Production Forecasting of Deep Shale Gas Wells Using the ROA-Transformer-Mamba Hybrid Network

Production Forecasting of Deep Shale Gas Wells Using the ROA-Transformer-Mamba Hybrid Network | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Production Forecasting of Deep Shale Gas Wells Using the ROA-Transformer-Mamba Hybrid Network Weikang He, Xizhe Li, Yujin Wan, Nan Wang, Honming Zhan, Xiangyang Pei, and 3 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7438168/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract The production of shale gas wells is influenced by multiple factors, including geological conditions, fracturing techniques, and production strategies. Its production patterns exhibit pronounced nonlinearity, temporal variability, and multi-scale fluctuations. In particular, deep shale gas wells often follow a complex trend characterized by “high initial production, rapid decline, and low stable output,”making it difficult for traditional decline models to accurately fit and predict their dynamic behavior.To improve the accuracy and generalization capability of production prediction in deep shale gas wells, this study proposes a hybrid deep learning model that integrates the Rabbit Optimization Algorithm (ROA), Transformer, and Mamba modules. The model first employs 1D convolution to extract local features, followed by a Transformer to capture global dependencies, and further utilizes the Mamba module to model long-term temporal dependencies and dynamic state transitions. The ROA is applied to optimize key hyperparameters, thereby enabling efficient analysis of complex nonlinear time-series data.Experimental results demonstrate that the proposed Transformer-Mamba model achieves favorable accuracy and stability, showing strong potential for generalization. It provides an effective technical foundation and valuable reference for production forecasting in deep shale gas reservoirs. Physical sciences/Energy science and technology Physical sciences/Mathematics and computing Earth and environmental sciences/Solid earth sciences Shale gas Production forecasting Transformer Mamba Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 1. Introduction Amid the ongoing restructuring of the global energy system and the escalating pursuit of low-carbon energy sources, natural gas—especially unconventional forms like shale gas—has emerged as a pivotal component in safeguarding national energy supply and promoting the evolution toward a cleaner energy mix. In recent years, China has accelerated the exploration and development of shale gas, achieving significant breakthroughs in regions such as the Sichuan Basin, Fuling, Changning–Weiyuan, and Zhaotong [1–3] . According to statistics from the National Energy Administration, China’s annual shale gas production exceeded 25 billion cubic meters in 2023, with its share in total domestic natural gas output continuing to increase. It has become a primary contributor to natural gas production growth during the 14th Five-Year Plan period (see Fig. 1 ). However, shale gas resources are typically characterized by deep burial, complex geological structures, and strong reservoir heterogeneity [5] , which makes their production behavior significantly different from that of conventional gas reservoirs and poses substantial challenges for accurate production forecasting. Traditional shale gas production forecasting methods are largely based on decline curve models—such as Arps, DCA curves, and SEPD—whose theoretical foundations lie in statistical or empirical equations. While these models provide useful guidance during the early stages of reservoir development, they often fail to capture the complex dynamic behaviors arising from multi-factor interactions. Their prediction accuracy declines significantly in scenarios involving sharp production fluctuations, gas–water co-production, or well control adjustments. Bu Tao et al. [4] proposed a rapid EUR estimation method based on flowback-phase dynamic data, enabling early-stage assessment of recoverable reserves and offering preliminary guidance for field development. Zhu Yuanchong et al. [5] , through statistical feature analysis of large-scale datasets, explored the correlations between shale gas productivity and production parameters, emphasizing the importance of data-driven approaches and laying a foundation for the introduction of artificial intelligence (AI) models. With the rapid advancement of AI and data-driven technologies, deep learning–based prediction models [5–10] have been increasingly applied to oil and gas production forecasting. Among them, Long Short-Term Memory (LSTM) networks have demonstrated excellent performance in modeling time-series data, and have been successfully implemented in daily gas production prediction for various shale gas blocks. Alolayan et al. [6] adopted a transfer learning strategy to enhance model generalization, effectively addressing the challenge of limited data in shale gas prediction. Nguyen-Le et al. [7] developed a multivariate input model that combines early production data to forecast the Barnett shale reservoir, yielding strong fitting results. Qiao Songbo et al. [11] proposed a hybrid model combining REMD, CNN, Transformer, and LSTM for carbon price prediction, demonstrating the potential of hybrid architectures in complex sequence modeling. Liang et al. [17] introduced a BiLSTM-RF-MPA integrated model tailored to the nonstationary nature of shale gas production, significantly improving forecasting accuracy and robustness. Meanwhile, the recent success of the Transformer architecture in natural language processing and time-series modeling has offered new perspectives for capturing long-range dependencies in complex sequential data. This study focuses on real production data from shale gas fields and proposes a multi-module prediction model that integrates deep feature learning with time-series modeling capabilities. Its predictive performance is evaluated by comparison with Transformer-LSTM and Mamba-based models, aiming to overcome the limitations of traditional approaches in handling production volatility and incomplete datasets. By comparing the model outputs with actual production data, we assess the model's applicability and accuracy, with the goal of providing practical guidance and reference for production forecasting in deep shale gas wells. 2. Research Methods 2.1 Transformer Model The Transformer model [11–16] introduces a self-attention mechanism that enables the modeling of dependencies between any two time points without relying on sequential order, thereby capturing global features and significantly enhancing both modeling capacity and generalization ability (as illustrated in Fig. 2 ). When dealing with oil and gas production data—which are characterized by multi-scale temporal patterns including periodic fluctuations, abrupt disturbances, production regime shifts, and long-term nonlinear trends—the Transformer model is capable of learning complex dynamic relationships. At the core of the Transformer lies the multi-head attention mechanism, which allows the model to extract key features from the sequence from multiple perspectives at each time step. This mechanism enables the handling of multi-dimensional inputs, making the model well-suited for capturing the intricate relationships between various operational parameters (e.g., casing pressure, tubing pressure, temperature, daily water production, flowback ratio, and choke pressure drop) and production rates in oil and gas operations. Unlike LSTM models, which must process sequences step by step, the Transformer employs parallel computation and positional encoding to compute attention weights for all time steps simultaneously, resulting in efficient training and forecasting. In the context of production forecasting for oil and gas wells, the Transformer model can effectively identify short-term fluctuations in wellbore behavior and capture long-term reservoir dynamics. It is particularly advantageous for datasets involving multiple wells, long time windows, and strong heterogeneity. 2.2 LSTM Model In oil and gas production forecasting—particularly in modeling shale gas well production data—traditional linear or decline curve models often fail to capture the nonlinear, time-varying, and multi-periodic patterns inherent in the data. Long Short-Term Memory (LSTM) networks [17–19] , an advanced variant of recurrent neural networks (RNNs), have been widely adopted for time series modeling tasks and are especially well-suited for processing and forecasting oil and gas production data with temporal dependencies. The core innovation of LSTM lies in its gated mechanism—comprising input, forget, and output gates—which effectively mitigates the vanishing or exploding gradient problems commonly encountered in standard RNNs. This enables the model to “remember” important information over long temporal spans during training, while filtering out irrelevant or noisy data. Architecturally, an LSTM model typically consists of a series of connected LSTM units (as shown in Fig. 3 ), each acting as a micro memory cell. The output of each unit is influenced not only by the current input but also by the hidden state and cell state from the previous time step. This “temporal context propagation” mechanism enhances the model’s ability to adapt to time-dependent factors such as wellbore dynamics, reservoir heterogeneity, and injection–production regime changes. LSTM emphasizes sequential dependency modeling at each time step. By integrating the temporal memory capacity of LSTM with the non-local modeling strength of Transformer architectures, the hybrid Transformer-LSTM model is expected to achieve improved prediction accuracy and generalization in shale gas wells under complex geological conditions. Such integration supports the optimization of development strategies and enhances decision-making in production management. The commonly used formulation for the LSTM model is given as follows: $$\:\begin{array}{c}{f}_{t}=\sigma\:\left({W}_{f}\cdot\:\left[{h}_{t-1},{x}_{t}\right]+{b}_{f}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（1）\:\\\:{i}_{t}=\sigma\:\left({W}_{f}\cdot\:\left[{h}_{t-1},{x}_{t}\right]+{b}_{i}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（2）\\\:{\stackrel{\sim}{C}}_{t}=tanh\left({W}_{C}\cdot\:\left[{h}_{t-1},{x}_{t}\right]+{b}_{C}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（3）\\\:{C}_{t}={f}_{t}*{C}_{t-1}+{i}_{t}*{\stackrel{\sim}{C}}_{t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（4）\\\:{o}_{t}=\sigma\:\left({W}_{o}\cdot\:\left[{h}_{t-1},{x}_{t}\right]+{b}_{o}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（5）\\\:{h}_{t}={o}_{t}*tanh\left({C}_{t}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:（6）\end{array}$$ Where \(\:{x}_{t}\) ​ denotes the input at time step t, I, F, and O represent the input gate, forget gate, and output gate, respectively. The symbol σ denotes the sigmoid activation function, and tanh denotes the hyperbolic tangent function. 2.3 Mamba Model Mamba(Multi-scale Adaptive Memory-Based Architecture) [20–25] is a novel framework based on Selective State Space Modeling, designed to address the trade-off between modeling efficiency and computational cost when handling ultra-long sequences. Compared with earlier models such as RNNs and LSTMs, Mamba eliminates the reliance on explicit temporal recurrence. Unlike the Transformer architecture, it also forgoes computationally intensive global self-attention mechanisms. Instead, Mamba leverages state space modeling (SSM) combined with a streaming state propagation mechanism (Scan) to construct a neural network architecture capable of efficiently capturing long-range dependencies while maintaining linear computational complexity. The core idea of Mamba is to integrate continuous-time state equations into a neural network framework. At each time step, a learnable selective gating mechanism dynamically regulates the contribution of input features to the current state update, thereby enabling deep memory selection and dynamic modeling of time series data. 2.4 Rabbit Optimization Algorithm The Rabbit Optimization Algorithm (ROA) [26–28] is a novel population-based metaheuristic optimization algorithm inspired by the dynamic survival strategies of wild rabbits in their natural habitats, including their foraging behaviors, reproductive mechanisms, and predator evasion tactics. Similar to other bio-inspired algorithms such as Grey Wolf Optimization, Whale Optimization, and Pigeon-Inspired Optimization, ROA emulates the instinctive behavior of biological organisms navigating uncertain environments to offer an adaptive and robust approach for solving complex, nonlinear, and high-dimensional optimization problems. Rabbits exhibit remarkable agility, flexibility, and adaptability. During foraging, they typically combine global exploration (long-range hopping) with local exploitation (targeted movement), and display strong defensive behaviors such as sudden escape or camouflage. These ecological traits are abstracted in the ROA algorithm as a dynamic switching mechanism between global exploration and local exploitation phases, enabling efficient and balanced search capabilities across the solution space. 2.5 Hybrid Model Architecture By sequentially integrating the global attention mechanism of the Transformer with the dynamic state modeling capability of the Mamba-like module, the hybrid architecture effectively enhances the dual modeling capacity for both short-term variations and long-term trends in daily gas production time series. Convolutional layers are employed for dimensional expansion to enhance local feature perception; the Transformer captures global dependencies; and the Mamba module reinforces temporal memory and dynamic pattern recognition. Residual connections and normalization further improve training stability. Overall, the architecture demonstrates superior performance in capturing nonlinear and multi-scale variations, thereby improving both predictive accuracy and generalization capability. The hybrid model architecture is illustrated in Fig. 4 . 2.6 Evaluation Metrics To systematically evaluate the performance of the proposed model in shale gas well production forecasting, four commonly used regression metrics are adopted in this study: Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and the Coefficient of Determination (R-squared, R 2 ). The corresponding formulas are as follows: $$\:\text{M}\text{S}\text{E}=\frac{1}{n}\sum\:_{i=1}^{n}\:({y}_{i}-{\widehat{y}}_{i}{)}^{2}$$ 7 $$\:\text{R}\text{M}\text{S}\text{E}=\sqrt{\frac{1}{n}\sum\:_{i=1}^{n}({y}_{i}-{\widehat{y}}_{i}{)}^{2}}$$ 8 $$\:\text{M}\text{A}\text{E}=\frac{1}{n}\sum\:_{i=1}^{n}\:|{y}_{i}-{\widehat{y}}_{i}|$$ 9 $$\:{\text{R}}^{2}=1-\frac{\sum\:_{i=1}^{n}\:({y}_{i}-{\widehat{y}}_{i}{)}^{2}}{\sum\:_{i=1}^{n}\:({y}_{i}-\stackrel{-}{y}{)}^{2}}$$ 10 Where \(\:{y}_{i}\) is the actual value, \(\:{\widehat{y}}_{i}\) ​ is the predicted value, \(\:\stackrel{-}{y\:}\) is the mean of the actual values, and n is the total number of samples. Ethical Approval and Compliance All methods were carried out in accordance with relevant guidelines and regulations. No human participants or animals were involved in this study. The use of shale gas production data was approved by Southwest Oil & Gasfield Company under a data use agreement. 3. Results and Discussion 3.1 Experimental Data Data Availability The data supporting the findings of this study are available from Southwest Oil & Gasfield Company, but restrictions apply to the availability of these data, which were used under license for the current study, and are therefore not publicly available. However, data may be obtained from the authors upon reasonable request and with permission from Southwest Oil & Gasfield Company. The research data were extracted from the oil–gas–water well production data management system of a domestic oil and gas company. The dataset comprises long-term production histories of multiple shale gas wells located in a key production block. To ensure representativeness and consistency, only wells within the same production horizon were selected. The dataset spans the entire production lifecycle—from initial commissioning onward—featuring long temporal coverage, high sampling frequency, and substantial dynamic variation, thus providing a robust foundation for model training and validation. 1.1 Data Preprocessing The collected production data consist of seven primary variables: daily gas production, average casing pressure, daily water production, flowback ratio, average tubing pressure, fracturing fluid volume, and wellhead temperature. Spearman’s rank correlation coefficient [29] was employed to analyze the relationships among these variables. The three features most strongly correlated with daily gas production were selected as auxiliary inputs for the forecasting model. The formula for calculating the Spearman correlation coefficient is given as follows: Where d i denotes the difference in ranks between two variables, and n is the number of samples. Where d i represents the rank difference between the two variables, and n is the sample size. The Spearman correlation coefficient is a statistical measure of the linear relationship between two variables, with values ranging from [−1，1]: ρ =1：Indicates perfect positive correlation, meaning that as one variable increases, the other increases proportionally. ρ =−1：Indicates perfect negative correlation, meaning that as one variable increases, the other decreases proportionally. ρ =0：Indicates no correlation, meaning there is no linear relationship between the two variables. The strength of the correlations is presented in Table 1. As shown in Figure 5, the analysis results indicate that daily gas production exhibits strong correlations with daily water production, flowback ratio, and average casing pressure. Therefore, in the multivariate forecasting model, these three variables are selected as input features to predict gas production. Table 1 Correlation Strength. 1.2 Prediction Results and Analysis The preprocessed data were fed into the constructed Transformer-Mamba model, and the model’s hyperparameters were optimized using the Rabbit Optimization Algorithm. The predicted daily gas production was then compared with the actual production data to evaluate the model’s effectiveness. As shown in Figure 6, the Transformer-Mamba model demonstrates good fitting performance and is capable of accurately forecasting future daily gas production. Figure 7 illustrates the convergence process of the loss function. As shown in the figure, the loss value decreases progressively with the increase in the number of epochs and approaches convergence around epoch 10. The model exhibits a relatively fast convergence rate, and no signs of overfitting or underfitting are observed during the training and prediction processes. In the comparative experiments, three model architectures were selected for production forecasting: the baseline Mamba model, a hybrid model (Transformer-LSTM), and the proposed hybrid model (Transformer-Mamba). To ensure fairness in comparison, all three models utilized the same input data format and underwent hyperparameter tuning using the Rabbit Optimization Algorithm. Figures 8 and 9 present the loss function convergence curves for the Mamba model and the hybrid Transformer-LSTM model during training. As observed, both models converge around the 10th epoch, indicating stable training and a relatively fast convergence rate. Throughout the training process, no significant signs of overfitting or underfitting were detected, suggesting that both models achieved a good balance between fitting the training data and maintaining generalization capability. Figures 10 and 11 show the comparison between the predicted results and the actual production data for the Mamba model and the Transformer-LSTM model. It can be observed that both models are capable of effectively capturing the overall trends in the production time series, with their prediction curves closely aligning with the actual data. Among the three models compared, the Transformer-Mamba model demonstrates the best performance in terms of peak prediction, inflection point detection, and fluctuation amplitude reconstruction, followed by the Transformer-LSTM model. To rigorously evaluate the predictive capability and ensure a comprehensive comparison of model accuracy and robustness, four statistical indicators—MAE, RMSE, MSE, and R²—were employed to measure the fitting effectiveness of each model. As illustrated in Figure 12 and Table 2, the Transformer-Mamba architecture demonstrates marginally superior performance over the Transformer-LSTM counterpart, exhibiting the lowest error values across MAE, RMSE, and MSE. These findings further suggest that embedding the Transformer structure contributes positively to the overall effectiveness of the Mamba-based framework. The experimental results clearly demonstrate that the proposed Transformer-Mamba model, which integrates the global attention mechanism of Transformer with the dynamic state modeling capability of a Mamba-like module in a serial configuration, is well-suited for shale gas well production forecasting. The model exhibits strong performance in terms of prediction accuracy, stability, and generalization ability, indicating its potential for engineering applications and its value in providing data-driven decision support for field operations. Table 2 Comparison of Evaluation Metrics Model MSE MAE RMSE Mamba 0.0012 0.0213 0.0352 0.929 Transformer-LSTM 0.0011 0.0183 0.0336 0.935 Transformer-Mamba 0.0011 0.0174 0.0328 0.938 4. Conclusion (1) A Spearman correlation analysis was first performed on the historical production data of shale gas wells. Three dynamic parameters—daily water production, flowback ratio, and average casing pressure—were identified as most strongly correlated with gas production, and were therefore selected as auxiliary input variables for the predictive model. (2) From the perspective of model design, this study proposes a novel Transformer–Mamba hybrid architecture, which integrates the global attention mechanism of the Transformer with the temporal dynamic modeling capacity of a Mamba-like module in a serial configuration. Furthermore, the model’s hyperparameters were tuned using the Rabbit Optimization Algorithm (ROA) to improve predictive performance. Comparative experiments demonstrate that the proposed hybrid model achieves higher accuracy than both Transformer–LSTM and standalone Mamba models, while also showing stronger generalization across different application scenarios. (3) The incorporation of Transformer’s global attention substantially enhances the performance of the Mamba-based architecture. Future research will aim to refine the Transformer component, and to explore the use of transfer learning and data augmentation strategies to further improve prediction accuracy and robustness under complex operating conditions. Declarations Funding: This research received no external funding. Author Contribution Author ContributionsWeikang He: Conceptualization, Methodology, Software, Data curation, Formal analysis, Visualization, Writing – original draft.Xizhe Li: Supervision, Project administration, Writing – review & editing.Yujin Wan: Validation, Investigation, Data acquisition.Nan Wang: Data curation, Resources, Writing – review & editing.Honming Zhan: Supervision, Funding acquisition, Writing – review & editing.Xiangyang Pei: Methodology, Validation, Writing – review & editing.Longyi Wang: Formal analysis, Visualization, Writing – review & editing.Wenxuan Yu: Data acquisition, Software support.Yuhang Zhou: Resources, Investigation, Writing – review & editing. Data Availability The data supporting the findings of this study are available from Southwest Oil & Gasfield Company, but restrictions apply to the availability of these data, which were used under license for the current study, and are therefore not publicly available. However, data may be obtained from the authors upon reasonable request and with permission from Southwest Oil & Gasfield Company. References Guo, T.L. (2025). Review and reflections on shale gas development in China—from the Silurian to the Cambrian. Oil & Gas Reservoir Evaluation and Development , 15(03), 339–348. https://doi.org/10.13809/j.cnki.cn32-1825/te.2025.03.001 Dong, D.Z., Zou, C.N., Dai, J.X., et al. (2016). Strategic recommendations for shale gas development in China. Natural Gas Geoscience , 27(03), 397–406. Nie, H.K., Dang, W, Zhang, K, et al (2024) 20 years of shale gas research and development in China: Review and prospects Natural Gas Industry , 44(03), 20–52 Liu, HY, Pu, XY, Zhang, LH, et al (2023) Profitable shale gas development in China: Theoretical logic, practical logic, and outlook Natural Gas Industry , 43(04), 177–183 Bu, T, Yan, XY, Wu, ZJ, et al (2023) Rapid evaluation method of shale gas EUR based on flowback period dynamic data Unconventional Oil & Gas , 10(03), 74–79+102 https://doiorg/1019901/jfcgyq20230310 Zhu, YC, Xian, YX, Li, QY, et al (2019) Shale gas productivity prediction based on big data Oil & Gas Well Testing , 28(01), 1–6 https://doiorg/1019680/jcnki1004-4388201901001 Alolayan, OS, Raymond, SJ, Montgomery, JB, & Williams, JR (2022) Towards better shale gas production forecasting using transfer learning Upstream Oil and Gas Technology , 9, 100072 https://doiorg/101016/jupstre2022100072 Nguyen-Le, V, Kim, M, Shin, H, & Little, E (2021) Multivariate approach to gas production forecast using early production data for Barnett shale reservoir Journal of Natural Gas Science and Engineering , 87, 103776 https://doiorg/101016/jjngse2020103776 Singh, S, Bansal, P, Hosen, M, & Bansal, SK (2023) Forecasting annual natural gas consumption in the USA: Application of machine learning techniques—ANN and SVM Resources Policy , 80, 103159 https://doiorg/101016/jresourpol2022103159 Meng, J, et al (2023) Hybrid data-driven framework for shale gas production performance analysis via game theory, machine learning, and optimization approaches Petroleum Science , 20(1), 277–294 https://doiorg/101016/jpetsci202209003 Chen, Y, et al (2024) Estimation of shale adsorption gas content based on machine learning algorithms Journal of Gas Science and Engineering , 127, 205349 https://doiorg/101016/jjgsce2024205349 Qiao, SB, Sun, Y, Hu, H, et al (2025) Carbon trading price prediction based on REMD-CNN-Transformer-LSTM hybrid model Journal of Xi’an University of Technology , 1–12 [Online] Available: http://knscnkinet/kcms/detail/611294N202504161617010html Liu, WT, & Lu, XM (2022) Research progress of Transformer based on computer vision Computer Engineering and Applications , 58(06), 1–16 Li, X, Zhang, T, Zhang, Z, et al (2023) A review of Transformer research in computer vision Computer Engineering and Applications , 59(01), 1–14 Mousa, R, Rezaei, B, Mahmoudi, L, & Abdollahi, J (2025) Multi-modal wound classification using wound image and location by Swin Transformer and Transformer Expert Systems with Applications , 280, 127077 https://doiorg/101016/jeswa2025127077 Xu, L, Gao, Z, Li, Y, & Gulliver, TA (2025) Cross-domain intelligent cooperative spectrum sensing algorithm based on Federated Learning and Swin-Transformer neural network Engineering Applications of Artificial Intelligence , 157, 111370 https://doiorg/101016/jengappai2025111370 Zhang, D, Wang, C, Wang, H, Fu, Q, & Li, Z (2025) An effective CNN and Transformer fusion network for camouflaged object detection Computer Vision and Image Understanding , 259, 104431 https://doiorg/101016/jcviu2025104431 Liang, B, et al (2024) A novel framework for predicting non-stationary production time series of shale gas based on BiLSTM-RF-MPA deep fusion model Petroleum Science , 21(5), 3326–3339 https://doiorg/101016/jpetsci202405012 Zhou, Q, et al (2024) Shale oil production prediction based on an empirical model-constrained CNN-LSTM Energy Geoscience , 5(2), 100252 https://doiorg/101016/jengeos2023100252 Fargalla, MAM, et al (2024) TimeNet: Time2Vec attention-based CNN-BiGRU neural network for predicting production in shale and sandstone gas reservoirs Energy , 290, 130184 https://doiorg/101016/jenergy2023130184 Wang, YZ, Yan, JH, Bu, FL, et al (2025) RMFKAN: A cyber army detection method based on improved graph Mamba Journal of Computer Science and Exploration , 19(05), 1365–1378 Wei, XJ, Liu, XY, Zhou, HL (2025) Seismic random noise suppression method driven by convolutional Mamba model Coal Geology & Exploration , 53(05), 196–206 Wang, MH, Gao, Y, Li, J (2025) Image deblurring network based on Mamba and frequency domain fusion Computer Measurement & Control , 33(06), 264–271 https://doiorg/1016526/jcnki11-4762/tp202506033 Ma, Z, Li, J, Jiang, K, & Wong, WK (2025) Integrating local and global correlations with Mamba-Transformer for multi-class anomaly detection Knowledge-Based Systems , 324, 113740 https://doiorg/101016/jknosys2025113740 Wu, Z, Lu, T, Zhang, Y, & Chai, X (2025) CMANet: A CNN-Mamba aggregation network for face super-resolution Pattern Recognition , 168, 111859 https://doiorg/101016/jpatcog2025111859 Zhang, J, Shi, X, Feng, Z, Gui, Y, & Wang, J (2025) TMCN: Text-guided Mamba-CNN dual-encoder network for infrared and visible image fusion Infrared Physics & Technology , 149, 105895 https://doiorg/101016/jinfrared2025105895 Xiang, B-W, Xiang, Y-X, & Zhang, T-Y (2024) Rabbit algorithm for global optimization Applied Mathematical Modelling , 115860 https://doiorg/101016/japm2024115860 Alsaiari, AO, Moustafa, EB, Alhumade, H, Abulkhair, H, & Elsheikh, A (2023) A coupled artificial neural network with artificial rabbits optimizer for predicting water productivity of different designs of solar stills Advances in Engineering Software , 175, 103315 https://doiorg/101016/jadvengsoft2022103315 Gülmez, B (2023) Stock price prediction with optimized deep LSTM network using artificial rabbits optimization algorithm Expert Systems with Applications , 227, 120346 https://doiorg/101016/jeswa2023120346 Shi, X, Xu, M, & Du, J (2023) Max-sum test based on Spearman's footrule for high-dimensional independence tests Computational Statistics & Data Analysis , 185, 107768 https://doiorg/101016/jcsda2023107768 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Apr, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 18 Dec, 2025 Reviews received at journal 02 Nov, 2025 Reviews received at journal 29 Oct, 2025 Reviews received at journal 26 Oct, 2025 Reviewers agreed at journal 20 Oct, 2025 Reviewers agreed at journal 17 Oct, 2025 Reviewers agreed at journal 17 Oct, 2025 Reviewers invited by journal 10 Sep, 2025 Editor assigned by journal 10 Sep, 2025 Editor invited by journal 10 Sep, 2025 Submission checks completed at journal 01 Sep, 2025 First submitted to journal 01 Sep, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7438168","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":515649076,"identity":"f729129d-960c-47ca-a342-42851aff26e6","order_by":0,"name":"Weikang He","email":"","orcid":"","institution":"Institute of Porous Flow and Fluid Mechanics, Chinese Academy of Sciences","correspondingAuthor":false,"prefix":"","firstName":"Weikang","middleName":"","lastName":"He","suffix":""},{"id":515649080,"identity":"4010ac24-b1c5-46f0-a8b4-fff1a90d22aa","order_by":1,"name":"Xizhe Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVRIiWNgGAWjYBACPgYGxgcfKiAcZgYDIrSwARUazjjDwMBDihY2Yd42mBZiAJtE+jNm3nl35OylDx/8XFBwR56B/fDRDQw1d3Br4TmQ9nDutmfGPHxpydIzDJ4ZNvCkpd1gOPYMtxb2huMGb7cdTuzh4TFj5jE4zNggwWN2g7HhMG4tzIxtErxzDtfDtNgT1sLezCbJ23A4gQeqJZGwFp5jwEA+9syw5wxbsjRQS3IbyC8Jx3Br4ZdIf/jgQ80defYe5oOfef4ctu1nP3zsxoca3Fqg4ACSvSAigZAGFC2jYBSMglEwCtABABJ3ToOtKYDhAAAAAElFTkSuQmCC","orcid":"","institution":"Research Institute of Petroleum Exploration and Development","correspondingAuthor":true,"prefix":"","firstName":"Xizhe","middleName":"","lastName":"Li","suffix":""},{"id":515649082,"identity":"0c94fef0-fb00-4f4e-94bd-d01bfc39aec0","order_by":2,"name":"Yujin Wan","email":"","orcid":"","institution":"Research Institute of Petroleum Exploration and Development","correspondingAuthor":false,"prefix":"","firstName":"Yujin","middleName":"","lastName":"Wan","suffix":""},{"id":515649084,"identity":"82bd5ce5-89a6-4126-bac1-bb7ccf35ad12","order_by":3,"name":"Nan Wang","email":"","orcid":"","institution":"Research Institute of Petroleum Exploration and Development","correspondingAuthor":false,"prefix":"","firstName":"Nan","middleName":"","lastName":"Wang","suffix":""},{"id":515649086,"identity":"6c10b42f-2d08-42a5-b646-ceb772fd88b1","order_by":4,"name":"Honming Zhan","email":"","orcid":"","institution":"Institute of Porous Flow and Fluid Mechanics, Chinese Academy of Sciences","correspondingAuthor":false,"prefix":"","firstName":"Honming","middleName":"","lastName":"Zhan","suffix":""},{"id":515649087,"identity":"e9eb7eea-f31c-4d1c-8b28-642e9b87fb8a","order_by":5,"name":"Xiangyang Pei","email":"","orcid":"","institution":"Research Institute of Petroleum Exploration and Development","correspondingAuthor":false,"prefix":"","firstName":"Xiangyang","middleName":"","lastName":"Pei","suffix":""},{"id":515649088,"identity":"93f841ca-74c6-40b6-8c94-67b8061efd85","order_by":6,"name":"Longyi Wang","email":"","orcid":"","institution":"Institute of Porous Flow and Fluid Mechanics, Chinese Academy of Sciences","correspondingAuthor":false,"prefix":"","firstName":"Longyi","middleName":"","lastName":"Wang","suffix":""},{"id":515649089,"identity":"c71935a9-1c8a-4786-ac25-b1482ecb2490","order_by":7,"name":"Wenxuan Yu","email":"","orcid":"","institution":"Research Institute of Petroleum Exploration and Development","correspondingAuthor":false,"prefix":"","firstName":"Wenxuan","middleName":"","lastName":"Yu","suffix":""},{"id":515649090,"identity":"7fbf473a-0053-41d1-a434-c00bdb3ea8ec","order_by":8,"name":"Yuhang Zhou","email":"","orcid":"","institution":"Institute of Porous Flow and Fluid Mechanics, Chinese Academy of Sciences","correspondingAuthor":false,"prefix":"","firstName":"Yuhang","middleName":"","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2025-08-23 03:08:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7438168/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7438168/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-026-45105-z","type":"published","date":"2026-04-03T15:58:15+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":91596561,"identity":"02b67edc-bd84-4ec9-ab25-8fe4364111af","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":45160,"visible":true,"origin":"","legend":"\u003cp\u003eAnnual Shale Gas Production in China (2015–2023), in Billion Cubic Meters.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/056f0904416a5e9b51ed4040.jpg"},{"id":91596568,"identity":"bc300957-6818-422b-85c9-c2669946de8c","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":93208,"visible":true,"origin":"","legend":"\u003cp\u003eTransformer Model Architecture\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/50592fbb8caf4d6aa658f633.jpg"},{"id":91596562,"identity":"3e32ea4f-9717-4a8d-b912-1b4cdc399110","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":58895,"visible":true,"origin":"","legend":"\u003cp\u003eBasic Unit of the LSTM Network\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/792b611eb4657e034b3044eb.jpg"},{"id":91597672,"identity":"8ef1d6ea-940f-4286-b481-fdfcd43a629c","added_by":"auto","created_at":"2025-09-18 08:00:29","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":91935,"visible":true,"origin":"","legend":"\u003cp\u003eHybrid Model Architecture\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/ee7b7468358632f9ac6a6476.jpg"},{"id":91596569,"identity":"97975d8a-4a47-4ee6-82e7-09964823e83b","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":138079,"visible":true,"origin":"","legend":"\u003cp\u003eSpearman Correlation Analysis.\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/563fe3c9a2451cf85b6a3bbe.jpg"},{"id":91596563,"identity":"584fd2c3-86ae-4640-89e9-3a43d202ae92","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":114260,"visible":true,"origin":"","legend":"\u003cp\u003ePrediction Results of the Transformer-Mamba Model for Well X\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/6ce1b428035b4b9361774e1a.jpg"},{"id":91596574,"identity":"23f3b3e1-f49b-406e-b417-c9e4c5819707","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":83381,"visible":true,"origin":"","legend":"\u003cp\u003eLoss Function Evolution of the Transformer-Mamba Model\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/a9027e72c7be9818c5f4bd88.jpg"},{"id":91596861,"identity":"4ade9d21-5323-450e-9289-1674ee65c19d","added_by":"auto","created_at":"2025-09-18 07:52:29","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":82025,"visible":true,"origin":"","legend":"\u003cp\u003eLoss Function Evolution of the Mamba Model\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/784a11d208dee958b989f666.jpg"},{"id":91596571,"identity":"d229e120-68b8-42cf-9ce3-c6e6d59f9715","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":85576,"visible":true,"origin":"","legend":"\u003cp\u003eLoss Function Evolution of the Transformer-LSTM Model\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/50fe91d30ffc3f3fd028380b.jpg"},{"id":91596865,"identity":"12264029-fabc-4108-b483-36ec5e0bb87b","added_by":"auto","created_at":"2025-09-18 07:52:29","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":115185,"visible":true,"origin":"","legend":"\u003cp\u003ePrediction Results of the Mamba Model\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/b4719e23de7ac8232c577eed.jpg"},{"id":91596863,"identity":"c2209d04-cc97-49a7-9b5a-8658cc1e2920","added_by":"auto","created_at":"2025-09-18 07:52:29","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":117110,"visible":true,"origin":"","legend":"\u003cp\u003ePrediction Results of the Transformer-LSTM Model\u003c/p\u003e","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/907e0492120c84c929c9e461.jpg"},{"id":91596577,"identity":"4d738cf5-d263-48d9-b875-57f31f97e531","added_by":"auto","created_at":"2025-09-18 07:44:29","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":75907,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of Evaluation Metrics for the Three Models\u003c/p\u003e","description":"","filename":"12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/a720f5f7e1e1dc07d282e581.jpg"},{"id":106344355,"identity":"f11a45b3-37b3-408d-91e9-984411dd6490","added_by":"auto","created_at":"2026-04-07 16:13:45","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1695752,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7438168/v1/1f41f329-2379-46fa-a0c0-c26056a8d2b9.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Production Forecasting of Deep Shale Gas Wells Using the ROA-Transformer-Mamba Hybrid Network","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAmid the ongoing restructuring of the global energy system and the escalating pursuit of low-carbon energy sources, natural gas\u0026mdash;especially unconventional forms like shale gas\u0026mdash;has emerged as a pivotal component in safeguarding national energy supply and promoting the evolution toward a cleaner energy mix.\u003c/p\u003e\u003cp\u003eIn recent years, China has accelerated the exploration and development of shale gas, achieving significant breakthroughs in regions such as the Sichuan Basin, Fuling, Changning\u0026ndash;Weiyuan, and Zhaotong \u003csup\u003e[1\u0026ndash;3]\u003c/sup\u003e. According to statistics from the National Energy Administration, China\u0026rsquo;s annual shale gas production exceeded 25\u0026nbsp;billion cubic meters in 2023, with its share in total domestic natural gas output continuing to increase. It has become a primary contributor to natural gas production growth during the 14th Five-Year Plan period (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eHowever, shale gas resources are typically characterized by deep burial, complex geological structures, and strong reservoir heterogeneity \u003csup\u003e[5]\u003c/sup\u003e, which makes their production behavior significantly different from that of conventional gas reservoirs and poses substantial challenges for accurate production forecasting.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTraditional shale gas production forecasting methods are largely based on decline curve models\u0026mdash;such as Arps, DCA curves, and SEPD\u0026mdash;whose theoretical foundations lie in statistical or empirical equations. While these models provide useful guidance during the early stages of reservoir development, they often fail to capture the complex dynamic behaviors arising from multi-factor interactions. Their prediction accuracy declines significantly in scenarios involving sharp production fluctuations, gas\u0026ndash;water co-production, or well control adjustments.\u003c/p\u003e\u003cp\u003eBu Tao et al. \u003csup\u003e[4]\u003c/sup\u003e proposed a rapid EUR estimation method based on flowback-phase dynamic data, enabling early-stage assessment of recoverable reserves and offering preliminary guidance for field development. Zhu Yuanchong et al. \u003csup\u003e[5]\u003c/sup\u003e, through statistical feature analysis of large-scale datasets, explored the correlations between shale gas productivity and production parameters, emphasizing the importance of data-driven approaches and laying a foundation for the introduction of artificial intelligence (AI) models.\u003c/p\u003e\u003cp\u003eWith the rapid advancement of AI and data-driven technologies, deep learning\u0026ndash;based prediction models \u003csup\u003e[5\u0026ndash;10]\u003c/sup\u003e have been increasingly applied to oil and gas production forecasting. Among them, Long Short-Term Memory (LSTM) networks have demonstrated excellent performance in modeling time-series data, and have been successfully implemented in daily gas production prediction for various shale gas blocks.\u003c/p\u003e\u003cp\u003eAlolayan et al. \u003csup\u003e[6]\u003c/sup\u003e adopted a transfer learning strategy to enhance model generalization, effectively addressing the challenge of limited data in shale gas prediction. Nguyen-Le et al. \u003csup\u003e[7]\u003c/sup\u003e developed a multivariate input model that combines early production data to forecast the Barnett shale reservoir, yielding strong fitting results.\u003c/p\u003e\u003cp\u003eQiao Songbo et al. \u003csup\u003e[11]\u003c/sup\u003e proposed a hybrid model combining REMD, CNN, Transformer, and LSTM for carbon price prediction, demonstrating the potential of hybrid architectures in complex sequence modeling. Liang et al. \u003csup\u003e[17]\u003c/sup\u003e introduced a BiLSTM-RF-MPA integrated model tailored to the nonstationary nature of shale gas production, significantly improving forecasting accuracy and robustness.\u003c/p\u003e\u003cp\u003eMeanwhile, the recent success of the Transformer architecture in natural language processing and time-series modeling has offered new perspectives for capturing long-range dependencies in complex sequential data.\u003c/p\u003e\u003cp\u003eThis study focuses on real production data from shale gas fields and proposes a multi-module prediction model that integrates deep feature learning with time-series modeling capabilities. Its predictive performance is evaluated by comparison with Transformer-LSTM and Mamba-based models, aiming to overcome the limitations of traditional approaches in handling production volatility and incomplete datasets.\u003c/p\u003e\u003cp\u003eBy comparing the model outputs with actual production data, we assess the model's applicability and accuracy, with the goal of providing practical guidance and reference for production forecasting in deep shale gas wells.\u003c/p\u003e"},{"header":"2. Research Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Transformer Model\u003c/h2\u003e\u003cp\u003eThe Transformer model \u003csup\u003e[11\u0026ndash;16]\u003c/sup\u003e introduces a self-attention mechanism that enables the modeling of dependencies between any two time points without relying on sequential order, thereby capturing global features and significantly enhancing both modeling capacity and generalization ability (as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). When dealing with oil and gas production data\u0026mdash;which are characterized by multi-scale temporal patterns including periodic fluctuations, abrupt disturbances, production regime shifts, and long-term nonlinear trends\u0026mdash;the Transformer model is capable of learning complex dynamic relationships.\u003c/p\u003e\u003cp\u003eAt the core of the Transformer lies the multi-head attention mechanism, which allows the model to extract key features from the sequence from multiple perspectives at each time step. This mechanism enables the handling of multi-dimensional inputs, making the model well-suited for capturing the intricate relationships between various operational parameters (e.g., casing pressure, tubing pressure, temperature, daily water production, flowback ratio, and choke pressure drop) and production rates in oil and gas operations. Unlike LSTM models, which must process sequences step by step, the Transformer employs parallel computation and positional encoding to compute attention weights for all time steps simultaneously, resulting in efficient training and forecasting.\u003c/p\u003e\u003cp\u003eIn the context of production forecasting for oil and gas wells, the Transformer model can effectively identify short-term fluctuations in wellbore behavior and capture long-term reservoir dynamics. It is particularly advantageous for datasets involving multiple wells, long time windows, and strong heterogeneity.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2 LSTM Model\u003c/h2\u003e\u003cp\u003eIn oil and gas production forecasting\u0026mdash;particularly in modeling shale gas well production data\u0026mdash;traditional linear or decline curve models often fail to capture the nonlinear, time-varying, and multi-periodic patterns inherent in the data.\u003c/p\u003e\u003cp\u003eLong Short-Term Memory (LSTM) networks \u003csup\u003e[17\u0026ndash;19]\u003c/sup\u003e, an advanced variant of recurrent neural networks (RNNs), have been widely adopted for time series modeling tasks and are especially well-suited for processing and forecasting oil and gas production data with temporal dependencies. The core innovation of LSTM lies in its gated mechanism\u0026mdash;comprising input, forget, and output gates\u0026mdash;which effectively mitigates the vanishing or exploding gradient problems commonly encountered in standard RNNs. This enables the model to \u0026ldquo;remember\u0026rdquo; important information over long temporal spans during training, while filtering out irrelevant or noisy data.\u003c/p\u003e\u003cp\u003eArchitecturally, an LSTM model typically consists of a series of connected LSTM units (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), each acting as a micro memory cell. The output of each unit is influenced not only by the current input but also by the hidden state and cell state from the previous time step. This \u0026ldquo;temporal context propagation\u0026rdquo; mechanism enhances the model\u0026rsquo;s ability to adapt to time-dependent factors such as wellbore dynamics, reservoir heterogeneity, and injection\u0026ndash;production regime changes.\u003c/p\u003e\u003cp\u003eLSTM emphasizes sequential dependency modeling at each time step. By integrating the temporal memory capacity of LSTM with the non-local modeling strength of Transformer architectures, the hybrid Transformer-LSTM model is expected to achieve improved prediction accuracy and generalization in shale gas wells under complex geological conditions. Such integration supports the optimization of development strategies and enhances decision-making in production management.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe commonly used formulation for the LSTM model is given as follows:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}{f}_{t}=\\sigma\\:\\left({W}_{f}\\cdot\\:\\left[{h}_{t-1},{x}_{t}\\right]+{b}_{f}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（1）\\:\\\\\\:{i}_{t}=\\sigma\\:\\left({W}_{f}\\cdot\\:\\left[{h}_{t-1},{x}_{t}\\right]+{b}_{i}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（2）\\\\\\:{\\stackrel{\\sim}{C}}_{t}=tanh\\left({W}_{C}\\cdot\\:\\left[{h}_{t-1},{x}_{t}\\right]+{b}_{C}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（3）\\\\\\:{C}_{t}={f}_{t}*{C}_{t-1}+{i}_{t}*{\\stackrel{\\sim}{C}}_{t}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（4）\\\\\\:{o}_{t}=\\sigma\\:\\left({W}_{o}\\cdot\\:\\left[{h}_{t-1},{x}_{t}\\right]+{b}_{o}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（5）\\\\\\:{h}_{t}={o}_{t}*tanh\\left({C}_{t}\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:（6）\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{t}\\)\u003c/span\u003e\u003c/span\u003e​ denotes the input at time step t, I, F, and O represent the input gate, forget gate, and output gate, respectively. The symbol σ denotes the sigmoid activation function, and tanh denotes the hyperbolic tangent function.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3 Mamba Model\u003c/h2\u003e\u003cp\u003eMamba(Multi-scale Adaptive Memory-Based Architecture) \u003csup\u003e[20\u0026ndash;25]\u003c/sup\u003e is a novel framework based on Selective State Space Modeling, designed to address the trade-off between modeling efficiency and computational cost when handling ultra-long sequences. Compared with earlier models such as RNNs and LSTMs, Mamba eliminates the reliance on explicit temporal recurrence. Unlike the Transformer architecture, it also forgoes computationally intensive global self-attention mechanisms. Instead, Mamba leverages state space modeling (SSM) combined with a streaming state propagation mechanism (Scan) to construct a neural network architecture capable of efficiently capturing long-range dependencies while maintaining linear computational complexity.\u003c/p\u003e\u003cp\u003eThe core idea of Mamba is to integrate continuous-time state equations into a neural network framework. At each time step, a learnable selective gating mechanism dynamically regulates the contribution of input features to the current state update, thereby enabling deep memory selection and dynamic modeling of time series data.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e2.4 Rabbit Optimization Algorithm\u003c/h2\u003e\u003cp\u003eThe Rabbit Optimization Algorithm (ROA) \u003csup\u003e[26\u0026ndash;28]\u003c/sup\u003e is a novel population-based metaheuristic optimization algorithm inspired by the dynamic survival strategies of wild rabbits in their natural habitats, including their foraging behaviors, reproductive mechanisms, and predator evasion tactics. Similar to other bio-inspired algorithms such as Grey Wolf Optimization, Whale Optimization, and Pigeon-Inspired Optimization, ROA emulates the instinctive behavior of biological organisms navigating uncertain environments to offer an adaptive and robust approach for solving complex, nonlinear, and high-dimensional optimization problems.\u003c/p\u003e\u003cp\u003eRabbits exhibit remarkable agility, flexibility, and adaptability. During foraging, they typically combine global exploration (long-range hopping) with local exploitation (targeted movement), and display strong defensive behaviors such as sudden escape or camouflage. These ecological traits are abstracted in the ROA algorithm as a dynamic switching mechanism between global exploration and local exploitation phases, enabling efficient and balanced search capabilities across the solution space.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e2.5 Hybrid Model Architecture\u003c/h2\u003e\u003cp\u003eBy sequentially integrating the global attention mechanism of the Transformer with the dynamic state modeling capability of the Mamba-like module, the hybrid architecture effectively enhances the dual modeling capacity for both short-term variations and long-term trends in daily gas production time series. Convolutional layers are employed for dimensional expansion to enhance local feature perception; the Transformer captures global dependencies; and the Mamba module reinforces temporal memory and dynamic pattern recognition. Residual connections and normalization further improve training stability. Overall, the architecture demonstrates superior performance in capturing nonlinear and multi-scale variations, thereby improving both predictive accuracy and generalization capability. The hybrid model architecture is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e2.6 Evaluation Metrics\u003c/h2\u003e\u003cp\u003eTo systematically evaluate the performance of the proposed model in shale gas well production forecasting, four commonly used regression metrics are adopted in this study: Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and the Coefficient of Determination (R-squared, R\u003csup\u003e2\u003c/sup\u003e). The corresponding formulas are as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\text{M}\\text{S}\\text{E}=\\frac{1}{n}\\sum\\:_{i=1}^{n}\\:({y}_{i}-{\\widehat{y}}_{i}{)}^{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\text{R}\\text{M}\\text{S}\\text{E}=\\sqrt{\\frac{1}{n}\\sum\\:_{i=1}^{n}({y}_{i}-{\\widehat{y}}_{i}{)}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\text{M}\\text{A}\\text{E}=\\frac{1}{n}\\sum\\:_{i=1}^{n}\\:|{y}_{i}-{\\widehat{y}}_{i}|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{\\text{R}}^{2}=1-\\frac{\\sum\\:_{i=1}^{n}\\:({y}_{i}-{\\widehat{y}}_{i}{)}^{2}}{\\sum\\:_{i=1}^{n}\\:({y}_{i}-\\stackrel{-}{y}{)}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the actual value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{y}}_{i}\\)\u003c/span\u003e\u003c/span\u003e​ is the predicted value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{-}{y\\:}\\)\u003c/span\u003e\u003c/span\u003eis the mean of the actual values, and \u003cem\u003en\u003c/em\u003e is the total number of samples.\u003c/p\u003e\u003cp\u003e\u003cb\u003eEthical Approval and Compliance\u003c/b\u003e\u003c/p\u003e\u003cp\u003eAll methods were carried out in accordance with relevant guidelines and regulations. No human participants or animals were involved in this study. The use of shale gas production data was approved by Southwest Oil \u0026amp; Gasfield Company under a data use agreement.\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003ch2\u003e3.1 \u0026nbsp;Experimental Data\u003c/h2\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data supporting the findings of this study are available from Southwest Oil \u0026amp; Gasfield Company, but restrictions apply to the availability of these data, which were used under license for the current study, and are therefore not publicly available. However, data may be obtained from the authors upon reasonable request and with permission from Southwest Oil \u0026amp; Gasfield Company.\u003c/p\u003e\n\u003cp\u003eThe research data were extracted from the oil\u0026ndash;gas\u0026ndash;water well production data management system of a domestic oil and gas company. The dataset comprises long-term production histories of multiple shale gas wells located in a key production block. To ensure representativeness and consistency, only wells within the same production horizon were selected. The dataset spans the entire production lifecycle\u0026mdash;from initial commissioning onward\u0026mdash;featuring long temporal coverage, high sampling frequency, and substantial dynamic variation, thus providing a robust foundation for model training and validation.\u003c/p\u003e\n\u003ch2\u003e1.1 Data Preprocessing\u003c/h2\u003e\n\u003cp\u003eThe collected production data consist of seven primary variables: daily gas production, average casing pressure, daily water production, flowback ratio, average tubing pressure, fracturing fluid volume, and wellhead temperature. Spearman\u0026rsquo;s rank correlation coefficient\u003csup\u003e\u0026nbsp;[29]\u003c/sup\u003e was employed to analyze the relationships among these variables. The three features most strongly correlated with daily gas production were selected as auxiliary inputs for the forecasting model. The formula for calculating the Spearman correlation coefficient is given as follows:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 349px; height: 59.1753px;\" width=\"349\" height=\"59.1753\"\u003e\u003c/p\u003e\n\u003cp\u003eWhere\u0026nbsp; \u003cem\u003ed\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e denotes the difference in ranks between two variables, and \u003cem\u003en\u003c/em\u003e is the number of samples.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Where \u003cem\u003ed\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e represents the rank difference between the two variables, and \u003cem\u003en\u003c/em\u003e is the sample size.\u003c/p\u003e\n\u003cp\u003eThe Spearman correlation coefficient is a statistical measure of the linear relationship between two variables, with values ranging from [\u0026minus;1，1]:\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026rho;\u003c/em\u003e=1：Indicates perfect positive correlation, meaning that as one variable increases, the other increases proportionally.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026rho;\u003c/em\u003e=\u0026minus;1：Indicates perfect negative correlation, meaning that as one variable increases, the other decreases proportionally.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026rho;\u003c/em\u003e=0：Indicates no correlation, meaning there is no linear relationship between the two variables.\u003c/p\u003e\n\u003cp\u003eThe strength of the correlations is presented in Table 1.\u003c/p\u003e\n\u003cp\u003eAs shown in Figure 5, the analysis results indicate that daily gas production exhibits strong correlations with daily water production, flowback ratio, and average casing pressure. Therefore, in the multivariate forecasting model, these three variables are selected as input features to predict gas production.\u003c/p\u003e\n\u003cp\u003eTable 1 Correlation Strength.\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 405px; height: 98.888px;\" width=\"405\" height=\"98.888\"\u003e\u003c/p\u003e\n\u003ch2\u003e1.2 Prediction Results and Analysis\u003c/h2\u003e\n\u003cp\u003eThe preprocessed data were fed into the constructed Transformer-Mamba model, and the model\u0026rsquo;s hyperparameters were optimized using the Rabbit Optimization Algorithm. The predicted daily gas production was then compared with the actual production data to evaluate the model\u0026rsquo;s effectiveness. As shown in Figure 6, the Transformer-Mamba model demonstrates good fitting performance and is capable of accurately forecasting future daily gas production.\u003c/p\u003e\n\u003cp\u003eFigure 7 illustrates the convergence process of the loss function. As shown in the figure, the loss value decreases progressively with the increase in the number of epochs and approaches convergence around epoch 10. The model exhibits a relatively fast convergence rate, and no signs of overfitting or underfitting are observed during the training and prediction processes.\u003c/p\u003e\n\u003cp\u003eIn the comparative experiments, three model architectures were selected for production forecasting: the baseline Mamba model, a hybrid model (Transformer-LSTM), and the proposed hybrid model (Transformer-Mamba). To ensure fairness in comparison, all three models utilized the same input data format and underwent hyperparameter tuning using the Rabbit Optimization Algorithm.\u003c/p\u003e\n\u003cp\u003eFigures 8 and 9 present the loss function convergence curves for the Mamba model and the hybrid Transformer-LSTM model during training. As observed, both models converge around the 10th epoch, indicating stable training and a relatively fast convergence rate. Throughout the training process, no significant signs of overfitting or underfitting were detected, suggesting that both models achieved a good balance between fitting the training data and maintaining generalization capability.\u003c/p\u003e\n\u003cp\u003eFigures 10 and 11 show the comparison between the predicted results and the actual production data for the Mamba model and the Transformer-LSTM model. It can be observed that both models are capable of effectively capturing the overall trends in the production time series, with their prediction curves closely aligning with the actual data. Among the three models compared, the Transformer-Mamba model demonstrates the best performance in terms of peak prediction, inflection point detection, and fluctuation amplitude reconstruction, followed by the Transformer-LSTM model.\u003c/p\u003e\n\u003cp\u003eTo rigorously evaluate the predictive capability and ensure a comprehensive comparison of model accuracy and robustness, four statistical indicators\u0026mdash;MAE, RMSE, MSE, and R\u0026sup2;\u0026mdash;were employed to measure the fitting effectiveness of each model. As illustrated in Figure 12 and Table 2, the Transformer-Mamba architecture demonstrates marginally superior performance over the Transformer-LSTM counterpart, exhibiting the lowest error values across MAE, RMSE, and MSE. These findings further suggest that embedding the Transformer structure contributes positively to the overall effectiveness of the Mamba-based framework.\u003c/p\u003e\n\u003cp\u003eThe experimental results clearly demonstrate that the proposed Transformer-Mamba model, which integrates the global attention mechanism of Transformer with the dynamic state modeling capability of a Mamba-like module in a serial configuration, is well-suited for shale gas well production forecasting.\u003c/p\u003e\n\u003cp\u003eThe model exhibits strong performance in terms of prediction accuracy, stability, and generalization ability, indicating its potential for engineering applications and its value in providing data-driven decision support for field operations.\u003c/p\u003e\n\u003cp\u003eTable 2 Comparison of Evaluation Metrics\u003c/p\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"100%\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eModel\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003eMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003eMAE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003eRMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13px;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eMamba\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0012\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0213\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0352\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13px;\"\u003e\n \u003cp\u003e0.929\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eTransformer-LSTM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0011\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0183\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0336\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13px;\"\u003e\n \u003cp\u003e0.935\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eTransformer-Mamba\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0011\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0174\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 15px;\"\u003e\n \u003cp\u003e0.0328\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13px;\"\u003e\n \u003cp\u003e0.938\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003e(1) A Spearman correlation analysis was first performed on the historical production data of shale gas wells. Three dynamic parameters\u0026mdash;daily water production, flowback ratio, and average casing pressure\u0026mdash;were identified as most strongly correlated with gas production, and were therefore selected as auxiliary input variables for the predictive model.\u003c/p\u003e\u003cp\u003e(2) From the perspective of model design, this study proposes a novel Transformer\u0026ndash;Mamba hybrid architecture, which integrates the global attention mechanism of the Transformer with the temporal dynamic modeling capacity of a Mamba-like module in a serial configuration. Furthermore, the model\u0026rsquo;s hyperparameters were tuned using the Rabbit Optimization Algorithm (ROA) to improve predictive performance. Comparative experiments demonstrate that the proposed hybrid model achieves higher accuracy than both Transformer\u0026ndash;LSTM and standalone Mamba models, while also showing stronger generalization across different application scenarios.\u003c/p\u003e\u003cp\u003e(3) The incorporation of Transformer\u0026rsquo;s global attention substantially enhances the performance of the Mamba-based architecture. Future research will aim to refine the Transformer component, and to explore the use of transfer learning and data augmentation strategies to further improve prediction accuracy and robustness under complex operating conditions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding:\u003c/h2\u003e\u003cp\u003eThis research received no external funding.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAuthor ContributionsWeikang He: Conceptualization, Methodology, Software, Data curation, Formal analysis, Visualization, Writing \u0026ndash; original draft.Xizhe Li: Supervision, Project administration, Writing \u0026ndash; review \u0026amp; editing.Yujin Wan: Validation, Investigation, Data acquisition.Nan Wang: Data curation, Resources, Writing \u0026ndash; review \u0026amp; editing.Honming Zhan: Supervision, Funding acquisition, Writing \u0026ndash; review \u0026amp; editing.Xiangyang Pei: Methodology, Validation, Writing \u0026ndash; review \u0026amp; editing.Longyi Wang: Formal analysis, Visualization, Writing \u0026ndash; review \u0026amp; editing.Wenxuan Yu: Data acquisition, Software support.Yuhang Zhou: Resources, Investigation, Writing \u0026ndash; review \u0026amp; editing.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data supporting the findings of this study are available from Southwest Oil \u0026amp; Gasfield Company, but restrictions apply to the availability of these data, which were used under license for the current study, and are therefore not publicly available. However, data may be obtained from the authors upon reasonable request and with permission from Southwest Oil \u0026amp; Gasfield Company.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eGuo, T.L. (2025). Review and reflections on shale gas development in China\u0026mdash;from the Silurian to the Cambrian. \u003cem\u003eOil \u0026amp; Gas Reservoir Evaluation and Development\u003c/em\u003e, 15(03), 339\u0026ndash;348. https://doi.org/10.13809/j.cnki.cn32-1825/te.2025.03.001\u003cbr\u003eDong, D.Z., Zou, C.N., Dai, J.X., et al. (2016). Strategic recommendations for shale gas development in China. \u003cem\u003eNatural Gas Geoscience\u003c/em\u003e, 27(03), 397\u0026ndash;406.\u003c/li\u003e\n\u003cli\u003eNie, H.K., Dang, W, Zhang, K, et al (2024) 20 years of shale gas research and development in China: Review and prospects \u003cem\u003eNatural Gas Industry\u003c/em\u003e, 44(03), 20\u0026ndash;52\u003c/li\u003e\n\u003cli\u003eLiu, HY, Pu, XY, Zhang, LH, et al (2023) Profitable shale gas development in China: Theoretical logic, practical logic, and outlook \u003cem\u003eNatural Gas Industry\u003c/em\u003e, 43(04), 177\u0026ndash;183\u003c/li\u003e\n\u003cli\u003eBu, T, Yan, XY, Wu, ZJ, et al (2023) Rapid evaluation method of shale gas EUR based on flowback period dynamic data \u003cem\u003eUnconventional Oil \u0026amp; Gas\u003c/em\u003e, 10(03), 74\u0026ndash;79+102 https://doiorg/1019901/jfcgyq20230310\u003c/li\u003e\n\u003cli\u003eZhu, YC, Xian, YX, Li, QY, et al (2019) Shale gas productivity prediction based on big data \u003cem\u003eOil \u0026amp; Gas Well Testing\u003c/em\u003e, 28(01), 1\u0026ndash;6 https://doiorg/1019680/jcnki1004-4388201901001\u003c/li\u003e\n\u003cli\u003eAlolayan, OS, Raymond, SJ, Montgomery, JB, \u0026amp; Williams, JR (2022) Towards better shale gas production forecasting using transfer learning \u003cem\u003eUpstream Oil and Gas Technology\u003c/em\u003e, 9, 100072 https://doiorg/101016/jupstre2022100072\u003c/li\u003e\n\u003cli\u003eNguyen-Le, V, Kim, M, Shin, H, \u0026amp; Little, E (2021) Multivariate approach to gas production forecast using early production data for Barnett shale reservoir \u003cem\u003eJournal of Natural Gas Science and Engineering\u003c/em\u003e, 87, 103776 https://doiorg/101016/jjngse2020103776\u003c/li\u003e\n\u003cli\u003eSingh, S, Bansal, P, Hosen, M, \u0026amp; Bansal, SK (2023) Forecasting annual natural gas consumption in the USA: Application of machine learning techniques\u0026mdash;ANN and SVM \u003cem\u003eResources Policy\u003c/em\u003e, 80, 103159 https://doiorg/101016/jresourpol2022103159\u003c/li\u003e\n\u003cli\u003eMeng, J, et al (2023) Hybrid data-driven framework for shale gas production performance analysis via game theory, machine learning, and optimization approaches \u003cem\u003ePetroleum Science\u003c/em\u003e, 20(1), 277\u0026ndash;294 https://doiorg/101016/jpetsci202209003\u003c/li\u003e\n\u003cli\u003eChen, Y, et al (2024) Estimation of shale adsorption gas content based on machine learning algorithms \u003cem\u003eJournal of Gas Science and Engineering\u003c/em\u003e, 127, 205349 https://doiorg/101016/jjgsce2024205349\u003c/li\u003e\n\u003cli\u003eQiao, SB, Sun, Y, Hu, H, et al (2025) Carbon trading price prediction based on REMD-CNN-Transformer-LSTM hybrid model \u003cem\u003eJournal of Xi\u0026rsquo;an University of Technology\u003c/em\u003e, 1\u0026ndash;12 [Online] Available: http://knscnkinet/kcms/detail/611294N202504161617010html\u003c/li\u003e\n\u003cli\u003eLiu, WT, \u0026amp; Lu, XM (2022) Research progress of Transformer based on computer vision \u003cem\u003eComputer Engineering and Applications\u003c/em\u003e, 58(06), 1\u0026ndash;16\u003c/li\u003e\n\u003cli\u003eLi, X, Zhang, T, Zhang, Z, et al (2023) A review of Transformer research in computer vision \u003cem\u003eComputer Engineering and Applications\u003c/em\u003e, 59(01), 1\u0026ndash;14\u003c/li\u003e\n\u003cli\u003eMousa, R, Rezaei, B, Mahmoudi, L, \u0026amp; Abdollahi, J (2025) Multi-modal wound classification using wound image and location by Swin Transformer and Transformer \u003cem\u003eExpert Systems with Applications\u003c/em\u003e, 280, 127077 https://doiorg/101016/jeswa2025127077\u003c/li\u003e\n\u003cli\u003eXu, L, Gao, Z, Li, Y, \u0026amp; Gulliver, TA (2025) Cross-domain intelligent cooperative spectrum sensing algorithm based on Federated Learning and Swin-Transformer neural network \u003cem\u003eEngineering Applications of Artificial Intelligence\u003c/em\u003e, 157, 111370 https://doiorg/101016/jengappai2025111370\u003c/li\u003e\n\u003cli\u003eZhang, D, Wang, C, Wang, H, Fu, Q, \u0026amp; Li, Z (2025) An effective CNN and Transformer fusion network for camouflaged object detection \u003cem\u003eComputer Vision and Image Understanding\u003c/em\u003e, 259, 104431 https://doiorg/101016/jcviu2025104431\u003c/li\u003e\n\u003cli\u003eLiang, B, et al (2024) A novel framework for predicting non-stationary production time series of shale gas based on BiLSTM-RF-MPA deep fusion model \u003cem\u003ePetroleum Science\u003c/em\u003e, 21(5), 3326\u0026ndash;3339 https://doiorg/101016/jpetsci202405012\u003c/li\u003e\n\u003cli\u003eZhou, Q, et al (2024) Shale oil production prediction based on an empirical model-constrained CNN-LSTM \u003cem\u003eEnergy Geoscience\u003c/em\u003e, 5(2), 100252 https://doiorg/101016/jengeos2023100252\u003c/li\u003e\n\u003cli\u003eFargalla, MAM, et al (2024) TimeNet: Time2Vec attention-based CNN-BiGRU neural network for predicting production in shale and sandstone gas reservoirs \u003cem\u003eEnergy\u003c/em\u003e, 290, 130184 https://doiorg/101016/jenergy2023130184\u003c/li\u003e\n\u003cli\u003eWang, YZ, Yan, JH, Bu, FL, et al (2025) RMFKAN: A cyber army detection method based on improved graph Mamba \u003cem\u003eJournal of Computer Science and Exploration\u003c/em\u003e, 19(05), 1365\u0026ndash;1378\u003c/li\u003e\n\u003cli\u003eWei, XJ, Liu, XY, Zhou, HL (2025) Seismic random noise suppression method driven by convolutional Mamba model \u003cem\u003eCoal Geology \u0026amp; Exploration\u003c/em\u003e, 53(05), 196\u0026ndash;206\u003c/li\u003e\n\u003cli\u003eWang, MH, Gao, Y, Li, J (2025) Image deblurring network based on Mamba and frequency domain fusion \u003cem\u003eComputer Measurement \u0026amp; Control\u003c/em\u003e, 33(06), 264\u0026ndash;271 https://doiorg/1016526/jcnki11-4762/tp202506033\u003c/li\u003e\n\u003cli\u003eMa, Z, Li, J, Jiang, K, \u0026amp; Wong, WK (2025) Integrating local and global correlations with Mamba-Transformer for multi-class anomaly detection \u003cem\u003eKnowledge-Based Systems\u003c/em\u003e, 324, 113740 https://doiorg/101016/jknosys2025113740\u003c/li\u003e\n\u003cli\u003eWu, Z, Lu, T, Zhang, Y, \u0026amp; Chai, X (2025) CMANet: A CNN-Mamba aggregation network for face super-resolution \u003cem\u003ePattern Recognition\u003c/em\u003e, 168, 111859 https://doiorg/101016/jpatcog2025111859\u003c/li\u003e\n\u003cli\u003eZhang, J, Shi, X, Feng, Z, Gui, Y, \u0026amp; Wang, J (2025) TMCN: Text-guided Mamba-CNN dual-encoder network for infrared and visible image fusion \u003cem\u003eInfrared Physics \u0026amp; Technology\u003c/em\u003e, 149, 105895 https://doiorg/101016/jinfrared2025105895\u003c/li\u003e\n\u003cli\u003eXiang, B-W, Xiang, Y-X, \u0026amp; Zhang, T-Y (2024) Rabbit algorithm for global optimization \u003cem\u003eApplied Mathematical Modelling\u003c/em\u003e, 115860 https://doiorg/101016/japm2024115860\u003c/li\u003e\n\u003cli\u003eAlsaiari, AO, Moustafa, EB, Alhumade, H, Abulkhair, H, \u0026amp; Elsheikh, A (2023) A coupled artificial neural network with artificial rabbits optimizer for predicting water productivity of different designs of solar stills \u003cem\u003eAdvances in Engineering Software\u003c/em\u003e, 175, 103315 https://doiorg/101016/jadvengsoft2022103315\u003c/li\u003e\n\u003cli\u003eG\u0026uuml;lmez, B (2023) Stock price prediction with optimized deep LSTM network using artificial rabbits optimization algorithm \u003cem\u003eExpert Systems with Applications\u003c/em\u003e, 227, 120346 https://doiorg/101016/jeswa2023120346\u003c/li\u003e\n\u003cli\u003eShi, X, Xu, M, \u0026amp; Du, J (2023) Max-sum test based on Spearman\u0026apos;s footrule for high-dimensional independence tests \u003cem\u003eComputational Statistics \u0026amp; Data Analysis\u003c/em\u003e, 185, 107768 https://doiorg/101016/jcsda2023107768\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Shale gas, Production forecasting, Transformer, Mamba","lastPublishedDoi":"10.21203/rs.3.rs-7438168/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7438168/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe production of shale gas wells is influenced by multiple factors, including geological conditions, fracturing techniques, and production strategies. Its production patterns exhibit pronounced nonlinearity, temporal variability, and multi-scale fluctuations. In particular, deep shale gas wells often follow a complex trend characterized by \u0026ldquo;high initial production, rapid decline, and low stable output,\u0026rdquo;making it difficult for traditional decline models to accurately fit and predict their dynamic behavior.To improve the accuracy and generalization capability of production prediction in deep shale gas wells, this study proposes a hybrid deep learning model that integrates the Rabbit Optimization Algorithm (ROA), Transformer, and Mamba modules. The model first employs 1D convolution to extract local features, followed by a Transformer to capture global dependencies, and further utilizes the Mamba module to model long-term temporal dependencies and dynamic state transitions. The ROA is applied to optimize key hyperparameters, thereby enabling efficient analysis of complex nonlinear time-series data.Experimental results demonstrate that the proposed Transformer-Mamba model achieves favorable accuracy and stability, showing strong potential for generalization. It provides an effective technical foundation and valuable reference for production forecasting in deep shale gas reservoirs.\u003c/p\u003e","manuscriptTitle":"Production Forecasting of Deep Shale Gas Wells Using the ROA-Transformer-Mamba Hybrid Network","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-18 07:44:24","doi":"10.21203/rs.3.rs-7438168/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-12-18T12:27:51+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-02T19:19:27+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-29T14:35:54+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-26T20:15:47+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"74734297026766549857977773554786775648","date":"2025-10-20T08:21:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"283668102866401371089567606338718469346","date":"2025-10-17T09:13:47+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"185768826112648967878301636810141539325","date":"2025-10-17T09:12:43+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-09-11T02:39:45+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-09-11T02:34:51+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-09-10T15:59:56+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-09-02T01:34:18+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-09-02T01:31:22+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0f47b8bf-ad8b-4dd7-80c1-38a2e2b2ae20","owner":[],"postedDate":"September 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":54775574,"name":"Physical sciences/Energy science and technology"},{"id":54775575,"name":"Physical sciences/Mathematics and computing"},{"id":54775576,"name":"Earth and environmental sciences/Solid earth sciences"}],"tags":[],"updatedAt":"2026-04-07T16:09:37+00:00","versionOfRecord":{"articleIdentity":"rs-7438168","link":"https://doi.org/10.1038/s41598-026-45105-z","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-04-03 15:58:15","publishedOnDateReadable":"April 3rd, 2026"},"versionCreatedAt":"2025-09-18 07:44:24","video":"","vorDoi":"10.1038/s41598-026-45105-z","vorDoiUrl":"https://doi.org/10.1038/s41598-026-45105-z","workflowStages":[]},"version":"v1","identity":"rs-7438168","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7438168","identity":"rs-7438168","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}