Issue 
Sci. Tech. Energ. Transition
Volume 79, 2024



Article Number  59  
Number of page(s)  9  
DOI  https://doi.org/10.2516/stet/2024063  
Published online  27 August 2024 
Regular Article
Realtime automatic identification methods for downhole whirling based on mechanical specific energy model of drill bit
^{1}
State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, PR China
^{2}
Research Institute of Petroleum Exploration and Development, CNPC, Beijing 100083, PR China
^{*} Corresponding author: yhy_skd@sdsut.edu.cn
Received:
26
October
2023
Accepted:
29
July
2024
In the drilling exploitation of hot dry rock for geothermal energy, whirling is one of the main lowefficiency conditions that affect the efficiency of Polycrystalline Diamond Compact (PDC) bit in horizontal well drilling. Realizing automatic and realtime identification with whirling is of great significance to save nonproductive time and ensure drilling safety and benefit. In this paper, we first constructed a realtime drilling mechanical specific energy (MSE) model combined with whiledrilling testing to reflect the realtime drilling conditions. The MSE model is used to normalize multisource parameters. Secondly, we constructed a Back Propagation (BP) artificial neural network (ANN), and then the normalization effect verification, optimization of network parameters, and identification effect verification of whirling were carried out. The final results show that the established MSE model has a favourable effect on data normalization, which could also reduce the complexity of the required network model, and shorten the training time by 20–30 s/step. The optimal algorithm is Trainscg, whose optimal number of hidden layer nodes is 5, and the optimal maximum number of iteration steps is 1000. The established ANN model can accurately identify whirling based on MSE, the accuracy is about 0.94, and the average relative error is 1.3%. The method established in this paper provides a reference for the automatic identification of various lowefficiency conditions based on MSE.
Key words: Whirl / Mechanical specific energy / BP artificial neural network / Machine learning / Drilling optimization
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nomenclature
v : Rate of penetration (ROP), m/h
A_{b}: The area of the drill bit, m^{2}
T : The torque of drill bit, N · m
N : The rotation in a minute (RPM), r/min
E_{m}: The energy efficiency, dimensionless
D_{b}: The diameter of drill bit or wellbore, m
μ : The resistance coefficient between rotational drill bit and stable bottomhole rock, dimensionless
W_{e}: Effective weight on bit, N
∆P_{b}: The pressure drop of drill bit, Pa
η : The efficiency factor, dimensionless
ρ_{d}: The density of drilling fluid, kg/m^{3}
A_{0}: The equivalent diameter of bit nozzles, m^{3}
T_{e}: Torque of bit while normal rotary drilling, N · m
T_{p}: Torque consumed by rotary drill pipes, N · m
T_{d}: Torque of Downhole Motor, N · m
ζ_{W}: Weight of bit efficiency coefficient, dimensionless
ζ_{T}: Torque efficiency coefficient, dimensionless
MSE_{e}: The effective Mechanical Specific Energy, Mpa
MSE: Mechanical Specific Energy, Mpa
λ_{W}: Influencing coefficient of weight of bit on specific energy, dimensionless
λ_{T}: Influencing coefficient of torque on specific energy, dimensionless
λ_{Q}: Influencing coefficient of hydraulic parameter on specific energy, dimensionless
1 Introduction
The dry hot rock reservoir contains abundant geothermal energy. The benefit of hot dry rock exploitation determines whether it can become strategic replacement energy. During the exploitation of hot dry rock for geothermal energy, drilling a sufficient length of horizontal section to carry out multistage volume fracturing is the key to ensuring industrial value and heat recovery rate The applicating of Polycrystalline Diamond Compact (PDC) bits in horizontal wells plays an important role in improving the rate of penetration (ROP) and ensuring the efficiency of horizontal section extension. Whirling is one of the main reasons affecting the fatigue life of PDC bits [1, 2]. Bit whirling refers to when the bit rotates in the hole, the instantaneous rotation centre displaces, deviates from the geometric centre, and hits the wall [3]. When whirling occurs, a large centrifugal force will be generated, and friction will be generated at the contact point incidentally intensifying the motion [4]. Whirling causes continuous partial contrarotation of PDC bit cutters, coupled with the cutter characteristic that the lateral and reverse stress intensity is weaker than that of the forward one, which is easy to causes bit damage [5, 6]. Timely identification and control of whirling can improve the economic benefit.
The Mechanical Specific Energy (MSE) Model reflects realtime drilling efficiency by establishing a functional correlation between drilling parameters and total rockbreaking power. The more stable/lower the specific energy value, the higher the drilling efficiency is [7–9]. Its specific response characteristics can well reflect certain drilling conditions, which contribute to the identification of whirling [10–12].
The identification and optimization of lowefficiency drilling conditions based on MSE has a relatively mature application in the field. However, the existing research methods mainly rely on manual judgment and verification based on experience after drilling [13, 14], It is of better significance to realize realtime automatic identification and optimization of whirling while drilling. At present, the researches on automatic working condition recognition mainly use machine learning models such as support vector machines, long and shortterm memory recurrent neural networks deep learning, etc. With multiple drilling parameter vectors as the input and cumbersome data processing, conduct the training and testing [15–18]. According to the principles of the existing theoretical model, MSE completes the normalization of weight on bit (WOB), rpm, torque, ROP, jet impact, bit diameter, flow rate, pump pressure, and even other parameters including the bit nozzle parameters, that is, its dimension is unified into Pa. Theoretically, the input vector group of multiparameter vectors can be simplified by using the MSE model, which provides an optimization scheme for automatic condition recognition based on machine learning. According to the characteristics of MSE, in particular, it has an important role in simplifying training for the accurate identification of lowefficiency conditions. Further, based on the MSE model, a simpler machine learning model and a smaller amount of training are enough to identify the lowefficiency conditions with high accuracy. In this paper, based on the construction of MSE model combined with testing while drilling and reflecting the realtime conditions of the downhole, a Back Propagation (BP) artificial neural network (ANN) is established and the network parameter value is optimized. After training and testing, the network can accurately identify the downhole whirling based on MSE.
2 Physical and mathematical model
2.1 Construction of realtime rock breaking specific energy model of drill bit
Based on the combined deformation of tension and torsion, and rock breaking power of drilling tools, Teale [19] proposed the MSE model,(1)where W is WOB, N; v stands for ROP, m/h; A_{b} represents the area of the drill bit, m^{2}, T stands for the torque of drill bit, N · m; N is the rotation in a minute (RPM), r/min.
Dupreest and Koederitz [20, 21] proposed that the highest drilling efficiency is usually around 35% when studying the use of the MSE model for drilling time prediction. Cherif and Bits [22] studied that the range of mechanical efficiency (E_{m}) values is generally between 0.26 and 0.64.(2) (3)
Meng et al. [23] considered the effect of hydraulic factors on rock breaking and bottom hole purification. Then corrected the MSE in WOB and cited the drill bit torque integration formula to correct the torque. Finally, formed a hydraulic MSE model under actual drilling conditions:(4)where ∆P_{b} is the pressure drop of the drill bit, Pa; Q stands for the displacement, m^{3}/h; η represents the efficiency factor, dimensionless.
In the Meng model, the effective WOB (W_{e}) is calculated by,(5)where the F_{j} is calculated by,(6)where ρ_{d} is the density of drilling fluid, kg/m^{3}. A_{0} represents the equivalent diameter of bit nozzles, m^{3}.
The principle is to subtract the hydraulic impact reaction force of the drill bit from the recorded drilling pressure value on the ground to obtain the effective WOB of the drill bit. In fact, during the actual drilling process, the frictional resistance F_{f} between the drilling tool and the wellbore during the lowering of the drilling tool also consumes a certain amount of WOB. Therefore, the effective weight on bit (W_{e}) can be calculated by,(7)where F_{f} is the frictional resistance of the drilling tool.
The frictional resistance here includes the frictional force, obstruction force, and other actual loss of WOB exerted by the wellbore wall on the drilling string during the drilling process, it is not completely equivalent to frictional force. We once proposed a method of segmented testing of drilling tool friction by lifting the drilling tools away from the bottom of the well, putting down drilling tools without real drilling and recording the hook load. The floating weight of drilling tools and the hook load during drilling will vary with the increase of well depth, but the frictional resistance F_{f} changes very little within a day normally. This method is also applicable to nonstraight well sections.
For torque, the actual torque required in engineering applications is also at the drill bit rather than at the ground. The calculation method for drill bit torque in the Meng model is:(8)where D_{b} stands for the diameter of the drill bit or wellbore, m; μ is defined as the resistance coefficient between the rotational drill bit and stable bottomhole rock, dimensionless. W_{e} is effective WOB, N.
The sliding friction coefficient μ in the formula is related to the rock strength under confining pressure, density of drilling fluid, and diameter of the drill bit. We also modified the existing model by getting torque of bit from field test rather than theoretical derivation. Lift the drilling tool away from the bottom of the well and keep it idling at the original speed, record the torque value T_{p}. Considering the T_{d} provided by the downhole motor, the torque at the bit can be calculated by,(9)
Furthermore, the WOB efficiency coefficient ζ_{W}, torque efficiency coefficient ζ_{T}, the effective Mechanical Specific Energy MSE_{e} and the expression for the effective MSE are formed:(10)
According to the mechanical efficiency coefficient E_{m}, we can see that:(15)
Introduce the influencing coefficient of WOB on specific energy λ_{W}, the influencing coefficient of torque on specific energy λ_{T}, and the influencing coefficient of hydraulic parameter on specific energy λ_{Q},(17) (18) (19)
Furthermore, the efficiency of MSE based on drilling parameters can be obtained,(20)
In this way, a MSE efficiency expression based on engineering segment testing is formed. F_{f} and T_{p} are obtained by sectional testing, and then the MSE efficiency E_{m} is calculated by the WOB efficiency coefficient and torque efficiency coefficient as the bridge. Based on sufficient E_{m} statistics, the MSE_{e} value of the drill bit can be obtained directly from ground parameters in the same formation/under similar conditions, reducing the workload and cost of testing drill string friction and torque, and making it easier to obtain the specific rock breaking energy of drill bit. The MSE model can reflect the downhole drilling conditions in realtime by measuring the rockbreaking efficiency.
2.2 Establishment of BP artificial neural network
BP algorithm ANN is the most common model of ANN, which is mainly composed of one input layer, one or more hidden layers, and one output layer. In an integrated training process, the connection weights between nodes are constantly updated. The excitation function introduces nonlinear factors into the neural network, so it is important to select the appropriate excitation function for the model training of the BP ANN. The excitation functions used in this paper are tansig and purelin. The Tansig function can effectively map the input value to another space and can compress the variation range of the input, making the input value easier to process. The Purelin function is differentiable everywhere in its domain and is often used as an excitation function for the output layer because it can map an input of any size to an output of any size. The BP neural network model used to study the lowefficiency conditions of downhole whirling is shown in Figure 1. The input layers are multidimensional vector sets of DEPTH, HOOKLOAD, BITDEP, WOB, TORQUE, RPM, FLOWIN or onedimensional vectors of MSE. The output layer includes whirling and normal conditions.
Fig. 1 Diagram of BP neural network structure for whirling identification. 
The training of the BP neural network mainly includes forward propagation and backpropagation. In the forward propagation process, the training sample data is input into the input layer of the ANN, and then the weight between the hidden layer is linearly transformed. The calculated result needs to be processed by the excitation function, and then the input feature vector processed by the activation function of the hidden layer is transmitted to the output layer. In this process, the output of each layer is the input of the next layer. In the process of reverse error propagation, the error between the predicted output and the actual output is backpropagated, the error is propagated layer by layer, and the weight and deviation of the network are adjusted until the accuracy reaches the set requirement or the number of iterations reaches the set number.
Although the training accuracy increases with the increase in the number of hidden layers, overfitting will occur when the number of layers is too large, which reduces the reliability of the training results. The number of hidden layers in this paper is chosen as 2 after testing. The initial values are determined as follows: the target error is 10^{−6}, the maximum momentum is 10^{10}, the learning rate is 0.1, and the minimum gradient is 10^{−6}. Goodness of fit and average relative error were used to evaluate the test results. The number of hidden layer nodes, algorithm, and maximum number of iteration steps were selected by testing, too. The Kfold crossvalidation method was used for training and testing, and the process is shown in Figure 2.
Fig. 2 Schematic diagram of Kfold crossvalidation. 
3 Training and testing procedures
3.1 Data processing
3.1.1 Data cleaning
Located in the low steep structure zone of the southeastern Sichuan Basin, China, a development evaluation well occurred with obvious whirling damage in multiple runs during the second open. The wear and collapse of the shoulder teeth/outer teeth are heavier than those of the nose teeth/inner teeth. The afterdrilling bit of some runs is shown in Figure 3. Given this lowefficiency condition, the data is selected for further processing.
Fig. 3 Whirling damage of bits in part of the runs. 
However, the original data recorded by the well log contains some parameters that are not meaningful for our research, and there are some missing and local burr values caused by downhole drilling conditions and fluctuations of signal transmission. Further processing of the data is required. First, within the threshold of depth and time near the whirling, continuous and complete logging data should be selected as far as possible. For the missing drilling data, the maximum likelihood estimation method, random forest method, Knearest neighbour method and other methods could be used to fill in the data [24–26]. In this paper, we used the mean filling method to fill in the missing parts. Further, to screen and smooth partial outliers caused by engineering environment, sensing or detection problems, the common methods include the Local Outlier Factor (LOF) method applicable to highdimensional data, and a wavelet noise reduction algorithm based on multilevel joint detection and reconstruction of data [27–29]. The method we used was the 3σ method based on mean value and variance. The original logging data consists of 16 parameters including date time, MSE, ROP and others. Part of the data of the first run firstly selected is shown in Table 1, and the process is shown in Table 2.
Part of the firstly selected logging data of 1^{#} run.
Processed logging data of 1^{#} run.
3.1.2 Data normalization
The orders of magnitude of the logging parameters are of large difference, which increases the difficulty of training and testing and affects the accuracy. In order to eliminate the influence of dimensionality between different features of the data, it is necessary to normalize the features. This step affects the accuracy of identification a lot, especially when the data set is built with multisource drilling parameters, which makes the ANN complicated and difficult to achieve reliable accuracy [30–33]. Based on constructing the MSE model to reflect the drilling conditions in realtime, the normalization of multisource drilling parameters is completed.
3.2 Analysis of the normalization effect of MSE
The BP ANN was constructed with the initial number of hidden layer nodes as 4, the training algorithm as Trainscg, and the maximum number of iteration steps as 1000. Firstly, well depth, hook load, bit depth, flowin, WOB, rotary torque, and rotary speed were used as common vectors for training and testing, and then MSE was used as an input vector alone. The results are shown in Figures 4 and 5. The true value of the well interval under normal working conditions is marked as 1, and the true value of the well interval under whirling is marked as 2. The higher the fitting degree between the output value after the BP neural network and the true value, the smaller the average relative error and the higher the recognition accuracy could be. It can be seen that the training of multiparameter input is ineffective when choosing a relatively simple ANN. However, the goodness of fit of MSE onedimensional input can reach 0.93424. The BP ANN can fully identify normal working conditions (1), and the error of identifying whirling (2) is pretty small. The results abovementioned verify the normalization effect of the constructed MSE model.
Fig. 4 Results of multiparameter input training test. 
Fig. 5 Results of MSE input training test. 
3.3 Optimize BP neural network parameters
In the preliminary test of the BP model, we found that the number of hidden layers and their nodes, the maximum number of iteration steps and the training algorithm have a significant impact on the final accuracy of the model. To this end, the optimal quantity tests of the above parameters were carried out respectively.
In order to optimize the number of the hidden layer and its nodes, the Trainscg algorithm was adopted, and the maximum number of iteration steps was set as 800 firstly. The network was trained with MSE as input under the number of the hidden layer as 1 and 2, and the nodes from 4 to 20 respectively. The test results are shown in Figure 6. It can be seen that the ANN model with two hidden layers (the same number of nodes in both hidden layers) has higher accuracy and better performance than the ANN model with one hidden layer. The optimal number of hidden layer nodes is 5.
Fig. 6 The influence of the number of hidden layers and its nodes on the training results. 
Further, with 2 hidden layers, and the same maximum number of iteration steps (800), the Trianlm and Traingd algorithms were replaced in optimizing the number of the hidden layer nodes respectively. The comparison of the three algorithms is shown in Figure 7. It can be seen that the optimal numbers of hidden layer nodes under Trainscg, Trainlm and Traingd algorithms respectively are 5, 6 and 12. Under the Traingd algorithm, with the increase in the number of hidden layer nodes, the fitting failure appears earlier. The overall goodness of fit of the Trainlm/LM algorithm is relatively stable, but the mean value is low. Although the goodness of fit fluctuates a little more obviously with the increase of the number of hidden layer nodes under the Trianscg algorithm, it has the highest goodness of fit under the optimal number of hidden layer nodes (5). Therefore, Trianscg is selected as the optimal algorithm, and the optimal number of hidden layer nodes is determined to be 5.
Fig. 7 The effect of algorithm on training results. 
Finally, under the optimal algorithm and the number of nodes, the maximum number of iterative steps from 100 to 1500 was selected to train the BP ANN respectively. The test results are shown in Figure 8. It is determined that the optimal value of the maximum number of iteration steps is 1000.
Fig. 8 The influence of the maximum number of iteration steps on the training results. 
3.4 Results of Kfold crossvalidation
Under the optimal model parameters, that is, the maximum number of iteration steps is 1000 and the number of hidden layer nodes is 5, the Trainscg algorithm is used to conduct 10fold crossvalidation training and testing. The results are shown in Table 3. It can be seen that when different sub datasets are used as test data sets, the differences in goodness of fit, relative error and time consumption are small. The time consumption is shortened by 20–30 s/step compared with the normal multiparameter input network for condition recognition, which is usually 31–40 s/step. The abovementioned indicates that the performance of the BP ANN model is stable and efficient. In addition, the average value of the goodness of fit is 0.94, and the average value of the average relative errors is 1.3%, which indicates that the BP ANN has a good ability to identify the whirling.
Result of 10fold crossvalidation.
4 Conclusions
Constructing the realtime rockbreaking specific energy model of bit, the multidimensional logging parameters are normalized. A BP ANN with tansig and purelin as excitation function is established to identify the whirling automatically. The following conclusions can be drawn:

The MSE model based on the whiledrilling testing can achieve the normalization of the multiparameters in logging data. Using MSE to simplify and do the reduction of the dimensionality of multisource parameter data can reduce the complexity and training difficulty of the required ANN model, and shorten the training time.

The optimal parameters of the established BP ANN read as follows: Trainscg algorithm has more advantages than Trainlm and Traingd algorithms. The maximum number of iteration steps is 1000, the number of hidden layers is 2 and the number of nodes is 5.

Machine learning based on the established MSE and BP ANNs can achieve accurate identification of whirling. More meaningfully, the method established in this paper can provide a reference for the automatic identification of other working conditions.
Acknowledgments
The financial support from the National Natural Science Foundation of China (52204009, 52174026 and 42102338), and the Natural Science Foundation of Shandong Province (ZR2021ME008) are highly appreciated.
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All Tables
All Figures
Fig. 1 Diagram of BP neural network structure for whirling identification. 

In the text 
Fig. 2 Schematic diagram of Kfold crossvalidation. 

In the text 
Fig. 3 Whirling damage of bits in part of the runs. 

In the text 
Fig. 4 Results of multiparameter input training test. 

In the text 
Fig. 5 Results of MSE input training test. 

In the text 
Fig. 6 The influence of the number of hidden layers and its nodes on the training results. 

In the text 
Fig. 7 The effect of algorithm on training results. 

In the text 
Fig. 8 The influence of the maximum number of iteration steps on the training results. 

In the text 
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