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Conventional fault diagnosis and production calculation of an oil well can be conducted with the surface dynamometer cards, which are obtained by load sensor installed on the horse head. This method to measure the dynamometer cards is limited by the sensor maintenance and calibration, battery replacement, and safety hazards for staff. As the basic parameter of the oil extraction industry, electric parameters have the advantages of low cost and high efficiency. So the inversions of dynamometer card with electric parameters are attracting more and more attention. In order to solve the problem of insufficient data and consider the real-time performance in the actual oil extraction process, this paper proposes a novel hybrid model which consists of two parts: the mechanism model of polished rod load and the suspension displacement calculated with the space vector equations of motor and a data-dependent kernel online sequential extreme learning machine (DDKOS-ELM) model proposed to correct the output error of the mechanism model, which improves the kernel function selection and makes it real-time. Thus, the highlights of this paper can be summed up in two points:

The beam pumping unit plays a predominant part in oil extraction, which is the major artificial lift method and employed by virtually more than 80% oil wells all over the world [

As the significant reference variables which are the direct and real-time reflection of the working condition of the beam pumping system, the induction motor electronic parameters are often ignored by the diagnostic staff. There are some calculation methods for the transformation from motor parameters to dynamometer card in recent years. Li et al. [

In consideration and summary of above discussion, our main contributions in this paper can be summed up as follows. For the part of the mechanism model, the relationship among voltage, current, and torque in the stationary second rectangular coordinate system was derived based on the mathematical model of AC induction motor. The instantaneous torque of AC induction motor was identified according to the stator line voltage and the line current of the motor, and then the polished rod load was calculated from the motor torque in consideration of the crank weight. Finally, the corresponding relation between the polished rod load and suspension displacement is obtained from the signal of the crank rotation angle. For the part of the data-driven model, the inputs of it are the mechanism and electric motor parameters including stroke, the crank radius, stroke frequency, pulley diameter, unbalanced weight, crank quality, balance block weight, two line voltages and line currents of the stator, motor speed, and other technical parameters, and the outputs are the error estimation between the actual measured values and the outputs of the mechanism model.

The typical structure diagram of beam pumping units is shown as Figure

Typical structure diagram of the beam pumping units.

In a stroke, the closed curve mapped out by the load and displacement of the suspension point is known as the dynamometer card, and the area of the curve is the size of the work done by the pump in a single stroke.

When only considering the liquid column static load above the rod string and the plunger surface and the elastic deformation of sucker rod and tubing, regardless of the sand, wax, gas, and interference of other environmental factors, the curve is obtained as the theoretical dynamometer card, which is similar to parallelogram. As shown in Figure

Theoretical dynamometer card.

When the load on the polished rod increases enough to pull the plunger, the elastic deformation of tubing and sucker rod ends, the high load BC section starts drawing, at this time the load of the polished rod stays the same, the plunger is pulled upward, standing valve is opened, and the pumped oil takes up the pump space until the top dead point.

In unloading section CD, the plunger reaches the top dead point and starts to go down, the travelling valve is opened and the standing valve is closed, elastic deformation of tubing and sucker rod occurs again, the tubing is elongated, and the sucker rod is shortened. During this time, although the polished rod is moving downward, the plunger relative to the pump barrel does not produce relative changes either.

When the load on the polished rod is reduced to the same as the weight of the plunger and the polished rod, the elastic deformation of tubing and sucker rod ends, the low load DA section starts drawing, at this time the load of the polished rod remains the same, the plunger and the polished rod move downward in a constant speed, travelling valve is opened, and the pumped oil flowed into the plunger from the pump barrel.

It is not hard to conclude from the oil pumping process introduced above that different shapes of the dynamometer cards may cause corresponding changes because of the combined variation of gas, liquid, and machinery.

In the actual oil production field, the induction motor is employed as the main equipment for the oil pumping motor, and the space vector equation of ac motor can be described as follows [

Then the

Considering the mechanical loss of the motor, the output torque and output power of the motor can be obtained from the following formulas:

The surface structure of the beam pumping unit consists of drive motor, reduction gearbox, and four-bar linkage. For the four-bar linkage mechanism, the fixed rod is the connection between the beam support point and the crankshaft center, and the three movable rods are crank, connecting rod, and backward beam. As shown in Figure

Four-bar linkage mechanism.

The angle between the backward beam and connecting rod can be calculated by applying the cosine theorem to triangle

The angle between the crank and connecting rod can be calculated as follows:

The angles of

Counting from the bottom dead point (12 o’clock of crank), the motor parameters are collected every 40ms,

where

When the beam pumping unit is working, the torque synthesized on the crankshaft by the suspension point load and the balance weight is in equilibrium with the output torque of motor.

Thus, the relation between the displacement and the crank angle

It can be observed from the description in Section

Extreme Learning Machine (ELM) [

Structure diagram of basic ELM model with single input and output.

Then the hidden layer output matrix

The training purpose of ELM model is to find the optimal output weight matrix

After solving the quadratic optimization problem above, the output of the ELM is obtained as follows:

If the nonlinear mapping function

Then the output function of kernel-based ELM can be expressed as follows:

Apparently, the selection and construction of kernel functions greatly influence the performance of K-ELM. Besides, as the volume of datasets increases, the training time complexity of

In the actual oil extraction process, the measured dynamometer cards data is arriving in a streaming fashion in the actual oil extraction process. Therefore, in order to ease the computation complexity of the inverse matrix in (

According to the OS-ELM, the dataset is divided into successive minibatches

In combination with recursive least squares (RLS), at each new minibatch

In the traditional kernel ELM for regression, the selection of kernel function mostly depends on experience and the optimization of kernel parameters generally depends on intelligent optimization algorithms such as genetic algorithm, clonal selection algorithm, and particle swarm optimization algorithm. To a certain extent, these methods improve the performance of kernel based learning algorithms. However, the impact of data is not taken into account, the distribution structure of data in the kernel mapping space does not change and the performance of kernel learning algorithm cannot be fundamentally raised to higher level because the kernel function is fixed in different training samples’ structures. Thus, a novel data-dependent kernel learning method is proposed to improve the performance by replacing the basic kernel with a data-dependent kernel function [

There are many methods to optimize the combination expansion coefficients and the Gaussian kernel parameter

As a major parameter of the algorithm, the search radius

Thus, the implementation steps of improved free search algorithm can be summarized as Algorithm

Step

Step

Step

When the termination condition is satisfied, output the optimal parameter

By applying the data-dependent kernel to the kernel online sequential extreme learning machine for regression, the training framework of the proposed IFS-DDKOS-ELM error model is summarized as Algorithm

Based on the above discussion, the parallel hybrid model is constructed in Figure

The prediction process of hybrid model.

The inputs of the hybrid model are the mechanism and electric motor parameters including stroke, the crank radius, stroke frequency, pulley diameter, unbalanced weight, crank quality, balance block weight, three line voltages and line currents of the stator, motor speed, and other technical parameters. The outputs of the hybrid model are expressed as follows:

In this section, the proposed hybrid model is firstly employed to predict the load and the displacement of the suspension point of the beam pumping unit. Secondly, the curves of polished rod load and suspension point displacement with respect to crank angle

Technical parameters of the beam pumping unit.

Number of oil wells | CYJ10-3-53HB |
---|---|

Stroke number per minute (min^{−1}) | 5 |

Transmission ratio of belt-pulley gear box | 150.82 |

Crank radius (m) | 1.134 |

Length of the link rod (m) | 3.60 |

Length of the beam forearm (m) | 3.8 |

Length of the beam backward arm (m) | 2.5 |

Vertical height H (m) | 3.2 |

Distance between the reference points K (m) | 4.53 |

Transmission efficiency | 0.95 |

Unbalanced weight (kN) | -1.12 |

Balance block weight (kN) | 1.40 |

Electronic parameters of the induction motor.

Number of oil wells | CYJ10-3-53HB |
---|---|

Rated power (kW) | 25 |

Rated voltage (V) | 380 |

Rated rotor speed ( | 585 |

Stator resistance ( | 0.36 |

Rotor resistance ( | 0.19 |

Stator inductance (mH) | 33 |

Rotor inductance (mH) | 35 |

Peak value of the mutual inductance between rotor and stator (mH) | 41 |

Moment of inertia ( | 1.71 |

Rotational resistance coefficient | 0.019 |

Pole-pairs number | 3 |

Motor transmission efficiency | 0.98 |

The data to establish the hybrid model is collected with the indicator instrument, the electric parameters measurement instrument, and two proximity switches. The indicator instrument installed at the suspension point collects the surface dynamometer cards periodically when the beam pumping units are working. The collected dynamometer cards are taken as the reference data and the historical training data for the IFS-DDK-ELM error predictive model. The electric parameters measurement instrument is installed in the electric control cabinet connecting to the induction motor with the sampling interval of 47.8ms, and the collected current, voltage, and power of the induction motor are taken as the input of the hybrid model. One proximity switch is installed at 12 o’clock on the pedestal of the crank shaft to measure the cycle of one stroke, when the crank is rotated to the 12 o’clock position, triggering the proximity switch action, and the timer of the electric parameters measurement instrument starts ticking until the crank is rotated again to 12 o’clock. Therefore, a periodic displacement curve can be obtained. The other one is installed on the motor shaft to measure the motor speed.

The data collection instruments are installed behind the electronic control cabinet (as shown in Figure

The data collection instruments installed behind the electronic control cabinet.

1800 groups of measurement data in total three months from Aug 9, 2017, to Nov 10, 2017, including the electric parameters and the dynamometer cards are collected to establish the data-driven model. Randomly select 1200 groups of the collected data as the training samples and the other 600 groups of data as the test samples. Figure

Comparison of performance indicators between predictive value and actual value.

Predictive value | Actual value | Test error (%) | |
---|---|---|---|

Maximum load (kN) | 82.87 | 82.22 | 0.7905 |

Minimum load (kN) | 44.49 | 44.32 | 0.3835 |

Area of dynamometer card | 67.0445 | 67.2565 | 0.3152 |

The suspension point displacement curve.

The torque factor curve.

Polished rod load curve.

The prediction result of dynamometer card.

In order to approve the effectiveness of the proposed hybrid prediction method, the mechanism predictive model without data-driven model and pure IFS-DDKOS-ELM model are chosen to compare the prediction performance, and the prediction performances are shown as Figure

The predictive contrast results of dynamometer card.

Root mean square error (RMSE) and mean absolute error (MAE) are used as the prediction evaluation indicators, which are defined as follows:

Comparison of the prediction results with the three models by RMSE and MAE.

Predictive methods | RMSE | MAE |
---|---|---|

Hybrid model | 0.2383 | 0.1731 |

Mechanism model | 1.4829 | 1.1775 |

Pure IFS-DDKOS-ELM | 1.4365 | 1.1780 |

Comparison of the prediction results with the three data-driven models by RMSE and MAE.

Predictive methods | RMSE | MAE |
---|---|---|

IFS-DDKOS-ELM model | 1.4365 | 1.1780 |

LS-SVM model | 3.4873 | 2.9140 |

KOS-ELM model | 2.5195 | 2.1492 |

From the prediction performance of Figure

In summary, the experiment results demonstrate that the proposed hybrid model can perform better dynamometer cards inversion effect than the pure mechanism model and the pure data-driven model. And it can produce precise dynamometer cards for fault diagnosis of the beam pumping unit.

In actual oil production, real-time and accurate collection of dynamometer cards has far-reaching influence on the oil well fault diagnosis. In view of the insufficient data in the actual oilfield, the data collection is not timely. This paper proposes a novel hybrid modeling for dynamometer cards inversion based on the electronic parameters collected from the motor. The simulation results can basically match with the practical measurement dynamometer cards. The advantages of the proposed hybrid model are summarized as follows.

As the project will not be finished until December 2019 and we have confidentiality agreement with Liaohe Oilfield, the data could not be released so far. For any information about the article, please contact us via weijingl_neu@163.com.

The authors declare that they have no conflicts of interest.

Financial support from the National Natural Science Foundation of China under Grants 61573088, 61573087, and 61433004 is acknowledged. The authors are also grateful for the support from the Liaohe Oilfield of China National Petroleum Corporation, providing them with research and experimental conditions.