An Efficient Method for Identifying Inter-Well Connectivity Using AP Clustering and Graphical Lasso: Validation with Tracer Test Results

Abstract

Identifying inter-well connectivity is crucial for optimizing reservoir development and facilitating informed adjustments. While current engineering methods are effective, they are often prohibitively expensive due to the complex nature of reservoir conditions. In contrast, methods that utilize historical production data to identify inter-well connectivity offer faster and more cost-effective alternatives. However, when faced with incomplete dynamic data—such as long-term shut-ins and data gaps—these methods may yield substantial errors in correlation results. To address this issue, we have developed an unsupervised machine learning algorithm that integrates sparse inverse covariance estimation with affinity propagation clustering to map and analyze dynamic oil field data. This methodology enables the extraction of inter-well topological structures, facilitating the automatic clustering of producers and the quantitative identification of connectivity between injectors and producers. To mitigate errors associated with sparse production data, our approach employs sparse inverse covariance estimation for preprocessing the production performance data of the wells. This preprocessing step enhances the robustness and accuracy of subsequent clustering and connectivity analyses. The algorithm’s stability and reliability were rigorously evaluated using long-term tracer test results from a test block in an actual reservoir, covering a span of over a decade. The results of the algorithm were compared with those of the tracer test to evaluate its accuracy, precision rate, recall rate, and correlation. The clustering results indicate that wells with similar characteristics and production systems are automatically grouped into distinct clusters, reflecting the underlying geological understanding. The algorithm successfully divided the test block into four macro-regions, consistent with geological interpretations. Furthermore, the algorithm effectively identified the inter-well connectivity between injectors and producers, with connectivity magnitudes aligning closely with actual tracer test data. Overall, the algorithm achieved a precision rate of 79.17%, a recall rate of 90.48%, and an accuracy of 91.07%. This congruence validates the algorithm’s effectiveness in the quantitative analysis of inter-well connectivity and demonstrates significant potential for enhancing the accuracy and efficiency of inter-well connectivity identification.

1. Introduction

Reservoir dynamic inter-well connectivity is one of the most important aspects of oilfield development evaluation. It can provide key technical support for identifying dominant channels, controlling profiles, describing remaining oil distribution, and understanding the relationship between different reservoirs and different wells [1]. The prediction and calculation of inter-well connectivity serve as an important basis for production planning and decision-making [2], and are significant for guiding the optimal design of injection-production schemes—such as water plugging, profile control, fracturing, and hole filling—ultimately improving the recovery efficiency of water and gas drive reservoirs.

Numerous methods exist for determining inter-well connectivity, including static connectivity analysis and dynamic connectivity analyses. Static connectivity analysis primarily pertains to the initial stage before the evaluation of reservoir development [3], while dynamic analysis focuses on studying the migration path and velocity of reservoir fluids under the condition of reservoir changes to clarify their behavioral patterns. Static connectivity analysis encompasses techniques such as logging exploration [4,5,6], seismic method [7,8], and reservoir parameter comparison method [9,10]. In contrast, dynamic connectivity analysis includes tracer testing and well test analysis methods. However, these methods have notable disadvantages, including high costs, lengthy operational times, and complex procedures. For instance, well test analysis is highly accurate but requires modifications to the working system of the target well, leading to elevated costs. Similarly, the tracer analysis method is complex and necessitates integration with numerical simulation techniques, making it less suitable for large-scale implementation.

Compared to the aforementioned engineering methods, the inter-well connectivity analysis based on production dynamics offers several advantages, including simplicity, reduced time requirements, and lower costs. The production dynamic data of producing wells contain valuable information regarding inter-well connectivity, as the dynamic changes in production between two connected wells exhibit a specific correlation. Thus, exploring the correlation within production dynamic data is crucial for identifying inter-well connectivity [11,12,13]. Production performance analysis typically requires only the injection and production history data of the wells, without necessitating changes to the production scheme of each well. Traditional methods based on production dynamic analysis primarily include Spearman Rank Correlation (SRC), Multiple Linear Regression (MLR), and the Capacitance-Resistance Model (CRM).

Based on the Spearman Rank Correlation coefficient, SRC reflects the inter-well connectivity through the fluctuation of inter-well injection and production data. Heffer [14] applied the SRC coefficient to propose that the inter-well connectivity can be reflected by the fluctuation relationship of the production data of injectors and producers and carried out the geomechanics-fluid flow coupling numerical simulation. Fedenczuk [15] compared the SRC coefficient between injectors and producers, and proposed a new method to respond to the characteristics of injectors and horizontal producers. Soeriawinata [16] makes use of the dynamic data to further apply the SRC model and describes the specific flow of inter-well connectivity analysis.

MLR is mainly used to establish test samples with different parameters, analyze the influence of different reservoir and production parameters on the degree of inter-well connectivity, perform multiple linear regression on different experimental samples, and identify inter-well connectivity based on a multi-factor regression model. Alejandro [17] proposed for the first time to quantitatively identify reservoir inter-well connectivity by using MLR processing on the actual injection rate and fluid production rate. Dinh [18] applies MLR to the analysis of inter-well connectivity in hydraulically fractured wells, horizontal wells, complex wells, and multiple complex wells at the same time. Wang [19] derived an MLR model which is suitable for ultra-low permeability reservoirs.

CRM is derived based on the material balance equation and signal processing theory. The injection rate is taken as the input signal, the production rate as the output signal, and the inter-well connectivity and the corresponding time delay constitute unknown system parameters. Yousef [20] proposed a CRM to quantitatively describe the connectivity between vertical wells in the reservoir based on the fluctuations of dynamic data and solved the problems of compressibility and time lag in previous inter-well connectivity research methods. Kaviani [21] studied the impact of unsteady pressure, proposing the segmented CRM and compensated CRM. Mirzayev [22] improved the applicability of the CRM in tight reservoirs.

In recent years, the application of machine learning in various industries has made rapid progress and development [23,24,25]. Based on statistics and computational science, machine learning can uncover hidden information from production data, thereby improving the accuracy and reliability of decision-making [26,27,28]. Using machine learning to process large amounts of complex production data and automatically extract useful features can overcome the error caused by human computations [29,30] and determine inter-well connectivity faster and cheaper. In addition, machine learning is more adaptable and flexible, and can automatically adjust and improve the model to suit different reservoir properties [31]. Liu [2] proposed a machine learning method for inferring inter-well connectivity using bottom-hole pressure data from fixed-rate wells. Jiang [30] developed a Physical Knowledge Fusion Neural Network (PKFNN) model that integrates physical laws with machine learning to characterize inter-well connectivity in water drive reservoirs and predict well productivity. Huang [32] employed various recurrent neural network (RNN) models to forecast production and inter-well connectivity in Steam Assisted Gravity Drainage (SAGD) operations for heavy oil reservoirs. Sinha [33] utilized machine learning to predict fluid production rates based solely on water injection rates, assessing various models for this purpose. Yu [34] highlighted the role of L1-norm regularization in identifying inter-well connectivity and proposed a neural network-based proxy model for predicting well production dynamics. Li [35] applied a modified Long- and Short-term Time-series network (LSTNet) combining CNN and RNN to perform virtual interference tests, focusing on physics-based constraints rather than handcrafted features to predict well pressure and identify well connectivity and sealing structures.

The production dynamics utilized in the aforementioned methods often stem from numerical simulations or short-term ideal complete datasets. However, issues such as extended production periods, loss of data, or prolonged shut-ins result in significant dynamic gaps. Traditional processing techniques, such as covariance matrix analysis, face limitations when addressing the sparse or incomplete fluctuations of oil well production. To address these challenges, this paper presents a set of unsupervised learning algorithms capable of accurately clustering producers based on characteristics and calculating inter-well connectivity, particularly in cases of extensive oilfield lifespans and sparse production dynamics due to long-term shut-ins. Specifically, we employ sparse inverse covariance estimation for dynamic preprocessing of production data, using the graphical lasso algorithm to solve the matrix. Producers are then grouped into clusters via Affinity Propagation Clustering based on production characteristics, and the connectivity between injectors and producers is derived from the responsibility matrix. We applied this methodology to analyze the production dynamics of an oilfield over the past 20 years, calculating the clustering and connectivity of 15 wells in the southern region of a block. Comparison with actual tracer results revealed that our clustering algorithm effectively identified wells with similar geological properties across faults, and the inter-well connectivity calculations closely aligned with field tracer tests. These findings demonstrate the algorithm’s capability to manage sparse or incomplete production data over extended periods, enabling precise clustering of producers and estimation of connectivity to specific injectors. This method is straightforward, rapid, and cost-effective, providing valuable insights for reservoir development adjustments and geological understanding.

The production dynamics employed in the aforementioned methods typically derive from numerical simulations or short-term ideal complete datasets. However, issues such as extended production periods, data loss, or prolonged shut-ins can lead to significant dynamic discrepancies. Traditional processing techniques, such as covariance matrix analysis, encounter limitations when addressing sparse or incomplete fluctuations in oil well production. To tackle these challenges, this paper presents a set of unsupervised learning algorithms capable of accurately clustering producers based on their characteristics and calculating inter-well connectivity, particularly in scenarios where the oilfield’s lifespan is long and production dynamics are sparse due to prolonged shut-ins. The main contributions of this method include the following: (1) stable operation under sparse or incomplete historical dynamics; (2) reliable calculation of inter-well connectivity using only injection and production rates; (3) effective identification and intuitive clustering of wells located in similar geological features or exhibiting analogous production variation characteristics; and (4) recognition of producers affected by the injector, providing a substitute for tracer tests and allowing for quantitative characterization of connectivity magnitude. Specifically, Section 2 outlines the background of the tracer tests for validation purposes, and Section 3 details the algorithm’s workflow. Section 3.1 discusses the dynamic preprocessing of production data using sparse inverse covariance estimation and the graphical lasso algorithm for matrix solution. Section 3.2 elaborates on the Affinity Propagation Clustering method for grouping producers based on production characteristics, with inter-well connectivity derived from the responsibility matrix. Section 3.3 presents the specific calculation formulas for the inter-well connectivity between injectors and producers. Section 4 provides an application analysis of the method, utilizing 20 years of production dynamics from a specific oilfield to compute the clustering and inter-well connectivity of 15 wells in the southern region of the block. Comparisons with actual tracer results indicate that the clustering algorithm effectively identifies wells with similar geological properties across faults, with calculated inter-well connectivity closely matching field tracer test outcomes.

2. Background

The oil field tracer test refers to the injection of tracer mixed displacement fluid into the oil reservoir, which can track the movement of fluid and get the movement direction and speed of the fluid in the reservoir. The tracer test can study the reservoir flow performance and reservoir properties, to guide the design of gas drive and water drive production systems. It can be divided into inter-well tracers and frac tracers by use. The connectivity verification data used in this paper are from the results of inter-well tracer tests performed on multiple injectors in a block in the last 10 years. The engineer implemented tracer monitoring in all producers in the test area, and collected data such as the time, well distance, speed, peak concentration, peak width, peak number, and current concentration of the tracer. Through the interpretation and analysis of the production of the tracer, the flow of the tracer fluid in the well can be revealed, and the connectivity between different wells in the well group can be judged. Oilfield tracer interpretation technology refers to the processing and analysis of fluid samples taken from the wellhead, usually requiring professional personnel and equipment to perform and analyze the data, combined with geological models and numerical simulation methods, and then reveal the reservoir information contained in the analysis. At present, tracer interpretation methods mainly include four categories, which are the numerical simulation method, analytical method, semi-analytical method, and comprehensive interpretation method [36,37,38]. The flow of connectivity is shown in Figure 1 [39]. In this paper, the obtained tracer test data, such as tracer concentration, tracer time, tracer speed, and peak tracer concentration, are compared with the calculation results of inter-well connectivity, to determine whether the connectivity is consistent with the actual well pattern tracer flow law. Combined with the tracer test results, we verify that the wells with strong connectivity identified by the algorithm also show strong connectivity in actual reservoir production. All the code programs in this work are written in Python Spyder (4.1.5) and algorithms are implemented based on and Scikit-learn (0.19.2) [40].

3. Methodology

In this paper, sparse inverse covariance estimation [41] is used to pretreat the daily fluid rate and daily water injection rate in production dynamics, aiming to mine information from data fluctuations and eliminate the interference of data incompleteness in the results. On this basis, cluster operation is carried out on the results of sparse inverse covariance estimation of daily fluid rate from producers, and the correlation between producers is calculated according to the similarity and difference of production characteristics. Cluster families are divided according to correlation size and the central wells and subordinate wells with similar production characteristics are obtained. Finally, the water injection rate of a certain injector is added to the matrix of sparse inverse covariance estimation, and its inter-well connectivity with other producers is obtained through responsibility matrix calculation., and the overall algorithm flow is shown in Figure 2.

Specifically, to obtain a graph structure between the production dynamics of different wells, in this paper, the graphical lasso algorithm [41] is used to solve the matrix consisting of daily fluid flow data of producers. The covariance matrix and sparse inverse covariance matrix of daily fluid production are obtained, and the interpretation and analysis of the two matrices can know which other production dynamics of each well are affected, and the degree of the impact. Then, the covariance matrix results of producers are clustered through affinity propagation clustering [42]. Producers that are in the same cluster family have similar impacts on the reservoir level, which reflects the similar geological characteristics and the impact of the production systems of these wells. Finally, a specific injector is added to process the responsibility matrix in the clustering algorithm. The degree of connectivity between the injector and other producers can be calculated. Figure 3 demonstrates the process of solving reservoir production data into an inverse covariance matrix and then visualizing inter-well connectivity [43].

3.1. Sparse Inverse Covariance Estimation

Sparsity in sparse inverse covariance means that most of the elements in the inverse covariance matrix are zero, which represents many conditional independent relationships between variables. By introducing sparsity, the model can be simplified, the stability of interpretation and estimation can be improved, and the computational complexity can be effectively reduced in high-dimensional data.

In high-dimensional data, sparsity can significantly reduce the complexity of the model because only a few non-zero elements need to be estimated and stored. This not only reduces computational costs but also reduces the risk of overfitting. By excluding unimportant relationships between variables, sparse inverse covariance estimation can better capture the main structure in the data and help to improve the stability and interpretability of estimates.

Sparse inverse covariance estimation is a method to learn the fluctuations of point connections in images, and the graphical lasso algorithm [41] is an accurate sparse inverse covariance matrix for estimating multivariate normal distribution, which is used to accurately estimate the correlation between multidimensional data. The algorithm has been shown to have high applicability against sparse or incomplete data [44]. Its advantage is that it has a very high accuracy, can accurately estimate the correlation between variables, and is suitable for large-scale and high-dimensional data sets. This paper studies the graphical structure of production behavior between different wells in a real oil field, and learns covariance and sparse accuracy from dynamic production history.

The first step of the algorithm assumes that the $p$ dimensional random variable $x$ follows a Gaussian distribution: its mean is $\mu$, its covariance matrix is $\Sigma$, and its probability density $f(x,\mu ,\Sigma )$ can be written as follows [41]:

$$f(x,\mu ,\Sigma )={(2\pi )}^{-p/2}\det {(\Sigma )}^{-1/2}\exp \left[-{\displaystyle \frac{1}{2}}{(x-\mu )}^T{\Sigma }^{-1}(x-\mu )\right]$$

(1)

where T is the symbol for the transposed matrix.

If there are m observed samples, the logarithmic likelihood function can be written as follows:

$$\log L(\mu ,\Sigma )=C+{\displaystyle \frac{m}{2}}\log \det ({\Sigma }^{-1})-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^m{\left(x^{(i)}-\mu \right)}^{\mathrm{T}}}{\Sigma }^{-1}\left(x^{(i)}-\mu \right)$$

(2)

In Equation (2), C represents a constant term that is independent of the parameters of the data and the model. This constant term does not affect the estimation of the parameter when maximizing the log-likelihood function because it does not depend on $\mu$ and $\Sigma$. Therefore, the constant term C can be ignored when maximizing the logarithmic likelihood function. When $\log L(\mu ,\Sigma )$ reaches its maximum, the optimal covariance matrix or precision matrix can be obtained. In the oil field production history, dynamic data are generally sparse, so L1-Penalty $\lambda {\displaystyle \sum _{j\ne k}\left|{{\Sigma }^{-1}}_{jk}\right|}$ is added to enhance the sparsity of the precision matrix, and the precision matrix to be solved is written as follows [41]:

$${\widehat{\Sigma }}^{-1}=\underset{{\Sigma }^{-1}\ge 0}{\mathrm{argmin}}\left({\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{\left(x^{(i)}-\mu \right)}^{\mathrm{T}}}{\Sigma }^{-1}\left(x^{(i)}-\mu \right)-\log \det ({\Sigma }^{-1})+\lambda {\displaystyle \sum _{j\ne k}\left|{{\Sigma }^{-1}}_{jk}\right|}\right)$$

(3)

If the two samples are conditionally independent, then the corresponding precision matrix coefficient is 0. In this case, ${\Sigma }^{-1}$ can be a positive semi-definite matrix. To find a sparse estimate of the inverse covariance matrix, the objective function can be simplified.

Make

$$S={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\left(x^{(i)}-\mu \right)}{\left(x^{(i)}-\mu \right)}^T$$

(4)

Then

$${\widehat{\Sigma }}^{-1}=\underset{{\Sigma }^{-1}\ge 0}{\mathrm{argmin}}\left(\mathrm{tr}(S{\Sigma }^{-1})-\log \det {\Sigma }^{-1}+\lambda {\displaystyle \sum _{j\ne k}\left|{{\Sigma }^{-1}}_{jk}\right|}\right), s.t. S\ge 0$$

(5)

In Equation (5), $S$ is the sample covariance; ${\Sigma }^{-1}$ is the inverse covariance matrix; $\mathrm{tr}(S{\Sigma }^{-1})$ is the trajectory of the product of sample covariance matrix and inverse covariance matrix; $\log \det {\Sigma }^{-1}$ is the logarithm of the determinant of the inverse covariance matrix, which guarantees the positive quality of the inverse covariance matrix; $\lambda {\displaystyle \sum _{j\ne k}\left|{{\Sigma }^{-1}}_{jk}\right|}$ is the L1 regularization term, which allows the sparsity of non-diagonal elements in the inverse covariance matrix; $\lambda$ is the parameter that controls the sparsity, and a larger $\lambda$ value results in more elements being estimated to be zero, resulting in a more sparse matrix. And ${\Sigma }^{-1}\ge 0$ guarantees that the inverse covariance matrix is positive semidefinite.

In this paper, the graphical lasso algorithm [41] is used to solve the objective function (5). The algorithm can effectively estimate the sparse inverse covariance matrix by combining the elements of the sample covariance matrix to ensure positive definiteness. By introducing the L1 regularization term, the graphical lasso enhances the sparsity to simplify and improve the model’s interpretability and estimation stability. The model can effectively reduce the computational complexity in the long production of dynamic data.

3.2. Affinity Propagation Clustering

Clustering is a method of dividing a data set into groups so that the data points in the same group are similar to each other, while the data points in different groups are quite different. Cluster analysis can help identify different types of production dynamics, classify producers with similar production characteristics, provide reference for production program design, and provide new geological understanding. More precise production control strategies can be developed for each type of well in the field to improve overall production efficiency. Common clustering methods include K-means clustering [45], hierarchical clustering [46], density clustering [47], spectral clustering [48], and Affinity Propagation (AP) clustering [42]. In this paper, AP clustering is used to cluster samples of sparse inverse covariance matrix. The reason is that compared with other clustering methods that need to specify the number of clusters in advance, affinity propagation clustering has the advantage that it does not need to specify the number of clusters in advance, but can automatically select the number and center of clusters by relying on algorithms to find the structure between data, which is in line with the actual demand of inter-well clustering in oil fields.

The basic idea of the AP clustering algorithm is to pass information between samples that will reflect the suitability of the samples as clustering centers. The information exchanged between the input date samples falls into two categories [49]: the availability matrix $A\left(i,k\right)$ and the responsibility matrix $R\left(i,k\right)$, which are calculated from the similarity matrix $S\left(i,k\right)$. The similarity matrix $S\left(i,k\right)$ represents the similarity between points i and k, and its value indicates the degree to which point k is suitable as the clustering center of point i, and the larger the value, the more suitable it is. The non-diagonal elements in the similarity matrix $S\left(i,k\right)(i\ne k)$ represent the negative distance between point i and point k, and the larger the value, the more similar the two points are. The diagonal element in the similarity matrix $S\left(k,k\right)$ represents the degree to which point k is suitable as the clustering center. When initialized, $S\left(k,k\right)$ is set by default to the median or minimum value in the similarity matrix except for diagonal elements. During the clustering process of oil field producers, the responsibility matrix $R\left(i,k\right)$ represents the degree to which producer k fits as the cluster center of producer i, and the availability matrix $A\left(i,k\right)$ represents the degree to which producer k is suitable for being the cluster center of points other than point i, as shown in Figure 4.

The calculation of the responsibility matrix is shown in Equation (6), which shows that the dynamic data of any producer as a cluster center candidate can affect other candidate producers, allowing the production dynamics of all wells to participate in the ownership of the target producer [42]:

$$R\left(i,k\right)\leftarrow S\left(i,k\right)-\underset{k^{\prime }\ne k}{\max }\left\{A\left(i,k^{\prime }\right)+S\left(i,k^{\prime }\right)\right\}$$

(6)

Specifically, the iterative update of the responsibility matrix $r\left(i,k\right)$ can be written as follows [42]:

$$R_{t+1}(i,k)=\left\{\begin{array}{l}S(i,k)-\underset{k^{\prime }\ne k}{\max }\left\{A_t(i,k^{\prime })+R_t(i,k^{\prime })\right\},i\ne k\hfill \\ S(i,k)-\underset{k^{\prime }\ne k}{\max }\{S(i,k^{\prime })\},i=k\hfill \end{array}\right.$$

(7)

The availability matrix $A(i,k)$ is calculated as shown in Equation (8), which means that the availability matrix $A(i,k)$ is equal to the self-responsibility or ‘self-appropriateness’ $R\left(k,k\right)$ of well k, which measures how suitable well k is to serve as its own exemplar, plus the positive responsibility obtained from other producers $i^{\prime }$, i.e., $R\left(i^{\prime },k\right)>0$. Only responsibility $R\left(i^{\prime },k\right)$ greater than 0 is added here because only a positive responsibility will support well k as the cluster center. If the self-responsibility $R(k,k)$ of producer k is negative, it means that the producer is more suitable for other producers than as a cluster center.

$$A(i,k)\leftarrow \min \left\{0,R(k,k)+{\displaystyle \sum _{ i^{\prime }\notin \{i,k\}}\max }\left\{0,R\left(i^{\prime },k\right)\right\}\right\}$$

(8)

The diagonal element $A(k,k)$ of the availability matrix is the self-availability of a well k, which is equal to the sum of the positive liability degrees $R\left(i^{\prime },k\right)$ obtained from other producers $i^{\prime }$:

$$A(k,k)\leftarrow {\displaystyle \sum _{i^{\prime }\ne k}\max }\left\{0,R\left(i^{\prime },k\right)\right\}$$

(9)

$$A_{t+1}(i,k)=\left\{\begin{array}{l}\min \left\{0,R_{t+1}(k,k)+{\displaystyle \sum _{i^{\prime }\ne i,k}\max }\left\{R_{t+1}(i^{\prime },k),0\right\}\right\},i\ne k\hfill \\ {\displaystyle \sum _{i^{\prime }\ne k}\max }\left\{R_{t+1}(i^{\prime },k),0\right\},i=k\hfill \end{array}\right.$$

(10)

The iterative formulas for the responsibility matrix $R\left(i,k\right)$ and the availability matrix $A(i,k)$ can be written as follows:

$$R_{t+1}(i,k)=\lambda \times R_t(i,k)+(1-\lambda )\times R_{t+1}(i,k)$$

(11)

$$A_{t+1}(i,k)=\lambda \times A_t(i,k)+(1-\lambda )\times A_{t+1}(i,k)$$

(12)

In the above equation, $\lambda$ is the update weight set as 0.5 in this work. At the end, the decision matrix $E(i, k)$ is calculated as Equation (13), which is used to determine whether to end the algorithm, and the corresponding elements greater than zero in the matrix $E(i, k)$ are listed as the current clustering center.

$$E(i, k)=R(i, k)+A(i, k)$$

(13)

Iterative Formulas (11) and (12) continuously update the matrix $R\left(i,k\right)$, $A(i,k)$ and $E(i, k)$, and if the cluster center does not change for several consecutive times, the cluster ends. If the number of iterations of the loop exceeds the maximum number of iterations, which is set to 200 in this paper, the clustering ends and will be displayed as clustering failure. According to the final decision matrix $E(i, k)$, the cluster attribution of each sample is determined.

3.3. Rapid Evaluation of Injection-Producer Connectivity

In order to quickly evaluate the inter-well connectivity between an injector and each producer in the pattern, the injection rate of target injector can be included in the calculation of the inverse covariance of production history. The responsibility matrix $r\left(i,k\right)$ can be obtained through the calculation in Section 3.2, for the target injector $k_{inj}$ which has a column vector:

$$\overrightarrow{X}=\left\{T,x_1,x_2,\dots ,x_i\right\}$$

(14)

In Equation (14), T is the value of the self-responsibility of the target injector $k_{inj}$, i.e., $T=R\left(k_{inj},k_{inj}\right)$; $x_i$ is the responsibility value between target injector $k_{inj}$ and producer $i$, i.e., $x_i=R\left(i,k_{inj}\right)$. $x_i$ indicates the degree of trend of fluid injected from injector $k_{inj}$ flowing to producer i and can be considered as the inter-connectivity value to be processed. If the responsibility value $x_i$ of producer $i$ is less than T, then it indicates that fluid from injector $k_{inj}$ will not flow to producer i, so there is no connectivity between them. For $x_i$ larger than the value T of the responsibility matrix, use the quadratic power to weight each value. The specific calculation steps of the evaluation of the connectivity between injector and producers are as follows:

(1) $x_i=R\left(i,k\right)$ represents the tendency of injection from $k_{inj}$ to flow to producer $i$. If $x_i$ is less than or equal to T, it means that the target injector does not have inter-well connectivity to producer i, so all less than T is set to zero:

$$x_i^{\prime }=\left\{\begin{array}{l}{x}_i \mathrm{if} x_i>T \\ 0 \mathrm{if} x_i\le T \end{array}\right.$$

(15)

(2) Calculate the square of the difference from T for each $x_i^{\prime }$ to get the connectivity squared value $y_i$:

$$y_i={\left(x_i^{\prime }-T\right)}^2$$

(16)

(3) In order to meet the oil field practice, the inter-well connectivity $C_i$ of injector $k_{inj}$ to each producer i is calculated as follows:

$$C_i={\displaystyle \frac{y_i}{{\displaystyle \sum _{j=1}^n{y}_j}}}$$

(17)

In the above formula, ${\displaystyle \sum _{j=1}^n{y}_j}$ is the sum of all the squared values of connectivity, with the aim of ensuring that the sum of all $C_i$ is 1.

4. Results and Discussion

4.1. Tracer Testing

The stratigraphic characteristics of the test area are in the Volgian and Kimmeridgian stages of the Mesozoic Jurassic period, and the sedimentary characteristics are deep-water turbidite sandstone reservoirs, mainly Subarkose sandstone, with high maturity. The reservoir buried depth is 2332–3064 m, the reservoir is a low-amplitude monoclinic structure, the faults are relatively developed, the plane distribution is different, the faults become complicated from south to north, and the trend is mainly east to west. The reservoir in this experimental area is of normal-temperature and high-pressure type, stratified, lithologic, edge water, and structural reservoir, and the formation pressure and saturation pressure differ greatly. The porosity is 15–30% with an average of 25%, the permeability is 100–18,000 mD with an average of 500–1500 mD, the initial oil saturation is 0.91 on average, the crude oil density is 0.772 to 0.778 at saturation pressure, and the oil viscosity is 1.40–1.67 cP at 4600 PSIG. The target reservoir has low sulfur content and the formation water type is CaCl₂.

The tracer test was conducted in the southern part of the test area, as shown below the black line in Figure 5. This area has good lateral reservoir connectivity, perfect well pattern, good injection-production correspondence, and a relatively uniform water drive sweep. The production history of this block has a long time span and there are long-term shut-ins and production data defaults, which is in line with the problem that the algorithm aims to solve. A large number of tracer tests have been carried out in this area in the past decade, and the actual inter-well connectivity results are supported by comparative verification. Therefore, this paper takes the southern area of the test area as an example to verify the algorithm.

4.2. Results of Reservoir Inter-Well Clustering

In order to verify that the proposed algorithm can still work effectively when processing missing data or data sets with limited sample size, the daily fluid rate of 14 producers in the southern part of the test area is now imported into the model, and the specific daily fluid rate data are shown in Figure 6. It can be intuitively seen that under the overall production time span of the reservoir, a large number of producers have zero daily fluid rate due to shut-in or adjustment of the production system.

The covariance matrix is an effective tool for analyzing the overall linear correlation between the daily fluid rates of producers and facilitates the identification of direct relationships between wells. Figure 7a illustrates the heat map of the covariance matrix, calculated based on the daily fluid rates of the producers. The color of the diagonal elements in the covariance matrix heat map represents the variance in the daily fluid rates of each well, reflecting the degree of fluctuation in daily output. The brighter the color of the grid, the greater the variance in daily fluid rates, indicating a higher degree of fluctuation. As depicted in Figure 7a, the diagonal elements of all 14 producers exhibit bright colors, indicating that the variance in daily fluid rates is relatively high for each well. The color of the non-diagonal elements represents the covariance values between producers; a brighter color indicates a higher covariance value, suggesting a strong positive correlation between the daily fluid rates of those wells. Conversely, darker colors indicate lower or negative covariance values, reflecting a weak or negative correlation, or no significant relationship. Producers 0, 4–8, 10, 12, and 13 exhibit several brightly colored non-diagonal elements, suggesting that their daily fluid rates are strongly positively correlated with those of other producers. In other words, an increase in production from one well is likely to be accompanied by an increase in production from another. By contrast, the non-diagonal elements for producers 1 and 9 are darker, indicating little to no correlation with the daily fluid rates of other wells.

The heat map of the sparse inverse covariance matrix of fluid production rates is shown in Figure 7b. Brighter non-diagonal elements indicate stronger conditional correlations between two producers, meaning that even when accounting for the influence of other wells, their daily production rates remain significantly correlated. A brighter color denotes stronger conditional correlation, while a darker color suggests conditional independence or near-independence. Wells with strong conditional correlations in the matrix should be considered together when optimizing production strategies, whereas wells with darker colors should be treated as independent, reflecting distinct geological or production characteristics. For instance, wells 0–4 show strong conditional correlations with wells 7–13, whereas the latter group exhibits darker colors among themselves, indicating weaker correlations.

Following the application of affinity propagation clustering, the covariance matrix and sparse inverse covariance matrix are reorganized according to the clustering results, aligning elements within the same cluster for proximity, thus facilitating the identification of clustering structures and correlation patterns in the heat maps. Figure 8 presents the heat maps for both matrices. The figure reflects only the clustering outcome; the numerical values indicate sample counts and do not correspond to the producer numbers prior to sorting. As shown in Figure 8a, distinct block structures emerge in the covariance matrix heat map after clustering, corresponding to producers within the same cluster. Bright-colored blocks indicate a high degree of correlation among producers in the same cluster, whereas darker blocks denote low correlation between producers across clusters. The bright diagonal blocks suggest strong intra-cluster correlations, potentially attributable to shared geological conditions or similar production strategies. Conversely, dark blocks indicate weak or negligible correlations between different producers. In Figure 8b, the sparse inverse covariance matrix is likewise reorganized. Bright blocks indicate that, even after controlling for the effects of other producers, significant conditional correlations persist among producers within the same cluster.

Affinity propagation clustering imposes a clear block structure on the covariance matrix and sparse inverse covariance matrix, making the correlations and conditional independencies between producers more visually distinct and intuitive. This facilitates easier analysis of intra-cluster correlations and inter-cluster conditional independencies. Such clustering and visualization methods enable field managers to better understand the relationships between producers, thereby optimizing field management strategies.

The result of the decision matrix $E(i, k)$ after clustering is shown in

Table 1. Clustering result of decision matrix.

	B2	B3	B7	B8	B10	B12	B13	B17	B22	B23Z	B25	B30Y	B31	B38
B2	−0.214	−0.345	−0.214	−0.268	0.204 1	−0.344	−0.204	−0.514	−0.519	−0.345	−0.529	−0.381	−0.268	−0.530
B3	−0.308	0.032	−0.033	−0.267	−0.054	−0.303	−0.227	−0.450	−0.453	−0.177	−0.470	−0.223	−0.177	−0.443
B7	−0.213	−0.033	−0.018	−0.252	0.018	−0.228	−0.092	−0.308	−0.307	−0.192	−0.344	−0.220	−0.040	−0.316
B8	−0.231	−0.267	−0.252	−0.068	0.068	−0.231	−0.249	−0.426	−0.353	−0.173	−0.374	−0.254	−0.162	−0.508
B10	−0.278	−0.370	−0.316	−0.316	0.276	−0.274	−0.324	−0.544	−0.536	−0.399	−0.517	−0.346	−0.334	−0.629
B12	−0.348	−0.344	−0.234	−0.273	0.028	−0.228	−0.188	−0.283	−0.281	−0.406	−0.321	−0.355	−0.028	−0.449
B13	−0.242	−0.302	−0.092	−0.324	−0.083	−0.228	−0.092	−0.283	−0.275	−0.250	−0.334	−0.319	0.083	−0.266
B17	−0.542	−0.516	−0.374	−0.492	−0.293	−0.255	−0.260	−0.283	−0.328	−0.495	−0.364	−0.436	0.254	−0.472
B22	−0.562	−0.533	−0.387	−0.433	−0.300	−0.320	−0.247	−0.343	−0.275	−0.441	−0.308	−0.436	0.247	−0.466
B23Z	−0.308	−0.177	−0.192	−0.173	−0.083	−0.365	−0.175	−0.430	−0.361	0.083	−0.331	−0.226	−0.163	−0.443
B25	−0.492	−0.470	−0.344	−0.374	−0.201	−0.280	−0.259	−0.299	−0.275	−0.331	−0.293	−0.345	0.201	−0.460
B30Y	−0.344	−0.223	−0.220	−0.254	−0.030	−0.314	−0.244	−0.371	−0.355	−0.226	−0.345	−0.046	0.030	−0.303
B31	−1.220	−1.166	−1.029	−1.150	−1.007	−0.975	−0.914	−0.923	−0.908	−1.152	−0.989	−0.989	0.921	−0.977
B38	−0.505	−0.455	−0.328	−0.520	−0.325	−0.420	−0.191	−0.419	−0.397	−0.455	−0.472	−0.315	0.191	−0.266

¹ The underscore bold indicates that well k is the cluster center of well i.

, where the corresponding element greater than zero in the matrix $E(i, k)$ is listed as the current clustering center. It can be seen that for each producer $i$ row, after successful iteration, there will only be one element greater than zero in each row, and the corresponding column of the element is the cluster center well $k$ of the current producer $i$. It can be seen from the results that the producers in the cluster center are B3, B10, B23Z, and B31.

The clustering results obtained according to decision matrix $E(i, k)$ are shown in

Table 2. Clustering result: wells group and center.

Cluster Group	Well Number	Cluster Center
1	B3	B3
2	B2, B7, B8, B10, B12	B10
3	B23Z	B23Z
4	B13, B17, B22, B25, B30Y, B31, B38	B31

. Producer B3 and B23Z exist alone as the cluster center of Cluster 1 and 3; they have no correlation with other producers and the reason for this will be discussed.

Based on the analysis of geological conditions in the southern oilfield, the possible reason is that there are undiscovered faults near B3, resulting in poor connectivity between B3 and all other injectors in the southern oilfield. This explanation can also be proved by the tracer analysis results below. For example, B3 and B7 are at the same distance as the injection of the S2 well, but the tracers injected at S2 do not reach B3 in large quantities; while the tracers injected at S1 and S3 do not result in tracers in B3, but can cross B3 and result in tracers in the B2 well, which is farther away from them. Under actual geological conditions, there are hyperpermeability bands in the formation, which make it difficult for injection fluids from injectors S2 and S3 to spread to areas near B3. As wells with similar production dynamics, B2, B7, B8, B10, and B12 are classified into Cluster 2. Indeed, they have similar properties in geological conditions, such as good connectivity in the horizontal and vertical reservoirs, complete well patterns, good injection-production correspondence, and relatively uniform water drive spread among them. In the actual management of the oilfield, these wells are also uniformly planned and designed for production. B23Z was individually clustered into Cluster 3. Subsequent tracer analysis of this well showed that it was only affected by S1, which was farthest away from it but not affected by S2, S3, and S4, which were closer to it. It shows different production characteristics from other producers, indicating that B23Z only had a dominant seepage channel with S1, but had poor connectivity with other injectors. The seven wells far away from the injectors are clustered into cluster group 4, with B31 as the cluster center. These wells have the same characteristics of low water drive sweep degree, relatively low formation pressure maintenance level, and insufficient formation pressure maintenance in some wells. In the actual production of oil fields, these wells show the characteristics of unidirectional displacement, low recovery degree, and low water content. To sum up, as shown in Figure 9, clustering can cluster all related wells together very intuitively, and the clustering results can reflect the actual geological characteristics of the reservoir, and can intuitively and appropriately show the differences and similarities between producers in geological structure, reservoir characteristics and fluid physical properties.

4.3. Evaluation of Inter-Well Connectivity Results

In the test block, the chemical tracer test of injectors S1, S2, S3, and S4 has been carried out for a long time, which results in a high reference value. Now, the inter-well connectivity value calculated by the algorithm is compared with the tracer experimental results.

The inter-well connectivity results of injector S1 and the field tracer flow are shown in Figure 10. On the whole, it can be seen that the calculated connectivity is consistent with the actual field tracer flow.

The inter-well connectivity results of injector well S1 to other producers were compared with field tracer tests, as shown in

Table 3. Comparison of inter-well connectivity with tracer test results of injector S1.

Well Number $\mathit{i}$	Responsibility $\mathit{r}\left(\mathit{i},{\mathit{k}}_{\mathbf{S}1}\right)$	Connectivity ${\mathit{C}}_{\mathit{i}}$	Time of Tracer Arrival d	Well Distance m	Tracer Access Rate m/d	Tracer Peak Concentration ppb	Tracer Peak Width d	Tracer Current Concentration ppb
S1	−0.16806	-	-	-	-	-	-	-
B2	−0.01502	0.367	1302 (3.6 years)	3314	2.55	200	27	0.91
B3	−0.23451	-	-	-	-	-	-	-
B7	−0.21688	-	-	-	-	-	-	-
B8	−0.24073	-	-	-	-	-	-	-
B10	−0.05243	0.210	1610 (4.4 years)	3280	2.04	80	38	0.81
B12	−0.10518	0.062	2603 (7.1 years)	3275	1.26	44	116	7.75
B13	−0.21440	-	-
B17	−0.27691	-	2781 (7.6 years)	3836	1.38	33	57	1.58
B22	−0.22857	-	-	-	-	-	-	-
B23Z	−0.07011	0.150	1724 (4.7 years)	3639	2.11	25	66	16
B25	−0.11096	0.051	2716 (7.4 years)	3679	1.35	30	587	1.73
B30Y	−0.08618	0.105	2520 (6.9 years)	2454	0.97	27	72	0.36
B31	−1.38504	-	-	-	-	-	-	-
B38	−0.10935	0.054	3007 (8.2 years)	2994	1.00	23	109	1.66

. The last six columns list the tracer test results in the target reservoir block, where “Time of tracer arrival” refers to the time required for the tracer to travel from the injector to the monitoring producers, which indirectly reflects the fluid flow capacity within the reservoir, measured in days (d); “Tracer peak concentration” represents the concentration of the tracer at its maximum value in the monitoring well, and is used to assess the degree of fluid mixing and the strength of inter-well connectivity, measured in parts per billion (ppb); ”Tracer peak width” refers to the width of the tracer concentration curve at its peak, typically corresponding to the time interval over which the concentration drops to a certain percentage of its maximum value, expressed in days (d); ”Tracer current concentration” denotes the real-time measured concentration of the tracer in the monitoring well, providing immediate feedback on fluid flow and assisting in reservoir dynamic monitoring, also measured in ppb. In the table, the second column $r\left(i,k_{\mathrm{S}1}\right)$ is the column vector of injector S1 in the responsibility matrix $r\left(i,k\right)$, the value of which is the degree of responsibility of injector S1 and each corresponding well; the third column $C_i$ is the calculation result of the connectivity value of S1 and each well. The algorithm calculated the connectivity of 14 producers to injector S1, identifying seven wells with connectivity to S1. All seven of these wells had detected tracers in the tracer test, indicating that the precision rate of the algorithm in determining connectivity to injector S1 was 100% (seven of seven). Compared with the tracer test results, a total of eight producers detected tracers at the wellhead, and the algorithm successfully identified seven of them, resulting in a recall rate of 87.5% (seven of eight) for connectivity between well S1 and the other wells.

The top three producers, B2, B10, and B23Z, were all effectively identified and accurately quantified by the algorithm. Among them, well B2 had the shortest time of agent detection, the fastest rate of agent detection, and the largest peak concentration. The calculated connectivity $C_i$ of well B2 was 0.367, which was consistent with the results. The algorithm calculated that the wells with good connectivity but inferior to B2 were B10 and B23Z, and their connectivity $C_i$ was 0.210 and 0.150, respectively. In the actual results, wells B10 and B23Z also showed a faster tracer access rate and a higher peak concentration.

The inter-well connectivity results of injector S2 and the field tracer flow direction are shown in Figure 11. On the whole, it can be seen that the calculated connectivity is consistent with the actual field tracer flow direction. There were four wells in which the tracer actually arrived as shown in Figure 11a, and nine wells in which the algorithm identified connectivity as shown in Figure 11b.

The inter-well connectivity of injector S2 was quantitatively compared with the tracer test, as shown in

Table 4. Comparison of inter-well connectivity with tracer test results of injector S2.

Well Number $\mathit{i}$	Responsibility $\mathit{r}\left(\mathit{i},{\mathit{k}}_{\mathbf{S}2}\right)$	Connectivity ${\mathit{C}}_{\mathit{i}}$	Time of Tracer Arrival d	Well Distance m	Tracer Access Rate m/d	Tracer Peak Concentration ppb	Tracer Peak Width d	Tracer Current Concentration ppb
S2	−0.26029	-	-	-	-	-	-	-
B2	−0.22552	0.003	-	-	-	-	-	-
B3	−0.39248		-	-	-	-	-	-
B7	−0.08248	0.400	1895 (5.2 years)	1558	0.82	11	1144	2.88
B8	−0.15862	0.075	-	-	-	-	-	-
B10	−0.21212	0.008	-	-	-	-	-	-
B12	−0.20017	0.015	4802 (13.2 years)	2688	0.56	6.61	59	0.31
B13	−0.10191	0.283	1288 (3.5 years)	2341	1.82	41	76	1.46
B17	−0.32331	-	-	-	-	-	-	-
B22	−0.33142	-	-	-	-	-	-	-
B23Z	−0.42384	-	-	-	-	-	-	-
B25	−0.30059	-	-	-	-	-	-	-
B30Y	−0.11645	0.212	3506 (9.6 years)	2006	0.57	5.6	79	0.66
B31	−0.76099	-	-	-	-	-	-	-
B38	−0.21986	0.005	-	-	-	-	-	-

. In comparison with the tracer test results, the algorithm identified connectivity between eight producers and injector S2, while tracers were detected in four of these wells, yielding a precision rate of 50% (four out of eight). Additionally, tracer tests detected tracers in four producers, all of which were correctly identified by the algorithm, resulting in a recall rate of 100% (four out of four).

Although the algorithm identified additional wells with connectivity, the wells B7, B12, B13, and B30Y that actually saw the tracer had high connectivity in the algorithm results. The values were 0.4, 0.015, 0.283, and 0.212, respectively, accounting for 91% of the total connectivity, while the other four wells with additional identification had low connectivity values, accounting for only 9% of the total, which did not affect the overall judgment. In addition, wells B7 and B13 with the highest agent velocity and peak concentration obtained better connectivity results. Therefore, it is thought that the algorithm and the tracer results are still consistent.

The inter-well connectivity results of injector S3 and the tracer flow direction of the field are shown in Figure 12. There are six wells in which the tracer injected at S3 arrived, and five of the wells with connectivity were identified by the algorithm, while the connectivity between wells S3 and B17 was not recognized, and the connectivity between S3 and B8 was additionally recognized.

The inter-well connectivity of injector S3 was quantitatively compared to the tracer test, as shown in

Table 5. Comparison of inter-well connectivity with tracer test results of injector S3.

Well Number $\mathit{i}$	Responsibility $\mathit{r}\left(\mathit{i},{\mathit{k}}_{\mathbf{S}3}\right)$	Connectivity ${\mathit{C}}_{\mathit{i}}$	Time of Tracer Arrival d	Well Distance m	Tracer Access Rate m/d	Tracer Peak Concentration ppb	Tracer Peak Width d	Tracer Current Concentration ppb
S3	−0.12755	-	-	-	-	-	-	-
B2	−0.07960	0.368	1118 (3.1 years)	2898	2.59	11	44	4.3
B3	−0.19022	-	-	-	-	-	-	-
B7	−0.09385	0.182	5533 (15.2 years)	2378	0.43	4.03	195	4.03
B8	−0.11838	0.013	-	-	-	-	-	-
B10	−0.09902	0.130	3779 (10.4 years)	2878	0.76	12	24	2.33
B12	−0.27936	-	-	-	-	-	-	-
B13	−0.09169	0.206	1520 (4.2 years)	3288	2.16	330	23	0.22
B17	−0.26696	-	5078 (13.9 years)	3415	0.67	10.85	144	10.85
B22	−0.27268	-	-	-	-	-	-	-
B23Z	−0.12762	-	-	-	-	-	-	-
B25	−0.10245	0.101	3880 (10.6 years)	3291	0.85	20.79	181	3.42
B30Y	−0.24667	-	-	-	-	-	-	-
B31	−0.32515	-	-	-	-	-	-	-
B38	−0.23590	-	-	-	-	-	-	-

. The algorithm has identified six producers that have connectivity to S3, while tracers were detected in five of these wells, yielding a precision rate of 83.33% (five out of six). Additionally, tracer tests detected tracers in six producers, five of which were correctly identified by the algorithm, resulting in a recall rate of 83.33% (five out of six).

B17 was not identified by the algorithm. The authors believe the reason that B17 was not identified is that B17 took a long time (13.9 years) to detect the tracer from injector S3 which indicates that the effect of S3 on B17 has only recently become effective, and the historical data do not show the effect of S3 on B17 before the tracer arrival. The production dynamics of B17 over a long-time span before seeing the tracer did not show an obvious positive correlation with S3, which is a normal calculation result. Among the wells with recognized connectivity, B2 and B3 were the two wells with the fastest rate of agent detection, with the rate of agent detection being 2.59 m/d and 2.16 m/d, respectively, obtaining 0.368 and 0.206 connectivity, while the wells with slower agent detection B7, B10, and B25 also obtained lower connectivity. The results are 0.182, 0.130, and 0.101, respectively, which are consistent with the test results.

The inter-well connectivity results of injector S4 and the tracer flow direction of the field are shown in Figure 13. Among them, there are three wells in which the tracer injected by S4 was detected, all of which were identified by the algorithm.

As shown in

Table 6. Comparison of inter-well connectivity with tracer test results of injector S4.

Well Number $\mathit{i}$	Responsibility $\mathit{r}\left(\mathit{i},{\mathit{k}}_{\mathbf{S}4}\right)$	Connectivity ${\mathit{C}}_{\mathit{i}}$	Time of Tracer Arrival d	Well Distance m	Tracer Access Rate m/d	Tracer Peak Concentration ppb	Tracer Peak Width d	Tracer Current Concentration ppb
S4	−0.09306	-	-	-	-	-	-	-
B2	−0.15851	-	-	-	-	-	-	-
B3	−0.13416	-	-	-	-	-	-	-
B7	−0.11343	-	-	-	-	-	-	-
B8	−0.12233	-	-	-	-	-	-	-
B10	−0.11047	0.435	4871 (13.3 years)	3184	0.65	16.44	60	0.13
B12	−0.04682	0.372	4871 (13.3 years)	3244	0.67	6.69	53	2.45
B13	−0.05027	-	-	-	-	-	-	-
B17	−0.12878	-	-	-	-	-	-	-
B22	−0.24865	-	-	-	-	-	-	-
B23Z	−0.23282	-	-	-	-	-	-	-
B25	−0.11565	-	-	-	-	-	-	-
B30Y	−0.22640	0.193	4165 (11.4 years)	2443	0.59	6.21	675	4.29
B31	−0.06222	-	-	-	-	-	-	-
B38	−0.55428	-	-	-	-	-	-	-

, the tracers from S4 were detected after a long time span in wells B10, B12, and B30Y, with tracer arrival times of 13.3, 13.3, and 11.4 years, respectively, and tracer access rates of 0.65, 0.67, and 0.59 m/d, respectively. The peak tracer concentration in well B10 was the highest at 16.44 ppb, while wells B12 and B30Y had similar peak concentrations, ranging between 6 and 7 ppb. The highest connectivity, as calculated by the algorithm, was observed in B10 at 0.435, followed by B12 and B30Y at 0.372 and 0.193, respectively. In the connectivity analysis for injector S4, both precision rate and recall rate of the algorithm are 100% (three of three), demonstrating strong precision and accuracy.

The access rate in the tracer experiment is an important parameter in the analysis of oil field fluid behavior, which can directly reflect the inter-well connectivity. A high access rate indicates that the fluid injected from one injector is able to move quickly and efficiently to the specific producer, meaning a smoother flow path between those two wells. Conversely, a low tracer access rate usually indicates slow fluid mobility between wells, possible flow barriers or reservoir heterogeneity, which suggest poor inter-well connectivity. For the producers identified by the algorithm, a correlation analysis of their inter-well connectivity and tracer access rate was conducted, in which regression line fitting and Pearson correlation coefficient [50] were considered, as shown in Figure 14. The result shows that the Pearson correlation coefficient of S1, S3 and S4 are above 0.8 (0.88, 0.81 and 0.88, respectively), which indicates that there is a strong positive correlation between inter-well connectivity and tracer access rate. Injector S2 has a slightly lower value of 0.4, showing a moderately positive correlation. This allows the conclusion that the correlation between the connectivity calculated by the algorithm and the tracer results is above medium positive correlation. Therefore, the algorithm can largely fulfill the function of tracer tests, providing a convenient and cost-effective method for the quantitative characterization of inter-well connectivity.

5. Conclusions

To enable a rapid and effective identification of inter-well connectivity in cases where there are long-term shut-ins or missing production data in historical dynamics, this paper combines sparse inverse covariance estimation and the affinity propagation clustering algorithm to establish a rapid evaluation method of inter-well connectivity. Combined with the tracer test results, the algorithm was verified and discussed, and the main conclusions were drawn as follows:

(1)

For the problem of data default caused by long-term shut-in and data loss, the proposed algorithm can still ensure stability and accuracy. In the validation of calculation results with the tracer test results for the four injectors, the algorithm achieved a precision rate of 79.17% (19 of 24) and a recall rate of 90.48% (19 of 21).

(2)

In the 56 calculations of inter-well connectivity, the algorithm was consistent with the tracer test results in 51 instances, yielding an accuracy of 91.07%. Furthermore, the erroneous inter-well connectivity values ranged from a maximum of 0.075 to a minimum of only 0.003, making the overall impact on the assessment of injector connectivity acceptable.

(3)

The inter-well connectivity calculated by the algorithm demonstrated a consistent correspondence with the tracer access rate from the tracer test. The Pearson correlation coefficients for the results of the four injectors ranged from 0.4 to 0.88, indicating a moderate to strong positive correlation. Notably, three of the injectors exhibited Pearson correlation coefficients above 0.8, demonstrating a strong positive correlation.

(4)

The algorithm demonstrates stable performance when processing production history data for water flooding and continuous CO₂ flooding. However, it faces limitations when handling production data under water-gas alternating injection patterns. This is primarily due to the presence of three-phase flow involving oil, gas, and water in the production rates. After gas breakthrough in the producers, gas production rate significantly increases, leading to itself and oil production rates being of different magnitudes. Consequently, fluctuations in oil production are masked, resulting in the inability to generate reliable results consistently.