M. Bubak et al. (Eds.): ICCS 2004, LNCS 3037, pp. 372379, 2004. Springer-Verlag Berlin Heidelberg 2004

A Uniform and Reduced Mathematical Modelfor Sucker Rod Pumping

Leiming Liu1, Chaonan Tong1, Jianqin Wang2, and Ranbing Liu3

1University of Science and Technology in Beijing, 100083, Chinaleiming.liu@163.com

2Laboratory of Remote Sensing Information Sciences, Institute of Remote SensingApplications, Chinese Academy of Sciences, P. Box 9718, Beijing 100101, China

3University of Petroleum in Beijing, 102249, China

Abstract. The dynamic mathematical models were greatly developed todescribe the motion of the sucker-rod pumping system since 1960s and variousmathematic models based on different assumptions were presented. This paperdoes a study in three sides. First, a mathematical model is presented to describethe sucker-rod string with longitudinal and transverse vibrations, and coupledwith longitudinal vibration of the tubing and fluid columns in a deviated well.Second, the relations of several kinds of mathematical models in sucker-rodpumping systems are discussed, and the model made in this paper, when basedon different assumptions, can become different models made by people theseyears, which are presented in important references. Third, a method ofcharacteristics is used to transform the set of partial differential equations whichdescribe the vibration of the sucker-rod string, and coupled with vibrations oftubing and liquid columns in the sucker-rod pumping system. Through thetransformation, a diagonal partial differential equations set is obtained. Hence arelatively complex model is transformed into a reduced model which is easy tosolve. This model has basic meaning for using the technique of patternrecognition to make automatic diagnosis of sucker rod pumping system.

1 Introduction

Rod pumping is still the most widely used means for artificial lift in oil wells, so eversince 1960s, people lay great emphasis on the mathematical methods of predicting,designing and diagnosis of the sucker-rod pumping systems.

Gibbs(1963)[1] made a 1D mathematical model for the vibration of the suckerrod, which is comprised with a second-order partial differential equation includingboundary conditions. Doty and Schmidt(1983)[2] presented a composite model inwhich both rod string dynamics and fluid dynamics are coupled to account for viscousdamping, which was comprised of four first-order equations with four unknownvariables, boundary conditions and initial conditions. In the paper of Wang et al.(1992)[3], a set of six equations governing the vibration of the sucker-rod string,tubing and liquid columns in the sucker-rod pumping system are presented.

All these three models stated above are for vertical wells. At present many wellsare designed as deviated wells. Lukasiewicz(1991)[4] presented a model of sucker-

A Uniform and Reduced Mathematical Model for Sucker Rod Pumping 373

rod string with longitudinal and transverse displacements in a inclined well. Gibbs(1992)[5] studied the situation that sucker rod is crooked, but he assumed thetransverse displacement was ignored, for the transverse displacement of the rod wasconstrained by the tubing. The wave equation about the rod with longitudinaldisplacement that Gibbs presented took the crooked wells into consideration. Xu andMo(1993, 1995)[6,7] made a 3D model of vibrations of sucker-rod string. Li et al.(1999)[8] made a set of equations describing the 3D vibration of the sucker rod beampumping system.

The models stated above didnt take the movement of the fluid column intoconsideration. On the basis of the work stated above, this paper presents a set of first-order partial differential equations and the concerning boundary and initial conditions.The equations describe the 2D vibration of rod string, the longitudinal vibration oftubing column and coupled with the displacement of the fluid column in thedirectional and horizontal wells. In these equations, three of them are geometryequations can be solved through the method dealing with non-dynamic problems, theother six first-order partial differential equations are a set of quasi-linear hyperbolicequations, they are dynamic equations, but can be reduced to diagonal equations, thatis to make each equation be a partial deviation equation containing only one unknownvariable.

2 Analysis of the Geometry Relations and Forces Acting on theSpatially Curved Rod

2.1 Geometry Equations of Spatial Curve

We build a spatial rectangular coordinates END (see Fig.1 )for the spatial crookedsucker-rod at the well bore, take a rod element of length s and build a naturalcoordinates t1 t2 t3 at the center ofs. t1 is the tangent unit vectors of the elements ,t2 is the principal unit normal vectorsand t3 is the subnormal unit normalvectors. The Frenet equations is:

23

312

21

1

/dd

/dd

/dd

/dd

tt

ttt

tt

rt

sks

ks

s

=

+=

=

=

(1)

where k , are the curvature andbuckling of the well axis curve andr the position vector of the point onthe curve .

Fig. 1. Forces on rod element

374 L. Liu et al.

2.2 The Analysis of Forces Acting on the Element of Curved Rod

Let the internal forces, force moments and external forces acting on element of thecurved rod and the displacement of the centroid of the element in the naturalcoordinates respectively be presented as follows:

==

==

TT

TT

uuuqqq

MMMfff

),,(,),,(

),,(,),,(

321321

321321

uq

Mf (2)

We can obtain the following force equilibrium equations and moment equilibriumequations:

==+

1tfM

uqf

s

tms

/

// 22 (3)

where m is the mass of unit length rod (kg) , and Am = , where A is the rod cross-sectional area (m2) , is the rod material density (kg/m3) .Let AE be draggingstiffness of the rod ( 2mN ), IE be bending stiffness of the rod (N). If the effects oftwisting moment are ignored, the constitutive equations describing the relationsbetween the internal forces acting on the rod element of length and deformation areobtained:

+

=

=

=

)(

)(

)(

322

2

3

223

2

2

21

ks

u

s

uEM

s

u

s

uEM

kus

uEf

I

I

A

(4)

3 Partial Differential Equations Describing the Movement ofSuck-Rod String and Tubing and Fluid Columns in theDirectional and Horizontal Wells

3.1 Fundamental Assumptions

(1) The well bore is considered as planar curve.(2) The sucker-rod only has longitudinal and transverse vibrations in a plane, i.e.u3=0.(3) The cross-section of the sucker rod is round.(4) The sucker rod string is concentric with the well bore.

According to these assumptions, from Eq. (4), we get M2 =0, from the secondequation of (3), we get 01 =

s

M.

A Uniform and Reduced Mathematical Model for Sucker Rod Pumping 375

3.2 Partial Differential Equations Describing Motions of Rod String andTubing Column

Based on fundamental assumptions, we expand Eq.(3) and take the derivative of thesecond equation of Eq.(3) with respect to s, then apply the principal of vectorderivation and Eq.(4), we obtain:

+

=

=

=+

=

=

=+

)(

)(,

)(

)(,

22

2

2

22

1

222

22

211

111

11

ks

u

sE

s

fkf

vt

u

t

vmq

s

f

vks

vE

t

f

vt

u

t

vmq

s

f

Irr

r

rrr

rrr

rrr

Arr

rrr

rrr

(5)

where rk is the plane curvature of the rod ( /m1 ), rm is mass of the unit length of thesucker rod ( kg ), ArE the dragging stiffness (N), IrE the bending stiffness ( 2mN ).

The curvature causes the lateral displacements of the rods between the two guides,so the relations between the transverse displacement and internal force of the suckerrod can be described by the last two equations of Eq.(5). As for the tubing, it has thesame expressions as Eq.(5).Considering the computing precision of the moment of the tubing, we ignore thetransverse displacement of the tubing, the last two equations of Eq.(5) can become ageometry equation. In the following discussion the subscribe t refer to tubingproperty, then we can obtain the differential equations describing vibrations of thetubing:

=+

=

=

=+

2

2

2

1

d

d

)(,

s

kEq

s

uEk

s

vE

t

f

vt

u

t

vmq

s

f

tItt

tAtt

tAt

t

ttt

ttt

(6)

3.3 1D Flow Equation of the Fluid

As for the flowing fluid of the tubing, we assume fluid column contains no gas,though the well bore is a curve, the movement of the fluid is still 1D flow. Let f bethe density of the fluid (kg/m3) at the point of s and at the time of t ; and the functionsof velocity (m/s) and pressure (N/m2) of the fluid are respectively represented by thevariables of fv and fp . So the equation of continuity of the fluid, Euler motionequation and the state equation are presented as follows:

376 L. Liu et al.

=

=

+

+

=

+

+

)(

)(

1

0

fff

rtf

ff

f

ff

f

ff

ff

f

pp

AA

q

s

p

s

vv

t

v

sv

s

v

t

(7)

where fq is the external forces acting on the fluid column of per unit length(including forces by the sucker rod string , rod coupling and tubing ), rA is the cross-sectional area of the sucker rod and tA the inner cross-sectional area of the tubing(m2). The equation of fluid state shows that the pressure fp is only related withdensity f , and according to the physical meaning, 0

d

d>

p

, so define

2)(d

df

f

f ap

= (8)

f has the representation:

)(0

0ppCf

f =

(9)

or the approximate expression of Eq. (9): )](1[ 00 ppC ff += (10)

where 0 , 0p are respectively the density and pressure of the fluid in the standardstate, C is the relative increment of the density when the pressure increases one unit,i.e. elastic condensation coefficient of the density of the fluid. Let

)(/ rtfff AAqQ = (11)Using Eqs. (8) and (11) , Eq. (7) becomes the following equations:

=

+

+

=

+

+

ff

f

fff

f

ff

ff

f

Qs

a

s

vv

s

vs

v

sv

t

)(

02 (12)

4 Analysis of External Forces Acting on the Rod String andTubing and Fluid Columns

Through the effects of the fluid viscosity resistance, lateral extrusion stress, coulombfriction and gravity caused by the periodic motion of the sucker rod, the sucker rod

A Uniform and Reduced Mathematical Model for Sucker Rod Pumping 377

string, tubing column and fluid column comprises a system of motion. These forcesare expressed as follows:

cos)( 111 mgvsignqvq rNrr += (13)

sincos22 mgqvq Nrr += (14)where is viscous damping coefficient of the fluid, rm is the mass of the unit lengthrod, is the angle between lateral extrusion stress and the principal normal vector, is friction factor, Nq is lateral extrusion stress between the rod and the tubing[7], is the angle between the tangent of the element of the curved rod and vertical. Herethe viscosity resistance is comprised of two parts, one is the viscosity resistancebetween fluid and the sucker rod, the other between the fluid and the coupling of therod, and also:

cos)(1 mgvsignqvq tNtt += (15)

sincos2 gmqq tNt = (16)where , , Nq is the same as which is stated above, but the difference is there is noterm of viscous resistance in Eq. (16), and tm is the mass of the unit length rod.The force fq in Eq. (7) has the representation as follows:

trrtff vvgAAq ++= 1)( (17)

1rv , tv of Eq. (17) has the contrary sign with the corresponding term of Eqs. (13)and (15), this shows that the viscous resistance acting on the rod string and tubingcolumn has the reaction to the fluid.

5 Boundary and Initial Conditions

The boundary and initial conditions are very important, the former determines if thesolution can correctly describe the motion stated, and the correct choice of the lattercan make the computing program converges to its periodic numeric solution ofstationary state as fast as possible. Papers [1], [2], [3] and [7] made deep study aboutthe boundary and initial conditions and presented useful conclusions. This paperapplies those results.As for the diagnosis, the surface boundary conditions are

====

====

)(),0(,0),(,0),0(,0),0(

0),0(,0),0(),(),0(),(),0(

02

221111

tptptsvtftv

tftvtvtvtftf

firtt

rrrr (18)

where )(1 tf is the measured loading function of the polished rod, )(1 tv is the velocityof the polished rod , is ( Ni ,,2,1 "= ) represents placing a rod guide where iss = , and

378 L. Liu et al.

0p represents tubing head pressure and is constant in most of the cases. As for thepredicted model, we should consider the boundary conditions in the place of oil-wellpump, see paper [3].

The correct choice of the initial conditions can make the computing programconverges to its periodic numeric solution of stationary state as fast as possible, sorefer to papers [2] and [3], we make discrete transform along the axes of the well forthe initial conditions given in the papers, and obtain the initial conditions for thismodel.

6 Characteristic Transform of the Partial Differential EquationsSet

The first two equations of Eq. (5), Eq. (6) and Eq.(12) comprise a equations set aboutthe unknown variables of 1rv , 1rf , tv , tf , fv and f . We apply the theory ofcharacteristic for the quasi-linear hyperbolic partial differential equations set totransform the six equations into a diagonal equations set about the variables of

1w , 2w , 3w , 4w , 5w and 6w .

=

+

=

++

=

+

=

=

+

=

ffff

tht

t

At

t

tht

t

At

t

rr

r

r

Ar

rr

r

r

Ar

Qs

wav

t

wQ

s

wav

t

wAA

q

s

w

Et

w

AA

q

s

w

Et

w

A

q

s

wE

t

w

A

q

s

wE

t

w

6655

4433

122111

)(,)(

)(,

)(

,

(19)

where 1111

r

Arrr

r fEA

vw

+= , 1121

r

Arrr

r fEA

vw

= , t

thAtt

t fAAE

vw)(

13

+=

,

t

thAtt

t fAAE

vw)(

14

=

, += f

f

zz

zavw f

0

d)(

5 , += ff

zz

zavw f

0

d)(

6 .

where 0f is the density of the fluid when 0pp = and 0p is the tubing head pressure.The Eq. (19) is easy to solve applying the difference method.

7 Conclusions

The model presented in this paper is the basic model for the directional and horizontalwells, it takes the longitudinal and transverse vibrations of the sucker rod intoconsideration, and also considers the coupled motions of rod string and tubing andfluid columns. So it is a relatively accurate model simulating the dynamic behavior ofthe sucker-rod system.

A Uniform and Reduced Mathematical Model for Sucker Rod Pumping 379

This is a basic model and concentrates the characters of some principalmathematical models of the sucker-rod system. When the curvature of the well bore

0=k , the mathematical model becomes the model for the coupled vibrations of therod string and tubing and fluid column in the vertical well[3]. If we further considerthe tubing is anchored, and assume the tubing is rigid, the model describing thecoupled vibrations of the rod string and fluid column in the vertical well isobtained[2]. If the fluid is assumed as non-condensable, the wave equation of thevibration of the sucker-rod in the vertical well is obtained[1].

As for the mathematical model about the directional and horizontal well, if weassume the fluid is non-condensable, and assume the tubing is anchored and rigid, wecan obtain the mathematical model about the 2D vibration of the sucker rod in thedirectional and horizontal well[4,5].

Finally, according to the characteristic deformation we have done, as the Eq. (19)shows, it is also easily-solving model.

Acknowledgement. Jianqin Wang thanks Dr. Yong Xue, his PhD supervisor, for hissupport from the "CAS Hundred Talents Program".

References

1. Gibbs, S.G., 1963. Predicting the behavior of sucker-rod systems. J. Pet. Technol. (Jul.), pp.769-778.

2. Doty, D.R. and Schmidt, Z., 1983. An improved model for sucker rod pumping. J. Soc. Pet.Eng. (Feb.), pp. 33-41.

3. Wang, G.W., Rahman,S.S. and Wang, G.Y.,1992.An improved model for sucker rodpumping systems. Proc. 11th. Australas Fluid Mech. Conf., Tasmania, 14-18 Dec., 1992,2:1137-1140.

4. Lukasiewicz, S.A., 1991.Dynamic behavior of the sucker rod string in the inclinedwell.SPE 21665,pp.313- 321.

5. Gibbs, S.G., 1992.Design and diagnosis of deviated rod-pumped wells.J. Pet. Technol.(Jul.), pp.774-781.

6. Xu, J., Hu, Y. and U.T.,1993.A method for designing and predicting the sucker rod string indeviated pumping wells.SPE 26929,pp.383-384.

7. Xu, J.and Mo, Y.,1995.Longitudinal and transverse vibration model of the sucker rod stringin directional wells and its application in diagnosis.J.Tongji.University.(Feb.), pp.26-30.

8. Li, Z. et al.,1999.Fundamental equations and its applications for dynamical analysis of rodand pipe string in oil and gas wells.ACTA.20(3),pp.87-90.

1 Introduction2 Analysis of the Gemetry Relations and Forces Acting on the Spatially Curved Rod2.1 Geometry Equations of Spatial Curve2.2 The Analysis of Forces Acting on the Element of Curved Rod

3 Partial Differential Equations Describing the Movement of Suck-Rod String and Tubing and Fluid Columns in the Directional and Horizontal Wells3.1 Fundamental Assumptions3.2 Partial Differential Equations Describing Motions of Rod String and Tubing Column3.3 1D Flow Equation of the Fluid

4 Analysis of External Forces Acting on the Rod String and Tubing and Fluid Columns5Boundary and Initial Conditions6Characteristic Transform of the Partial Differential Equations Set7ConclusionsAcknowledgement. Jianqin Wang thanks Dr. Yong Xue, his PhD supervisor, for his support from the "CAS Hundred Talents Program".References