Drilling Parameters Optimization for Horizontal Wells Based on a Multiobjective Genetic Algorithm to Improve the Rate of Penetration and Reduce Drill String Drag

Abstract

With the development of China’s oil and gas exploration and development to complex oil and gas fields, the drilling efficiency and safety of complex formations with large hardness and strong abrasiveness have become increasingly significant. Optimizing drilling parameters is an effective means to increase the rate of penetration (ROP) and improve drilling efficiency. However, traditional drilling parameter optimization methods with only a single objective of increasing the ROP lack consideration of the drill string’s drag which may also be increased when drilling parameters change. When drilling a horizontal well, increased drag can reduce drilling efficiency. Aiming at this problem, this paper uses the logging data of the oil field as the data source, establishes an intelligent ROP prediction model through the random forest algorithm, and calculates the string drag using the “hard-string” model. Finally, the nondominant sorting genetic algorithm-II (NSGA-II), which is a domination-based multiobjective optimization algorithm, is used to optimize the drilling parameters to increase the ROP and reduce the drag at the same time. The optimized drilling parameters guide the drilling operations. We used the proposed method to optimize the parameters during the drilling of a new horizontal well. The results show that the ROP of the horizontal section of the new well increases by 10.3%, and the drag reduces by 4.5% on average compared with the adjacent well.

1. Introduction

The horizontal well is a well type formed by drilling the wellbore parallel to the oil reservoir for a certain length after drilling to the oil reservoir vertically or inclined. The horizontal wellbore has a larger contact area with the oil reservoir, which can greatly improve oil and gas production. However, the drilling operation of horizontal wells is difficult with slow ROP, which seriously affects the drilling efficiency [1]. Optimization of drilling parameters is an important means to improve drilling efficiency. With the main goal of high ROP and low cost, drilling parameters (e.g., weight on bit (WOB), revolutions per minute (RPM)), and hydraulic parameters (e.g., pump pressure (PP), pumping flow per minute (GPM)) are optimized [2].

So far, researchers have proposed a variety of methods for optimizing drilling parameters. Some previous methods optimize parameters with a single objective. Chen. [3], Bahari and Hankins [4,5], Hegde, Gray, and Momeni [6,7,8] took the minimum mechanical specific energy (MSE), the minimum drilling cost, and the maximum ROP, respectively, as single optimization objectives to optimize the drilling parameters. This single-objective optimization method takes into account fewer factors affecting drilling efficiency and has very limited speedup effects. In actual drilling operations, different optimization objectives often conflict with each other, and the increase in one objective is often accompanied by a decrease in other objectives. Therefore, researchers came to use multiple indicators as optimization objectives at the same time for the synergistic optimization of drilling parameters, as shown in

Table 1. Related works of multi-objective optimization of drilling parameters.

Work	Obj. Function	Opt. Parameters
Abughaban et al. [9]	max ROP, min MSE	WOB, RPM
Payette et al. [10]	max ROP, min stick–slip risk	WOB, RPM, GPM
Gidh et al. [11]	max ROP, max bit life	WOB, RPM
Guria et al. [12]	max ROP, max bit life	WOB, RPM, PP
Ammar et al. [13]	max ROP, min nonproductive time (NPT)	WOB, RPM, GPM

. These works achieved better results than single objective optimization. It can be seen that the method of multiobjective optimization of drilling parameters considering the restriction effect between objectives is more scientific.

Although drilling parameter optimization methods are developing rapidly, there is still a lack of research on drilling parameter optimization of horizontal wells [14], and the existing drilling parameter optimization methods do not consider the impact of horizontal well drag on drilling efficiency. For horizontal well drilling, the change in drilling parameters will affect the drag, and the drag will also affect ROP. In addition, large drag will also lead to complex downhole accidents such as stick–slip and stuck [15].

In this paper, according to the characteristics of horizontal well drilling, using oil field big data and machine learning algorithm, the maximum ROP and minimum drag are simultaneously used as optimization objectives of drilling parameter, and the parameter combination of horizontal well drilling is given more scientifically to improve drilling efficiency. Field tests show that the proposed optimization method of drilling parameters can improve the ROP of the horizontal section under the requirement of reducing drag.

2. Data and Methods

This study establishes data-driven ROP prediction models based on the big data from the oil field and machine learning algorithms. The “hard-string” drag calculation model is also established. Since both ROP and drag are affected by drilling parameters such as WOB and torque, we optimized both models simultaneously using NSGA-Ⅱalgorithm, a multitarget genetic optimization algorithm. The optimized drilling parameters can increase the ROP and reduce the string drag within a certain range. The specific work includes the four works shown in Figure 1.

Work 1: Merge and clean data from different sources, including mud logging, well logging, and bit parameters data. Clean data can improve the prediction accuracy of models;

Work 2: Establish the ROP prediction models based on SVM, random forest, and BP neural network algorithms, and compare the prediction effect of the models;

Work 3: Establish drag calculation model by “hard-string” drag calculation model;

Work 4: The ROP prediction model and drag calculation model are used as the objective functions, and the weight on bit (WOB), revolutions per minute (RPM), and pump flow rate (GPM) are optimized by NSGA-II multiobjective optimization algorithm.

2.1. Data

ROP is the result of engineering factors and formation factors. To make our intelligent ROP prediction model take into account engineering and formation factors, we collected bit data, mud logging data, and well logging data from six wells in the oil field and then merged the data existing in different documents into a data table by depth. The table contains a total of 22,317 pieces of data, and each piece contains 15 features, as shown in

Table 2. Oilfield multi-source data.

Data Sources	Features
Mud logging	Depth, rate of penetration (ROP), weight on bit (WOB), handing load, revolutions per minute (RPM), torque(T), pumping flow per minute (GPM), equivalent density of drilling fluid
Well logging	Well deviation, azimuth, acoustic logging, density logging
Bit parameters	Diameter, cutting-tooth diameter, number of cutting tooth

.

2.2. Data-Driven Algorithms

ROP is one of the important indicators to measure drilling efficiency. Therefore, the maximum ROP is selected as one of the optimization objectives in this study, and the ROP prediction model is taken as the optimization function in the multiobjective optimization algorithm. The traditional ROP model is based on laboratory experiments studying the relationship between parameters such as WOB, RPM, and ROP. The ROP prediction equations are established linear or nonlinear combinations of these parameters shown in

Table 3. Traditional ROP equations and coefficients.

Work	Equation	Coefficients
Bingham [16]	$\mathrm{ROP}=k{(\frac{W}{d})}^{a}N$	$k,a$
B and Y [17]	$\mathrm{ROP}=w_f\cdot k\cdot c\cdot W\cdot N^{\lambda }$	$w_f,k,c,\lambda$
Hareland [18]	$\mathrm{ROP}=w_f\frac{G\cdot N^{\gamma }\cdot W^{\alpha }}{d\cdot S}$	$w_f,\gamma ,\alpha$
Al-abduljabbar [19]	$\mathrm{ROP}=16.96\frac{W^a\cdot N\cdot T\cdot SSP\cdot Q}{d^2\cdot \rho \cdot PV\cdot UC{S}^b}$	$a,b$

Where W is the weight on the bit, N is the rotary speed, T is the torque, Q is theflow rate, d is the drill-bit diameter, SSP is the standpipe pressure, PV is the plastic viscosity, and UCS is the uniaxial compressive strength.

[16,17,18,19]. However, there are some uncertain coefficients in the traditional ROP model, which need to be adjusted according to different formations, different types of circulating media, and different drilling tool combinations.

With the digital transformation of the oil field, MWD and other measurement tools are widely used. The oil field recorded and stored a large amount of drilling and logging data. Nowadays, many data-driven ROP models have been proposed. Researchers used the random forest regression method (RF) [20,21], gradient boosting tree [22], BP neural network [23,24,25], Support Vector Machine Regression (SVM) [26,27], and other algorithms to establish the ROP prediction models. These ROP prediction models have higher accuracy and timeliness than traditional ROP models.

In this work, we establish three drill-rate prediction models based on SVM, RF, and BP neural network algorithms and compare their prediction accuracy within the same dataset. The principles of the three algorithms are below.

(1)

SVM algorithm

Support vector machine (SVM) is a machine learning method based on statistical learning theory. For regression problems, conventional regression models usually calculate the loss directly on the basis of the difference between the predicted values of the model and the true values, and the loss is 0 if and only if the predicted value of the model is exactly equal to the ground truth. The loss is 0 only if the predicted value of the model is exactly equal to the true values. However, the SVM regression model assumes that an error ε between the predicted value and the true value can be tolerated. The loss is calculated only if the absolute value of the difference between the predicted value and the true value is greater than 2ε, as shown in Figure 2. The training problem of the SVR model is an optimization problem under a given objective function and constraint conditions.

The objective function is as follows:

$$\underset{w,b}{\min }\frac{1}{2}||w|{|}^2+C{\displaystyle {\sum }_{i=1}^m{\xi }_i},{\xi }_i\ge 0$$

(1)

The constraint is as follows:

$$\left|\right|f(x_i)\left|\right|-y_i=\epsilon +{\xi }_i$$

(2)

where w is weight, b is the bias, C is the penalty factor, $\epsilon$ is the error tolerance interval, and ${\xi }_i$ is the slack variable.

(2)

Random forest algorithm

Random forest is a classical machine learning algorithm that has been widely used in the industry since it was first proposed by Breiman in 2001 [28]. In this algorithm, decision trees are selected as weak learners. Decision trees can be classified into classification trees and regression trees. In this paper, we established the ROP regression prediction model, so the regression decision tree is selected as the weak learner. The schematic diagram of the random forest algorithm is shown in Figure 3.

When training the binary decision tree model, we need to consider how to select the cut variables and cut points and how to measure the goodness of a cut variable and cut point. The impurity of the cut nodes is generally used to measure the goodness. In Breiman’s work (Equations (3) and (4)) [28], the impurity is calculated as follows:

$$G\left(x_i,v_i\right)=\frac{n_l}{N_s}H\left(X_l\right)+\frac{n_r}{N_s}H\left(X_r\right)$$

(3)

where $x_i$ is a cut variable, $v_i$ is a cut point, $n_l$, $n_r$, and $N_S$ are the number of samples of the left child node, the number of samples of the right child node, and the number of samples of the parent node, $X_l$, $X_r$ is the set of samples of the left and right child nodes, and H(X) is the impurity function. The regression task generally uses the squared mean error (MSE) as the impurity function. In this study, the random forest algorithm was used to establish the ROP regression model, so MSE was chosen as the impurity function, and the impurity formula was calculated as follows:

$$H\left(X_m\right)=\frac{1}{N_m}{{\displaystyle \sum }}_{i\in N_m}{\left(y-\overline{y}\right)}^2$$

(4)

(3)

Backpropagation (BP) neural network algorithm

A backpropagation (BP) neural network is a kind of backpropagation algorithm according to error, shown in Figure 4. By using the steepest descent method and the learning rule of backpropagation, the weights and thresholds of the network are adjusted continuously so as to train a multilayer neural network with minimum errors. The training of the BP neural network consists of two processes: forward propagation and backward propagation. In the forward propagation, the training samples are input from the input layer, calculated step by step by each hidden layer, and passed to the output layer. Then, the error between the output results and the real results is calculated. If the error exceeds the given threshold, the backward propagation stage will be carried out. In the backward feedback phase, all neurons in each layer will correct the weights according to the error. The BP neural network model will repeat the above process until the error is less than the set threshold.

2.3. The “Hard-String” Drag Calculation Method

The classic drilling string drag torque models are the “soft-string” model and the “hard-string” model. The ”soft-string” model used earlier is simple and has certain accuracy, but it does not take into account the rigidity of the drilling string and is suitable for straight wells and wells with small slope angles. The “hard-string” model makes up for this shortcoming. The intelligent analysis method of drag and torque proposed by Zhu [29] uses the hard-string model to calculate the drag and has achieved good results. Therefore, this study adopts H.-S. Ho’s ‘’hard-string’’ model (Equations (5)–(13)) [30] for the real-time calculation of the drilling string drag.

The differential equation for the force of the overall drill string is:

$$\frac{d\left(-F\right)}{ds}=-EI{k}_b\frac{d{k}_b}{ds}-qcos\alpha \mp {\mu }_1·{\mu }_t$$

(5)

$$\frac{dM}{ds}={\mu }_1·{\mu }_t·\frac{D_o}{2}$$

(6)

where in the “∓”, the ‘‘−’’ stands for pulling up the string, the ‘’+’’ stands for pressing down the string; F is the axial pressure on the drill string N, s is the depth of the well, m; q is the gravity of the drill string per unit length, N/m; α is the inclination angle of the well, rad; EI is the bending stiffness of the drill string, N m²; $n_t$ is the contact drag force between the drill string and the well wall, N/m; ${\mu }_1$ is the axial drag coefficient of the drill string; $k_b$ is the wellbore axis curvature, m⁻¹.

The borehole curvature $k_b$ is calculated as:

$$K_b=\sqrt{{\left(\frac{d\alpha }{ds}\right)}^2+si{n}^2\alpha {\left(\frac{d\phi }{ds}\right)}^2}$$

(7)

where $\phi$ is the azimuth angle of the well.

The contact drag force $n_t$ on the drill column is calculated by:

$$n_t=\sqrt{\frac{A^2+B^2}{1+{\mu }_1^2}}$$

(8)

$$A=EI\frac{d^2{k}_b}{d{s}^2}+k_{b}F-k_n\left(-k_b{M}_T+EI·k_b{k}_n\right)+\frac{q}{k_b}\frac{d\alpha }{ds}sin\alpha$$

(9)

$$B=\frac{d}{ds}\left(-k_b{M}_T+EI·k_b{k}_n\right)+EI·k_n\frac{d{k}_b}{ds}-\frac{q}{k_b}\frac{d\phi }{ds}si{n}^2\alpha$$

(10)

$$k_n=\frac{sin\alpha }{k_b^2}\left(\frac{d\alpha }{ds}\frac{d^2\phi }{d{s}^2}-\frac{d\phi }{ds}\frac{d^2\alpha }{d{s}^2}\right)+cos\alpha ·\left(\frac{1}{k_b^2}{\left(\frac{d\alpha }{ds}\right)}^2+1\right)\frac{d\phi }{ds}$$

(11)

We use the finite difference method to solve for the forces on the entire drill string. The drill string can be divided into several small drill string sections, as shown in Figure 5. The forces on the small drill string sections can be obtained by the following equations, with the pressure as positive:

$$F_i=F_{i+1}+\frac{1}{2}E{I}_i\left(K_{bi}^2-K_{bi+1}^2\right)+\left(-q_{i}cos{\alpha }_i\mp {\mu }_1·{\mu }_{ti}\right)\Delta s_i$$

(12)

$$M_{Ti}=\frac{1}{2}{\mu }_1·n_{ti}·D_{bi}\Delta s_i+M_{Ti+1}$$

(13)

Among them, $F_i$ and $F_{i+1}$ are the axial force at the end of the i-th drill string near the ground and near the bit, respectively, N; $M_{Ti}$ and $M_{Ti+1}$ are the torque at both ends of the i-th drill string, respectively, in N·m; $K_{bi}^2$ and $K_{bi+1}^2$ are the borehole curvature at both ends of the i-th section, respectively, m-1; $q_i$, $E{I}_i$, $n_{ti},$ $\Delta s_i$, and $D_{bi}$ are the line weight (N/m) and bending stiffness (N·m²) of the drill string in the i-th section, respectively, and the contact with the borehole wall force (N/m), length (m), and outer diameter (m) of small drill string sections.

2.4. Nondominant Sorting Genetic Algorithm-II

In this study, NSGA-Ⅱ was used to solve the optimization parameters. NSGA-Ⅱ is an improved algorithm of NSGA proposed by K. Deb [31], which is one of the best evolutionary algorithms to solve multiobjective optimization problems. It searches for the optimal solution by simulating the natural evolution process and can find the Pareto front quickly and keep the diversity of the population.

Firstly, the algorithm will randomly generate the initial population N for nondominant sorting. Then the next generation population M is obtained by genetic operation (Mutation, Crossover, Competition). The elite preservation strategy is used to merge the populations M and N to form a new population, and the new population is sorted by fast nondominant sorting. The crowding distance is assigned to the individuals belonging to the same nondominant layer. According to the crowding distance value between individuals, N individuals are selected to form a new population. The next generation population M is obtained by genetic operation. The above process is repeated until the termination condition is reached. The program flow chart of the NSGA-II algorithm is shown in Figure 6.

3. Results and Discussion

3.1. Abnormal Data Processing

Because of the limited accuracy of the sensors used for data acquisition in the field, abnormal points will be collected in the drilling process, as shown in Figure 7. These outliers will increase the difficulty of training the model. The quality of the data determines the upper limit of the accuracy of the model. Since we aim at the prediction of mechanical drilling rate and the optimization of drilling parameters in the normal drilling process, detecting and eliminating these abnormal data is more conducive to improving the training accuracy of the model.

The machines that record data in the field take one piece of data every second. The ground engineering parameters remain in a relatively stable range in a short period of time (5 s, 10 s). In such a short time interval, it can be considered that the formation has not changed. When drilling into a uniform formation, the drilling parameters and formation should not change much in a short time, and the ROP should be maintained in a small range. If abnormal data points occur in a small time window, the data variance will increase.

Based on this regulation, we select a data window with a fixed time step and calculate the variance of the engineering data in the window. When new data appear in the window, if the variance of the data in the window changes in a range exceeding the set value, the new data are judged as abnormal data and vice versa as normal data. This outlier detection process was programmed in python, and good detection results were achieved on the collected data, as shown in Figure 8.

3.2. Correlation Analysis between Drilling Parameters and ROP

The Pearson correlation coefficient is used to measure the degree of linear correlation between two variables, X and Y, and the calculation formula is as follows [32]:

$$\mathsf{\rho }=\frac{{{\displaystyle \sum }}_{\mathrm{i}}\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)\left({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}}\right)}{\sqrt{{{\displaystyle \sum }}_{\mathrm{i}}{\left({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}}\right)}^2{{\displaystyle \sum }}_{\mathrm{i}}{\left({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}}\right)}^2}}$$

(14)

where x is one of the drilling variables and y is the ROP.

Input variables are critical to the performance of ROP models, especially to the data-driven ROP models. The correlation between drilling parameters and ROP is an important index for selecting model input variables. Drilling parameters with high correlation with ROP (such as Torque, WOB, etc.) were used as key data for model input. Drilling parameters with little correlation with ROP (such as Acoustic) can be selectively used as input to the ROP model. Figure 9 shows the Pearson correlation coefficient between each parameter.

3.3. Rate of Penetration Prediction Model

In this study, SVM, BP, and RF methods are selected to establish an ROP model. The model optimization work was carried out by comparing the mean absolute percentage error (MAPE) of each model.

$$MAPE=\frac{100\%}{n}{{\displaystyle \sum }}_{i=1}^n\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|$$

(15)

where $\widehat{y}$ is predicted ROP, and $\mathrm{y}$ is measured ROP.

The modeling process of the three data-driven ROP prediction models is as follows:

Data Partition: The data of five wells collected from the oil field have 18,143 remaining after outlier processing. Among them, 80% is used as the training set and 20% as the test set.

Model Training: 14 features in Table 2 are used as inputs except for the ROP, and the ROP is the output. After training, the MAPE of the training set was 5%. Hyperparameters of the random forest, SVM, and BP model were selected in

Table 4. Superparameter of the ROP prediction data-driven models.

Model	Hyperparameter Type	Value
RF	number of decision trees	185
	maximum depth of decision tree	130
	minimum number of samples required to split decision tree nodes	2
	finding the optimal number of feature variables to consider for a node	9
	impurity evaluation function	MSE
SVM	kernel function	rbf
	penalty parameter	100
	gamma parameter	0.01
BP	number of hidden layers	3
	number of neurons	[64, 128, 32]
	optimizer	Adam
	learning rate	0.001
	loss function	MSE

.

Model Prediction: The test set was imported into the trained model, and the prediction results were output. The prediction results of the test set are shown in Figure 10.

The random forest ROP model performs best in both test and training sets. Therefore, we choose to use the random forest ROP prediction model as one of the multiobjective optimization functions.

3.4. Drag Calculation Model

The above Equations (5)–(13) can be used to calculate the drag of the drill string, which is used as another optimization function of the multiobjective optimization algorithm. The equations were calculated using the finite difference method and programmed through the python computer language. Figure 11 shows the calculated drag distribution on the drill string of well A:

3.5. Multiobjective Collaborative Optimization Model

The random forest rate of the penetration prediction model and drag calculation model are used as the optimization functions of NSGA-II, and the maximum ROP and minimum drag are used as the optimization objectives. Giving the optimization boundary of drilling parameters and setting the hyperparameters of NSGA-II, the maximum ROP and minimum drag within the parameter boundary can be obtained using NSGA-II, as well as their corresponding parameter combinations (WOB, RPM, and GPM). The hyperparameters are selected in

Table 5. NSGA-II Multiobjective optimization algorithm hyperparameters selection.

Hyperparameters Type	Value
number of individuals in each population	20
number of population iterations	25
probability of individual crossover	0.7
probability of individual mutation	0.2

.

We used the model to provide drilling parameters for another neighboring well under drilling.

Table 6. The ROP increase and drag reduction effect at test points.

Depth (m)	Values	WOB (kN)	RPM (r/min)	GPM (l/min)	ROP (m/h)	Improve	Drag (kN)	Reduce
1181	Initial	206.3	60.5	25.0	18.26	16.31%	47.2	4.0%
	Recommended	188.2	66.8	27.0	21.24		45.3
1591	Initial	95.4	61.2	34.0	20.08	12.50%	53.5	3.0%
	Recommended	89.5	68.0	37.0	22.59		51.9
2304	Initial	85.2	52.5	38.0	15.11	12.37%	104.7	0.1%
	Recommended	80.0	59.2	42.0	16.98		103.6
3004	Initial	144.9	56.0	43.0	10.23	19.66%	118.2	7.0%
	Recommended	133.2	63.0	46.0	12.22		109.9
3636	Initial	170.0	48.2	48.0	3.26	19.63%	120.6	5.0%
	Recommended	163.0	56.8	52.0	3.9		114.6

shows the drilling parameters before and after the optimization of the well’s five test points, as well as the resulting ROP increase and drag reduction. Compared with the neighboring well, the average ROP in the horizontal section of the new well is increased by 10.3% on average, and the average drag of the drill string is reduced by 4.5%. The final test results show that the optimized parameters achieve the purpose of increasing ROP and reducing drag.

4. Conclusions

In this paper, the random forest algorithm is used to establish a data-driven ROP prediction model of horizontal wells. The MSE of the training set is 5%, and the test set is 9%, which can accurately predict the ROP of horizontal wells.

The random forest ROP model and the “hard-string” drag calculation model are used as the optimization function, and the maximum ROP and minimum drag are used as the optimization objectives. In the drilling process, the NSGA-II multiobjective optimization model is used to optimize the drilling parameters of the horizontal well. The oil field experiment results show that the multiobjective optimization model for drilling parameters is reliable. Compared with the neighboring well, the ROP of the test well is increased by 10.3%, and the drag is reduced by 4.5% on average. The drilling time is effectively shortened.

However, the training of the ROP prediction models is purely data driven. Similar to other machine learning models, the interpretability of the model is low, and the reasons why the optimized parameters can improve ROP and reduce drill string drag could not be well explained. Improving the interpretability of data-driven models is an important direction for future research.