Mudstone interbeds are composed of layers with high mud content, such as gray mudstone, silty mudstone, and light gray muddy fine sandstone. These interbeds typically have low permeability and strong sealing ability, and they are widely and stably developed in J1S2² with a certain degree of continuity. The distribution and scale of most mudstone interbeds are closely related to sedimentary microfacies ( Deng, 2016 ), and they are mostly developed in sedimentary microfacies such as frontal mud, underwater natural levees, interdistributary bays, and distal bars. Due to the high mud content of mudstone interbeds, their logging curve response characteristics are similar to those of mudstone, with the most notable manifestation being a high natural gamma value, and the curve showing medium-to-high amplitude tongue-shaped or high-amplitude finger-shaped protrusions; the spontaneous potential amplitude decreases without obvious anomalies, and the curve is close to the mudstone baseline; the resistivity index is relatively low, with little or no difference in the amplitude between deep and shallow induction resistivity; the density value is low, and the curve appears as a low-amplitude tongue shape; both acoustic time difference and compensated neutron values are relatively high.

Physical property barriers composed of fine sandstone mainly develop in low-permeability intervals within the formation, with lithologies mainly being light gray massive fine sandstone and siltstone containing multiple fine-grained materials and poor sorting. Although these barriers are mainly low-permeability, they still have a certain degree of permeability and porosity, but their physical properties are relatively poor. The logging curve response characteristics of physical property barriers composed of fine sandstone fall between those of mudstone and calcareous interbeds, with the natural gamma being moderate and the curve showing medium-amplitude protrusions; the spontaneous potential exhibits a decrease in negative anomaly amplitude; the density value is low but higher than that of mudstone interbeds; the wellbore diameter curve shows no significant expansion; the spontaneous potential curve shows a reversal or slight reversal phenomenon.

Calcareous sandstone interbeds mainly develop in sandstone intervals with better internal physical properties. Under the control of secondary diagenesis, strong calcitic cementation within the sandstone intervals clogging pores, resulting in extremely poor porosity development. At the same time, the development degree of these interbeds in the area is poor, and their distribution is relatively random. The most significant logging curve response characteristic of calcareous interbeds is that the deep and shallow resistivity show abnormally high values, with the curve appearing as sharp peak finger shapes; the wellbore diameter curve does not change significantly; due to pore cementation by calcium, the acoustic time difference shows abnormally low values, but the density is high, generally greater than 2.2 g/cm³ (as shown in Figures 2(a)-(c) ).

Using cross-plot technology, we analyzed the response characteristics of various logging curves, including natural gamma (GR), deep induction resistivity (RILD), density (DEN), spontaneous potential (SP), acoustic time difference (AC), and compensated neutron logging (CNL). These analyses were calibrated with core data ( Yi et al., 2021 ). It was found that different types of interbedded shales and barriers exhibited strong responses to logging curves, such as natural gamma, acoustic time difference, resistivity, and density ( Figure 3 ). As shown in the figure, muddy interbedded shales are characterized by high acoustic time difference, high natural gamma, and low resistivity in logging responses, while calcareous interbedded shales show low acoustic time difference, low natural gamma, and high resistivity. The physical property interbedded shales fall between these two types. Finally, eliminate parameters that are correlated. Choose either the sonic transit time or the density curve to be included in the selection of natural gamma, deep induction resistivity, sonic transit time, and permeability as the sensitive logging parameters for identifying interbed types. Establish a logging response sample set for each lithological interbed.

The Full Connected Neural Network (FCNN) is a neural network based on the single-layer perceptron network, which introduces multiple hidden layers to enhance its ability to learn and capture complex data features ( Karande & Kalbande, 2018 ). Theoretically, the FCNN possesses powerful function approximation capabilities, meaning it can theoretically fit any function, with a particularly strong ability to fit nonlinear functions. As the number of hidden layers increases, the expressive power of the neural network also strengthens ( Jin et al., 2024 ). It can recognize the feature relationships between input data and labels, thereby enabling complex information processing and reasoning functions. Through learning, the neural network can continuously optimize its performance and accuracy ( Bai et al., 2024 ). The basic structure of the FCNN includes an input layer, hidden layers, and an output layer. Each layer is interconnected by several neurons, with no neuron connections within the same layer. The input layer is mapped to the output layer through simple arithmetic operations represented by neurons ( Zhong et al., 2024 ) ( Figure 4 ). The basic structure of the FCNN is as follows:

1) Input Layer: Relevant feature vectors from the training sample data are input. The number of feature vectors determines the number of input nodes, and no function processing is performed at this layer.

2) Hidden Layers: The hidden layers are the core of the FCNN. The number of hidden layers and the number of neurons in each layer are adjusted according to specific tasks. These layers perform nonlinear transformations and feature extraction on the training sample data. Subsequently, feature information is exchanged between neurons and passed to the activation function for processing. After processing, the new feature information is transmitted to the next layer. To effectively prevent overfitting of the model, this paper adopts the Rectified Linear Unit (ReLU) activation function, which is defined as follows:

$\begin{array}{c}\text{ReLU}\left(x\right)=\max \left(0,x\right)\end{array}$(1)

The calculation formula for the hidden layer is:

$\begin{array}{c}y=R\left(W^{l}t+b^l\right)\end{array}$(2)

In the formula: y represents the output value of the hidden layer; $t$ represents the input value of the hidden layer; $W^l$ denotes the weights of the L-th hidden layer; $b^l$ represents the bias vector of the L-th hidden layer; $R$ stands for the ReLU function.

3) Output Layer: This layer is the last layer of the neural network, responsible for converting the learned features or patterns into final prediction results or classification labels.

Grey Relation Analysis (GRA) is employed to calculate comprehensive discriminant indicators. This technique is primarily used to quantify the grey relational degree among various parameters. This method is often applied to analyze the degree of association between different factors or indicators ( Lee et al., 2012 ). In grey systems, due to incomplete or uncertain information, GRA provides a way to quantify these degrees of association. The application of this analysis method to the study of interbeds essentially involves a comprehensive evaluation of multiple parameters. The core essence lies in determining the weight coefficients of multiple parameters for interbeds, ultimately obtaining a comprehensive discriminant indicator, and relying on it to classify interbeds.

The current identification and analysis are primarily conducted on the basis of preprocessed research data, including routine core analysis, well logging data normalization, and relevant geological information. Firstly, a comprehensive discriminant indicator for interbeds is calculated. Subsequently, a Fully Connected Neural Network (FNCC) is used to learn from this comprehensive discriminant indicator, enhancing the ability of multi-well logging data to characterize differences in responses to various interbeds. Meanwhile, an interbed identification network model is constructed. The specific process is illustrated in Figure 5 . The well logging data from the Moxizhuang Oilfield in the study area is complex, with missing and abnormal values. Additionally, the nature of the well logging data results in inconsistent units and orders of magnitude. Therefore, preprocessing of the well logging data is carried out, including normalization, which can be expressed by the formula:

$\begin{array}{c}Y=\frac{X-X_{\min }}{X_{\max }-X_{\min }}\end{array}$ (3)

In the formula: $Y$ represents the normalized structure; $X$ represents the original well logging data; $X_{\max }$ and $X_{\min }$ represents the maximum and minimum values of the original well logging data, respectively.

1) Comprehensive discriminant indicator: Natural gamma, acoustic time difference, and resistivity are selected as the sensitive parameters for well logging. The grey relational degree theory is employed for calculation. The calculation formula and process are shown in Figure 6 . Firstly, the original data is dimensionless through averaging. Then, the grey relational coefficient of each data is calculated, followed by the calculation of the relational degree of each parameter ( Yi et al. 2021 ). Ultimately, the weight coefficients of natural gamma, deep lateral resistivity, and acoustic time difference for identifying interbeds in the Moxizhuang j2S2 formation are determined to be 0.32, 0.34, and 0.3, respectively. When the discriminant parameters and their weight coefficients for interbeds are obtained, the obtained weight parameters are multiplied by the sensitive parameters to calculate the comprehensive discriminant indicator for interbeds (I-interlayer = 0.32 × GR + 0.34 × RILD + 0.3 × AC) for qualitative identification of interbeds. By calibrating the calculated I-interlayer value of the cored wells with the core samples from the cored intervals, the comprehensive discriminant indicators for the target layer can be classified, with I-interlayer ≥ 60 indicating a shale interbed, 55 ≤ I-interlayer ≤ 60 indicating a calcareous interbed, and I-interlayer ≤ 55 indicating a physical property interbed.

2) Construction of a full neural network: The comprehensive discriminant indicators for interbeds from four wells (Z1, Z101, Z103, Z107) in the study area are selected for training and divided into a training set and a validation set in a ratio of 8:2. Additionally, the comprehensive discriminant indicators for interbeds from two other wells (Z109, Z110) are selected as a test set. Subsequently, the training set and validation set are used to train the fully connected neural network model. To achieve optimal results for the neural network model, the mean squared error loss function (Equation 3) is employed during forward propagation to calculate the distance between the true value and the predicted output value, thereby obtaining the error of the current model. The gradient descent method ( Hochreiter et al., 2001 ) is used for training the backpropagation of errors to gradually adjust and update the network weights and bias parameters between layers, thereby optimizing the parameters of the neural network model. After multiple iterations, the convergence of the loss function is observed, as shown in Figure 7(a) , where the loss function is small and convergent. At this point, the optimal fully connected neural network prediction model is constructed. After construction, the validation set is input into the model for recognition, analysis, and processing to assess whether the neural network model meets the expected accuracy requirements, as shown in Figure 7(b) . Once the recognition accuracy requirements are met, an interbed identification network model is obtained. Finally, the test set is used to verify the recognition effect, achieving the effect of interbed identification, as shown in Figure 7(c) .

(4)

In the formula: $y_i$ represents the true value of the i-th sample; ${y^{\prime }}_i$ represents the predicted value of thei-th sample; $n$ represents the number of samples.

The identification of interbeds in some wells in the study area was validated using core data, classification criteria for interbed identification, and reservoir intervals interpreted from well logging results. Meanwhile, conduct a performance comparison of similar algorithms, and use the K-means and KNN algorithms to train and test the data samples respectively. 1) K-means algorithm for interbed division: With k = 3 as the clustering center, the training accuracy of the interbed division using this algorithm reaches 94.5%, and the prediction accuracy for the test data reaches 92.1%. 2) KNN algorithm for interbed division: With k = 8 and using Euclidean distance, the KNN algorithm is applied to train the interbed sample data, achieving a training accuracy of 93.7%, and the prediction accuracy for the test data is 91.5%. The use of K-means and KNN algorithms for interbed division in the study area shows that both the training and testing accuracies are lower than those of the GRA-FCNN algorithm, proving that the GRA-FCNN algorithm performs better in the study area. A comprehensive evaluation of the effectiveness of interbed identification and classification was conducted on the vertical plane of individual wells ( Figure 8 ). Through comparative analysis of core data and well logging identification results, the coincidence rate for identifying the types and quantities of interbeds exceeded 90%. According to the identification statistics, the thickness of shale-type interbeds in the second member of the Sangonghe Formation (J1S2²) in the study area ranged from 4.4 to 19.6 meters, with a distribution frequency of 0.016 to 0.129 layers per meter. These shale-type interbeds are widely and stably developed at the top and within J1S2² ( Table 1 ). The development of fine-grained sandstone physical property interbeds and calcareous sandstone interbeds is generally poor, with similar distribution frequencies and densities. Both types belong to intra-layer interbeds and their development varies in different well areas. The sealing and permeability capabilities of these two types of interbeds are much lower than those of shale interbeds. Their higher distribution frequencies and densities within the reservoir will greatly restrict fluid migration and accumulation.

Traditional methods for identifying and classifying interbeds often face issues of insufficient accuracy and low efficiency. To address this problem, a quantitative identification method based on a fully connected neural network combined with grey relational analysis is proposed. This method has demonstrated a high degree of effectiveness in practical applications, but there are still some key issues to be addressed and resolved. For neural networks, feature learning is often used for prediction and classification labeling. Since its fully connected neural network cannot automatically extract features, it usually relies on manually designed features or specific data preprocessing steps for feature extraction. Insufficient data quantity and the presence of erroneous or misleading learning features in the actual data will inevitably introduce some errors. Although data normalization is

performed, errors will still exist. For example, the logging environment of each well leads to differences in logging data quality, and the heterogeneity of reservoir geological characteristics, such as wellbore conditions affecting GR and other radioactive logging methods, different resistivity logging methods affecting RILD and other resistivity logging curves, and formation water salinity affecting SP curve amplitudes ( Si et al., 2023 ). Therefore, when applying this method to different regions and reservoirs with varying degrees of heterogeneity, besides standardizing the necessary logging curve data, corresponding adjustments and corrections can also be made to the logging environment and sensitive logging parameters.

From the results, the overall identification effect for thicker shale-type interbeds is good, with errors mainly distributed in the identification of thin interbeds, such as physical property interbeds and calcareous sandstone interbeds. This method can still identify and classify thin interbeds within sand bodies that were not previously classified by original logging interpretations or were not obviously observed in cores, and the identified and classified interbeds within the reservoir are more detailed, which is the main reason for the identification errors in thin interbeds.