same as that of the sucker rod during the up strokeThe sucker rodmoves downward but the direction of net flowof oil is upward Then the average velocity of oil is oppositeto the direction of sucker rod movement on the occasionof the down stroke Therefore when calculating the viscousenergy consumption along the tubing the viscous energyconsumption of the upstroke and the downstroke should becalculated respectively according to the net flow rate of oil

41 Couette Flow of Circular Tube In the process of pumpingthe tubing is always in a static state and the sucker rod isin a moving state Therefore the oil flow can be regarded asCouette flow when analyzing the flow of oil in the tubing-rodstring annulus [24]

Mathematical Problems in Engineering 3

Polished rodStuffing box

Sucker rod

Produced oil

Tubing

GasLiquid level

Tubing anchor

Pump

Perforation

Upstroke

Rod string

Rod element

Resting positionu (x t)

u(x + Δx t)

g

Figure 1 The diagram of system and microelement of sucker rod string

The momentum equation of incompressible fluid withconstant physical properties in cylindrical-coordinates sys-tem (Figure 2) can be expressed as

120597119881119909120597119905 + 119881119903 120597119881119909120597119903 + 119881120593119903 120597119881119909120597120593 + 119881119909 120597119881119909120597119909= minus1120588 120597119901120597119909 + 120592(120597

21198811199091205971199032 + 11199032 12059721198811199091205971205932 + 120597

21198811199091205971199092 + 1119903 120597119881119909120597119903 )+ 119892

(8)

The function of the energy consumption along the tubingcan be expressed as

Φ = 120588120592[2(120597119881119903120597119903 )2 + 2(1119903

120597119881120593120597120593 + 1119903119881119903)2 + 2(120597119881119909120597119909 )

2

+ (1119903 120597119881119903120597120593 + 120597119881120593120597119903 minus 1119903119881120593)2 + (1119903 120597119881119909120597120593 + 120597119881120593120597119909 )

2

+ (120597119881119903120597119909 + 120597119881119909120597119903 )2]

(9)

where if 119903 = 1199030 then 119881119903 lt infin 119881120593 lt infin and 119881119909 lt infin If119903 = 1199031 then 119881119903 = 119881120593 = 119881119909 = 0To solve the above equations the oil is regarded as steady

one-way parallel flow along the x-axis which is uniform inthe direction of flow and axisymmetric in the axial direction

119881119909 = 119908119881119903 = 119881120593 = 0

z

y

r

oil

tubing

Sucker rod

Figure 2 Sucker rod string in cylindrical-coordinate system

120597120597119905 = 0120597120597120593 = 0

120597119881119909120597119909 = 0(10)

4 Mathematical Problems in Engineering

r0 r1

(a)

r0 r1

(b)

Figure 3 Velocity distribution of crude along section (a) Upstroke and (b) downstroke

Substituting the above boundary conditions equation(10) the momentum equation (8) can be simplified to

minus1120588 120597119901120597119909 + 120592(12059721199081205971199032 + 1119903 120597119908120597119903 ) + 119892 = 0 (11)

where 119901 is a function of x and 120597119901120597119909 = 119889119901119889119909Define 119875 = 119889119901119889119909 minus 120588119892 119875 is the baric gradient of fluid

flow except gravity And 120583 = 120588120592Themomentum equation (11) can be further simplified to

119875 = 120583(11988921199081198891199032 + 1119903 119889119908119889119903 ) (12)

Substituting the above boundary conditions equation(10) the energy consumption function along the tubing canbe obtained

Φ = 120583(119889119908119889119903 )2

(13)

By integrating the momentum equation (13) the oilvelocity equation can be obtained

119908 = 141205831198751199032 + 1198621 ln 119903 + 1198622 (14)

Substitute the boundary conditions 119903 = 1199030 119908 = V 119903 =1199031 119908 = 0 The coefficient 1198621 and 1198622 can be obtained

1198621 = V + (1198754120583) (11990321 minus 11990320)ln (11990301199031) (15)

1198622 = minus 119875412058311990321 minusV + (1198754120583) (11990321 minus 11990320)

ln (11990301199031) ln 1199031 (16)

Cross-section flow in X direction is as follows

119876 = int21205870int11990311199030

119908119903119889119903119889120579

= 120587119903401198758120583 [[1 minus 1198984 + (1198982 minus 1)

2

ln119898 ]]

+ 12058711990320V(1198982 minus 12 ln119898 minus 1)

(17)

119898 = 11990311199030 gt 1 (18)

According to (17) the cross-section flow in the X direc-tion is the vector sum of differential pressure flow and shearflowThe first half of the formula is differential pressure flowand the second half is shear flow

According to (13)-(16) and (18) the equation of energydissipation per time per length of oil can be derived

119873 = int21205870int11990311199030

Φ119903119889119903119889120579

= 1205871198752119903408120583 [[1198984 minus 1 minus (1198982 minus 1)

2

ln119898 ]]+ 2120587120583V2

ln119898(19)

42 Viscous Energy Consumption of the Upstroke The direc-tion of oil movement is the same as that of the sucker rodin the upstroke at the same time the sucker rod and the

Mathematical Problems in Engineering 5

oil should be regarded as an integral whole in considerationof the viscous energy consumption The viscous energyconsumption is caused by the work loss by the oil shear forceand the hydraulic loss along the oil flow The actual flowrate of the oil pumping system is equal to the volume of theliquid discharged by the pump per time When calculatingthe viscous energy consumption the flow rate should beconsidered as the actual flow rate of the oil pumping system

Without considering the elastic deformation of the suckerrod the volume of the liquid discharged from the upstrokeper time can be expressed as

119902 = 120587 (1199032119901 minus 11990320) V119909=119871 (20)

where 119903119901 is the radius of the downhole pump plungerAccording to the continuity equation of oil flow the cross-

section flow rate in the X direction is equal to the volume ofliquid discharged by the pump per time Combine (17) and(20) P can be given by

119875 = 4120583119903202 (11989822 minus 1) V119909=119871 ln119898 minus V (1198982 minus 1 minus 2 ln119898)

(1 minus 1198984) ln119898 + (1198982 minus 1)2 (21)

1198982 = 1199031199011199030 (22)

On the upstroke (0 -1199051) according to (19) and (21) theviscous energy consumption along the oil flow can be givenby

1198821198891 = int1198710int11990510119873119889119905119889119909

= [[2120587120583ln119898 + 2120587120583 (1198982 minus 1 minus 2 ln119898)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898]]sdot int1198710int11990510V2119889119905119889119909 + 8120587120583 (11989822 minus 1)2 ln119898

(1198984 minus 1) ln119898 minus (1198982 minus 1)2sdot int1198710int11990510V2119909=119871119889119905119889119909 minus 8120587120583

sdot (1198982 minus 1 minus 2 ln119898) (11989822 minus 1)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int119871

0int11990510V119909=119871119906119889119905119889119909

(23)

43 Viscous EnergyConsumption of theDownstroke The inletvalve of defueling pump is closed and the outlet valve isopened during downstroke at the same time the sucker rod

enters the tubing and discharges corresponding volume ofoil Therefore without considering leakage the actual outputflow of the pumping system is equal to the volume of thesucker rod entering the tubing per time

The volume of the sucker rod entering the tubing in pertime can be expressed as

119902 = minus12058711990320V119909=119871 (24)Combine (17) and (24) P can be given by

119875 = 412058311990320(1198982 minus 1 minus 2 ln119898) V + 2V119909=119871 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2 (25)

On the downstroke (1199051- T) according to (18) and (25) theviscous energy consumption along the tubing can be given by

1198821198892 = int1198710int1198791199051

119873119889119905119889119909 = 2120587120583

sdot (1198982 minus 1 minus 2 ln119898)2 + (1198984 minus 1) ln119898 minus (1198982 minus 1)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898sdot int1198710int1198791199051

V2119889119905119889119909 + 8120587120583 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2

sdot int1198710int1198791199051

V2119909=119871119889119905119889119909 + 8120587120583

sdot (1198982 minus 1 minus 2 ln119898)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int

119871

0int1198791199051

V119909=119871V119889119905119889119909

(26)

5 Local Viscous Energy Consumption of Oil

Without considering the eccentricity of the sucker rod theannulus area between sucker rod and tubing is small at thecoupling since diameter of the coupling is larger than thesucker rod After the oil passing through the coupling theflow regime is changed and local resistance is lost Similarlylocal resistance loss will also occur at position of rod guides

51 Local Viscous Energy Consumption at Couplings Calcula-tion formula of local damping force at coupling can be givenby [25]

119865ℎ = 121205871205821205881199082ℎ (1199032ℎ minus 11990320) sgn (Vℎ) (27)

where 119903ℎ is the radius of coupling119908ℎ is the average veloc-ity of oil passing through coupling 120582 is the dimensionlesscoefficient which is defined as

120582 = 52 times 104 times (119903ℎ1199031 minus 0381)257 [277 minus 169 (119873Re1015840119873Re) sgn (Vℎ)]119873Re(28)

6 Mathematical Problems in Engineering

While on the upstroke sgn(Vℎ) is equal to 1 when Vℎ islarger than 0 on the downstroke sgn(Vℎ) is equal to -1 whenVℎ is less than 0

119873Re = 2119908ℎ120588 (1199031 minus 1199030)120583 (29)

119873Re1015840 = 2Vℎ120588 (1199031 minus 1199030)120583 (30)

Local resistance pressure drop can be defined as shownbelow

Δ119875ℎ = 119865ℎ120587 (1199032ℎminus 11990320) =

121205821205881199082ℎ sgn (Vℎ) (31)

Local viscous energy consumption at coupling

119882ℎ = 119899sum1

int1199050Δ119875ℎ119876119889119905 (32)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908ℎ) and the rodspeed (Vℎ) can be obtained

119908ℎ = 1199032119901 minus 1199032011990321 minus 11990320 Vℎ (33)

According to (28)-(33) the local viscous energy con-sumption of upstroke at coupling can be given by

119882ℎ1 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1) 1199030(119898 minus 1) (1198982 minus 1)

119899sum1

int11990510VℎV119909=119871119889119905 (34)

On the downstroke the relationship between the velocityof oil (119908ℎ) and the rod speed (Vℎ) can be obtained

119908ℎ = 11 minus 1198982 Vℎ (35)

According to (28)-(32) and (35) the local viscous energyconsumption of downstroke at coupling can be given by

119882ℎ2= 13120587120583 times 104 times (119903ℎ1199031 minus 0381)

257 (108 minus 1691198982) 1199030(119898 minus 1) (1 minus 1198982)sdot 119899sum1

int1198791199051

VℎV119909=119871119889119905(36)

52 Local Viscous Energy Consumption at Rod Guides Fric-tion caused by the resistance along rod guides can beexpressed as [26]

1198651198921 = 2120587120583120594119897119892119899119892119908119892 (37)

Friction caused by local hydraulic loss at rod guides canbe expressed as

1198651198922 = 2120587120583120594119897119892119890119899119892119908119892 (38)

According to (37) and (38) the calculation formula oflocal damping force at rod guides can be given by

119865119892 = 2120587120583120594119899119892119908119892 (119897119892 + 119897119892119890) (39)

where 119897119892 is the length of rod guides 119899119892 is the number ofrod guides 119908g is the average velocity of oil passing throughrod guides120594 is the dimensionless coefficient which is definedas

120594 = 1ln (21199031119889119892119890) (40)

119897119892119890 is the equivalent length of rod guides

119897119892119890 = 120576119892120582119892 119889119892119890 (41)

120576119892 is the local resistance coefficient of rod guides 120582119892is the resistance coefficient along the rod guides 119889119892119890 is theequivalent diameter of rod guides

119889119892119890 = minus2119860119892119905 + 2radic1198602119892119905 + (11986211989211990511990321 minus 21198601198921199051199031) 119862119892119905119862119892119905 (42)

where 119860119892119905 is the flow surface between rod guides andtubing 119862119892119905 is the wet circumference of flow surface betweenrod guides and tubing

Local resistance pressure drop can be defined as shownbelow

Δ119875119892 = 119865119892119860119892119905 (43)

Local viscous energy consumption at rod guides is asfollows

119882119892 = int1199050Δ119875119892119876119889119905 (44)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908g) and the rodspeed (Vℎ) can be obtained

119908119892 = 120587 (1199032119901 minus 11990320)119860119892119905 Vℎ (45)

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 3

Polished rodStuffing box

Sucker rod

Produced oil

Tubing

GasLiquid level

Tubing anchor

Pump

Perforation

Upstroke

Rod string

Rod element

Resting positionu (x t)

u(x + Δx t)

g

Figure 1 The diagram of system and microelement of sucker rod string

The momentum equation of incompressible fluid withconstant physical properties in cylindrical-coordinates sys-tem (Figure 2) can be expressed as

120597119881119909120597119905 + 119881119903 120597119881119909120597119903 + 119881120593119903 120597119881119909120597120593 + 119881119909 120597119881119909120597119909= minus1120588 120597119901120597119909 + 120592(120597

21198811199091205971199032 + 11199032 12059721198811199091205971205932 + 120597

21198811199091205971199092 + 1119903 120597119881119909120597119903 )+ 119892

(8)

The function of the energy consumption along the tubingcan be expressed as

Φ = 120588120592[2(120597119881119903120597119903 )2 + 2(1119903

120597119881120593120597120593 + 1119903119881119903)2 + 2(120597119881119909120597119909 )

2

+ (1119903 120597119881119903120597120593 + 120597119881120593120597119903 minus 1119903119881120593)2 + (1119903 120597119881119909120597120593 + 120597119881120593120597119909 )

2

+ (120597119881119903120597119909 + 120597119881119909120597119903 )2]

(9)

where if 119903 = 1199030 then 119881119903 lt infin 119881120593 lt infin and 119881119909 lt infin If119903 = 1199031 then 119881119903 = 119881120593 = 119881119909 = 0To solve the above equations the oil is regarded as steady

one-way parallel flow along the x-axis which is uniform inthe direction of flow and axisymmetric in the axial direction

119881119909 = 119908119881119903 = 119881120593 = 0

z

y

r

oil

tubing

Sucker rod

Figure 2 Sucker rod string in cylindrical-coordinate system

120597120597119905 = 0120597120597120593 = 0

120597119881119909120597119909 = 0(10)

4 Mathematical Problems in Engineering

r0 r1

(a)

r0 r1

(b)

Figure 3 Velocity distribution of crude along section (a) Upstroke and (b) downstroke

Substituting the above boundary conditions equation(10) the momentum equation (8) can be simplified to

minus1120588 120597119901120597119909 + 120592(12059721199081205971199032 + 1119903 120597119908120597119903 ) + 119892 = 0 (11)

where 119901 is a function of x and 120597119901120597119909 = 119889119901119889119909Define 119875 = 119889119901119889119909 minus 120588119892 119875 is the baric gradient of fluid

flow except gravity And 120583 = 120588120592Themomentum equation (11) can be further simplified to

119875 = 120583(11988921199081198891199032 + 1119903 119889119908119889119903 ) (12)

Substituting the above boundary conditions equation(10) the energy consumption function along the tubing canbe obtained

Φ = 120583(119889119908119889119903 )2

(13)

By integrating the momentum equation (13) the oilvelocity equation can be obtained

119908 = 141205831198751199032 + 1198621 ln 119903 + 1198622 (14)

Substitute the boundary conditions 119903 = 1199030 119908 = V 119903 =1199031 119908 = 0 The coefficient 1198621 and 1198622 can be obtained

1198621 = V + (1198754120583) (11990321 minus 11990320)ln (11990301199031) (15)

1198622 = minus 119875412058311990321 minusV + (1198754120583) (11990321 minus 11990320)

ln (11990301199031) ln 1199031 (16)

Cross-section flow in X direction is as follows

119876 = int21205870int11990311199030

119908119903119889119903119889120579

= 120587119903401198758120583 [[1 minus 1198984 + (1198982 minus 1)

2

ln119898 ]]

+ 12058711990320V(1198982 minus 12 ln119898 minus 1)

(17)

119898 = 11990311199030 gt 1 (18)

According to (17) the cross-section flow in the X direc-tion is the vector sum of differential pressure flow and shearflowThe first half of the formula is differential pressure flowand the second half is shear flow

According to (13)-(16) and (18) the equation of energydissipation per time per length of oil can be derived

119873 = int21205870int11990311199030

Φ119903119889119903119889120579

= 1205871198752119903408120583 [[1198984 minus 1 minus (1198982 minus 1)

2

ln119898 ]]+ 2120587120583V2

ln119898(19)

42 Viscous Energy Consumption of the Upstroke The direc-tion of oil movement is the same as that of the sucker rodin the upstroke at the same time the sucker rod and the

Mathematical Problems in Engineering 5

oil should be regarded as an integral whole in considerationof the viscous energy consumption The viscous energyconsumption is caused by the work loss by the oil shear forceand the hydraulic loss along the oil flow The actual flowrate of the oil pumping system is equal to the volume of theliquid discharged by the pump per time When calculatingthe viscous energy consumption the flow rate should beconsidered as the actual flow rate of the oil pumping system

Without considering the elastic deformation of the suckerrod the volume of the liquid discharged from the upstrokeper time can be expressed as

119902 = 120587 (1199032119901 minus 11990320) V119909=119871 (20)

where 119903119901 is the radius of the downhole pump plungerAccording to the continuity equation of oil flow the cross-

section flow rate in the X direction is equal to the volume ofliquid discharged by the pump per time Combine (17) and(20) P can be given by

119875 = 4120583119903202 (11989822 minus 1) V119909=119871 ln119898 minus V (1198982 minus 1 minus 2 ln119898)

(1 minus 1198984) ln119898 + (1198982 minus 1)2 (21)

1198982 = 1199031199011199030 (22)

On the upstroke (0 -1199051) according to (19) and (21) theviscous energy consumption along the oil flow can be givenby

1198821198891 = int1198710int11990510119873119889119905119889119909

= [[2120587120583ln119898 + 2120587120583 (1198982 minus 1 minus 2 ln119898)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898]]sdot int1198710int11990510V2119889119905119889119909 + 8120587120583 (11989822 minus 1)2 ln119898

(1198984 minus 1) ln119898 minus (1198982 minus 1)2sdot int1198710int11990510V2119909=119871119889119905119889119909 minus 8120587120583

sdot (1198982 minus 1 minus 2 ln119898) (11989822 minus 1)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int119871

0int11990510V119909=119871119906119889119905119889119909

(23)

43 Viscous EnergyConsumption of theDownstroke The inletvalve of defueling pump is closed and the outlet valve isopened during downstroke at the same time the sucker rod

enters the tubing and discharges corresponding volume ofoil Therefore without considering leakage the actual outputflow of the pumping system is equal to the volume of thesucker rod entering the tubing per time

The volume of the sucker rod entering the tubing in pertime can be expressed as

119902 = minus12058711990320V119909=119871 (24)Combine (17) and (24) P can be given by

119875 = 412058311990320(1198982 minus 1 minus 2 ln119898) V + 2V119909=119871 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2 (25)

On the downstroke (1199051- T) according to (18) and (25) theviscous energy consumption along the tubing can be given by

1198821198892 = int1198710int1198791199051

119873119889119905119889119909 = 2120587120583

sdot (1198982 minus 1 minus 2 ln119898)2 + (1198984 minus 1) ln119898 minus (1198982 minus 1)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898sdot int1198710int1198791199051

V2119889119905119889119909 + 8120587120583 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2

sdot int1198710int1198791199051

V2119909=119871119889119905119889119909 + 8120587120583

sdot (1198982 minus 1 minus 2 ln119898)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int

119871

0int1198791199051

V119909=119871V119889119905119889119909

(26)

5 Local Viscous Energy Consumption of Oil

Without considering the eccentricity of the sucker rod theannulus area between sucker rod and tubing is small at thecoupling since diameter of the coupling is larger than thesucker rod After the oil passing through the coupling theflow regime is changed and local resistance is lost Similarlylocal resistance loss will also occur at position of rod guides

51 Local Viscous Energy Consumption at Couplings Calcula-tion formula of local damping force at coupling can be givenby [25]

119865ℎ = 121205871205821205881199082ℎ (1199032ℎ minus 11990320) sgn (Vℎ) (27)

where 119903ℎ is the radius of coupling119908ℎ is the average veloc-ity of oil passing through coupling 120582 is the dimensionlesscoefficient which is defined as

120582 = 52 times 104 times (119903ℎ1199031 minus 0381)257 [277 minus 169 (119873Re1015840119873Re) sgn (Vℎ)]119873Re(28)

6 Mathematical Problems in Engineering

While on the upstroke sgn(Vℎ) is equal to 1 when Vℎ islarger than 0 on the downstroke sgn(Vℎ) is equal to -1 whenVℎ is less than 0

119873Re = 2119908ℎ120588 (1199031 minus 1199030)120583 (29)

119873Re1015840 = 2Vℎ120588 (1199031 minus 1199030)120583 (30)

Local resistance pressure drop can be defined as shownbelow

Δ119875ℎ = 119865ℎ120587 (1199032ℎminus 11990320) =

121205821205881199082ℎ sgn (Vℎ) (31)

Local viscous energy consumption at coupling

119882ℎ = 119899sum1

int1199050Δ119875ℎ119876119889119905 (32)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908ℎ) and the rodspeed (Vℎ) can be obtained

119908ℎ = 1199032119901 minus 1199032011990321 minus 11990320 Vℎ (33)

According to (28)-(33) the local viscous energy con-sumption of upstroke at coupling can be given by

119882ℎ1 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1) 1199030(119898 minus 1) (1198982 minus 1)

119899sum1

int11990510VℎV119909=119871119889119905 (34)

On the downstroke the relationship between the velocityof oil (119908ℎ) and the rod speed (Vℎ) can be obtained

119908ℎ = 11 minus 1198982 Vℎ (35)

According to (28)-(32) and (35) the local viscous energyconsumption of downstroke at coupling can be given by

119882ℎ2= 13120587120583 times 104 times (119903ℎ1199031 minus 0381)

257 (108 minus 1691198982) 1199030(119898 minus 1) (1 minus 1198982)sdot 119899sum1

int1198791199051

VℎV119909=119871119889119905(36)

52 Local Viscous Energy Consumption at Rod Guides Fric-tion caused by the resistance along rod guides can beexpressed as [26]

1198651198921 = 2120587120583120594119897119892119899119892119908119892 (37)

Friction caused by local hydraulic loss at rod guides canbe expressed as

1198651198922 = 2120587120583120594119897119892119890119899119892119908119892 (38)

According to (37) and (38) the calculation formula oflocal damping force at rod guides can be given by

119865119892 = 2120587120583120594119899119892119908119892 (119897119892 + 119897119892119890) (39)

where 119897119892 is the length of rod guides 119899119892 is the number ofrod guides 119908g is the average velocity of oil passing throughrod guides120594 is the dimensionless coefficient which is definedas

120594 = 1ln (21199031119889119892119890) (40)

119897119892119890 is the equivalent length of rod guides

119897119892119890 = 120576119892120582119892 119889119892119890 (41)

120576119892 is the local resistance coefficient of rod guides 120582119892is the resistance coefficient along the rod guides 119889119892119890 is theequivalent diameter of rod guides

119889119892119890 = minus2119860119892119905 + 2radic1198602119892119905 + (11986211989211990511990321 minus 21198601198921199051199031) 119862119892119905119862119892119905 (42)

where 119860119892119905 is the flow surface between rod guides andtubing 119862119892119905 is the wet circumference of flow surface betweenrod guides and tubing

Local resistance pressure drop can be defined as shownbelow

Δ119875119892 = 119865119892119860119892119905 (43)

Local viscous energy consumption at rod guides is asfollows

119882119892 = int1199050Δ119875119892119876119889119905 (44)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908g) and the rodspeed (Vℎ) can be obtained

119908119892 = 120587 (1199032119901 minus 11990320)119860119892119905 Vℎ (45)

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

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4 Mathematical Problems in Engineering

r0 r1

(a)

r0 r1

(b)

Figure 3 Velocity distribution of crude along section (a) Upstroke and (b) downstroke

Substituting the above boundary conditions equation(10) the momentum equation (8) can be simplified to

minus1120588 120597119901120597119909 + 120592(12059721199081205971199032 + 1119903 120597119908120597119903 ) + 119892 = 0 (11)

where 119901 is a function of x and 120597119901120597119909 = 119889119901119889119909Define 119875 = 119889119901119889119909 minus 120588119892 119875 is the baric gradient of fluid

flow except gravity And 120583 = 120588120592Themomentum equation (11) can be further simplified to

119875 = 120583(11988921199081198891199032 + 1119903 119889119908119889119903 ) (12)

Substituting the above boundary conditions equation(10) the energy consumption function along the tubing canbe obtained

Φ = 120583(119889119908119889119903 )2

(13)

By integrating the momentum equation (13) the oilvelocity equation can be obtained

119908 = 141205831198751199032 + 1198621 ln 119903 + 1198622 (14)

Substitute the boundary conditions 119903 = 1199030 119908 = V 119903 =1199031 119908 = 0 The coefficient 1198621 and 1198622 can be obtained

1198621 = V + (1198754120583) (11990321 minus 11990320)ln (11990301199031) (15)

1198622 = minus 119875412058311990321 minusV + (1198754120583) (11990321 minus 11990320)

ln (11990301199031) ln 1199031 (16)

Cross-section flow in X direction is as follows

119876 = int21205870int11990311199030

119908119903119889119903119889120579

= 120587119903401198758120583 [[1 minus 1198984 + (1198982 minus 1)

2

ln119898 ]]

+ 12058711990320V(1198982 minus 12 ln119898 minus 1)

(17)

119898 = 11990311199030 gt 1 (18)

According to (17) the cross-section flow in the X direc-tion is the vector sum of differential pressure flow and shearflowThe first half of the formula is differential pressure flowand the second half is shear flow

According to (13)-(16) and (18) the equation of energydissipation per time per length of oil can be derived

119873 = int21205870int11990311199030

Φ119903119889119903119889120579

= 1205871198752119903408120583 [[1198984 minus 1 minus (1198982 minus 1)

2

ln119898 ]]+ 2120587120583V2

ln119898(19)

42 Viscous Energy Consumption of the Upstroke The direc-tion of oil movement is the same as that of the sucker rodin the upstroke at the same time the sucker rod and the

Mathematical Problems in Engineering 5

oil should be regarded as an integral whole in considerationof the viscous energy consumption The viscous energyconsumption is caused by the work loss by the oil shear forceand the hydraulic loss along the oil flow The actual flowrate of the oil pumping system is equal to the volume of theliquid discharged by the pump per time When calculatingthe viscous energy consumption the flow rate should beconsidered as the actual flow rate of the oil pumping system

Without considering the elastic deformation of the suckerrod the volume of the liquid discharged from the upstrokeper time can be expressed as

119902 = 120587 (1199032119901 minus 11990320) V119909=119871 (20)

where 119903119901 is the radius of the downhole pump plungerAccording to the continuity equation of oil flow the cross-

section flow rate in the X direction is equal to the volume ofliquid discharged by the pump per time Combine (17) and(20) P can be given by

119875 = 4120583119903202 (11989822 minus 1) V119909=119871 ln119898 minus V (1198982 minus 1 minus 2 ln119898)

(1 minus 1198984) ln119898 + (1198982 minus 1)2 (21)

1198982 = 1199031199011199030 (22)

On the upstroke (0 -1199051) according to (19) and (21) theviscous energy consumption along the oil flow can be givenby

1198821198891 = int1198710int11990510119873119889119905119889119909

= [[2120587120583ln119898 + 2120587120583 (1198982 minus 1 minus 2 ln119898)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898]]sdot int1198710int11990510V2119889119905119889119909 + 8120587120583 (11989822 minus 1)2 ln119898

(1198984 minus 1) ln119898 minus (1198982 minus 1)2sdot int1198710int11990510V2119909=119871119889119905119889119909 minus 8120587120583

sdot (1198982 minus 1 minus 2 ln119898) (11989822 minus 1)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int119871

0int11990510V119909=119871119906119889119905119889119909

(23)

43 Viscous EnergyConsumption of theDownstroke The inletvalve of defueling pump is closed and the outlet valve isopened during downstroke at the same time the sucker rod

enters the tubing and discharges corresponding volume ofoil Therefore without considering leakage the actual outputflow of the pumping system is equal to the volume of thesucker rod entering the tubing per time

The volume of the sucker rod entering the tubing in pertime can be expressed as

119902 = minus12058711990320V119909=119871 (24)Combine (17) and (24) P can be given by

119875 = 412058311990320(1198982 minus 1 minus 2 ln119898) V + 2V119909=119871 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2 (25)

On the downstroke (1199051- T) according to (18) and (25) theviscous energy consumption along the tubing can be given by

1198821198892 = int1198710int1198791199051

119873119889119905119889119909 = 2120587120583

sdot (1198982 minus 1 minus 2 ln119898)2 + (1198984 minus 1) ln119898 minus (1198982 minus 1)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898sdot int1198710int1198791199051

V2119889119905119889119909 + 8120587120583 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2

sdot int1198710int1198791199051

V2119909=119871119889119905119889119909 + 8120587120583

sdot (1198982 minus 1 minus 2 ln119898)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int

119871

0int1198791199051

V119909=119871V119889119905119889119909

(26)

5 Local Viscous Energy Consumption of Oil

Without considering the eccentricity of the sucker rod theannulus area between sucker rod and tubing is small at thecoupling since diameter of the coupling is larger than thesucker rod After the oil passing through the coupling theflow regime is changed and local resistance is lost Similarlylocal resistance loss will also occur at position of rod guides

51 Local Viscous Energy Consumption at Couplings Calcula-tion formula of local damping force at coupling can be givenby [25]

119865ℎ = 121205871205821205881199082ℎ (1199032ℎ minus 11990320) sgn (Vℎ) (27)

where 119903ℎ is the radius of coupling119908ℎ is the average veloc-ity of oil passing through coupling 120582 is the dimensionlesscoefficient which is defined as

120582 = 52 times 104 times (119903ℎ1199031 minus 0381)257 [277 minus 169 (119873Re1015840119873Re) sgn (Vℎ)]119873Re(28)

6 Mathematical Problems in Engineering

While on the upstroke sgn(Vℎ) is equal to 1 when Vℎ islarger than 0 on the downstroke sgn(Vℎ) is equal to -1 whenVℎ is less than 0

119873Re = 2119908ℎ120588 (1199031 minus 1199030)120583 (29)

119873Re1015840 = 2Vℎ120588 (1199031 minus 1199030)120583 (30)

Local resistance pressure drop can be defined as shownbelow

Δ119875ℎ = 119865ℎ120587 (1199032ℎminus 11990320) =

121205821205881199082ℎ sgn (Vℎ) (31)

Local viscous energy consumption at coupling

119882ℎ = 119899sum1

int1199050Δ119875ℎ119876119889119905 (32)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908ℎ) and the rodspeed (Vℎ) can be obtained

119908ℎ = 1199032119901 minus 1199032011990321 minus 11990320 Vℎ (33)

According to (28)-(33) the local viscous energy con-sumption of upstroke at coupling can be given by

119882ℎ1 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1) 1199030(119898 minus 1) (1198982 minus 1)

119899sum1

int11990510VℎV119909=119871119889119905 (34)

On the downstroke the relationship between the velocityof oil (119908ℎ) and the rod speed (Vℎ) can be obtained

119908ℎ = 11 minus 1198982 Vℎ (35)

According to (28)-(32) and (35) the local viscous energyconsumption of downstroke at coupling can be given by

119882ℎ2= 13120587120583 times 104 times (119903ℎ1199031 minus 0381)

257 (108 minus 1691198982) 1199030(119898 minus 1) (1 minus 1198982)sdot 119899sum1

int1198791199051

VℎV119909=119871119889119905(36)

52 Local Viscous Energy Consumption at Rod Guides Fric-tion caused by the resistance along rod guides can beexpressed as [26]

1198651198921 = 2120587120583120594119897119892119899119892119908119892 (37)

Friction caused by local hydraulic loss at rod guides canbe expressed as

1198651198922 = 2120587120583120594119897119892119890119899119892119908119892 (38)

According to (37) and (38) the calculation formula oflocal damping force at rod guides can be given by

119865119892 = 2120587120583120594119899119892119908119892 (119897119892 + 119897119892119890) (39)

where 119897119892 is the length of rod guides 119899119892 is the number ofrod guides 119908g is the average velocity of oil passing throughrod guides120594 is the dimensionless coefficient which is definedas

120594 = 1ln (21199031119889119892119890) (40)

119897119892119890 is the equivalent length of rod guides

119897119892119890 = 120576119892120582119892 119889119892119890 (41)

120576119892 is the local resistance coefficient of rod guides 120582119892is the resistance coefficient along the rod guides 119889119892119890 is theequivalent diameter of rod guides

119889119892119890 = minus2119860119892119905 + 2radic1198602119892119905 + (11986211989211990511990321 minus 21198601198921199051199031) 119862119892119905119862119892119905 (42)

where 119860119892119905 is the flow surface between rod guides andtubing 119862119892119905 is the wet circumference of flow surface betweenrod guides and tubing

Local resistance pressure drop can be defined as shownbelow

Δ119875119892 = 119865119892119860119892119905 (43)

Local viscous energy consumption at rod guides is asfollows

119882119892 = int1199050Δ119875119892119876119889119905 (44)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908g) and the rodspeed (Vℎ) can be obtained

119908119892 = 120587 (1199032119901 minus 11990320)119860119892119905 Vℎ (45)

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

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Mathematical Problems in Engineering 5

oil should be regarded as an integral whole in considerationof the viscous energy consumption The viscous energyconsumption is caused by the work loss by the oil shear forceand the hydraulic loss along the oil flow The actual flowrate of the oil pumping system is equal to the volume of theliquid discharged by the pump per time When calculatingthe viscous energy consumption the flow rate should beconsidered as the actual flow rate of the oil pumping system

Without considering the elastic deformation of the suckerrod the volume of the liquid discharged from the upstrokeper time can be expressed as

119902 = 120587 (1199032119901 minus 11990320) V119909=119871 (20)

where 119903119901 is the radius of the downhole pump plungerAccording to the continuity equation of oil flow the cross-

section flow rate in the X direction is equal to the volume ofliquid discharged by the pump per time Combine (17) and(20) P can be given by

119875 = 4120583119903202 (11989822 minus 1) V119909=119871 ln119898 minus V (1198982 minus 1 minus 2 ln119898)

(1 minus 1198984) ln119898 + (1198982 minus 1)2 (21)

1198982 = 1199031199011199030 (22)

On the upstroke (0 -1199051) according to (19) and (21) theviscous energy consumption along the oil flow can be givenby

1198821198891 = int1198710int11990510119873119889119905119889119909

= [[2120587120583ln119898 + 2120587120583 (1198982 minus 1 minus 2 ln119898)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898]]sdot int1198710int11990510V2119889119905119889119909 + 8120587120583 (11989822 minus 1)2 ln119898

(1198984 minus 1) ln119898 minus (1198982 minus 1)2sdot int1198710int11990510V2119909=119871119889119905119889119909 minus 8120587120583

sdot (1198982 minus 1 minus 2 ln119898) (11989822 minus 1)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int119871

0int11990510V119909=119871119906119889119905119889119909

(23)

43 Viscous EnergyConsumption of theDownstroke The inletvalve of defueling pump is closed and the outlet valve isopened during downstroke at the same time the sucker rod

enters the tubing and discharges corresponding volume ofoil Therefore without considering leakage the actual outputflow of the pumping system is equal to the volume of thesucker rod entering the tubing per time

The volume of the sucker rod entering the tubing in pertime can be expressed as

119902 = minus12058711990320V119909=119871 (24)Combine (17) and (24) P can be given by

119875 = 412058311990320(1198982 minus 1 minus 2 ln119898) V + 2V119909=119871 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2 (25)

On the downstroke (1199051- T) according to (18) and (25) theviscous energy consumption along the tubing can be given by

1198821198892 = int1198710int1198791199051

119873119889119905119889119909 = 2120587120583

sdot (1198982 minus 1 minus 2 ln119898)2 + (1198984 minus 1) ln119898 minus (1198982 minus 1)2

(1198984 minus 1) ln2119898 minus (1198982 minus 1)2 ln119898sdot int1198710int1198791199051

V2119889119905119889119909 + 8120587120583 ln119898(1198984 minus 1) ln119898 minus (1198982 minus 1)2

sdot int1198710int1198791199051

V2119909=119871119889119905119889119909 + 8120587120583

sdot (1198982 minus 1 minus 2 ln119898)(1198984 minus 1) ln119898 minus (1198982 minus 1)2 int

119871

0int1198791199051

V119909=119871V119889119905119889119909

(26)

5 Local Viscous Energy Consumption of Oil

Without considering the eccentricity of the sucker rod theannulus area between sucker rod and tubing is small at thecoupling since diameter of the coupling is larger than thesucker rod After the oil passing through the coupling theflow regime is changed and local resistance is lost Similarlylocal resistance loss will also occur at position of rod guides

51 Local Viscous Energy Consumption at Couplings Calcula-tion formula of local damping force at coupling can be givenby [25]

119865ℎ = 121205871205821205881199082ℎ (1199032ℎ minus 11990320) sgn (Vℎ) (27)

where 119903ℎ is the radius of coupling119908ℎ is the average veloc-ity of oil passing through coupling 120582 is the dimensionlesscoefficient which is defined as

120582 = 52 times 104 times (119903ℎ1199031 minus 0381)257 [277 minus 169 (119873Re1015840119873Re) sgn (Vℎ)]119873Re(28)

6 Mathematical Problems in Engineering

While on the upstroke sgn(Vℎ) is equal to 1 when Vℎ islarger than 0 on the downstroke sgn(Vℎ) is equal to -1 whenVℎ is less than 0

119873Re = 2119908ℎ120588 (1199031 minus 1199030)120583 (29)

119873Re1015840 = 2Vℎ120588 (1199031 minus 1199030)120583 (30)

Local resistance pressure drop can be defined as shownbelow

Δ119875ℎ = 119865ℎ120587 (1199032ℎminus 11990320) =

121205821205881199082ℎ sgn (Vℎ) (31)

Local viscous energy consumption at coupling

119882ℎ = 119899sum1

int1199050Δ119875ℎ119876119889119905 (32)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908ℎ) and the rodspeed (Vℎ) can be obtained

119908ℎ = 1199032119901 minus 1199032011990321 minus 11990320 Vℎ (33)

According to (28)-(33) the local viscous energy con-sumption of upstroke at coupling can be given by

119882ℎ1 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1) 1199030(119898 minus 1) (1198982 minus 1)

119899sum1

int11990510VℎV119909=119871119889119905 (34)

On the downstroke the relationship between the velocityof oil (119908ℎ) and the rod speed (Vℎ) can be obtained

119908ℎ = 11 minus 1198982 Vℎ (35)

According to (28)-(32) and (35) the local viscous energyconsumption of downstroke at coupling can be given by

119882ℎ2= 13120587120583 times 104 times (119903ℎ1199031 minus 0381)

257 (108 minus 1691198982) 1199030(119898 minus 1) (1 minus 1198982)sdot 119899sum1

int1198791199051

VℎV119909=119871119889119905(36)

52 Local Viscous Energy Consumption at Rod Guides Fric-tion caused by the resistance along rod guides can beexpressed as [26]

1198651198921 = 2120587120583120594119897119892119899119892119908119892 (37)

Friction caused by local hydraulic loss at rod guides canbe expressed as

1198651198922 = 2120587120583120594119897119892119890119899119892119908119892 (38)

According to (37) and (38) the calculation formula oflocal damping force at rod guides can be given by

119865119892 = 2120587120583120594119899119892119908119892 (119897119892 + 119897119892119890) (39)

where 119897119892 is the length of rod guides 119899119892 is the number ofrod guides 119908g is the average velocity of oil passing throughrod guides120594 is the dimensionless coefficient which is definedas

120594 = 1ln (21199031119889119892119890) (40)

119897119892119890 is the equivalent length of rod guides

119897119892119890 = 120576119892120582119892 119889119892119890 (41)

120576119892 is the local resistance coefficient of rod guides 120582119892is the resistance coefficient along the rod guides 119889119892119890 is theequivalent diameter of rod guides

119889119892119890 = minus2119860119892119905 + 2radic1198602119892119905 + (11986211989211990511990321 minus 21198601198921199051199031) 119862119892119905119862119892119905 (42)

where 119860119892119905 is the flow surface between rod guides andtubing 119862119892119905 is the wet circumference of flow surface betweenrod guides and tubing

Local resistance pressure drop can be defined as shownbelow

Δ119875119892 = 119865119892119860119892119905 (43)

Local viscous energy consumption at rod guides is asfollows

119882119892 = int1199050Δ119875119892119876119889119905 (44)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908g) and the rodspeed (Vℎ) can be obtained

119908119892 = 120587 (1199032119901 minus 11990320)119860119892119905 Vℎ (45)

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

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6 Mathematical Problems in Engineering

While on the upstroke sgn(Vℎ) is equal to 1 when Vℎ islarger than 0 on the downstroke sgn(Vℎ) is equal to -1 whenVℎ is less than 0

119873Re = 2119908ℎ120588 (1199031 minus 1199030)120583 (29)

119873Re1015840 = 2Vℎ120588 (1199031 minus 1199030)120583 (30)

Local resistance pressure drop can be defined as shownbelow

Δ119875ℎ = 119865ℎ120587 (1199032ℎminus 11990320) =

121205821205881199082ℎ sgn (Vℎ) (31)

Local viscous energy consumption at coupling

119882ℎ = 119899sum1

int1199050Δ119875ℎ119876119889119905 (32)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908ℎ) and the rodspeed (Vℎ) can be obtained

119908ℎ = 1199032119901 minus 1199032011990321 minus 11990320 Vℎ (33)

According to (28)-(33) the local viscous energy con-sumption of upstroke at coupling can be given by

119882ℎ1 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1) 1199030(119898 minus 1) (1198982 minus 1)

119899sum1

int11990510VℎV119909=119871119889119905 (34)

On the downstroke the relationship between the velocityof oil (119908ℎ) and the rod speed (Vℎ) can be obtained

119908ℎ = 11 minus 1198982 Vℎ (35)

According to (28)-(32) and (35) the local viscous energyconsumption of downstroke at coupling can be given by

119882ℎ2= 13120587120583 times 104 times (119903ℎ1199031 minus 0381)

257 (108 minus 1691198982) 1199030(119898 minus 1) (1 minus 1198982)sdot 119899sum1

int1198791199051

VℎV119909=119871119889119905(36)

52 Local Viscous Energy Consumption at Rod Guides Fric-tion caused by the resistance along rod guides can beexpressed as [26]

1198651198921 = 2120587120583120594119897119892119899119892119908119892 (37)

Friction caused by local hydraulic loss at rod guides canbe expressed as

1198651198922 = 2120587120583120594119897119892119890119899119892119908119892 (38)

According to (37) and (38) the calculation formula oflocal damping force at rod guides can be given by

119865119892 = 2120587120583120594119899119892119908119892 (119897119892 + 119897119892119890) (39)

where 119897119892 is the length of rod guides 119899119892 is the number ofrod guides 119908g is the average velocity of oil passing throughrod guides120594 is the dimensionless coefficient which is definedas

120594 = 1ln (21199031119889119892119890) (40)

119897119892119890 is the equivalent length of rod guides

119897119892119890 = 120576119892120582119892 119889119892119890 (41)

120576119892 is the local resistance coefficient of rod guides 120582119892is the resistance coefficient along the rod guides 119889119892119890 is theequivalent diameter of rod guides

119889119892119890 = minus2119860119892119905 + 2radic1198602119892119905 + (11986211989211990511990321 minus 21198601198921199051199031) 119862119892119905119862119892119905 (42)

where 119860119892119905 is the flow surface between rod guides andtubing 119862119892119905 is the wet circumference of flow surface betweenrod guides and tubing

Local resistance pressure drop can be defined as shownbelow

Δ119875119892 = 119865119892119860119892119905 (43)

Local viscous energy consumption at rod guides is asfollows

119882119892 = int1199050Δ119875119892119876119889119905 (44)

On the upstroke according to the continuity equationthe relationship between the velocity of oil (119908g) and the rodspeed (Vℎ) can be obtained

119908119892 = 120587 (1199032119901 minus 11990320)119860119892119905 Vℎ (45)

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

According to (37)-(45) the local viscous energy con-sumption of upstroke at rod guides can be given by

119882g1 = 21205873120583120594 (119897119892 + 119897119892119890) 119899119892 (1199032119901 minus 11990320)2 int11990510 V2ℎ119889119905119860119892119905 (46)

On the downstroke the relationship between the velocityof oil (119908119892) and the rod speed (Vℎ) can be obtained

119908119892 = minus12058711990320119860119892119905 Vℎ (47)

According to (37)-(44) and (47) the local viscous energyconsumption of downstroke at rod guides can be given by

119882g2 = 21205873120583120594 (119897119892 + 119897119892119890) 11989911989211990340 int1198791199051 V2ℎ119889119905119860119892119905 (48)

6 Equivalent Viscous Damping Coefficient

61 Theoretical Calculation During the actual movementof the sucker rod the energy consumption includes theresistance consumption along the sucker rod and the localenergy consumption which can be respectively expressed by(23) (26) (34) (36) (46) and (48) These energy consump-tion equations all contain the velocity of sucker rod Whencalculating the equivalent viscous damping coefficient it isnecessary to simplify the term of velocity in the energy con-sumption equation at the final calculation of the equivalentviscous damping coefficient

According to (1)-(4) the velocity equation of pumpingrod can be obtained by Fourier series expansion

V = V119909=0 + 119899sum119899=1

(119886119899119909 cos 119899120596119905 + 119887119899119909 sin 119899120596119905) (49)

The first half of (49) represents the suspension velocitythe second half represents the wave velocity 119886119899119909 and 119887119899119909 arethe Fourier coefficient

The wave velocity is a periodic function and is far lessthan the suspension velocityTherefore the influence of wavevelocity can be neglected in calculating energy consumption

V asymp V119909=0 (50)

int1198710int1198790V2119889119905119889119909 asymp int119871

0int1198790V119909=119871V119889119905119889119909 asymp int119871

0int1198790V2119909=119871119889119905119889119909

asymp int1198710int1198790V2119909=0119889119905119889119909

(51)

In themotion period of the sucker rod the upstroke speedand downstroke speed can be regarded as equal and (51) canbe simplified to

int1198710int11990510V2119889119905119889119909 asymp int119871

0int1198791199051

V119909=119871V119889119905119889119909asymp 12 int

119871

0int1198790V2119909=119871119889119905119889119909

(52)

The velocity of integral terms in (34) and (36) can beapproximated to

119899sum1

int11990510V2ℎ119889119905 asymp 119899sum

1

int1198791199051

V2ℎ119889119905 asymp 121198711 int119871

0int1198790V2119889119905119889119909 (53)

The velocity of integral terms in (46) and (48) can beapproximated to

int11990510V2ℎ119889119905 asymp int119879

1199051

V2ℎ119889119905 asymp 12119871 int119871

0int1198790V2119889119905119889119909 (54)

where 1198711 is the length of per sucker rodSubstituting (23) (26) (34) (36) (46) (48) and (50)-(54)

into (7) the viscous damping coefficient can be given by

119888 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711 + 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871

(55)

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Equation (55) can be decomposed into

119888 = 1198881 + 1198882 + 1198883 (56)

1198881 = 120587120583 (1198982 minus 1 minus 2 ln119898) (21198982 minus 41198982 ln119898 + 4 ln119898 minus 2)12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

+ 2120587120583 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 + 2ln2119898(11989822 minus 1)2 + 2ln2119898]12058712058811990311990320 [(1198984 minus 1) ln119898 minus (1198982 minus 1)2 ln119898]

(57)

1198882 = 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (27711989822 minus 1691198982 minus 108) (11989822 minus 1)21205871205881199031199030 (119898 minus 1) (1198982 minus 1) 1198711

+ 13120587120583 times 104 times (119903ℎ1199031 minus 0381)257 (1691198982 minus 108)

21205871205881199031199030 (1198982 minus 1) (119898 minus 1) 1198711(58)

1198883 = 1205872120583120594 ((1199032119901 minus 11990320)2 + 11990340) (119897119892 + 119897119892119890) 119899119892

11986011989211990512058811990311990320119871 (59)

1198881 is the equivalent damping coefficient caused by theviscous energy consumption in the process of sucker rodmovement 1198882 is the equivalent viscous damping coefficientcaused by the coupling energy consumption in the process ofsucker rod movement and 1198883 is the equivalent viscous damp-ing coefficient caused by the rod guides energy consumptionin the process of sucker rod movement The equivalentdamping coefficient of the sucker rod system is equal to thelinear superposition of the three parts

The equivalent damping coefficient of the sucker rodcan be calculated by (55) According to the actual measureof downhole pump dynamometer card and the surfacedynamometer card the actual equivalent damping coefficientof the sucker rod can be estimated Next the relevantparameters of oil wells in Xinjiang KaramayOilfield are takeninto (55) and Zhang Qi formula to estimate the equivalentdamping coefficient of sucker rods Compared with themeasured equivalent damping coefficient of sucker rods theresults are shown in Table 1

In this paper the energy consumption of coupling androd guides is considered The predicted value of the equiv-alent damping coefficient derived from this method is largerthan that predicted by Zhang Qi formula and is closer to thecalculated value of the measured parameters (Table 1)

62 Application and Regulation Analysis The theoreticalformula of equivalent damping coefficient is showed in (55)Themain factors affecting the equivalent damping coefficientare the radius of sucker rod the radius of tubing and oilviscosity When calculating the equivalent damping coeffi-cient the oil well parameters (Table 2) of Xinjiang KaramayOilfield are referred The radius of sucker rod is regarded asthe independent variable under different tubing radius The

equivalent damping coefficient caused by the viscous energyconsumption along with the sucker rod movement the localenergy consumption and the sucker rod system is calculatedrespectively

In Figure 4 the equivalent damping coefficient of thesucker rod system is between 001 and 0281When the tubingsize is fixed the equivalent damping coefficient of the suckerrod system and the equivalent damping coefficient causedby the viscous energy consumption along the sucker rodsystemdecreasewith the increasing of the radius of the suckerrod Subsequently the reduction rate of equivalent dampingcoefficient decreases graduallyThe equivalent damping coef-ficient caused by coupling energy consumption raises withthe increasing of sucker rod radius and the rate is graduallyraised The equivalent damping coefficient caused by rodguides energy consumption is between 000042 and 00060The rod guides have little effect on the equivalent dampingcoefficientThe larger the radius of the sucker rod the smallerthe influence of the rod guide In practical application thisphenomenon is usuallymanifested as follows due to the largedownhole load in deep well and heavy oil well it is necessaryto select large size sucker rod in order to meet the strengthwhich results in the decrease of annulus area between tubingand sucker rodThe velocity of oil is quickenedwhen the oil ispassing through the coupling and the rod guide the suddenchange of the sectional area in coupling and rod guidesincreases the influence on flow state of oil and the energyconsumption of coupling increases rapidly under certainconditions of oil well production Therefore it is necessaryto ensure that there is sufficient annulus area between suckerrod and tubing and select sucker rod with smaller size whenstrength is sufficient in the design andmatching of sucker rodand tubing

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Table 1 Main parameters and calculated results of 119888r0 rp r1 120583(N∙sm2) Pump setting depthm well depthm 119888119886 119888119887 119888119888mm mm mm8 16 253 0033 1148 1239 03845 04085 0347795 16 253 0028 1415 1504 03604 03807 0298495 16 253 0022 1836 1963 03221 03689 0234595 19 31 0029 1168 1253 03708 04605 0309195 19 31 0023 1724 1804 03369 03653 02451Notes 119888119886 is the value calculated by measured parameters 119888119887 is the value calculated by (55) 119888119888 is the value calculated by Zhang Qi formula

Table 2 Parameters in field application

r0 mm r1 mm rp mm 120588 (kgm3) 120583 (N∙sm2)65-95 205 16 7850 002065-115 253 19 7850 002065-115 31 22 7850 00208-145 38 285 7850 002095-145 442 35 7850 002011-145 503 415 7850 0020

Compared with Figures 4(a)ndash4(f) the ratio of equivalentdamping coefficient caused by coupling energy consumptionto equivalent damping coefficient of sucker rod systemdecreases gradually with the increasing of radius of thetubing The equivalent damping coefficient of sucker rodsystem is mainly produced by viscous energy consumptionalong the tubing When the radius of the tubing increasesto a value the equivalent damping coefficient caused by theenergy consumption of the coupling can be neglected Withthe increase of the radius of tubing the effect of rod guideson equivalent damping coefficient increases gradually butthe increase is not obvious The proportion of equivalentdamping coefficient caused by rod guides to the equivalentdamping coefficient of sucker rod system is less than 4At the scene of the oil field the main reason is that theannulus area between the tubing and the sucker rod amplifiesand the sudden change of the sectional area in sucker rodcoupling weakens the influence on flow state of oil with theincreasing of the tubing radius Therefore during the designand manufacture of sucker rod reducing the sudden changeof cross-section area at sucker rod coupling has remarkableeffect on reducing damping force on the basis of satisfyingthe strength of sucker rod coupling

In order to more accurately analyze the proportion ofequivalent damping coefficient caused by viscous energyconsumption in the total equivalent damping coefficient ofsucker rod system the proportional coefficient 119870 is intro-duced K represents the ratio of area of cross-section in thesucker rod and tubing

119870 = 1199032011990321 (60)

The smaller the ratio k of sectional area of tubing to suckerrod is (Figure 5) the larger the damping coefficient causedby viscous energy consumption in the sucker rod system is

For 119870 lt 0095 the damping coefficient caused by viscousenergy consumption in the sucker rod system accounts formore than 90 of the equivalent damping coefficient Inthe pumping field the larger the annulus area between thetubing and the sucker rod is the smaller the damping of thesucker rod system by coupling energy consumption is In slimhole oil well the tubing size is small and coupling energyconsumption is an important part of energy consumptionof sucker rod system Therefore to calculate the equivalentdamping coefficient of sucker rod system the local dampingat the coupling should be considered However for the largersize of oil tube and smaller size of sucker rod the dampingof sucker rod system is mainly caused by viscous energyconsumption So the influence of coupling and rod guidescan be neglected for the sake of simplified calculation

7 Conclusions

(1) The function of energy consumption along the suckerrod stroke is derived when taking the fluid in the tubing asCouette flow and the viscous energy consumption along thesucker rod downstroke and upstroke is deduced Consideringthe influence of sucker rod coupling and rod guides thefunction of the local energy consumption is derived under thecondition of oil passing through the coupling and rod guides

(2) Based on the principle of equal friction loss thetheoretical formula of equivalent damping coefficient ofsucker rod system is deduced by considering the viscousenergy consumption along the sucker rod and the localenergy consumption of the coupling and rod guides Theeffect of equivalent damping coefficient on sucker rod sys-tem is analyzed combined with the parameters in fieldapplication The results show that with the increase of theradius of rod the annulus area of sucker rod and tubingthe equivalent damping coefficient the equivalent dampingcoefficient caused by viscous energy consumption and the

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

C2 C1

C3 C

00065 00070 00075 00080 00085 00090 00095000002004006008010012014016018020022

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(a) r1= 205mm

C2 C1

C3 C

0006 0007 0008 0009 0010 0011 0012

000

002

004

006

008

010

012

014

016

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

(b) r1= 253mm

0006 0007 0008 0009 0010 0011 0012

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(c) r1= 31mm

0007 0008 0009 0010 0011 0012 0013 0014 0015

000002004006008010012014016018020

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(d) r1= 38mm

0009 0010 0011 0012 0013 0014 0015

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(e) r1= 442mm

00110 00115 00120 00125 00130 00135 00140 00145

000

005

010

015

020

025

030

Equi

vale

nt v

iscou

s dam

ping

coeffi

cien

t

Radius of sucker rod(mm)

C2 C1

C3 C

(f) r1= 503mm

Figure 4 Relationship between damping coefficient and radius of sucker rod (a) r1= 205mm (b) r1= 253mm (c) r1= 31mm (d) r1= 38mm(e) r1= 442mm and (f) r1= 503mm

equivalent damping coefficient caused by rod guides energyconsumption all decrease on the contrary the equivalentdamping coefficient caused by coupling energy consumptionincreases During the design and matching of sucker rod andtubing it is helpful to reduce the energy consumption ofcoupling by selecting the smaller sucker rod with sufficientstrength

(3) With the increase of the radius of tubing the ratio ofthe equivalent damping coefficient caused by local viscous

energy consumption to the equivalent damping coefficientof the sucker rod system decreases gradually The equivalentdamping coefficient of the sucker rod system is mainlyproduced by the viscous energy consumption along thesucker rod The smaller the ratio k of sectional area of tubingto sucker rod is the larger the damping coefficient caused byviscous energy consumption in the sucker rod system is For119870 lt 0095 the damping coefficient caused by viscous energyconsumption in the sucker rod system accounts formore than

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

004 006 008 010 012 014 016 018 020 022253035404550556065707580859095

100105

e r

atio

of e

quiv

alen

t dam

ping

coeffi

cien

t cau

sed

by en

ergy

cons

umpt

ion

alon

g th

e dist

ance

(

)

Proportional coefficient K

Ｌ1=205mmＬ1=253mmＬ1=31mm

Ｌ1=38mmＬ1=442mmＬ1=503mm

Figure 5 Relationship between equivalent damping coefficientalong the tubing and proportional coefficient K

90 of the equivalent damping coefficient In the design andmanufacture of sucker rod reducing the sudden change ofcross-section area of sucker rod coupling has a remarkableeffect on reducing damping force

Nomenclature

119906 Displacement of the sucker rod node (m)119909 Distance between the node and thewellhead (m)119888 Equivalent viscous damping coefficient119908g Average flow velocity of oil before andafter rod guides (ms)119889119892119890 Equivalent diameter of rod guides (m)119862119892119905 Wet circumference of flow surfacebetween rod guides and tubing (m)119882119892 Local viscous energy consumption at rodguides (Nsdotm)119882119889 Total energy consumption (Nsdotm)119871 Vertical depth (m)120583 Viscous coefficient of oil (Pasdots)1198711 Length of per sucker rod119898 Ratio of radius of tubing to sucker rod1198821198891 Viscous energy consumption alongupstroke (Nsdotm)1198821198892 Viscous energy consumption alongdownstroke (Nsdotm)119882ℎ1 Coupling energy consumption alongupstroke (Nsdotm)1199051 Upstroke time (s)1199030 Radius of sucker rod (m)119892 Gravity acceleration (ms2)119903119901 Radius of the pump plunger (m)

119870 Ratio of sectional area of tubing to sucker rod119897119892 Length of rod guides (m)119860119892119905 Flow surface between rod guides and tubing(m2)119899119892 Number of rod guidesΔ119875119892 Local resistance pressure drop at rod guides(Pa)119882ℎ Frictional work of sucker rod string (Nsdotm)119908ℎ Velocity of oil at coupling (ms)119876 Cross-section flow (m3s)119879 Period of the sucker rod motion (s)

Vℎ Speed of sucker rod (ms)119905 Time (s)119882g1 Local viscous energy consumption ofupstroke at rod guides (Nsdotm)119882g2 Local viscous energy consumption ofdownstroke at rod guides (Nsdotm)119882ℎ2 Coupling energy consumption alongdownstroke (Nsdotm)120588119903 Density of the rod string (Kgm3)1199031 Radius of tubing (m)120588 Density of the fluid (Kgm3)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors acknowledge the financial support by MianyangSaien New Energy Technology Co Ltd

References

[1] D J Schafer and J W Jennings An Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical ModelsTexas A ampM University 1987

[2] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[3] I N Shardakov and I N Wasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[4] J Xu A New Approach to the Analysis of Deviated Rod-PumpedWells Society of Petroleum Engineers Richardson Tx 1994

[5] L Zifeng ldquoYanshan University Research advances and debateson tubular mechanics in oil and gas wellsrdquo Journal of PetroleumScience amp Engineering 2016

[6] K Palka and J A Czyz ldquoOptimizing downhole fluid productionof sucker-rod pumps with variable motor speedrdquo SPE Produc-tion and Operations vol 24 no 2 pp 346ndash352 2009

[7] C H Liu ldquoSucker rod string design of the pumping systemsrdquoIngenieria e Investigacion vol 35 no 2 pp 6ndash14 2015

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

[8] G D Reges L Schnitman R Reis and F Mota ldquoA newapproach to diagnosis of Sucker Rod Pump Systems by analyz-ing segments of downhole dynamometer cardsrdquo in Proceedingsof the 2015 SPE Artificial Lift Conference - Latin America andCaribbean pp 414ndash426 Brazil May 2015

[9] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[10] M Xing and S Dong ldquoAn improved longitudinal vibrationmodel and dynamic characteristic of sucker rod stringrdquo Journalof Vibroengineering vol 16 no 7 pp 3432ndash3448 2014

[11] K Li X Gao Z Tian and Z Qiu ldquoUsing the curve momentand the PSO-SVMmethod to diagnose downhole conditions ofa sucker rod pumping unitrdquo Petroleum Science vol 10 no 1 pp73ndash80 2013

[12] S Miska ldquo9800858 a simple model for computer-aided opti-mization and design of sucker-rod pumping systemsrdquo Fuel ampEnergy Abstracts vol 39 no 3 pp 303ndash312 1997

[13] S Gibbs and A Neely ldquoComputer diagnosis of down-holeconditions in sucker rod pumping wellsrdquo Journal of PetroleumTechnology vol 18 no 01 pp 91ndash98 2013

[14] S Gibbs ldquoPredicting the behavior of sucker-rod pumpingsystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 2013

[15] S G Gibbs ldquoMethod of determining sucker rod pump perfor-mancerdquo US US3343409 1967

[16] S D L Lekia and R D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystems - part 1 model theory and solution methodologyrdquo SPEProduction and Facilities vol 10 no 1 pp 26ndash33 1995

[17] D Rubio and L San Andres ldquoStructural stiffness dry frictioncoefficient and equivalent viscous damping in a bump-type foilgas bearingrdquo Journal of Engineering for Gas Turbines and Powervol 129 no 2 pp 737ndash746 2007

[18] Q Zhang ldquoPumping well diagnostic technique and its appli-cationrdquo Journal of the University of Petroleum China 1984(Chinese)

[19] M J Bastian A two equation method for calculating down-holedynomometer cards with a stady of damping effects [MSThesis]Texas A ampM University Colleges Station TX USA 1989

[20] T A Everitt and J W Jennings ldquoImproved finite-differencecalculation of downhole dynamometer cards for sucker-rodpumpsrdquo SPE Production Engineering vol 7 no 1 pp 121ndash1271992

[21] R Sun Y Y Zhang F K Fan et al ldquoA new model andcalculatingmethod for pump dynamograph diagnosis of suckerrod pumping systemsrdquo Applied Mathematics amp Mechanics vol35 no 4 pp 423ndash431 2014

[22] D Wang and H Liu ldquoDynamic Modeling and Analysis ofSucker Rod Pumping System in a Directional Wellrdquo in Mecha-nism andMachine Science vol 408 of Lecture Notes in ElectricalEngineering pp 1115ndash1127 Springer Singapore Singapore 2017

[23] I Steliga J Grydzhuk and A Dzhus ldquoAn experimental andtheoretical method of calculating the damping ratio of thesucker rod column oscillationrdquo EasternEuropean Journal ofEnterprise Technologies vol 2 no 7 pp 20ndash25 2016

[24] W G Zhou and Z He Viscous Fluid Mechanics NationalDefense Industry Press 2013

[25] D Doty R and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo Society of Petroleum Engineers AIME vol 23 no 1pp 33ndash41 1983

[26] L X Du Lifting System and Simulation Optimization of High-Angle Pumping Wells Yanshan University 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom