The expert layer in this study has the same structure as the expert layer of the stacking model (Wolpert, 1992). It is an intelligent system composed of multiple algorithms (experts), and all experts handle the same tasks. During the model training process, the data set is divided into k folds, and all of the experts output simulation prediction results through K simulation training and prediction processes as the input data set of the meta-learner. Finally, all data sets are used to complete the training of the expert. In the actual prediction process, the test set is input to each expert, and their prediction results are input to the meta-learner for combination.

Permeability prediction is an important aspect of formation evaluation. In the stacking model, using strong learners as experts can improve the accuracy and robustness of the model. Therefore, we selected five experts to construct the expert layer: multilayer perceptron (MLP), support vector regression (SVR), ElasticNet (EN), light gradient boosting machine (LightGBM), and category boosting (CatBoost). The characteristics and advantages of the five experts are discussed in detail below:

An MLP is a feed-forward neural network composed of multiple layers of nodes and neurons. It performs nonlinear mapping through activation functions to complete classification and regression tasks. The error between the predicted and actual results is then measured based on the loss function, and a backpropagation algorithm is applied to update the weights and biases of the neural network. The MLP can adapt to different datasets and tasks, has good capabilities for solving complex function-fitting problems, and has certain generalization capabilities.

SVR is a regression algorithm based on a support vector machine (SVM). It maps the original features to a high-dimensional feature space, transforms the regression problem into a convex optimization problem by determining the optimal hyperplane, and solves the optimization problem to determine the best hyperplane. SVR has strong nonlinear modeling capabilities, can solve complex regression tasks involving nonlinear relationships and high-dimensional data, is robust to noise and outliers in the training data, and can effectively avoid overfitting.

ElasticNet regression is an extended form of linear regression. It uses both L1 and L2 regularization terms in the objective function. The L1 regularization term judges the importance of features through the size of the coefficient, such that for unimportant features the coefficient tends to zero, which is suitable for dealing with problems with redundant features. The L2 regularization term controls the complexity of the model and reduces the impact of the correlation between features on the model. The model is more stable when highly correlated features are present. ElasticNet regression has good robustness when dealing with regression problems with redundant features or fewer samples, and it is relatively insensitive to noise and outliers in the data.

LightGBM iteratively trains multiple weak learners using the gradient boosting algorithm and optimizes and adjusts the newly generated weak learners according to the objective function to improve the accuracy of the model. It retains samples with larger gradients to accelerate the training process and uses a leaf-wise growth algorithm with depth restrictions to prevent overfitting. LightGBM sorts features according to their importance and performs feature selection based on thresholds, thereby improving the generalization ability and interpretability of the model. Further, LightGBM is suitable for processing large-scale data sets with uneven feature distributions.

CatBoost is a gradient boosting algorithm that is specially designed to effectively handle categorical features. The impact of most noise and outliers is diluted in the entire tree structure, thereby reducing their impact on individual nodes. CatBoost dynamically adjusts the learning rate according to the distribution of data and the complexity of the model and has strong accuracy and robustness when processing datasets with noise and outliers.

In the classic stacking model, the meta-learner is between the expert and output layers. Its main function is to combine the prediction results of different experts in a manner that minimizes errors, as shown in Figure 1. By generalizing the outputs of multiple expert models, the prediction accuracy of the overall model is improved, and the final prediction result is output. When choosing a meta-learner, the capacity and complexity should be considered. Stronger meta-learners may overfit the model, whereas weaker meta-learners may fail to capture complex relationships. Because the meta-learner only learns the combination of expert models, only simple machine learning models (weak learners) can be used; when dealing with high-dimensional feature sets output by multiple expert models, the fitting effect is often poor.

In this study, the AdaBoost model (Freund and Schapire, 1997) is used as the meta-learner, and the weak learner in the model is a linear regression model. During the training process for each learner, the sample weight is adjusted according to the error rate of the previous round, and greater weights are assigned to the misclassified samples to iteratively learn and correct the errors, thereby improving the accuracy of the meta-learner model. The GNS model proposed in this study uses a gate network to assign different weights to the input data of the meta-learner and adds physical model constraints to allow the meta-learner to better find the best combination of expert models while complying with the physical model constraints.

A gate network is a neural network structure that is used to control the flow and processing of information in a model. By adding a gate network mechanism, the input features can be selectively filtered, amplified, or suppressed, thus forming a constraint function for the model. In the classic stacking model, differences in expert performance cause the relationship between the features and learning objectives to become more complex. In addition, the impact of the added experts on the overall model cannot be measured. Therefore, this study introduces gate networks to solve the aforementioned problems, adjust the model architecture, and improve the accuracy and interpretability of the overall model.

In the GNS model, the role of the gate network includes three main aspects.

a. In the input layer, the gate network extracts the input petrophysical model and adds physical constraints to the input data of all expert models and meta-learners to improve the prediction performance and stability of the model and achieve model driving.

For the formation permeability prediction problem, the aforementioned components are used in this study to form a GNS model. First, a dataset is created and labeled. The logging dataset includes the natural gamma-ray log (GR), spectral gamma-ray log (K-TH-U), compressional slow-ness log (DT), neutron log (CN), and density log (DEN). Lab-measured core permeability data points are used as labels. In addition to the logging data mentioned above, V s h , N D , and ϕ D , calculated using Eqs 6–9, are also input into the GNS, called petrophysical constraints. Finally, a petrophysical and data-driven intelligent model is developed.

In the input layer, the optimal combination of the corresponding well log curves is selected for each expert model based on the SHAP value, and the petrophysical constraints are input into the gate network. In the expert layer, petrophysical constraints are added to the input data of the expert model through the gate network, and the hyperparameters of the model are self-optimized. The Bayesian optimization method is used in this study to optimize the parameters; it estimates the posterior distribution of the objective function by constructing a Gaussian process (GP) model and determines the hyperparameter value for the next sampling, thereby quickly finding the global optimal solution. Cross-validation partitioning the dataset into multiple subsets, training the model on some of these subsets, and evaluating its performance on the remaining subsets. After model optimization is completed, each expert model is simulated and predicted using a five-fold cross-validation method. The performance of the expert model is evaluated according to the MSE of the model and stored in the gate network. Each expert model is then trained using all of the datasets.

According to the ranking of expert performance indicators, the expert model simulation prediction results are added to the meta-learner and given weights to form the input set of the meta-learner. At the same time, the gate network adds petrophysical constraints.

Finally, the meta-learner is trained. We evaluate the output results, retain the expert models that positively impact the overall model performance, eliminate the expert models that negatively impact the overall model, and finally output the prediction results of the best model combination.

Because the expert model adds dynamic weight constraints, it is more robust and interpretable than the classic stacking algorithm. Figure 2 shows the GNS workflow based on petrophysical and data-driven methods.

To verify the reliability and accuracy of the proposed GNS model, three statistical parameters are introduced as performance evaluation indicators. The equations used to calculate each parameter are listed in Table 1.

The Dibei tight gas reservoir in the Kuqa Depression of the Tarim Basin has become a key exploration area because of its large gas reserve potential. The main production and storage unit is the Jurassic Ahe formation. The lithology is mainly coarse sandstone, and the sedimentary type is a braided river delta plain channel. The rock type in the sedimentation is mainly lithic sandstone, followed by feldspathic lithic sandstone, with a quartz content of more than 60%. Feldspar, clay minerals, calcite, and dolomite are all developed. The Dibei gas reservoir has low porosity, a complex pore throat structure, and a wide range of permeability changes. Multiple factors jointly control the permeability of the reservoir, and the predictive effect of the empirical model is poor. It is necessary to apply a model that can identify nonlinear characteristics between well logs and core data. Therefore, we apply the GNS model to predict the permeability.

The stability and accuracy of the model depend on the reliability of the training data. In this study, 1,088 samples were collected from five wells in the Dibei area. The locations of the core wells are shown in Figure 3. The input variables include seven conventional well logs (GR, K, TH, U, DT, DEN, and CN) and three petrophysical constraints (ND, V s h , and ϕ D ); the permeability value is the desired output. To ensure that the GNS model has the best possible performance, the training data were normalized. Based on the data normalization, we calculated the SHAP value of each input feature, as shown in Figure 4, which reflects the contribution of each feature to the model prediction, further indicating the relative importance of each feature to the prediction results.

The bar chart in Figure 4 shows that the contributions of the SHAP values in descending order are DEN, GR, TH, CN, K, DT, and U. Five different expert models were trained based on the SHAP value contribution of the well log, and the MSE of the training results is shown in the line chart in Figure 4. The MSE reduction is defined as a log that has a positive contribution to the model, while conversely, when it has a negative contribution to the model, the positive contribution log is retained as the input of the expert model, and different log combinations are input for different expert models. Together with the permeability label, these constitute the training set, as summarized in Table 2.

The feature combinations in Table 2 are input into the expert model for training, and the Bayesian optimization method is used to find the optimal hyperparameter values such that the model can achieve the best performance on the target task. A performance comparison of the expert model using optimal hyperparameter values and the expert model using default parameters is summarized in Table 3.

The petrophysical constraints stored in the gate network are added to the feature combination of each expert to form the input dataset for each expert. Targeting the core permeability, multiple random sampling and no-replacement iterative predictions are conducted on experts to obtain a set of simulation prediction results, record the expert model MSE, save it to the gate network, and use all datasets to train each expert model. The expert combination is formed iteratively based on the MSE of the expert model (from small to large), the weight of each expert in the combination is calculated (Eq. 8), and the corresponding simulation prediction results are weighted (Eq. 9). Next, the weighted prediction results and petrophysical constraints are input into the meta-learner for training to determine the optimal expert model combination strategy. To demonstrate the effectiveness and advantages of the GNS algorithm, a classic stacking model is used for comparison. To ensure consistency, the same experts and meta-learners are used in the models. A performance comparison of the two models is shown in Figure 5.

It can be seen that the addition of the gate network clearly improves the overall model. In contrast, the classic stacking model lacks petrophysical constraints and expert evaluation mechanisms and cannot optimize over-fitting, coupling correlation, and other problems existing in the integration process of experts; as a result, the overall performance of the stacking model is inferior to that of the GNS. For example, in the fourth iteration shown in Figure 5, the MLP model may have problems with dataset misfit or poor coupling with other expert results, resulting in negative improvements to the overall model when an expert is added to the stacking model. In the GNS algorithm, the addition of this expert is judged to be a negative improvement. Therefore, the prediction results of this expert are eliminated from the overall model, ultimately maintaining the quality of the overall expert group.

The above analysis shows that the performance of the GNS model is better than that of the stacking model, indicating that the improvement in the gate network is effective. To discuss the superiority of the model architecture, this study conducts a horizontal comparison with other ensemble learning models. We selected three representative integration strategies: two heterogeneous integration models (RCM and Voting) and a bagging integration model constructed using LightGBM. Three evaluation indicators—MSE, MAE, and R2—are used to evaluate the performance of the model. The results are summarized in Table 4. The GNS and RCM models have better prediction performance, and the GNS model achieves the best prediction results.

Figure 6A shows the error between the core permeability measured in the laboratory and that predicted using the GNS model. To compare the prediction effect more intuitively, the permeability is plotted on a cross diagram (Figure 6B), where the 45° diagonal line represents a perfect match between the predicted and true penetration rates. The prediction error of the model conforms to a normal distribution; therefore, the accuracy of the model can be judged based on the variance and MSE (Helle et al., 2001; Zhong et al., 2019). Figures 7–9 show the prediction results of the RCM, bagging, and voting models, respectively.

The GNS model has a variance of 0.0931 and MSE of 0.1364. It has the highest degree of fit with the core permeability values. The error does not exceed one order of magnitude for any of the core samples, showing the best performance (Figure 6). The variance of the RCM model is 0.1109 and the MSE is 0.1762. The prediction results are slightly lower in the high-permeability interval, and the performance is slightly weaker than that of the GNS (Figure 7). The variance of the bagging model is 0.1272, and the MSE is 0.2230. Some sample points exhibit large errors at different permeability intervals. This may be because random sampling will lead to a loss of some useful information (Figure 8). Finally, the voting model has the largest variance and MSE, and the predicted permeability has a larger error than the core permeability (Figure 9). These studies demonstrate that the GNS model is consistent with the measured permeability of the cores.

Well N4 is a core well in the Dibei gas reservoir in the Tarim Basin. The target reservoir is the Jurassic Ahe Formation. Logging data includes natural gamma ray log (GR), porosity logs (DEN-CN-DT), and natural gamma spectrum logs (K-TH-U). In this study, we selected the above seven well logs and three petrophysical constraints (VSH, PHID, and ND) as inputs, used the core permeability measured in the laboratory as label data, and entered it into the GNS model for prediction. For comparison, the results of several other ensemble learning models were obtained. Figure 10 shows the prediction results for each model in Well N4. The first track is the natural gamma curve, the second track is the natural gamma spectrum curve, the third track is the porosity log curve, and the fourth track is the petrophysical constraints. Tracks 6, 7, 8, and 9 are the comparisons between the prediction results of the GNS model, RCM model, bagging model, and voting model and the true value of the core permeability, respectively. The prediction results of the GNS model exhibit the best agreement with the core measurement results. The RCM and bagging models have certain errors at lower permeabilities. The voting model exhibits the largest difference from the core measurement results. The case study shows that the GNS model has advantages in predicting permeability. Importantly, we applied the gate network to integrate the petrophysical constraints with the model, rather than simply taking the petrophysical constraints as inputs. In addition, we controlled the weight of the model through the gate network, avoiding the influence of weak experts. Thus, the proposed model is more flexible than previously established empirical equations and other machine learning models.