Stacked generalization (David, 1992), as a typical learning method in ensemble learning (Lü et al., 2019), uses a certain learner to integrate the classification results of different base learners. The stacked generalization model can increase the nonlinear ability of the model by increasing the number of layers, but at the same time, it will also lead to the phenomenon of model overfitting. Therefore, the general stacked generalization model is a two-layer structure (Niyogisubizo et al., 2022). The stacked generalization model ensemble machine learning algorithm models with better identification effects, such as decision trees, nearest neighbors, and support vector machine algorithms. It makes full use of the different characteristics of different algorithms and then uses a new meta-learner to combine the prediction results of individual base learners to finally complete the prediction of the task results. The algorithm implementation process is illustrated in Figure 2. First, the dataset is divided into multiple subsets, which serve as inputs for the basic learners. Each basic learner generates a prediction result based on its corresponding subset. The outputs of all the basic learners are then used as inputs for the meta-learner, which produces the final prediction result. By stacking and generalizing the outputs of multiple base learners, the overall task’s prediction accuracy is improved.

In order to prevent the meta-learner from directly learning the training set of the base learner, which causes too much risk of overfitting, the learning of the base learner is carried out by means of cross-validation (Geisser, 1975). The original training set D is randomly divided into k sets D 1 , D 2 , D 3 , D 4 ,…, D k of the same size or with little difference. The j-th set D j = x 1 , y 1 , x 2 , y 2 , … , x m , y m has m pieces of data. Let the test set of the j-th base learner be D j , and the training set is D ¯ j = D − D j . Denoting the total number of base learners with T, each base learner algorithm is expressed as L 1 , L 2 , … , L t . Let the t-th base learner algorithm L t learn at D ¯ j ; for the i-th example x i , y i of D j , a secondary training sample z i t = h t x i is generated, and the output set of all base learners L 1 , L 2 , … , L t for i examples is z i 1 , z i 2 , … , z i T , y i ). D ′ = D ′ ∪ z i 1 , z i 2 , … , z i T , y i is obtained throughout the cross-validation process as a training instance of the meta-learner L, and the final meta-learner’s prediction H x is the final output of the model.

Considering the diverse principles underlying the base learners and their varying interpretations of the data space dimensions in well logging information (Zhou H et al., 2021), classic algorithms such as the SVM, KNN, decision trees, and stochastic gradient descent classification method have been widely used in the field of machine learning for many years and have been extensively validated in practical applications. These algorithms have broad application domains and are supported by a rich body of research, demonstrating their ability to achieve robust performance in various problem domains (Wu et al., 2017; Zhou X et al., 2021; Jung and Kim, 2023; Pałczyński et al., 2023). Therefore, in this article, they are used as the base learner, and the fully connected neural network is used as the meta-learner (detailed in the subsequent section). The pseudocode is shown in Table 4.

Among them, h t represents the learner obtained after L t is trained by D ¯ t , and H x is the output of the meta-learner, which is also the final output of the whole model.

Based on the introduction provided in the previous section, the SVM, KNN, decision tree, and stochastic gradient descent classification method with different mathematical principles are used as the base learners, and the fully connected neural network commonly used in deep learning is used as the meta-learner to control the final output. Among them, the fully connected neural network is also called multilayer perceptron, which is a kind of artificial neural network with forward feedback (Du et al., 2020), known as the deep feedforward network. A feedforward neural network includes an input layer, a hidden layer, and an output layer. The input layer mainly associates the feature information with the input nodes, transmits information for the hidden nodes in the next step, and provides data support for the calculation in the next step. The hidden layer is a node layer that further processes the feature information, calculates the data transmitted by the input node, and transmits the calculated information to the output node to improve the nonlinear ability of the model. The output layer is the last link in the transmission of information, which further calculates the data and transmits information to the outside of the network (Figure 3).

In the forward propagation stage, it can be seen that the input layers x ₁ , x ₂ .x _n form the X vector. Therefore, the hidden layer neurons are activated by the hyperbolic function tanh to obtain the output value of this layer: a j = tanh ∑ i = 1 n w i j x i j , (3) where a _j represents the output value of the hidden layer neuron, tanh represents the hidden layer activation function, g _j represents the input weighted sum of the j-th hidden layer neuron node, n represents the number of neurons in the input layer, i represents the subscript of the input layer neuron, j represents the subscript of the hidden layer neuron, and w_ij is the weight of the hidden layer neuron j.

Similarly, the output value of the output layer neuron is activated by the normalized exponential function softmax, which can be expressed as y = a k = softmax ∑ j = 1 m w j k x j k , (4) where y (a _k ) represents the output value of the hidden layer neuron, softmax represents the activation function of the output layer, m is the number of neurons in the hidden layer, w _jk is the weight of the output layer neuron k, and x _jk represents the output value of the neurons in the hidden layer, which is equal to the input value of the neurons in the output layer.

Let h k = ∑ j = 1 m w j k x j k . (5)

From the definition of the softmax function, it can be known that softmax h k = s h k = e h k ∑ l = 1 p e h l , (6) where p represents the number of neurons in the output layer.

The loss function can be expressed as L = − ln e h k ∑ l = 1 p e h l = − h k − ln ∑ l = 1 p e h l . (7)

In the backpropagation stage, when the loss function is passed from the output layer to the input layer, the stochastic gradient descent algorithm (SGD) is used as the optimizer to iteratively adjust the model weight. This method is the most basic iterative algorithm for optimizing neural networks at present, and it is simple and easy to implement (Li et al., 2022).

The gradient is also the partial derivative of the loss function L with respect to the weight w, and the calculation based on the chain rule can be expressed as ∂ L ∂ w i k = ∂ L ∂ h k ∂ h k ∂ w i k . (8)

The weight of the updated output layer can be obtained from the aforementioned formula, which is expressed as w j k = w j k − η ∂ L ∂ w j k , (9) where η represents the model learning rate that controls the update speed of the parameters.

Conventional well logging curves RT, CNL, DEN, AC, and QL are used as sample points, respectively, where QL can be calculated from the gas measurement data. According to the oil test data, 121 oil test intervals of 38 exploration wells were selected, and the corresponding fluid-sensitive logging parameters (deep lateral resistivity (RT), neutron (CNL), acoustic time difference (AC), formation density (DEN), and total hydrocarbon index (QL)) were used as the input data of the basic learner. The training dataset for this study consists of 100 samples, while the testing dataset comprises 26 samples. Among these, there are 76 training samples and 18 prediction samples for gas reservoirs. For water reservoirs, there are eight training samples and three prediction samples. Finally, for the gas–water layers, there are 16 training samples and 5 prediction samples. The distribution of sample data is shown in Table 5.

Based on the earlier discussion, where the cross-plot method demonstrated effective identification of dry formations, we will not specifically focus on the identification of dry formations in this study. The fluid data are mainly divided into the gas layer, water layer, and gas–water layer. The label is set to 0 for the water layer, 1 for the gas layer, and 2 for the same layer of gas–water.

The dimension and order of magnitude of different logging information are different. To eliminate the influence of unit and scale difference between logging information, it is necessary to normalize the features and normalize the input logging response feature data. The formula can be expressed as X norm = X − X min X max − X min . (11)

For nonlinear logarithmic characteristic curves, such as RT curves, logarithmic transformation should be performed before entering the model: X norm = lgX − lgX min lgX max − lgX min . (12)

In the formula, X n o r m represents the normalized data, X represents the original sample data, and X max 、and X min , respectively, represent the maximum and minimum values of the sample data.

After data standardization, the dataset was randomly divided into the training set and test set in the ratio of 8:2, and the training set was further divided into the sub-training set and sub-validation set using the five-fold cross-validation method. Using the sub-training set to train the model and the verification set to verify the training effect, the size of the model parameters is adjusted based on the effect of the verification. This process helps in selecting the optimal parameters for the basic learner. Stochastic gradient descent, KNN algorithm, decision tree algorithm, and support vector machine algorithm were selected as the basic learners, and different parameters in the algorithm were set. The relationship between different parameters of the experiment and the accuracy rate is shown in Figure 4.

Among them, the maximum depth and the maximum number of leaf nodes are used as the decision parameters of the decision tree. Since the number of features is selected as 5, the maximum leaf child nodes of the decision tree are set to 3, 4, and 5. The larger the maximum depth of the decision tree is, the more the tree is split, the ability of the model to capture the nonlinearity of the data is enhanced, the probability of the model overfitting is increased, and the generalization ability of the model is worse. It can be seen from Figure 4A that with the increase in the maximum depth of the decision tree, the accuracy of the model shows a fluctuating trend. When the maximum depth is 5 and the maximum number of child nodes is 4, the model accuracy is the highest, which is 79.8%.

The SVM algorithm completes the classification by maximizing the classification interface and finding a hyperplane to separate the classification targets as much as possible. When using SVM to identify fluids, it is necessary to pay attention to the penalty term coefficient C and the corresponding kernel function. Increasing the penalty coefficient C in a model strengthens its ability to constrain and suppress the parameters, which can lead to a higher degree of regularization. However, if the penalty coefficient C is set to an excessively large value, it can result in excessive parameter suppression and potentially cause underfitting. Kernel functions are generally used for higher-dimensional target tasks and can also provide models with different nonlinear capabilities. It can be seen from Figure 4B that in the SVM classification algorithm, the accuracy of the results using the polynomial and radial basis kernel functions is higher than that of the linear kernel function, and the accuracy of the polynomial kernel function is slightly higher than that of the radial basis kernel function. When the penalty term coefficient C is 1.1 and the kernel function is a polynomial optimizer, the model has the best accuracy of 81.5%.

It can be seen from Figure 4C that the stochastic gradient descent method is affected by the maximum number of iterations and the loss function. The log, hing, and square loss functions are selected as experimental comparisons. When the loss function is log loss, the stochastic gradient descent method evolves into a logistic regression algorithm with fast convergence and stable overall accuracy, with the highest accuracy reaching 81.2%. Other loss functions are more volatile and unstable, among which the square loss function has the most violent fluctuation.

As seen from Figure 4D, the KNN algorithm is affected by the number of nearest neighbors and the construction method of the tree. The construction method of the tree is divided into brute, kd-tree, and ball-tree. It can be found that the construction method of the tree has no obvious influence on the algorithm. When experimenting on the number of nearest neighbors, it is found that the number of nearest neighbor algorithms has a significant impact on the generalization effect of the model. When the number of nearest neighbors is 5, the highest accuracy is 79.7%. When the number of nearest neighbors is 15–20, the accuracy rate is not much different. When the number of nearest neighbors is larger, the model is more prone to overfitting, so the optimal number of nearest neighbors is 5.

The initial learning rate of the meta-learner is set to 0.001. Considering that there are too few data sample points, the model is prone to overfitting, and the regularization parameter penalty factor is set to 0.0005. The input layer is the secondary training set constructed by the output of the four basic learners after cross-validation training. Two hidden layers are set in the middle of the meta-learner, the number of neurons is set to 128 and 128, respectively, and the number of output neurons is set to 3. The learning rate decay factor is 0.8. The meta-learner is trained for 10,000 rounds, and the training loss and test loss are reduced to 0.04 and 0.03, respectively (Figure 5). It can be seen from the curve that the training set loss and the validation set loss have a stable and consistent downward trend, and there is no overfitting phenomenon.

The accuracy rate obtained by inferring the test set with the meta-learner model is used as the evaluation index, and the calculation method of the accuracy rate is as follows: P = T P T P + F P , (13) where P is the accuracy rate and TP is the number of positive samples where the sample itself is positive. The model also predicts the number of positive samples, and FP is the number of negative samples that are predicted to be positive samples.

The hyperparameter training batch, the number of intermediate hidden layers, the learning rate, and the regularization parameters in the experiment are modified. Based on the experimental method, the optimal parameters of the fully connected neural network model were selected, the accuracy of the model test set was used as the measurement index, and judging whether the model is overfitting is supplemented by the accuracy of the training set. The training batch-accuracy, learning rate-accuracy, number of hidden layers-accuracy, and regularization coefficient-accuracy graphs are made, respectively (Figures 6A–D).

When the other parameters are fixed at a specific value, the training batches are set in the range of 500–3,000, with an interval of 500 for each training iteration (Figure 6A). It is found that the training batch tends to be stable from 2,500 to 3,000, and the accuracy trend of the training and test sets are consistent, with no overfitting. At this time, the batch is 2,500 and has the best accuracy. When the model learning rate is too small, the network model is prone to fall into local optimum, and when the model is too large, it is not stable enough and lacks robustness. It can be seen from Figure 6B that when the learning rate is 0.009, the model training accuracy is high. When the number of neurons in the hidden layer is small, the model is prone to underfitting and lacks nonlinear ability. As the number of neurons increases, the nonlinear ability of the model increases. When the number of neurons is 512, the training accuracy increases and the test accuracy decreases significantly, and overfitting occurs at this time (Figure 6C). Therefore, the number of neurons should be 256 for the best accuracy. As the regularization coefficient increases, the penalty for the parameters is larger, and the ability to suppress overfitting is stronger. When the regularization coefficient is 0.004, the model test accuracy is the highest (Figure 6D).