This invention relates to techniques for damping lateral vibrations in electric submersible pumps (ESPs) of the type typically used in the oil industry. In particular, the invention relates to techniques for controlling motor speed to damp such vibrations.
ESPs are used in the oil industry to provide artificial lift in oil wells that do not have enough pressure to produce to the surface. ESPs typically comprise a motor section and a pump section, often separated by a protector section including a crossover and pump inlet. Because of the typical dimensions of an oil well, it is necessary for the motor (and pump) to be formed as a relatively long and thin unit. A shaft (or combination of shafts) extends through the motor so as to support a rotor inside as stator section; and then extends into the pump section where it supports a series of impellers which, together with corresponding diffusers fixed to the pump housing define a centrifugal pump. Such pumps typically operate at speeds of up to 3000 rpm, although higher speeds have been proposed.
While the pumps are designed to operate at substantially constant speed, there is a time during startup where operation will be taking place at other speeds for periods of time. It is generally attempted to balance the rotating structure of the ESP at the desired operational speed to avoid unwanted lateral vibrations.
It is known that lateral vibration of ESP shafts may reach undesirable values. This can happen, for example, during the startup and shutdown, when the ESP has to pass through a number of resonant frequencies; or during normal fluctuations of the operational speed if the ESP spectrum contains resonances close to this speed. Increases in vibrations may occur because of fluctuations of the ESP parameters during operation. Typical causes include:
The use of vibration dampers is not considered appropriate in case of ESPs as there is no possibility to tune the frequency band of a vibration damper since the shift of critical frequencies is generally unknown.
This invention seeks to provide an alternative method of damping based on operational control of the ESP.
A first aspect of the invention provides a method of controlling operation of an ESP, comprising:
Preferably, the periodic modulation comprises a harmonic additive of the determined operating speed, the method typically including determining values for amplitude and phase shift of the harmonic additive at which lateral vibrations are damped. The periodic modulation to be applied can be determined according to the relationship:
Ω+AnΩ cos (nΩt+ω0)
where Ω is the frequency of rotations, A and ω0 are the amplitude and the phase shift of the harmonic additive, t is the time, n is a number.
The periodic modulation is preferably selected so as to minimize torsional vibrations in the ESP.
The periodic modulation is typically selected so as to cause a variation of operating speed of less than 5%.
The operating speed at which the ESP experiences unacceptable levels of lateral vibrations can be determined by mathematical modelling and/or experimentation.
The method can also comprise monitoring operation of the ESP to detect unacceptable levels of lateral vibration and applying the modulation to the operating speed when such levels are detected.
A second aspect of the invention provides an ESP comprising a control system which operates according to the method of the first aspect of the invention.
Further aspects of the invention will be apparent from the following description.
According to the method of the invention, as soon as ESP lateral vibration reaches a critical value, generation of additional harmonic component to the constant frequency of rotations (for example, simply using frequency generator) can provide drastic decrease of such vibration. Thus, instead of rpm with frequency Ω, ESP rpm would have the form
Ω+AnΩ cos(nΩt+ω0)
where Ω is the frequency of rotation, A and ω0 are the amplitude and the phase shift of the harmonic additive, t is the time, n is a number. By choosing A and n one can provide different degrees of damping. Generally n can be any number: integer, fractional or even irrational. The pair of parameters should be chosen to ensure that the harmonic additive is among the allowable ones because certain changes in rpm may lead to an increase in torsional vibrations. The parameters can be determined both using mathematical modelling and experimentally.
To determine when the damping system must be activated (and/or to deactivated), it is desirable to provide feedback from the ESP relating to vibration levels. This can be done, for example, by measuring vibrations directly or inferring them from other measured operating parameters.
An example of the form of harmonic additive for the case Ω=100 Hz (i.e. 100 rpm), n=0.1, A=2.4, ω0=0 (Ω+AnΩcos(nΩt+ω0)=100+0.24 cos(10t)) is shown in
To illustrate how much vibration can be damped, normalized maximal radial displacement is shown in
where c is the bending stiffness of the shaft and m is the mass of the disk). Note that the shaft with one disk is quite a simple model; a more complex model might be needed to predict accurately vibrations of an ESP that has tens of masses (impellers) and a number of radial bearings quantitatively. Radial displacements for this model tend to come out much higher than those for real ESP because all masses of impellers are reduced just to the one large mass. Analytical solutions are difficult for the multi-mass model and so is not used here. However, the simplified model provides a good qualitative estimation of how much the radial vibration can be damped compared to the highest amplitudes that occur at first natural frequency
Within a range of variation of values of A and n for the cases shown in
By analyzing
The higher the parameter n, the better the damping (see
There is a number of values of A, for which damping is optimal (see two minima of the function shown in
The calculations presented above are based on a linear model in the approximation of first (“main”) resonance mode. Dynamics of real ESPs are governed by a large number of nonlinear equations accounting not only for first resonance mode but for the higher ones. An alternative to dealing with complex mathematics, is to perform a lab test aiming to:
Before testing a complex structure as an ESP, it is customary to perform a lab test.
In summary, the invention has the following features and benefits:
Mathematical modelling of the ESP vibration damping in the case of non-steady rpm can be based on consideration of a dynamic system that governs bending vibrations of a rotor consisting of a shaft with one imbalanced disk driven by an infinite power supply (see Error! Reference source not found.). The disk has an eccentricity (centre of gravity is placed at certain distance apart from the disc's centre). The shaft is fixed by bearings at its ends. Point W designates the disk geometrical centre, point S designates its gravity centre and the point O designates the axis of the unperturbed shaft.
When the rotor passes through a resonance domain, a harmonic component is added to a constant torque. Such a harmonic additive may also be used in the normal operating regime in the case when a control system shows undesirable vibration increase. In both cases, constant frequency of rotation experience harmonic modulation and governing equations have the form:
Here ε and κ are the external and internal damping, respectively, c is the bending stiffness of the shaft, φ is the angle of rotation, ω0 is the natural frequency of the rotor, Ω is the frequency of rotations, A and ω0 are the amplitude and the phase shift of the harmonic additive, n is a number. For convenience, variables and parameters in the system above are dimensionless; Ω=ω/ω0, differentiation is made over a dimensionless time ω0t=τ.
Suppose that coefficients of external and internal damping, and rotor eccentricity are small:
c/m=ω
0
2, εω0/c=μh, kω0/c=μh1, e=μv
where μ□ 1 is a small parameter. In this case, initial system takes the form
{dot over (x)}=x1,
{dot over (x)}1 =−Ω2x+μF1,
{dot over (y)}=y
1,
{dot over (y)}=−Ω2y+μF2,
{dot over (ψ)}=Ω.
Here F1=ν cos φ−(h+h1)x1−h1{dot over (100 )}y+Δx, F2=ν sin φ−(h+h1)y1+h1{dot over (φ)}x+Δy, φ=ψ+A sin(nψ+ψ0).
The goal is to answer the following question: would it be possible to choose the parameters of modulation A, n, ψ0 in such way that the amplitude of bending vibrations would be minimized and how these parameters should be chosen?
A system of two non-autonomous linear oscillators is obtained. Each of these oscillators represents a resonance filter of the frequency Ω, which corresponds to the first resonance harmonic cos φ=cos(ψ+A sin(nψ+ψ0)), sin φ=sin(ψ+A sin(nψ+ψ0)), ψ=Ωτ.
These functions may be decomposed into the Fourier series with coefficients Jk (A) representing Bessel functions of the first kind of integer argument. Naturally, since the model under consideration is linear, solutions can also be represented as such series. If the first harmonic that has the highest amplitude would be damped by a proper choice of the parameters of modulation, then amplitudes of the remaining harmonics will have values of the order ˜μ<<1. This is explained by filtering properties of the oscillators. It means that maximum radial displacements xmax, ymax will have the same order: xmax, ymax ˜μ (note that x(σ), y(σ) are the mutifrequency functions). Thus, the objective is to provide damping of the first harmonic.
A direct solution of this problem is: search for the solution of the initial system as a Fourier series with undetermined coefficients; determination of these coefficients and minimization of the amplitude of the first harmonic. At the same time, stability of the solution must be ensured. Both parts are standard if quite intricate. To combine both problems into the one problem, the averaging method can be used.
Using changes of the variables of the form
x=μ
1 sin ψ+ν1 cos ψ,
x
1=(μ1 cos ψ−ν1 sin ψ)Ω,
y=μ
2 sin ψ+ν2 cos ψ,
y
1=(μ2 cos ψ−ν2 sin ψ)Ω,
one can obtain a system of the form:
{dot over (μ)}1=μF1 cos ψ,
{dot over (ν)}1=−μF1 sin ψ,
{dot over (μ)}2=μF2 cos ψ,
{dot over (ν)}2=−μF2 sin ψ,
Values of the variables
{dot over (φ)}y sin ψψ, {dot over (φ)}y cos ψψ, {dot over (φ)}x sin ψψ, {dot over (φ)}x sin ψψ, cos(ψ+A sin(nψ+ψ0))sin ψψ, cos(ψ+A sin(nψ+ψ0))cos ψψ, sin ψ(ψ+A sin(nψ+ψ0))sin ψψ, sin(ψ+A sin(nψ+ψ0))cos ψψ strongly depend on the parametern . Namely, right-hand sides of the equations of the averaged system are be different for different n.
For this reason, it is necessary to consider three qualitatively different cases. Case 1. n is either any integer number of the interval n>2, any fractional number of the interval 0<n<1 or any irrational one.
Averaging system above over a fast-spinning phase yr and transforming time: μτ=τnew, one obtains equations in the first approximation with respect to a small parameter of the form
Here J0 (A) is the Bessel function of the first kind (see
Value A*=√{square root over (μ102+μ202+ν102ν202)}, where μ10, μ20 , ν10, ν20 are the coordinates of the equilibrium of this linear system is the amplitude of the first harmonic. The goal is to minimize this value.
The system thus obtained has a property: in the phase space of this system there exists a stable invariant manifold M={μ1=−ν2, μ2=ν1}. Indeed, system of equations with respect to the variables μ1+ν2=x,ν1−μ2=y has the form
The derivative of the Lyapunov function V=x2+y2, calculated in account of the system above is {dot over (V)}=−(h+2h1)(x2+y2)≦0, ∀(μ1, μ2, ν1, ν2, ξ)εG. Thus, equilibrium x=0,y=0, and, therefore, integral manifold M={μ1=−ν2, ν1=μ2} is stable.
This property allows consideration of the system at the manifold M={μ1=−ν2=μ, μ2=ν1=ν} that has the form
Values of the coordinates of equilibrium (μ0,ν0) of the system above are proportional to J0(A). The amplitude of the first harmonic is minimal for minimal values of |J0(A)| from the interval of allowed values of the amplitude of modulation A. The amplitude of the first harmonic is equal to zero (full damping) for all A, for which J0(A)=0 . Bessel function J0(A) has an infinite number of zeroes. In particular, the first zero corresponds to A=2.4 (minimal value). Having substituted A=2.4 into system above, a stable equilibrium is obtained.
Thus, having chosen A=2.4 , any integer n>2 or any irrational n, and any value of the phase shift ψ0 (for instance, ψ0=0), we obtain the full damping of the first harmonic of rotor bending vibration. In this case, amplitude of bending vibration becomes of the order ˜μ<<1.
Hereinafter, the following dimensionless parameters are used: h=h1=ω=ω0=ν=1, φ=0, μ=0.1
Case 2. n=1 (modulation at the rotor frequency). In this case, the averaged system takes the form
The equilibrium of system above is stable (the corresponding homogeneous system has a stable integral manifold with stable system, placed at this manifold, see above). This equilibrium has zero coordinates independently of the phase v0 , if J, (A)=J2(A)=0 . However, this system is inconsistent. In contrast to the previous case, for n=1 there are no values of A , for which the damping of the first harmonic would be full.
Case 3. n=2 (modulation at the doubled rotor frequency) In this case, averaged system takes the form
Note that in the case of integer n=1, 2 , damped vibrations are quasi-harmonic, while in the case of fractional n and n=3, damped vibrations are periodic but not harmonic.
Further variations within the scope of the invention will be apparent.
Filing Document | Filing Date | Country | Kind | 371c Date |
---|---|---|---|---|
PCT/RU08/00052 | 1/31/2008 | WO | 00 | 9/20/2010 |