rod pumping: modern methods of design, diagnosis, and

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  • R O D P U M P I N GModern Methods of Design, Diagnosis

    and Surveillance

    . . . wave equation in the form used in rod pumping

    @

    2y (x, t)

    @ t

    2= v2 @

    2y (x, t)

    @x

    2 c @y (x, t)

    @ t

    + g

    by

    Sam Gavin Gibbs, Ph.D.

  • 2THE WAVE EQUATION ASAPPLIED TO ROD PUMPING

    Contents

    2.1 Derivation of Wave Equation 342.2 Modeling of Downhole Friction 362.3 Speed of Stress Wave Propagation in Rod Material 372.4 Separating Solutions of the Wave Equation into

    Static and Dynamic Parts 392.4.1 Rod Buoyancy 46

    2.5 Design and Diagnostic Solutions 51Closure 51Exercises 54Glossary of Terms 56References 58

    33

  • 34 THE WAVE EQUATION AS APPLIED TO ROD PUMPING

    In this chapter. . .

    Wave equation as fundamental equation of rodpumping

    Derivation that includes the role of friction and gravity

    Examining the forces F (x, t) and F (x+x, t), theaxial forces along the rod

    A surprising connection with Archimedes principle ofbuoyancy

    Design and Diagnostic solutions

    2.1 Derivation of Wave Equation

    The purpose of this chapter is to establish the wave equation as thefundamental equation of rod pumping. The wave equation is usedto model the elastic behavior of the rod string. This famous equationarises from Newtonian dynamics and Hookes law of elasticity. Itsderivation is shown in many texts but usually without considerationof friction (Reference 1).

    Since downhole friction in a rod pumping system cannot be ig-nored, a derivation is given here which includes friction. Figure 2.1shows the forces acting on a rod element of length x with measureddepth x increasing downward. Velocities and forces are also consid-ered positive when oriented downward. The term Ax / 144g

    c

    rep-resents rod mass. Presuming that the element is moving downward,a friction force c0x@y(x, t) / @t acts in the upward direction.

    The forces F (x, t) and F (x + x, t) are axial forces along the rod.Newtons law states that unbalanced forces on the rod element causean acceleration of the element. Use of this law requires that velocitiesand accelerations be referred to a fixed coordinate system, say relative

  • 2.1 DERIVATION OF WAVE EQUATION 35

    Figure 2.1. Forces on rod element.

    to the well casing. Thus

    Ax

    144 gc

    @2 y (x, t)

    @ t2= F (x+x, t) F (x, t) c0x@y (x, t)

    @t+

    Agx

    144 gc

    .

    Using the definition of the partial derivative and passing to the limitas x ! 0, we obtain

    A

    144 gc

    @2y (x, t)

    @ t2=

    @ F (x, t)

    @ x c0 @ y (x, t)

    @ t+

    Ag

    144 gc

    .

    Introducing the one-dimensional form of Hookes law,

    F (x, t) = E A@ y (x, t)

    @ x,

    we obtain the wave equation in the form used in rod pumping,

    @2y (x, t)

    @ t2= v2

    @2y (x, t)

    @ x2 c @ y (x, t)

    @ t+ g , (2.1)

    in which

    v =

    s144E g

    c

    (2.2)

    andc =

    144 c0 gc

    A. (2.3)

  • 36 THE WAVE EQUATION AS APPLIED TO ROD PUMPING

    2.2 Modeling of Downhole Friction

    In Equation 2.1, friction along the rod string is presumed to be pro-portional to local velocity relative to the casing (Reference 2). Thisis not strictly correct. In reality, friction is a complicated blend ofCoulomb (rodtubing drag) and fluid friction effects. Fluid frictiondepends on relative velocity between rods and fluid (not rods andcasing). Coulomb friction is not considered to be velocity-dependentat all. The role of the friction term is to simulate removal of energyfrom the rod string. Experience shows that the same steady-state dy-namometer card results regardless of the manner in which the pump-ing equipment is started from rest. Downhole friction damps out theinitial transients in only a few strokes of the pumping system, and thedynamometer cards begin to repeat. We want the simulated pumpingsystem to exhibit this same behavior. In spite of its theoretical short-comings, the friction law postulated in Equation 2.1 has been found tobe useful and precise enough for practical purposes, at least in verticalwells. The friction law in Equation 2.1 is called equivalent friction. Thecriterion for equivalence is that the coefficient c is chosen to remove asmuch energy as do the real frictional forces.

    It is convenient to introduce a non-dimensional factor bymeans of

    c = v

    2L(2.4)

    The non-dimensional factor varies over narrow limits and is easier toremember than the dimensional quantities c and c0. The dimensionalquantity c0 and the non-dimensional factor are related through

    c0 = v A

    288gc

    L. (2.5)

    To get an impression of the magnitude of downhole friction in a verti-cal well, consider

    Example 2.1Using the dimensional quantity above, determine the dampingforce on 2000 ft of 0.875 in. rods moving at a uniform speed of10 ft/sec given the basic information:

  • 2.3 SPEED OF STRESS WAVE PROPAGATION IN ROD MATERIAL 37

    = 0.1 (a commonly used non-dimensional dampingvalue for low-viscosity pumping installations)

    v = 16000 ft /sec (velocity of stress waves in the rod material)

    = 490 lbm

    / ft3 (density of steel rods)

    A = 0.6012 in.2 (area of 0.875 in. rod)

    gc

    = 32.17 lbm

    ft /lbf

    sec2 (gravitational conversion constant)

    L= 4000 ft (pump depth) F

    Solution.

    Using Equation 2.5, convert from the nondimensional dampingformat to obtain the dimensional coefficient

    c0 = (16000) (0.1) (490) (0.6012)

    288 (32.17) (4000)= 0.0399 lb / ft / ft / sec .

    Expressed in words, the dimensional quantity means that a fric-tional force of 0.0399 lb

    f

    is exerted per foot of rod length perfoot per second of rod velocity. Thus, for 2000 ft of 0.875 in.rods moving at an average velocity of 10 ft/sec, the frictionalforce would be 798 lb, i.e., [0.0399 (2000) (10) = 798]. Average ve-locity is used for illustrative purposes only. The wave equationpredicts local velocity so that the appropriate damping force isapplied at each infinitesimal rod increment according to its ve-locity. N

    2.3 Speed of Stress Wave Propagation in RodMaterial

    The wave equation models the elastic behavior of the rod string.Mechanical stresses are propagated along the rod string at a certainspeed. An event which occurs at the downhole pump does not man-ifest itself immediately at the surface. A finite interval of time is re-

  • 38 THE WAVE EQUATION AS APPLIED TO ROD PUMPING

    quired for the stress wave to propagate the length of the rod string.In simulating the sucker rod system, the appropriate speed must becalculated and used in the model. The speed of stress wave travel isgiven by Equation 2.2. This is implied by the units of v, i.e., ft/sec.Another indicator that represents velocity is derived from one of theoldest solutions to the undamped wave equation, i.e., DAlembertssolution. This 18th century scientist showed that

    y (x, t) = f (x+ v t) + g (x v t) (2.6)

    is a solution to the undamped wave equation (obtained by setting c= 0in Equation 2.1). The arbitrary function f represents a wave travelingupward and g represents a wave traveling downward. If we presumeat t

    0

    that the f wave front is at x0

    , its value is f(x0

    + vt0

    ). At a latertime t

    1

    = t0

    + t, the f wave front has moved upward to positionx1

    = x0

    x. Thus, f now has the value f(x0

    x+ v(t0

    +t)). Inorder that the magnitude of f be the same at t

    0

    and t1

    , i.e., the positionon the wave front is maintained as the wave travels upward, we musthave x+ vt = 0. Thus, v = x/t, which implies that the waveprogresses at velocity v. Similar manipulations can be applied to thedownward-moving wave g.

    It is instructive to compute propagation velocity in a practical casefrom Equation 2.2. For steel rods (E = 30000000 psi and = 490 lb

    m

    /ft3),

    v =

    r144 (30000000) (32.17)

    490

    = 16841 ft/sec .

    Actual measurements indicate that velocity is about 16000 ft/sec insteel rods. Thus the value computed above is too fast. Although notintuitively obvious, Equation 2.2 applies to rods without couplings.Consideration of the added mass of the couplings in the computationsdiminishes the velocity. Since the distance between couplings is smallin relation to the length of the rod string, the effect of the couplingscan be approximated by increasing the density sufficiently to make arod without couplings as massive as a rod of the same body diameterwith couplings. Thus

    =144 g

    c

    w

    gA,

  • 2.4 SEPARATING SOLUTIONS OF THE WAVE EQUATION 39

    which leads to an alternate formula for propagation velocity,

    v =d

    2

    rE g

    w. (2.7)

    An example will clarify the use of Equation 2.7.

    Example 2.2Determine the speed of stress wave propagation in 1 in. rodsthat weigh 2.904 lb/ft. Assume that the local acceleration ofgravity is 32 ft/sec2. F

    Solution.From Equation 2.7,

    v =1

    2

    r (30000000) ( 32)

    2.904= 16014 ft/sec.

    This value better agrees with actual measurements of velocity.These measurements can be made by deliberately causing thepump to hit down. The round trip time of the stress wave socreated is measured and velocity is calculated from the distancetraveled. Equation 2.7 also applies to rod materials other thansteel. N

    2.4 Separating Solutions of the Wave Equation intoStatic and Dynamic Parts

    The g term in Equation 2.1 makes that equation non-homogeneous. Achange of variable will be made which makes the equation homoge-neous and at the same time reveals facts about buoyant rod weightand static stretch. For a rod string with only one interv

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