Intelligent Prediction of Rate of Penetration Using Mechanism-Data Fusion and Transfer Learning

Abstract

Rate of penetration (ROP) is crucial for evaluating drilling efficiency, with accurate prediction essential for enhancing performance and optimizing parameters. In practice, complex and variable downhole environments pose significant challenges for mechanistic ROP equations, resulting in prediction difficulties and low accuracy. Recently, data-driven machine learning models have been widely applied to ROP prediction. However, these models often lack mechanistic constraints, limiting their performance to specific conditions and reducing their real-world applicability. Additionally, geological variability across wells further hinders the transferability of conventional intelligent models. Thus, combining mechanistic knowledge with intelligent models and enhancing model stability and transferability are key challenges in ROP prediction research. To address these challenges, this paper proposes a Mechanism-Data Fusion and Transfer Learning method to construct an intelligent prediction model for ROP, achieving accurate ROP predictions. A multilayer perceptron (MLP) was selected as the base model, and training was performed using data from neighboring wells and partial data from the target well. The Two-stage TrAdaBoost.R2 algorithm was employed to enhance model transferability. Additionally, drilling mechanistic knowledge was incorporated into the model’s loss function as a constraint to achieve a fusion of mechanistic knowledge and data-driven approaches. Using MAPE as the measure of accuracy, compared with conventional intelligent models, the proposed ROP prediction model improved prediction accuracy on the target well by 64.51%. The model transfer method proposed in this paper has a field test accuracy of 89.71% in an oilfield in China. These results demonstrate the effectiveness and feasibility of the proposed transfer learning method and mechanistic–data integration approach.

1. Introduction

Currently, the exploration and development of oil and gas resources face numerous challenges and difficulties [1]. During the drilling process, the rate of penetration effectively reflects drilling efficiency and provides guidance for optimizing drilling parameters, making accurate predictions of rate of penetration highly significant [2]. At present, mechanism-based models for rate of penetration prediction are relatively common and mature [3], but they have certain limitations. In contrast, intelligent models are developing rapidly and are widely used in the drilling field, showing particularly significant results in predicting rate of penetration.

Traditional mechanistic models, such as the classic Maurer rate equation [4], Warren rate equation [5], Motahhari rate equation [6], and the modified Yang rate equation, are less effective in predicting rate of penetration due to their inability to handle complex downhole variations and thus fail to meet practical operational needs. In contrast, intelligent models have significant advantages in handling complex drilling conditions and large-scale data, with prediction performance markedly superior to mechanistic models, better adapting to changes in the actual operational environment.

With the development of artificial intelligence technology, data-driven models represented by machine learning have rapidly advanced in rate of penetration prediction [7]. Intelligent prediction models for rate of penetration are based on a large amount of actual drilling data and can uncover the nonlinear relationships between rate of penetration and various influencing factors, thereby accurately predicting rate of penetration. Research has found that scholars have established rate of penetration prediction models based on traditional machine learning models such as random forests [8], and also developed rate of penetration prediction methods based on deep learning models such as recurrent neural network [9] by extracting time series features from drilling data. These models have improved prediction accuracy to some extent; however, in recent years, pure data-driven models have exposed many limitations, such as overfitting and sensitivity to anomalous data. Therefore, scholars have begun to explore various innovative approaches to further enhance the prediction performance and adaptability of the models. For example, in 2018, Hegde et al. introduced a hybrid modeling approach combining mechanistic and data-driven models [10]. This method enhanced prediction accuracy and interpretability in rate of penetration forecasting, outperforming both traditional mechanistic models and purely data-driven models. In 2021, Ren et al. proposed the integration of mechanistic models with machine learning models to address the drawbacks of using a single model [11]. By using machine learning models for rate of penetration prediction and mechanistic models to compensate for the biases of machine learning models, they achieved coupling of mechanistic and intelligent models, resulting in higher prediction accuracy. In 2022, Yang et al. introduced a rate of penetration prediction method based on ensemble transfer learning, developing an integrated transfer regression model with physical constraints to predict rate of penetration in new oilfields [12]. Experiments on real drilling datasets indicated that their proposed method for rate of penetration prediction is effective. In 2022, Liu et al. extracted the temporal characteristics of drilling data based on the temporal features of LSTM long short-term memory neural networks and the nonlinear fitting ability of FNN, and established an intelligent prediction model for rate of penetration based on LSTM-FNN [13]. In 2022, to address the problem of predicting logging reservoir parameters, Shao et al. employed transfer learning methods to achieve geophysical logging reservoir parameter prediction [14]. In 2022, Zhang et al. proposed a real-time prediction of rate of penetration using a gating recurrent unit network with an attention mechanism and a fully connected neural network. Results indicate that the model can provide accurate and robust predictions [15]. In 2023, Liu et al. introduced a rate of penetration prediction model based on a self-attention mechanism, achieving good prediction results using an RNN model [16]. Pan et al. proposed an intelligent prediction method for horizontal well rate of penetration based on Di-GRU, with the model showing an average absolute percentage error of 11.8%, demonstrating high prediction accuracy [17]. In 2024, Bingrui Tu et al. proposed a GRU-Informer model for real-time prediction of rate of penetration [18]. This model combines the short-term correlations of GRU neural networks with the long-term dependencies of the Informer model. Experimental results show that it outperforms traditional models in real-time ROP prediction. In 2024, Zheng et al. summarized the relationship between transfer learning and traditional machine learning, conducted a feasibility analysis of rate of penetration prediction models based on transfer learning theory, and completed model design [19]. In the same year, Ye et al. improved rate of penetration prediction accuracy by processing anomalous drilling data, and the results indicated that this method helps enhance the prediction accuracy of rate of penetration prediction models [20].

Currently, machine learning and deep neural networks have become the mainstream methods for predicting rate of penetration (ROP). Although these methods have achieved good results for specific wells, they still fall short in addressing the issue of model transferability. Particularly in actual drilling operations, due to the complexity of geological conditions and significant differences between regions, the generalization capability of existing models is limited, leading to substantial variations in prediction performance across different wells. Furthermore, there is a lack of research exploring how transfer learning can address these challenges.

To address the gap in industry research on solving the transferability issues of ROP prediction models, this paper proposes a transfer learning model that integrates mechanistic knowledge with data-driven approaches. The model effectively combines the knowledge from adjacent wells and the target well through transfer learning, reducing data distribution differences between different blocks and wells, thereby enhancing the model’s transferability. Experimental results show that this model not only significantly improves prediction accuracy for the target well but also enhances the model’s stability and generalization ability, demonstrating superior performance compared to traditional methods.

2. Data Preprocessing and Feature Selection

The data sample comes from the original data of two domestic wells, including drilling data, logging data, etc. with 30 types of parameters. The data are indexed by well depth with a sampling interval of 0.1 m. High-quality data samples are crucial for improving model prediction performance and enhancing generalization ability. Therefore, it is necessary to preprocess the existing raw well data, including data cleaning, feature encoding, feature selection, and normalization, to ultimately provide high-quality data support for the mechanism–data fusion rate of penetration prediction model.

2.1. Data Cleaning

During the analysis of the sample data from multiple wells, it was discovered that the dataset contains 30 parameters (

Table 1. Parameters.

Category	Parameters
mud logging parameters	RPM, WOB, Hook Load, Torque, SPP, TOTSPM, TOTPITVOL
bit parameters	Bit Types, Footage Per Bit
drilling fluid parameters	MFI, MFO, MDI, MDO, MTI, MTO, MCI, MCO
well logging parameters	AZIM, BIT, CAL, CNL, DEN, DEVI, DT, GR, RM, RT, SP
other parameters	Depth, Weight of Drill String

). These parameters cover various aspects of downhole operations, including geological features, engineering parameters, and other factors that might influence downhole activities. Each well’s data is unique, and through in-depth analysis of each sample we can gain a better understanding of the key factors and potential issues in the downhole operation process.

Before data cleaning, all data were first merged to ensure effective integration of data from different sources, maintaining consistency, and improving the reliability of subsequent analysis and modeling. However, raw data typically contain missing values and outliers, which, if not properly addressed, could significantly affect the model’s predictive performance [21]. Therefore, two data cleaning steps—missing value imputation and data denoising—were applied to improve data quality. After cleaning, the data more accurately reflect the actual conditions of downhole operations, enhancing the model’s generalization ability and ensuring it maintains strong predictive performance even when processing new data.

After completing the data cleaning and organization, we ended up with 29,678 high-quality data points available for model training and testing.

2.2. Feature Analysis and Selection

Machine learning has the capability to handle large-scale data, but with data having up to 30 types of parameters, irrelevant variables can significantly impact the model’s performance. Therefore, it is necessary to remove irrelevant variables to simplify the model and reduce the risk of overfitting, which in turn enhances the model’s generalization ability and shortens the training time. Consequently, it is essential to perform correlation analysis on each parameter. Common methods are divided into three categories: Filter, Wrapper, and Embedded [22]. To thoroughly explore the linear and nonlinear correlations between parameters and the target variable without causing excessive computational load, this paper chooses the distance correlation method from the Filter approach for correlation analysis.

$$dcorr(X,Y)=\left\{\begin{array}{c}{\displaystyle \frac{dcov\left(X,Y\right)}{\sqrt{dcov\left(X,X\right)dcov\left(Y,Y\right)}}} , dcov(X,X)dcov(Y,Y)>0\\ 0 , dcov(X,X)dcov(Y,Y)=0\end{array}\right.$$

(1)

$dcorr\left(X,Y\right)$ is the distance correlation coefficient between X and Y; $dcov(X,Y)$ is the distance covariance between X and Y; $dcov\left(X,X\right)$ is the distance variance of X; and $dcov(Y,Y)$ is the distance variance of Y [23].

Based on preliminary experience, several engineering parameters were initially selected, and the distance correlation method was applied to calculate the correlation coefficients between each parameter and the rate of penetration (ROP). The distance correlation method was chosen over traditional correlation metrics such as Pearson or Spearman because it is capable of detecting both linear and nonlinear dependencies between variables. This is particularly useful in the context of drilling, where the relationship between parameters like weight on bit (WOB), revolutions per minute (RPM), and ROP may exhibit complex, nonlinear patterns.

As shown in Figure 1, the distance correlation coefficients were used to identify parameters that exhibit strong correlations with ROP while having weaker correlations with other input features, thus ensuring minimal multicollinearity. This approach helps to enhance the model’s generalization ability and robustness. Additionally, the correlation coefficients of WOB and RPM with ROP were 0.82 and 0.47, respectively, indicating their significant influence on ROP.

The distance correlation method not only allowed for a more nuanced understanding of the relationships between drilling parameters and ROP but also guided the selection of optimal input features for the predictive model. The ability to capture nonlinear relationships was crucial in identifying key parameters like WOB and RPM, which may not have been as evident using simpler linear correlation measures.

At the same time, ROP is influenced by multiple factors, and the distance correlation analysis further informed the selection of critical parameters. Based on the correlation results shown in Figure 1, the final set of selected engineering parameters includes revolutions per minute (RPM), weight on bit (WOB), torque (Torque), standpipe pressure (SPP), mud flow in (MFI), mud flow out (MFO), mud density in (MDI), mud density out (MDO), mud temperature in (MTI), mud temperature out (MTO), total strokes per minute (TOTSPM), and rate of penetration (ROP). By using the distance correlation method, the model benefits from a more refined selection of input features, enhancing both predictive accuracy and robustness.

2.3. Data Denoising

Due to the various environmental influences on drilling data, this study employs Kalman filtering to denoise the data. Kalman filtering [24] is a recursive algorithm used to estimate the state of dynamic systems. It excels in forecasting and correcting time series data and is widely applied in fields such as signal processing, control systems, navigation, and tracking. Kalman filtering provides optimal state estimates by utilizing both the system’s physical model and observation data. The core idea involves continuously refining state estimates through two steps, prediction and update, thereby reducing errors. The prediction step forecasts the next state based on the current state estimate and system model; the update step adjusts the prediction results using new observation data to improve accuracy. The advantage of Kalman filtering lies in its ability to provide efficient and accurate state estimates in noisy and uncertain environments.

The formula and noise reduction steps for the Kalman filter [25] are as follows:

• State prediction and covariance prediction.

$${\widehat{x}}_k^{-}=F_k{\widehat{x}}_{k-1}+B_k{u}_k$$

(2)

${\widehat{x}}_k^{-}$ represents the predicted state, $F_k$ is the state transition matrix, $B_k$ is the control input matrix, and $u_k$ denotes the control input.

$$P_k^{-}=F_k{P}_{k-1}{F}_k^T+Q_k$$

(3)

$P_k^{-}$ is the predicted covariance, $P_{k-1}$ is the covariance from the previous state, and $Q_k$ is the process noise covariance.

2.

Kalman gain, state update, and covariance update.

$$K_k=P_k^{-}{H}_k^T{\left(H_k{P}_k^{-}{H}_k^T+R_k\right)}^{-1}$$

(4)

$K_k$ is the Kalman gain, $H_k$ is the observation matrix, and $R_k$ is the observation noise covariance.

$${\widehat{x}}_k={\widehat{x}}_k^{-}+K_k\left(z_k-H_k{\widehat{x}}_k^{-}\right)$$

(5)

${\widehat{x}}_k$ is the updated status and $z_k$ is the observed value.

$$P_k=\left(I-K_k{H}_k\right)P_k^{-}$$

(6)

$P_k$ represents the updated covariance matrix and $I$ is the identity matrix.

After the aforementioned data processing steps, the data quality was significantly improved, as shown in Figure 2.

2.4. Data Normalization

The scales of different features in the original data vary significantly, which can easily lead to gradient issues. Normalizing the data using min–max scaling or z-score standardization [26] can eliminate the impact of dimensionality, avoid gradient issues, and speed up model training. The data are sequential with well depth as an index and does not show a clear normal distribution. Therefore, compared to z-score standardization, the data are more suitable for normalization.

$$x_i^{\prime }={\displaystyle \frac{x_i-\mathit{min}(x_i)}{\mathit{max}(x_i)-\mathit{min}(x_i)}}$$

(7)

In the formula, $x_i^{\prime }$ represents the normalized result of the i-th variable; $x_i$ denotes the i-th variable; $\mathit{min}(x_i)$ is the minimum value of the i-th variable; and $\mathit{max}(x_i)$ is the maximum value of the i-th variable.

2.5. Model Evaluation Metrics

The evaluation metrics for the prediction model in this article are as follows: mean squared error (MSE), mean absolute error (MAE), coefficient of determination (R²), and mean absolute percentage error (MAPE).

• Mean absolute error (MSE) [27]

Mean squared error (MSE) is a commonly used metric for evaluating the performance of regression models. MSE measures the average squared difference between predicted values and actual values and is calculated using the following formula:

$$MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$$

(8)

$n$ represents the number of samples, $y_i$ represents the actual value of the rate of penetration, and ${\widehat{y}}_i$ represents the predicted value of the rate of penetration. The smaller the MSE, the closer the model’s predicted results are to the actual values, indicating higher predictive accuracy of the model. The unit of MSE is the same as the unit of the target variable, which means it accounts for the scale of the errors to some extent.

2.

Mean absolute error (MAE)

$$MAE={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left|y_i-y_{pre}\right|$$

(9)

$m$ represents the number of sample points, $y_i$ denotes the mechanical ROP, and $y_{pre}$ is the predicted ROP. The mean absolute error is used to measure the absolute deviation between the model’s predictions and the actual results.

3.

Coefficient of determination (R²) [28]

The coefficient of determination (R-squared, R²) measures the proportion of the total variation that is explained by the model. The closer the R² value is to 1, the more variation the model explains, indicating a better model.

$$R^2=1-[{\displaystyle \frac{{\sum }_{i=1}^N(y_{ipre}-y_{itrue}{)}^2}{{\sum }_{i=1}^N(y_{ipre}-y_{ave}{)}^2}}]$$

(10)

4.

Mean absolute percentage error (MAPE)

$$MAPE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N|{\displaystyle \frac{y_{ipre}-y_{itrue}}{y_{itrue}}}|$$

(11)

This metric is also known as the average relative error, where $m$ represents the number of sample points, $y_i$ represents the rate of penetration, and $y_{pre}$ represents the predicted rate of penetration.

3. Model Building

3.1. MLP

The multilayer perceptron (MLP) [29] is a fundamental yet powerful artificial neural network model widely used in tasks such as classification and regression. An MLP consists of multiple layers of neurons, typically including an input layer, one or more hidden layers, and an output layer. Each neuron receives inputs from the previous layer through full connections, undergoes a linear transformation (weights and biases), and then passes through an activation function, introducing nonlinearity. This nonlinearity allows the MLP to capture complex patterns and relationships in the input data, making it highly effective for handling nonlinear problems. For tasks like rate of penetration prediction, the MLP can learn from large amounts of data to establish complex mapping relationships between input variables (such as WOB, RPM, etc.) and the output target (ROP). The MLP network structure is shown in Figure 3.

The learning process of the MLP is achieved through the backpropagation algorithm. During training, the model calculates the error between the predicted output and the true values, using a loss function to quantify this error. The backpropagation algorithm then propagates the error backward through each layer, adjusting the weights and biases in the network by computing gradients. This adjustment process is known as gradient descent, which aims to gradually reduce the prediction error, making the model’s predictions on the training data more accurate. To improve the model’s generalization ability and avoid overfitting, the MLP often incorporates regularization techniques, such as L2 regularization or dropout, to constrain model complexity. Additionally, the choice of activation function is crucial, with ReLU commonly used in hidden layer neurons because it alleviates the vanishing gradient problem, while different activation functions are typically chosen for the output layer based on the task type.

3.2. Transfer Learning Algorithms

The Two-stage TrAdaBoost.R2 algorithm is a supervised transfer learning method based on instances, suitable for regression tasks. This method is based on the “reverse boosting” principle, where the weight of source instances with poorer predictions is reduced in each boosting iteration, while the weight of target instances is increased.

The algorithm steps [30] are as follows:

1.

Normalize weights: initialize the weights of source domain $w_S$ and target domain $w_T$ data so that their total sums up to 1.

$$\sum w_S+\sum w_T=1$$

(12)

2.

Fit the AdaBoostR2 estimator: Use the AdaBoostR2 estimator $f$ on labeled data from both the source domain $\left(X_S,y_S\right)$ and the target domain $\left(X_{T,}{Y}_T\right)$, with weights of $w_S{ \mathrm{a}\mathrm{n}\mathrm{d} w}_T$. During training, the weights of the source domain $w_S$ are fixed.

3.

Calculate cross-validation scores: compute the cross-validation scores on the target domain data $\left(X_{T,}{Y}_T\right)$.

4.

Compute the error vectors for the training examples: Calculate the error vectors for the source domain and the target domain separately as ${ϵ}_S$ and ${ϵ}_T$:

$${ϵ}_S=L\left(f\left(X_S\right),y_S\right) {ϵ}_T=L\left(f\left(X_{T_{lab}}\right),y_{T_{lab}}\right)$$

(13)

$L$ represents the loss function.

5.

Normalize error vector: normalize the error vector and remove the maximum error term:

$${ϵ}_S={\displaystyle \frac{{ϵ}_S}{\underset{ϵ\in {ϵ}_S\cup {ϵ}_T}{\mathit{max}}ϵ}} {ϵ}_T={\displaystyle \frac{{ϵ}_T}{\underset{{ϵ\in ϵ}_S\cup {ϵ}_T}{\mathit{max}}ϵ}}$$

(14)

6.

Update the weights of the source domain and target domain: Based on the normalized error vector, update the data weights for both the source and target domains:

$$w_S=w_S{\beta }_S^{{ϵ}_S}/Z w_T=w_T/Z$$

(15)

The selection of ${\beta }_t$ ensures that the weight of the final target instances is ${\displaystyle \frac{m}{n+m}}+{\displaystyle \frac{t}{S-1}}(1-{\displaystyle \frac{m}{n+m}})$. $Z$ is the normalization constant.

7.

Return to step 2 and repeat the process until the set number of iterations, N, is reached.

Through the above process, the Two-stage TrAdaBoost.R2 algorithm can give predictions based on the best cross-validation score (Figure 4).

3.3. Mechanistic Constraints

Traditional neural network models continuously update and adjust network weights and biases based on input and output data, aiming to fit the complex nonlinear mapping relationship between inputs and outputs as much as possible. Such trained networks only address the statistical relationships between rate of penetration, drilling pressure, torque, and other factors, while neglecting the underlying physical relationships between these factors. To address this limitation, physical constraints can be introduced to enhance the model’s predictive capability. For rate of penetration prediction problems, an empirical inequality constraint can be established to better incorporate physical principles, thereby improving the model’s reliability and accuracy.

Based on Bingham’s fundamental ROP model [31], with known parameters such as ROP, WOB, and bit diameter $D_b$, the predicted value of the rate of penetration can be calculated using the following formula.

$$ROP= \alpha RPM{\left({\displaystyle \frac{WOB}{D_b}}\right)}^{\gamma }$$

(16)

According to this physical model, the greater the drilling pressure, the faster the rate of penetration, meaning that the partial derivative of rate of penetration with respect to the drilling pressure in the neural network model is greater than zero. Mechanistic constraints are generally integrated into the neural network loss function in the form of a penalty function, serving as a constraint on the neural network. Therefore, a penalty term is added to the original MSE loss function to form the model’s loss function:

$$LOSS={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(Y_i-{\widehat{Y}}_i)}^2+\mu L$$

(17)

$\mu$ is the penalty factor and $L$ can be expressed as:

$$L=\left\{\begin{array}{c}0, {\displaystyle \frac{\delta ROP}{\delta WOB}}>0\\ -{\displaystyle \frac{\delta ROP}{\delta WOB}}, {\displaystyle \frac{\delta ROP}{\delta WOB}}<0\end{array}\right.$$

(18)

4. Results and Discussion

4.1. Model Performance

Based on the use of transfer learning methods and the presence or absence of mechanistic constraints, the models are divided into four groups, each comparing the effect of mechanistic constraints on model performance, as shown in

Table 2. Model categories.

Model	A	B	C	D	E	F	G	H
transfer learning	×	×	×	×	×	×	√	√
mechanistic constraints	×	√	×	√	×	√	×	√
neighboring well data	100%	100%	0	0	100%	100%	100%	100%
target well data	0	0	70%	70%	70%	70%	70%	70%

. Model A and Model B use 100% neighboring well data and do not use target well data for model training; Model C and Model D do not use neighboring well data and use 70% target well data for model training; Model E and Model F use 100% neighboring well data and 70% target well data for model training; Model G and Model H, based on Model E and Model F, use the transfer learning algorithm Two-stage TrAdaBoost.R2 and test 30% of the target well data.

Comparing the rate of penetration prediction results and evaluation metrics of the four models mentioned above, under the conditions of with and without mechanistic constraints and using different transfer learning methods, using MAPE as the measure of accuracy, the following conclusions can be drawn: (1) using partial target well data for model training improves model accuracy by 30.56% compared with using data from all neighboring wells and all target wells, with a 4.19% increase; (2) using the Two-stage TrAdaBoost.R2 transfer learning algorithm improves model prediction accuracy by 32.29% compared with the transfer method using partial target well data; and (3) adding mechanistic constraints significantly improves model prediction performance, with a 24.50% increase in prediction accuracy compared with the unconstrained model (

Table 3. Model evaluation metrics.

Model	A	B	C	D	E	F	G	H
MSE	4.24	4.65	1.34	1.09	1.15	0.90	0.57	0.39
MAE	1.73	2.00	0.96	0.86	0.92	0.77	0.64	0.51
R2	−0.90	−1.09	0.40	0.51	0.48	0.60	0.74	0.83
MAPE	28.99%	36.61%	21.01%	18.85%	20.13%	17.24%	13.63%	10.29%

).

4.2. Analysis of the Effectiveness of Transfer Learning

This text explores two migration methods. First, by comparing the use of different training data, it was found that training the model with data from adjacent wells and the target well significantly enhances the migration effectiveness. This indicates that improving model transferability requires breaking down the data barriers between adjacent and target wells. Next, by employing transfer learning algorithms, the model can learn features from adjacent wells, further improving transferability.

Comparing the model evaluation metrics for Model A, Model C, and Model E, as shown in Figure 5, the results demonstrate that using data from neighboring wells and partial target wells positively impacts the model’s prediction accuracy. Analysis reveals that due to significant differences between neighboring and target well data, Model A, which was trained using only neighboring well data, exhibits poor transferability, which is a current issue with the model. Additionally, using only target well data does not effectively reflect the model’s transfer performance. Therefore, compared with Model A and Model C, Model E, which uses all neighboring well data and partial target well data, showed a significant improvement in prediction accuracy, indicating that the model’s transferability has been enhanced.

As shown in Figure 6, Model A, constructed using neighboring well data, showed an underestimation in its predicted values compared to the actual values, which was caused by the differences in data distribution between the neighboring wells and the target well. In contrast, Model E, which integrated target well data into the training process alongside neighboring well data, demonstrated a smaller discrepancy between predicted and actual values. This improvement was due to the inclusion of target well data that shared the same distribution as the data used during the prediction stage. Additionally, both Model C, developed solely from neighboring well data, and Model G, which was based on a transfer learning algorithm, exhibited a close match between predicted and actual values. Notably, Model G significantly outperformed all other models in terms of predictive performance metrics. By effectively utilizing knowledge from neighboring wells and adapting to the specific characteristics of the target well, Model G showcased exceptional generalization capability and adaptability, further validating the superior effectiveness of the transfer learning algorithm in transferring knowledge from neighboring well data.

In summary, the rate of penetration prediction model trained using data from neighboring wells and some target wells showed a significant improvement in accuracy. After employing the Two-stage TrAdaBoost.R2 transfer learning algorithm, the model’s prediction accuracy further increased, with Model G’s MAPE reaching 13.63%, which is a 32.29% reduction compared to the non-transfer learning Model E. The experimental results validate the feasibility and effectiveness of the transfer learning approach.

4.3. Analysis of the Effects of Mechanistic Constraints

Based on the experimental results, adding mechanistic constraints can significantly improve the predictive accuracy of the model. First, by comparing the model evaluation metrics of Model A and Model B, it can be observed that the predictive accuracy of Model B, after adding mechanistic constraints, is actually lower, and both have negative R² values. Further analysis of Model C and Model D, Model E and Model F, and Model G and Model H, shows that after adding mechanistic constraints, the predictive accuracy of these models has improved to varying degrees. This may be due to the significant distribution differences between the training data (neighboring wells) and the test data (target wells) in Model A and Model B. Although they share the same physical rules, the difference in data distribution introduces bias, leading to poor predictive performance. In contrast, the other three groups of models show significant improvements after adding mechanistic constraints. Mechanistic constraints help guide the models to focus more on useful information in the data during training while suppressing the impact of noise and outliers, thereby enhancing the generalization ability of the models and improving predictive accuracy. For example, a comparison of Model G and Model H shows that after adding mechanistic constraints, the mean squared error (MSE) of Model H decreased by 31.58%, the mean absolute error (MAE) decreased by 20.31%, the R² value increased by 12.16%, and the mean absolute percentage error (MAPE) decreased by 24.50%, resulting in a significant improvement in the model’s predictive accuracy (Figure 7).

The results above indicate that the method proposed in this paper, which incorporates drilling mechanism knowledge into the model loss, can significantly improve the model’s prediction accuracy, with a marked enhancement in performance.

5. Conclusions

This text proposes practical solutions to address the issue of poor transferability in rate of penetration models.

Firstly, compare the prediction performance and evaluation metrics of the rate of penetration prediction model by differentiating between the source domain and target domain in the training and test sets. When only using source domain data as the training set, the model’s training performance is poor. By training the model with all source domain data and some target domain data, the model’s prediction accuracy improved by 30.56%, proving that at the data level, the model learns features from the source domain and positively impacts predictions for the target domain.

Secondly, compare the predictive performance of the model before and after using transfer learning algorithms, and optimize the transfer learning algorithm. It is found that the Two-stage TrAdaBoost.R2 transfer learning algorithm performs better, with a 32.29% improvement in prediction accuracy compared with models that did not use transfer learning. This indicates that transfer learning algorithms significantly enhance the model’s performance in learning neighboring well features and predicting target wells.

Finally, by comparing the model’s prediction performance before and after adding mechanistic constraints, it is evident that the prediction accuracy has significantly improved. When using the latest transfer learning algorithms, the prediction accuracy increased by 24.50%, indicating that mechanistic constraints enhance the accuracy of rate of penetration predictions.

Therefore, training a high-performance, intelligent rate of penetration prediction model requires a large amount of data and a wide variety of working conditions. Incorporating mechanistic constraints can smooth the prediction curve, accelerate training speed, and improve convergence, thereby reducing the error rate. By embedding transfer learning models and utilizing transfer learning algorithms, the model’s generalization ability can be effectively enhanced. This paper analyzes the mechanistic knowledge and characteristics of neural networks in rate of penetration prediction, incorporating MLP networks with mechanistic constraints. The mechanistic-constrained MLP is used as a weak learner, and the Two-stage TrAdaBoost.R2 algorithm is applied to integrate and establish a mechanistic-constrained intelligent rate of penetration prediction transfer model. The final model achieves a correlation coefficient of 0.82 and a MAPE of 10.29%. By using Mechanism-Data Fusion and Transfer Learning, the model’s generalization ability and prediction accuracy have been effectively improved, providing new ideas and methods for ROP prediction, and further exploring the theories and methods of intelligent drilling.