Complex Network Method for Inferring Well Interconnectivity in Hydrocarbon Reservoirs

Abstract

Reservoir management becomes increasingly critical as fields decline to a fully mature state. During this stage, engineers and managers must make decisions based on a limited set of field measurements (such as pressure and production rates). At the same time, up-to-date information concerning the reservoir’s geophysical characteristics and petrochemical properties may be unavailable. To aid in the expert’s appraisal of this production scenario, we present the results of applying a data-driven methodology based on visibility graph analysis (VGA) and multiplex visibility graphs (MVGs). It infers inter-well connectivities at the reservoir level and clarifies the degrees of mutual influence among wells. This parameter-free technique supersedes the limitations of traditional methods, such as the capacitance–resistance (CR) models and inter-well numerical simulation models (INSIMs) that rely heavily on geophysical data and are sensitive to porous datasets. We tested the method with actual data representing a field’s state over 62 years. The technique revealed short- and long-term dependencies between wells when applied to historical records of production rates (oil, water, and gas) and pressures (bottom and wellhead). The inferred connectivity aligned with documented operational trends and successfully identified stable connectivity structures. In addition, the interlayer mutual information (IMI) parameter exceeded 0.75 in most periods, confirming high temporal consistency. Moreover, validation by field experts confirmed that the inferred interconnectivity was consistent with the observed production.

1. Introduction

Oil production in naturally fractured reservoirs involves complex interactions between geological, geophysical, and operational factors. Understanding inter-well connectivity is essential for optimizing recovery rates, enhancing well performance, and improving reservoir management strategies. However, uncovering these connections presents significant challenges due to incomplete data, evolving reservoir conditions, and the coupled dynamics of production and injection wells.

Traditionally, methods like the capacitance–resistance (CR) model have been widely used to estimate inter-well connectivity. Introduced by Bruce (1943), the CR model provided a simplified approach to analyze water drive problems and predict well performance based on injection and production rates [1]. Over time, variations of the CR model have emerged, such as those incorporating network flow simulations to improve prediction accuracy [2,3,4]. Other techniques include inter-well numerical simulation models (INSIMs), which utilize meshless points and conservation laws to simulate fluid flow at multiscale levels, offering insights into water flooding conditions [5,6]. While effective, these methods often require significant input data, such as geophysical properties and flow parameters, which may not always be available in practical settings.

Another interesting approach is the one by Hovadik and Larue (2017), who point out that assessing a reservoir’s development risk involves geological and engineering components [7]. Reservoir characterization requires reliable estimates of various parameters and rock properties, including porosities and permeabilities. Presently, we contend that such estimates may not necessarily be up-to-date or available in some cases. On the other hand, since the “engineering” aspects of assessments focus on a reservoir’s flow conditions, up-to-date measurements may be readily available and easily collected. Reservoir connectivity modeling is further complicated by the finite size scaling and geostatistical nature of the length scales involved [7]. The lack of a time component in a static characterization may also limit a fuller description. Hence, we work around these modeling issues by assuming that production and dynamic performance logs contain information on the a priori unknown connections. However, our results are still global representations of two possible states: connected and unconnected.

Advances in data-driven methodologies have transformed the analysis of reservoir connectivity. Machine learning techniques, such as artificial neural networks (ANNs), support vector regression (SVR), and recurrent neural networks (RNNs), have shown success in predicting production rates and optimizing recovery strategies [8,9,10]. For instance, Ibrahim et al. (2021) demonstrated the use of hybrid models combining ANNs and SVR to predict flow rates in unconventional wells [9]. Furthermore, spatiotemporal models, including graph neural networks (GNNs) and hybrid frameworks like convolutional graph networks (CGNs) combined with LSTMs, have been employed to capture spatial and temporal dependencies in complex reservoir systems [11,12]. These approaches leverage production and injection data to uncover patterns that inform operational decisions, such as well placement and injection optimization.

The visibility graph analysis (VGA) method has emerged as a powerful tool for transforming time-series production and injection data into complex networks, revealing hidden temporal and spatial relationships [13]. Unlike static approaches, VGA captures evolving states and flow pathways, making it particularly suitable for dynamic reservoir systems. To enhance this methodology, we employ a multiplex network representation, where multiple layers encode the evolution of well interconnections over time. This approach integrates information from different temporal states, capturing both short- and long-term dependencies. Metrics such as average edge overlap and interlayer mutual information provide insights into dynamic inter-well connectivity, uncovering patterns that evolve with the reservoir’s operational history [14,15].

Other methods, such as grey relational analysis (GRA), have also been applied to evaluate inter-well connectivity, particularly in systems with sparse or uncertain data [16]. GRA quantifies static relationships using grey relational coefficients, providing valuable insights into production–injection dependencies. However, its reliance on static snapshots limits its ability to capture temporal evolution, underscoring the need for dynamic approaches like VGA combined with multiplex networks.

This study introduces a methodology that leverages multiplex visibility graph analysis to uncover evolving inter-well connectivity patterns in a naturally fractured hydrocarbon reservoir. By representing production data as complex networks and multiplex layers, the proposed methodology captures the reservoir’s dynamic behavior without relying on predefined assumptions about connectivity. The inferred interconnections provide actionable insights into the influence of injection and production wells, guiding decisions such as optimizing injection schedules, identifying potential drilling locations, and understanding the reservoir’s long-term evolution.

Beyond its immediate applications, the inferred connectivity data hold potential for training machine learning models to forecast production trends based on historical management decisions. By integrating the adjacency matrices generated from the multiplex network analysis into frameworks such as message-passing neural networks (MPNNs) or recurrent neural networks (RNNs), stakeholders can simulate various operational scenarios. This predictive capability can assist in planning field development strategies, optimizing resource allocation, and mitigating risks associated with reservoir management.

The proposed methodology builds on and complements existing research in reservoir engineering. Recent advances, such as GNNs for the qualitative classification of hydrocarbon zones [17], hybrid graph simulators for subsurface flow optimization [18], and GeoDIN frameworks for predicting reservoir performance [19], highlight the utility of graph-based approaches. By extending these methodologies with multiplex VG network representations, this work offers a novel framework for analyzing complex, adaptive systems and refining connectivity predictions, with implications for both practical applications and future research.

The inferred interconnections between wells serve as a crucial tool for reservoir management by providing insights into the permeability of different regions. Areas with low interconnectivity suggest the presence of untapped hydrocarbon accumulations, where additional production wells could be drilled to enhance recovery. Conversely, regions with high interconnectivity indicate that fewer production wells are needed to efficiently drain the area. This methodology also aids in optimizing injection strategies by guiding the selection of injection well locations, fluid injection schedules, and flow regulation practices. By analyzing the density and strength of interconnections over time, operators can adjust injection rates, prevent the premature breakthrough of injected fluids, and mitigate potential well interference effects. Additionally, the evolving connectivity patterns provide a data-driven approach for identifying areas prone to pressure depletion and predicting future well interactions, thereby improving decision-making for long-term reservoir exploitation. Connectivity data can serve as input for predictive ML models, such as message-passing neural networks (MPNNs) and graph neural networks (GNNs), to forecast production trends based on management decisions. Furthermore, the approach can be extended beyond hydrocarbon reservoirs to applications in geothermal energy extraction and groundwater resource management.

2. Methodology

The hydrodynamic interactions in the reservoir were determined using historical data concerning oil production and injection flow rates, spanning 62 years (1958–2020). Within the scope of this work, other data, such as the bottom-hole pressures, fluid properties, and rock properties, were not considered. This dataset included oil and gas production rates and water injection rates, which were key to constructing an abstract representation of inter-well relationships as a network of interlinked nodes. The edges and weights in this network were defined by fluid flow rates and well types while accounting for the fraction of time each well remained operational. Importantly, this approach relied exclusively on production and injection logs, minimizing the need for additional data and thereby optimizing production costs. Despite inherent challenges, such as missing records and noncontinuous datasets, pre-processing ensured the reliable mapping of interwell connectivity. This data-driven method indirectly reflected the underlying geophysical conditions that governed the behavior of the reservoir. The data log issue with respect to discontinuity in the data collected, as suggested in recent studies on dynamic production data management [20], was also addressed in this methodology. Figure 1 shows the production and injection data for the years after the aggregation process.

2.1. Data Collection and Preprocessing

Hydrodynamic interactions within a specific reservoir were analyzed using real historical production data spanning 62 years, from 1958 to 2020 (Figure 1). The dataset included oil, gas, and water flow rates from both production and injection wells. Since production logs contained irregular time gaps and missing records, the one-hot encoding approach was applied to normalize the time series length across wells, ensuring a consistent input format for VG/MVG. This preprocessing step allowed the methodology to capture long-term dependencies without being affected by the non-uniform temporal distribution of production measurements. As field operations were subject to logistical constraints, production logs exhibited discontinuities and missing records. To address this, we employed a one-hot encoding approach to pre-process non-periodic time-series data, ensuring consistency across all wells. After the normalization, those measurements were mapped to a sequence of “events”, a technique that we used to address the non-periodicity of the data and apply the VG approach without losing the long-term dependencies information.

2.2. Transforming Time Series into Networks

To uncover inter-well connectivity patterns, we employed the visibility graph (VG) and multiplex visibility graph (MVG) methodologies. These approaches convert time series data into network representations, preserving both temporal and structural information. The transformation process followed the below steps.

2.2.1. Visibility Graph Mapping

Each well’s production time series (in sequence of events representation) was mapped onto a graph, where each data point became a node. Two nodes were connected if they satisfied the visibility criterion [21]:

$$y_i>y_i+(y_k-y_i){\displaystyle \frac{(t_j-t_i)}{(t_k-t_i)}}.$$

(1)

This ensured that local trends and dynamic fluctuations were preserved.

2.2.2. Multiplex Network Construction

Since well interactions evolve over time, individual VG layers were stacked onto an MVG representation. Each layer represented a different well measurement history in which each node represented an event (observable), enabling the detection of evolving connectivity structures.

2.3. Justification for Using VG and MVG

VG/MVG outperforms recurrence networks (RNs) and transition networks (TNs) by eliminating the need for predefined thresholds or Markovian assumptions, making it more adaptable to complex reservoir behaviors. As a parameter-free approach, it captures both short- and long-term dependencies without requiring additional tuning (

Table 1. Characteristics of network-based methods.

Method	Advantages	Disadvantages
Visibility Graph (VG)	Captures both local and global dependencies; parameter-free	Sensitive to noise in time series
Multiplex VG (MVG)	Preserves evolving interconnections over time	Requires higher computational cost
Recurrence Networks (RNs)	Detects recurrent patterns in time series	Requires an embedding dimension
Transition Networks (TNs)	Models state transitions probabilistically	Assumes discrete state changes
VG/MVG (This Study)	Adapts to evolving reservoir dynamics; suitable for real production data	Computationally intensive at high resolution

).

Given the non-stationary nature of reservoir operations, VG and MVG were well suited to this study as they do not assume periodicity and naturally adapt to changing conditions.

Unlike conventional ML approaches, which require labeled training data and predefined model assumptions, VG/MVG is a parameter-free method that identifies interconnectivity without prior knowledge of well relationships. This makes it particularly advantageous in settings where geophysical properties are incomplete or uncertain. Additionally, unlike recurrence networks, which rely on threshold-dependent recurrence quantification, VG/MVG naturally captures hierarchical and evolving dependencies over time.

2.4. Applying VG to This Specific Use Case

2.4.1. From Non-Periodic Time Series to Event Framework

Consider a set of wells $\omega =\{{\omega }_i|i\in \mathbb{N}\le W\}$, where each well ${\omega }_i$ possesses a time series of nonperiodic flow measurements $y_t^{w_i}=\{(t,y^{w_i})|t\in \mathbb{N}\le n_{w_i}\}$ with $n_{{\omega }_i}$ total observations. Since the wells started operating at different times, the $y_t^{{\omega }_i}$ series were preprocessed to align and standardize the data into equal-sized sets. The resultant time series were treated as indexed sequences of events ${(E_k^{{\omega }_i})}_{k\in \mathbb{N}\le n_{{\omega }_i}}$ to develop the proposed methodology. Finally, the dataset was conveniently represented as a matrix of labeled sequences of size $(W\times n_{{\omega }_i})$, which facilitated the computational procedures.

2.4.2. Event to Network Representation Transformation

Various methodologies can be applied to transform time series into networks (e.g., [21,22]). However, to capture long-term dependencies, we adopted the visibility graph (VG) algorithm introduced by Lacasa et al. [13]. This parameter-free algorithm transforms a time series into a network, preserving key features such as periodicity, self-similarity, and invariance under translations or linear trends. Compared to recursive networks (RNs), transfer networks (TNs), and similar methods, VG is scale-free and does not require parameter optimization, enhancing its robustness for this application. Figure 2 illustrates the transformation process from the time series to the complex graph representation of the network.

For a given sequence of events $(E_k^{{\omega }_i})$, the VG algorithm yielded a graph with k nodes. Any pair of nodes i and j in the graph were connected if they satisfied the so-called visibility criterion.

$$E_k^{w_i}<min(E_i^{w_i},E_j^{w_i}),\forall k|(i<k<j)\wedge (j<k<i).$$

(2)

This meant that that node i could “see” node j whenever any other node (k) located between them had a lower relative value. If the condition was met, the respective nodes were connected by an edge (as shown in Figure 2a). These connections could be remapped onto the network’s graph (Figure 2b). The resulting graph (Figure 2b) simplified the identification of connections and dependencies among events, providing a robust framework for analyzing complex temporal patterns.

2.4.3. Multivariate Times Series for Multiplex Visibility Graph Transformation

During the productive lifespan of an oil field, well interactions evolve, resulting in multiple sequences of events. These were effectively represented as a multivariate network using a multiplex visibility graph (MVG), as illustrated in Figure 3. The MVG retained the essential characteristics of the simpler VG method, such as its ability to capture long-term dependencies and topological features [14]. Multivariate event sequences could be expressed as $Y=\{Y_{\omega ,j}\}$, with the corresponding graph set represented as $G=\{G_{\omega ,j}\}$.

To simplify numerical implementation, these layered networks were recast as vectors of adjacency matrices $A=\{A_1,A_2,$ …, $A_{\omega }\}$, where $A_{\omega }={\{a_{i,j}\}}_{\omega }$ for each MVG layer $\omega$. Here, $a_{i,j}$ denotes the presence or absence of connectivity between nodes i and j [15].

Special events with long-term dependencies, referred to as “states”, were identified by analyzing edge overlaps across layers. The average edge overlap $(\omega )$ metric quantified these overlaps, highlighting persistent patterns over time [14]:

$${\omega }_0={\displaystyle \frac{{\Sigma }_i{\Sigma }_{j>i}{\Sigma }_{\lambda }{a}_{i,j}^{\lambda }}{W{\Sigma }_i{\Sigma }_{j>i}\left(1-{\sigma }_{0,{\Sigma }_{\lambda }{a}_{i,j}^{[\lambda ]}}\right)}}.$$

(3)

Applying this metric to the original event set yielded a subset of states, capturing key connectivity patterns across the reservoir’s operational timeline.

2.5. State Graphs Based on Fluid Behavior

States can be defined to identify relevant relationships in oil production rates. Using the VG technique, graphs were constructed with a modified visibility criterion focused on flow records. Here, a well i could “see” well k based on the flow rate $y_k$ of well k during its steady-state operation S. The condition for establishing an edge between wells i and k was given by

$$Y_k^{w_i}<min(Y_i^{w_i},Y_j^{w_i}),\forall k|(i<k<j)\wedge (j<k<i),$$

(4)

where i, j, and k denote different wells. An initial state $s_0$, derived from historical records, served as the baseline reference for this criterion.

Under this framework, first-, second-, and third-order interactions could be defined, indicating the connectivity relevance among the states. These interactions were represented in an S-dimensional (multilayered) network, denoted as ${\mathcal{G}}^{\lambda }=\{{\mathcal{G}}_{Y_{s,i}}^{w_i}|Y_{s,i}\in Y_s^{w_i},s\le S\}$. The adjacency matrices for these layers were $[A^{[1]},A^{[2]},$…$A^{[S]}]$, with each $A^{[s]}={\{a_{ij}^{\lambda }\}}^s$, where $a_{ij}^{\lambda }$ reflects connections based on production rates captured by the VG approach (Figure 3).

To evaluate connectivity across states, the interlayer mutual information metric was employed. For two layers $\eta$ and $ϵ$, the metric was defined as [14]

$$I_{\eta ,ϵ}={\Sigma }_{{\kappa }^{[\eta ]}}{\Sigma }_{{\kappa }^{[ϵ]}}P({\kappa }^{[\eta ]},{\kappa }^{[ϵ]})log{\displaystyle \frac{P({\kappa }^{[\eta ]},{\kappa }^{[ϵ]})}{P({\kappa }^{[\eta ]})P({\kappa }^{[ϵ]})}}.$$

(5)

Here, $\kappa$ represents the degree distribution across layers $\eta$ and $ϵ$, while $P({\kappa }^{[\eta ]})$ and $P({\kappa }^{[ϵ]})$ are their respective probability distributions. $P({\kappa }^{[ϵ]},{\kappa }^{[ϵ]})$ denotes the joint probability distribution, providing insights into the correlation of nodes across layers.

This analysis relied on curated production logs of oil and injection flow rates. Figure 4 highlights state configurations for a system of 16 production and injection wells, emphasizing the importance of interconnections within stable states. Figure 5 provides an overview of the methodology.

2.6. Validation of Inferred Connectivity Patterns

To ensure the reliability of the inferred inter-well connections, we validated our results using two key approaches:

• Alignment with Operational Data: Historical production logs were analyzed to verify whether inferred connections corresponded to known injection–production relationships.
• Temporal Consistency Analysis: We computed interlayer mutual information $(I_{MI})$ across MVG layers to assess the stability of connectivity over time.

$$I_{\eta ,ϵ}={\Sigma }_{{\kappa }^{[\eta ]}}{\Sigma }_{{\kappa }^{[ϵ]}}P({\kappa }^{[\eta ]},{\kappa }^{[ϵ]})log{\displaystyle \frac{P({\kappa }^{[\eta ]},{\kappa }^{[ϵ]})}{P({\kappa }^{[\eta ]})P({\kappa }^{[ϵ]})}},$$

(6)

where $\kappa$ represents the degree distribution of nodes in each layer $\eta$ and $ϵ$. High values of $I_{MI}$ indicated strong temporal consistency in well interconnectivity patterns.

Expert’s Validation and Field Data Comparison

To further validate the inferred connectivity patterns, the results were compared against historical field observations and expert assessments. Domain experts reviewed the connectivity graphs and confirmed that identified interconnections aligned with known reservoir behaviors and fluid migration trends. Additionally, major connectivity transitions detected in the network analysis were cross-referenced with operational changes, such as shifts in production rates, well interventions, and injection scheduling adjustments. This qualitative validation, combined with the quantitative interlayer mutual information $(I_{MI})$ metric, strengthened confidence in the inferred well interactions.

2.7. Advantages of the Proposed Methodology

Our approach offers the following advantages over existing models:

• Uncovers evolving interconnections without assuming prior knowledge of well relationships.
• Requires only historical production data, reducing dependence on costly geophysical surveys.
• Captures both short- and long-term dependencies, essential for adaptive reservoir management.
• Provides a framework for integrating ML techniques (e.g., GNNs, MPNNs) for future forecasting applications.

By leveraging VG and MVG, this study provides a novel, data-driven approach to analyze inter-well connectivity, with implications for optimizing reservoir management and improving predictive modeling.

3. Results and Discussion

The spatial distribution of the wells analyzed in this study is shown in Figure 6. Some wells functioned alternately as production wells (black dots) and injection wells (blue arrows) over the years.

The initial graph representation was constructed using productivity and flow rates from the initial events, applying the visibility graph (VG) methodology.

This data-driven approach identified temporal and functional dependencies rather than relying solely on spatial proximity. As the dataset spanned six decades, it captured the reservoir’s evolving operational strategies. The observed connectivity reflected the interplay between long-term production behaviors, operational planning, and reservoir dynamics, offering insights into functional rather than geographic topological relationships.

Using the proposed VG methodology, the results of the evolving dynamic interactions among the wells were revealed. Figure 7 highlights three representative steady states during the year 2020. In the first state, well $I_{16}$ exhibited the highest interactivity (weight $\omega =75$, the weight of a steady state was defined by the functioning over the time of that given state), followed by well $P_{12}$ with moderate interconnectivity. Over time, the system transitioned to a second steady state dominated by $P_5$, which displayed the highest production rate and significant interaction with $I_{16}$. The last steady state, observed in 2020, featured a basal configuration with significant interconnectivity between $P_5$ and $I_{13}$ $(\omega =185)$.

The network representations obtained via VG and MVG revealed a clear evolution in inter-well connectivity. In the first steady state (Figure 7, left), well $I_{16}$ exhibited the highest degree of centrality, with an average edge weight of $\omega =75$, indicating strong connectivity within this phase. The second steady state (Figure 7, center) was dominated by well $P_5$, which emerged as the highest-producing well, with a connectivity score of $\omega =19$, reflecting its evolving role within the reservoir. In the final steady state of 2020 (Figure 7, right), $P_5$ and $I_{13}$ exhibited the strongest interconnectivity $(\omega =185)$, highlighting the significant impact of injection activity in reshaping the production network.

Key transition periods in the reservoir’s evolution are summarized in Figure 8. To establish the robustness of the methodology, the first transition periods were used as training periods to develop and validate this methodology, leaving the last year, 2020, as the test period of the methodology, thereby retaining information on the previous behavior and evolution of the system to establish the interconnections of the wells that would remain thorough the lifetime of the reservoir.

During 1958–1979, only wells $P_1$ and $P_2$ were active, alternating in activity, with $P_2$ ultimately dominating. Limited connectivity insights were available due to the small number of active wells (see Figure A1 in Appendix A). Between 1980 and 1984, $P_3$ became operational, exhibiting moderate connectivity with $P_1$ and $P_2$ (see Figure A2 in Appendix A). From 1985 to 2004, new wells such as $P_4$ and $P_6$ were introduced, with $P_6$ showing increasing activity and connectivity with $P_3$ (Figure A3 and Figure A4).

In the period 2005–2009, multiple wells, including $P_1$ through $P_8$, were active, though the activity of well $P_5$ started after $P_8$ in 2010 (Figure A5). $P_3$ demonstrated a consistent triggering effect on $P_7$, indicating a strong functional relationship. Between 2010 and 2014, wells $P_1$–$P_{12}$ were active, with $P_9$ and $P_{12}$ displaying an alternating activity pattern, suggesting their interconnection (Figure A6). For a complete overview of the evolution during the years 1958–2009 with their respective states transition, see Figure 9.

From 2015 to 2019, the addition of injection wells $I_{13}$ and $I_{14}$ introduced new dynamics. $P_5$ exhibited strong connectivity with $I_{13}$ and $P_{12}$, while emerging second-order connections between $P_4$ and $P_{10}$ became evident (Figure A7). The global evolution for the years 2010 to 2019, together with the state transitions, can be appreciated in Figure 10.

Taking into account all previous years, different persistent patterns of interconnectivity were found. The same methodology was applied to the data for the year 2020. Notably, in 2020, well $P_{10}$ transitioned from production to injection as $I_{16}$. This transformation reinforced its functional interconnection with $P_5$, corroborating earlier findings (Figure 7).

To evaluate the temporal consistency of these interconnections, Interlayer Mutual Information $(I_{MI})$ was computed across those different time windows with Equation (6). The $I_{MI}$ values obtained for 1985–2004 and 2010–2014 were 0.82 and 0.76, respectively, indicating the high temporal stability of the interconnectivity patterns. However, the 2015–2019 period showed a lower $I_{MI}$ of 0.58, suggesting an increase in well interactions due to operational changes (

Table 2. Metric assessment values.

Period	First-Degree Connectivity	IMI Value	Edge–Weight Distribution $(\mathit{\omega })$
1985–2004	P3, P6	0.82	30–75
2010–2014	P7, P9, P12	0.76	50–100
2015–2019	P5, I13, P12	0.58	80–120
2020 (Test)	P5, I16	0.78	185

).

The inferred inter-well connectivity remained stable across different operational phases, as demonstrated by the interlayer mutual information $(I_{MI})$ values exceeding 0.75 in key production stages. Notably, in 2020, well $P_5$ exhibited the highest interconnectivity $(\omega =185)$, indicating its central role in reservoir dynamics. Similarly, wells $I_{16}$ and $P_{12}$ showed strong interconnections, particularly in response to injection activities. These observations highlight the ability of VG/MVG to capture evolving reservoir behaviors. Table 2 summarizes these findings, providing a quantitative assessment of the connectivity strengths between the production and injection wells.

The inferred connectivity patterns revealed key insights into well interactions, with dominant production hubs emerging over different operational periods. To ensure the reliability of these findings, we assessed their consistency through both quantitative metrics and expert validation, examining whether the inferred patterns aligned with known production histories and operational decisions.

3.1. Validation of Results and Robustness Assessment

To validate inferred connectivity, we consulted experienced reservoir engineers who compared the VG-/MVG-derived well interactions with historical operational records. Experts assessed whether the detected interconnections were consistent with known injection-induced responses, production rate fluctuations, and previously observed flow behaviors. The validation process confirmed that key inferred relationships—such as the strong connectivity between wells $P_5$ and $I_{16}$—matched historical well performance data. Furthermore, the methodology accurately detected the shifting role of $P_5$ as a dominant well in 2020, matching expert observations. Discrepancies were found in the connectivity strength of $P_9$–$P_{12}$, where operational records suggested stronger interactions than inferred by the model, highlighting potential improvements in weight assignment techniques. The model also revealed previously unrecognized interconnections, which experts suggested could indicate hidden permeability pathways not captured in conventional analyses. This validation strengthened confidence in the methodology’s ability to uncover both expected and novel reservoir behaviors.

3.1.1. Verification Metrics

• Consistency with Historical Operational Data: Connectivity patterns corresponded to documented operational changes, such as well conversions and periods of high activity. For instance, the identified link between $P_5$ and $I_{16}$ aligned with historical injection-driven responses; see Figure 11 and Figure 12 for more detail.
• Temporal Consistency: Connectivity patterns persisted across different time intervals, reflecting stable reservoir dynamics and reinforcing reliability.

In addition to these practical validations, the results demonstrated key characteristics of complex adaptive systems:

• Emergence: The interconnections revealed were emergent properties of the reservoir’s evolution, naturally arising from the interplay of historical production and injection activities.
• Self-Similarity: The recurring patterns of connectivity across temporal intervals reflected self-similarity, a feature that underscores the robustness of the VG methodology in capturing consistent and meaningful relationships within an evolving system.
• Reduction of Information Entropy: By transforming disorganized time-series data into structured network representations, the VG methodology reduced uncertainty, enabling stakeholders to uncover hidden patterns and dependencies critical for informed decision-making.

The dynamic nature of the reservoir added complexity as evolving flow pathways and occlusions were shaped by pressure depletion, erosion, and injection activities. These factors challenge static validation methods like cross-validation. Instead, metrics like temporal consistency, alignment with operational data, and the application of concepts from complex systems provide more meaningful evaluations of robustness.

3.1.2. Consideration of Reservoir’s Evolving Nature

The reservoir’s dynamic evolution over the decades highlights the need for methodologies that capture temporal changes. The VG methodology excels in adapting to these dynamics, uncovering both short- and long-term dependencies and emerging patterns. By leveraging insights from complex, adaptive systems, this approach provides stakeholders with a powerful tool to understand and manage reservoir behavior in real-time (see Figure 11 and Figure 12).

3.1.3. Comparison with Traditional Approaches

The visibility graph methodology differs from traditional capacitance–resistance (CR) models and INSIM-based numerical simulations in key ways. The CR model requires an explicit formulation of fluid flow equations and historical injection–production data but often fails to capture nonlinear dynamic interactions between wells. Similarly, INSIM simulations provide flow field approximations but require extensive parameterization, making them computationally expensive.

In contrast, the VG approach identifies inter-well connectivity directly from time series data without requiring explicit assumptions about reservoir properties. This is particularly advantageous for reservoirs with incomplete geophysical data. Additionally, the multiplex visibility graph (MVG) framework captures evolving connectivity over time, a feature not present in static CR models. The high $I_{MI}$ values (above 0.75 for most periods) indicated that our methodology provided robust connectivity patterns that align with historical operational trends, reinforcing its reliability for decision-making.

3.2. Limitations and Future Work

The results confirm that the VG/MVG framework provides a robust means of analyzing inter-well connectivity in complex reservoir systems. By leveraging only production and injection data, this approach offers a cost-effective, assumption-free, and adaptive method for understanding flow dynamics. The methodology’s ability to capture evolving connectivity patterns suggests its potential applicability beyond this study, particularly for real-time reservoir monitoring and decision-making in dynamic production environments.

While the proposed VG/MVG methodology successfully uncovered inter-well connectivity using historical production data, some assumptions may have impacted its predictive accuracy. First, the methodology did not incorporate direct geophysical measurements, meaning that it relied on production trends to infer connectivity. Additionally, the visibility criterion in VG may have introduced sensitivity to short-term fluctuations, which could be mitigated by integrating additional smoothing techniques or hybrid models (e.g., VG + ML). Future work will focus on refining edge-weighting schemes and incorporating machine learning techniques like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and graph neural networks (GNNs) to enhance predictive capabilities.

4. Conclusions

This study introduced a novel application of visibility graph analysis (VGA) and multiplex visibility graphs (MVGs) for inferring dynamic inter-well connectivity in naturally fractured hydrocarbon reservoirs. Unlike traditional approaches such as capacitance–resistance (CR) models and INSIMs, which require detailed geophysical assumptions, the proposed methodology relies solely on historical production data, making it particularly valuable in scenarios where subsurface data are incomplete or unavailable.

The key findings include the following:

• Major connectivity hubs were identified through steady-state analysis, with well P5 exhibiting the highest interconnectivity $(\omega =185)$ in 2020, confirming its dominant influence in the field.
• Temporal consistency evaluation using interlayer mutual information $(I_{MI})$ demonstrated stable well interactions over multiple decades $(I_{MI}>0.75)$, reinforcing the robustness of the inferred connectivity patterns.
• VG/MVG required fewer assumptions compared to CR models and INSIMs, offering a scalable, data-driven approach adaptable to diverse reservoir conditions.

The inferred connectivity patterns provided quantitative insights into well placement, injection scheduling, and field management strategies. Furthermore, validation through expert review confirmed that the discovered inter-well relationships aligned with known production trends and operational history, further strengthening the credibility of the method.

Future Work and Broader Implications

The methodology’s adaptability makes it a strong candidate for integration with advanced machine learning models to enhance predictive capabilities. Future developments may include the following:

• Graph neural networks (GNNs) and message-passing neural networks (MPNNs) to refine inferred connectivity and predict future well interactions.
• Recurrent neural networks (RNNs) (e.g., LSTM, GRU) to model long-term production trends and optimize reservoir management strategies.
• CNN-based hybrid models to integrate spatial features alongside time-series connectivity.

Beyond hydrocarbon reservoirs, the proposed VG/MVG framework has broader applicability in groundwater modeling, thermal reservoir engineering, and water networks, where complex interconnections evolve over time.

By uncovering hidden connectivity patterns and providing an adaptive, data-driven alternative to traditional reservoir modeling, this study contributes a valuable tool for reservoir engineers and decision-makers, paving the way for more efficient and cost-effective field development strategies.