Post-fracturing performance varies significantly across different fracturing intervals, stimulation techniques, and displacement mechanisms due to distinct underlying physical mechanisms. The workflow involves three key steps: (as shown in Figure 1):

1. Data Collection and preprocessing: Sample data from wells and target fracturing intervals are collected and preprocessed.

During fracturing design, the target interval is classified using a clustering model based on its characteristic features. The classified interval category is then matched with the optimal productivity prediction model tailored for that specific interval type. By activating the selected model and inputting relevant interval data and stimulation parameters, the post-fracturing production behavior—including production trends and decline patterns—can be predicted for the target interval. As shown in Figure 2.

Based on the physical mechanisms of fracturing-induced productivity enhancement, 12 characteristic parameters were selected, encompassing pre-fracturing geological parameters, production parameters, and fracturing operational parameters, along with two additional features: displacement mechanism and fracturing technique type. The displacement mechanisms primarily include water flooding, polymer flooding, and ASP flooding, while fracturing techniques are categorized into conventional fracturing, multi-fracture stimulation, and selective zonal fracturing. A dataset comprising 5,268 fractured well samples was established, as detailed in Table 1. The distribution of sample data across these categories is presented in Table 2. The dataset structured enables physics-informed machine learning while maintaining operational reality constraints.

For clustering analysis of fracturing intervals, numerical data are required. In addition to standard preprocessing steps such as normalization, the original dataset contains two categorical/textual variables—displacement mechanisms and fracturing techniques—which were converted into numerical representations using dummy variable encoding (Jin et al., 2023). For displacement mechanisms, ASP flooding was selected as the reference category, with water flooding and polymer flooding encoded as “01” and “10,” respectively. For fracturing techniques, selective zonal fracturing served as the reference category, while conventional fracturing and multi-fracture stimulation were encoded as “01” and “10.” The dataset before and after transformation is presented in Table 3.

The dataset is partitioned into subsets of similar characteristics based on the similarity between different intervals. Each subset contains samples with closely aligned properties, while maintaining distinct differences between subsets. This approach facilitates a comprehensive understanding of key information patterns within the fracturing well data. The K-means clustering algorithm (Wang, 2018; Chen and Xiao, 2004) — an iterative analytical method—provides interpretable results where cluster centroids represent the characteristic attributes of each group. As clustering outcomes critically depend on the selection of cluster numbers (k), determining the optimal k-value is pivotal, particularly when fracturing interval categories lack predefined definitions.

For this analysis, k-values ranging from 2 to 13 were systematically tested on the fracturing well dataset. Clustering performance was evaluated using the silhouette coefficient metric, which quantifies both intra-cluster cohesion (a(i)) and inter-cluster separation (b(i)) for each k-value. The silhouette score s(i) — calculated as: s i = b i − a i max ⁡ a i , b i

Serves as a composite evaluation criterion. Higher s(i) values indicate superior clustering configurations, with the k-value yielding the maximum s(i) identified as the optimal classification scheme for multidimensional fracturing interval features.

As shown in Figure 3, the silhouette coefficient initially increases with the number of clusters (k) and gradually plateaus. The maximum silhouette coefficient is achieved at k = 9, indicating optimal clustering results. Therefore, the fracturing intervals in the dataset exhibit the best clustering performance when classified into 9 categories.

The Maximal Information Coefficient (MIC) is a nonparametric statistical method rooted in mutual information theory, designed to quantify association strength between variables, particularly adept at capturing complex linear and nonlinear relationships in high-dimensional data. Its core principle involves dynamically partitioning data grids to compute the maximum mutual information across varying resolutions. Compared to Pearson’s correlation coefficient, which only identifies linear associations, MIC demonstrates significantly enhanced sensitivity to nonlinear patterns such as exponential, periodic, and piecewise relationships, while maintaining robustness against noise and outliers.

In hydraulic fracturing engineering, nonlinear characteristics frequently govern interactions between reservoir parameters (porosity ϕ, permeability k), operational parameters (proppant volume m, fracture count N), and productivity. Certain parameters exhibit threshold effects on productivity enhancement—where exceeding critical values leads to stabilized stimulation effects—while others follow power-law relationships with production outcomes. Traditional regression models struggle to characterize such complexities, whereas MIC enables precise identification of dominant factors through global optimization of variable association patterns.

Based on geomechanical and flow theory, the following multidimensional parameters were analyzed:

Geological Parameters:

Porosity (ϕ)

Operational Parameters:

Proppant volume (m)

Dynamic Parameters:

Current reservoir pressure (P)

Target Variable:

Post-fracturing productivity (Q)

Z-score standardization applied to eliminate dimensional heterogeneity. For each variable pair (X _i , Q), dynamic grid partitioning was performed in 2D space. Mutual information maxima were computed across grid resolutions. Normalized MIC values (0 ≤ MIC ≤1) were derived, characteristics with a MIC value greater than 0.4 can be considered as primary control factors, with results visualized in Figure 4.

The results demonstrate that distinct geological characteristics and microscale mechanistic variations across reservoir categories lead to divergent macro-scale dominant factors governing post-fracturing productivity within each fracturing interval type.

The training dataset for each category of fractured well productivity is denoted as D = {(X_i, Q_i)} (i = 1,2,3, … ,n), where X_i∈Rp represents the multidimensional feature vector determined by dominant governing factors, and Q_i∈R corresponds to the post-fracturing productivity. The dataset D was split into training and testing sets in an 8:2 ratio. Machine learning was performed on the training set to develop post-fracturing productivity prediction models for each fracturing interval category using Gradient Boosted Regression Trees (GBRT), Random Forests (RF), and Bagging. The post-fracturing productivity prediction models were optimized after comparing and analyzing the evaluation metrics.

The proppant volume is one of the critical parameters influencing fractured well productivity. Designing an appropriate proppant volume prior to fracturing operations not only maximizes the oil-enhancement effects of stimulation but also effectively controls single-well operational costs and improves the cost-benefit ratio (Guo et al., 2024). Insufficient proppant volume leads to inadequate fracture width and uneven proppant distribution, reducing the effective stimulated reservoir volume and reservoir permeability. Conversely, excessive proppant volume may cause proppant flowback or poor packing, hindering the formation of high-conductivity fractures and even resulting in fracturing failure. Therefore, optimizing proppant volume is essential for ensuring fracturing efficacy and enhancing hydrocarbon productivity.

The impact of proppant volume on post-fracturing productivity can be analyzed using predictive models. Sensitivity analysis was conducted on proppant volume for three fracturing intervals from three wells using the productivity prediction model, as shown in Figure 8. Key findings include that proppant volume exhibits distinct impact patterns on productivity across different intervals, yet each interval possesses an optimal proppant volume range. Within this range, fracturing achieves peak production enhancement for the specific interval.

A total of 72 oil wells were randomly selected from out-of-sample datasets to validate the post-fracturing productivity GBRT model. The model predicted production rates for the first month after stimulation, which were then compared with actual field data. As illustrated in Figure 9, the dashed lines represent actual daily incremental oil production, while the solid lines denote predicted values.

The performance metrics across the nine model categories in Table 7.