A Bayesian Network Model for Risk Management during Hydraulic Fracturing Process

Abstract

The escalating production of shale gas and oil, witnessed prominently in developed nations over the past decade, has sparked interest in prospective development, even in developing countries like Algeria. However, this growth is accompanied by significant opposition, particularly concerning the method of extraction: hydraulic fracturing, or ‘fracking’. Concerns regarding its environmental impact, water contamination, greenhouse gas emissions, and potential health effects have sparked widespread debate. This study thoroughly examines these concerns, employing an innovative approach to assess the risks associated with hydraulic fracturing operations in shale gas reservoirs. Through the integration of diverse data sources, including quantitative and qualitative data, observational records, expert judgments, and global sensitivity analysis using the Sobol method, a comprehensive risk assessment model, was developed. This model carefully considered multiple condition indicators and extreme working conditions, such as pressures exceeding 110 MPa and temperatures surpassing 180° F. The integration of these varied data streams enabled the development of a robust Bayesian belief network. This network served as a powerful tool for the accurate identification of process vulnerabilities and the formulation of optimal development strategies. Remarkably, this study’s results showed that this approach led to a notable 12% reduction in operational costs, demonstrating its practical efficacy. Moreover, this study subjected its model to rigorous uncertainty and sensitivity analyses, pinpointing the most severe risks and outlining optimal measures for their reduction. By empowering decision-makers to make informed choices, this methodology not only enhances environmental sustainability and safety standards but also ensures prolonged well longevity while maximizing productivity in hydraulic fracturing operations.

1. Introduction

Commercial production of shale gas dates back to the 1970s in the United States and Canada, and more recently, in 2016, it began in the United Kingdom. The estimated technically recoverable shale gas resources in the UK stand at approximately 0.57 trillion cubic meters. Other countries, including China, Argentina, and Algeria, possess substantial technically recoverable shale gas reserves, with estimated volumes of 31, 22, and 11 trillion cubic meters, respectively, positioning them as potential future contributors to global shale gas production. However, the comprehensive impacts of widespread shale gas utilization remain contentious, sparking intense debates among stakeholders. Opponents contend that potential environmental repercussions, such as drinking water contamination, and challenges in wastewater management could outweigh the perceived benefits, especially in areas that suffer from water scarcity. It is especially in the realm of water-related concerns that special attention is warranted. Extensive research on shale gas exploration has been conducted in various developed and developing countries, shedding light on shared challenges and deficiencies. These studies span diverse geographical regions, consistently highlighting environmental risks, regulatory insufficiencies, and social consequences associated with shale gas operations. Over the past five years, global research efforts have intensified, leading to advancements in best practices and innovative technologies for shale gas extraction. Turning our attention to Algeria, an emerging player in the shale gas arena, offers valuable insights into this nation’s unique context and potential within the dynamic shale gas sector (Figure 1) [1,2,3,4,5,6].

This research work involves attending field operations and conducting quantitative analyses through various tools used in the oil and gas field. Starting from hazard identification using HAZOP analysis, a fault tree analysis (FTA) is created, and initial results are analyzed to determine the level of risks through an event tree (ET). This approach allows for linking the causes and consequences of the operation, including failure or destruction of the well and reservoir. BBN models have been proposed as an approach to analyze risks associated with hydraulic fracturing system accidents. BBN facilitates the modeling of complex and dynamic behavior of real-world systems by structuring the integrated simulation based on input nodes and resultant nodes. One useful approach for providing input data to BBN models is the use of matrices generating Conditional Probability Tables (CPT). The results obtained from a BBN model enable the creation of CPTs, which can be used to evaluate the model and calculate sensitivity through large-scale analyses of uncertainty to attain more accurate results. The primary goals of hydraulic fracturing operations include increasing oil and gas flow, eliminating formation damage near the borehole wall, and interconnecting high-permeability zones. Success in hydraulic fracturing depends on selecting the appropriate hydraulic penetration technology, creating complex gel-fracture volumes with higher flow conductivity, and deploying fracture volumes rationally along the hole section. The process includes preflight, gel injection, and over flush stages. Overall, viscous slick water is used in the perforation stage; meanwhile, pads and gels are injected in different stages to propagate and etch fractures. Over flush is used to enhance gels’ penetration distance. Figure 2 illustrates the intricate process of hydraulic fracturing operations. In this detailed visual representation, key elements of the fracturing process come to life. In recent years, the rapid expansion of hydraulic fracturing, or ‘fracking’, for shale gas and oil extraction has sparked global interest and debate. While developed nations have embraced this technology, concerns regarding its environmental impact, water contamination, and potential health risks have raised significant opposition. Moreover, there is a lack of comprehensive studies that address these concerns using innovative and integrative approaches, especially in the context of developing countries like Algeria. Recognizing these gaps, this study aims to bridge the existing knowledge deficit by conducting a thorough examination of the risks associated with hydraulic fracturing operations in shale gas reservoirs. By integrating diverse data sources, employing advanced analytical methods, and conducting rigorous sensitivity analyses, this research endeavors to provide a nuanced understanding of the challenges posed by hydraulic fracturing [3,4,5,8].

In contrast to conventional techniques such as manual calibration and heuristic approaches, Bayesian networks offer a superior methodology for evaluating hydraulic fracturing risks. Manual methods, often lacking precision and time efficiency, are eclipsed by Bayesian networks’ structured and automated approach. With the advent of artificial intelligence technologies, leveraging extensive data processing capacities, reliance on sensors for real-time data transmission and Bayesian network processing has become integral. This establishes Bayesian networks as a preferred choice for assessing potential risks in hydraulic fracturing, particularly in challenging shale gas reservoir conditions. This shift towards automated and sophisticated approaches ensures more accurate and efficient decision-making, ultimately contributing to the sustainable and effective exploitation of shale gas resources [8,9].

This paper focuses on assessing the potential risks associated with hydraulic fracturing in a shale reservoir located in Algeria, which is an area grappling with water scarcity. The reservoir’s unique characteristics, including low permeability and narrow dimensions, present challenges typical of such formations. Employing a Bayesian network, the study explores various scenarios and uncertainties that could lead to operational failures or increased environmental risks. The goal is to enhance the management of the hydraulic fracturing process, mitigating these risks while optimizing costs. By doing so, the paper aims to contribute to the sustainability and effectiveness of shale gas exploitation in this region. Through rigorous analysis and innovative techniques, this research endeavors to pave the way for more efficient and responsible practices in the realm of hydraulic fracturing [4,10,11,12,13].

This paper is organized into six sections. Section 1 and Section 2 set out the introduction and methodology of the work, which formulates hazard identification and development of an event tree (ET) and determines the quantitative assessment of potential risks using BBN. Section 3 demonstrates the results and main findings of the hydraulic fracturing process with the elicitation of the BBN model, an evaluation of sensitivity analyses and validation of the BBN model. Section 4, provides a discussion of the analyses and Finally, Section 5 delineates summary of paper, limitation, and future recommendations.

2. Methodology

2.1. Case Study

This study delves into the intricacies of hydraulic fracturing in different shale gas wells located in the In Salah region of Algeria. The reservoir at a depth of 3600 m is characterized by its heterogeneous clayey nature. Geographical details of this location are depicted in Figure 3. This paper systematically explores various catastrophic scenarios inherent in the hydraulic fracturing process, meticulously analyzing results to mitigate risks and enhance professional implementation [7,9,14]

2.2. Formulation, Hazard Identification, and Development of an Fault Tree

In this study, a fault tree (FT) model is employed for system safety analysis within hydraulic fracturing operations, integrating hazard identification and an event tree setup based on field data. The methodology enables a comprehensive assessment of qualitative risks by utilizing the combined fault tree and event tree to represent various scenarios. Figure 4 illustrates the actual hydraulic fracturing process and the equipment used at the surface level for this operation. The necessary field data, crucial for this study, were obtained from different shale gas wells related to this process. Expert opinions in the field were collected, translated, and analyzed through the error tree, with the assistance of the Top Event FTA program. The methodology allows for a comprehensive assessment of qualitative risks, using the combined fault tree and event tree to represent multiple scenarios. The risk assessments rely on the Bayesian belief network (see Figure 5) [15,16,17,18].

The process involves two key steps: hazard identification, which evaluates risk factors and potential hazards that could cause harm, and fault tree development, which is a systematic quantitative analysis that is conducted through the development of a fault tree (Figure 6). The proposed framework integrates results from both fault tree and event tree analyses, establishing links between causes and results for scenario analysis during the hydraulic fracturing process. This integrated approach forms the foundation for Bayesian belief network (BBN) modeling, as demonstrated in

Table 1. Integrating results from fault tree and event tree analyses.

Type of Risks	Description Event	Select Nodes	Nodes	State
Characteristics of the static system of the environment	Strata with high in situ stress in shale reservoirs	Root node	RN1	Yes–no
	High permeability zones	Root node	RN2	Yes–no
	Poor physical properties	Root node	RN3	Yes–no
	High temperature	Root node	RN4	High–Med–Low
	Strongly heterogeneous reservoir	Root node	RN5	Yes–no
The resulting effects of the fracturing process on the well and the surrounding environment	Reservoir contamination (max treating)	Consequence node	CN1	High–Med–Low
	Restricting the further penetration of gels in the reservoir	Consequence node	CN2	Yes–no
	Effective distance of live additive	Consequence node	CN3	High–Med–Low
	Rapid acid rock reaction rate	Consequence node	CN4	Yes–no
	Limited gel-etched fracture conductivity	Root node	RN6	High–Med–Low
	Well production declines rapidly and fails to stabilize	Consequence node	CN5	Yes–no
	Shortens the gels penetration distance	Consequence node	CN6	Yes–no
	Difficult to achieve rational distribution of gels	Root node	RN7	Yes–no
	Near-wellbore friction	Root node	RN8	High–Med–Low
	Exceeding limitations of wellhead completion	Consequence node	CN7	Yes–no
The developments and changes that occur during the process	High pressure of the fracturing process	Root node	RN9	High–Med–Low
	Uncontrolled additive leak off	Consequence node	CN8	Yes–no
	Errors of the well trajectory control	safety node	SN1	Yes–no
	Fluid loss	Consequence node	CN9	Yes–no
	Gel flow velocity in the cracks	Root node	RN10	High–Med–Low
	Error gel system	safety node	SN2	Yes–no
	Gel dissolving power	Root node	RN11	High–Med–Low
	Gel and additive efficiency	Root node	RN12	High–Med–Low
	Gels amount additive loss	Root node	RN13	High–Med–Low
	Exceeding equipment loading capacity	Consequence node	CN10	Yes–no
	Large injection rate	Root node	RN14	Yes–no
	Uncontrolled in gels fracturing pumps	Consequence node	CN11	High–Med–Low
	High pressure inside primary pipeline	Root node	RN15	High–Med–Low
	Perforation friction	Root node	RN16	High–Med–Low
	Failure of gels fracturing process	Top event node	TEN	Yes–no
	Damage in the reservoir and shortening the life of the well	Top event node	TEN	Yes–no

, which serves as the basis for linkage in this study [9,10,19,20].

2.3. Building a BBN Model

A Bayesian belief network (BBN) emerges as a potent diagnostic tool for identifying system faults in the context of hydraulic fracturing. Utilizing hierarchical graphs, the BBN adeptly captures probabilistic causal relationships among uncertain variables, which are crucial when confronted with limited data. This dynamic model learns continuously from both historical and real-time data, adapting its understanding during hydraulic fracturing processes based on observed changes [6,12,21]. The categorization of nodes into root nodes (RN), consequence nodes (CN), and safety nodes (SN) enhances the system’s fault identification capabilities. Links in the BBN symbolize associations, reflecting variable states and conditional probabilities for consequence nodes. When safety or consequence nodes lack root nodes, probabilities are considered unconditional. A fault tree, visually represented in Figure 6, encapsulates this hierarchical structure, aiding in the comprehension of fault scenarios. This adaptive model, tailored for hydraulic fracturing intricacies, ensures continual learning and updates, contributing to effective fault identification and, consequently, heightened system safety and performance. In essence, the BBN model provides a comprehensive framework that amalgamates historical and real-time data, categorizes nodes meaningfully, and utilizes conditional probabilities to evaluate fault scenarios, ultimately enhancing system resilience in hydraulic fracturing [12,14,22].

In Figure 7, the initiation of modeling and probabilistic programming is demonstrated using Software GeNIe Academic Version 4.0.1922.0, which is specifically tailored for Bayesian belief networks (BBN) in the context of real-time hydraulic fracturing process data. The model’s beliefs align with fault tree analysis and event tree, while the links signify causal connections between them. Notably, the model encapsulates intricate relationships, such as the consequence node linking strata with high in situ stress in shale gas reservoirs to high permeability zones and near-wellbore friction. This representation, as depicted in Figure 8, provides a visual roadmap for understanding the interplay of variables, facilitating effective fault analysis and enhancing the system’s adaptive learning in hydraulic fracturing scenarios [6,13,15,23]. In addition, Figure 7 illustrates a Bayesian network model through groups defined as G = RN, SN, CN, which correspond to FTA and ET. It can be simplified into variables and expressed in the results as G = V, A, where V refers to node groups of different variables and the group of their related A, giving V = {1,.....n) and $A=\left\{\left(i,j\right)i,j\in V,i=j\right\}$. Meanwhile, the output result node i relates to the value of the variable and root nodes. The equation $R\left(i\right)=\{j\in V|\left(j,i\right)\in A\}$R(i) represents the root nodes for each of the consequence nodes, and the safety node is represented as i.

A path $D=\left(d_1\dots ..d_n\right)$ is a link vector for each state of $d_i\in D_i$ for each variable $\in V$ that has the same probability distribution:

$$P\left(d\right) {\displaystyle \prod }_{i=1}^{n}P(\raisebox{1ex}{$d_i$}\!\left/ \!\raisebox{-1ex}{$D_{R\left(i\right)}$}\right.)$$

(1)

This is given if the probability of the variable is without a root $\left(D_{R\left(i\right)}=\varnothing \right)$ and the negligible probabilities are $P\left(D_i\right)=P\left(X_i=d_i\right)$.

Equation (2) expresses the state of variables with the root of each variable through conditional probabilities

$$P\left(\frac{d_i}{D_I}\right)=P\left(X_i=d_i|X_j=D_j,j\in R_{\left(i\right)}\right)$$

(2)

where $D{R}_{\left(i\right)}$ represents the situation group of root nodes of i.

2.4. A BBN for Hydraulic Fracturing

The Bayesian belief network (BBN) model for hydraulic fracturing intricately connects nodes through probability distributions, articulating the maximum likelihood of various scenarios. Non-root nodes are characterized by continuous probability distributions, while conditional probabilities govern other nodes. Given the specificity and complexity of shale gas exploitation, extensive research and fieldwork are undertaken, as are collaborations with experts to collect and analyze data. This comprehensive approach, integrating artificial intelligence, focuses on developing a Bayesian belief network (BBN) capable of modeling and real-time prediction. Particularly, the model focuses on the propagation of Pressure Wave Peaks (PWPs) within coal rock fissures during hydraulic fracturing, as illustrated in Figure 8. The utilization of MATLAB software for modeling estimates the likelihood probability of high permeability zones and near-wellbore friction, showcasing the BBN model’s predictive capabilities in this intricate process [24,25,26].

These pressure waves are pivotal in assessing the effectiveness of the fracturing process [27]. To model PWP propagation, we utilize the following equations:

Equation (1)—original pressure wave equation:

$$P=A\sin \left(h\right)2\pi f\left(\frac{x}{c}t\right)+4i$$

(3)

where

$P$ represents the transient pressure loaded on the coal in kPa;

$A$ is the PWP peak in kPa;

$f$ is the PWP frequency in Hz;

$c$ is the velocity of the wave in m/s;

$p$ is the position in m; and

$t$ is the transient time in s.

Equation (2)—corrected pressure wave equation:

$$P=P_0+ \mathrm{A} \sin \left(h\right)2\pi f\left(\frac{x}{c}t\right)+4i$$

(4)

$P_0$ represents the average of the PWP that did not change during the PWP propagation period. Equation (1) represents the initial pressure wave equation, while Equation (2) corrects it by accounting for a constant offset represented as $P_0$ [18]. Additionally, we examine the synthesized wave resulting from incident and reflected waves using Equation (3)—Synthesized Wave Equation:

$$A_0=\sqrt{A_1^2+A_2^2+2A_1{A}_2\cos \left(4\theta \right)}$$

(5)

where

$A_0$ is the PWP peak of the synthesized wave;

$A_1$ and $A_2$ are the amplitudes of the incident and reflected waves, respectively; and

$\theta$ is the phase difference between the incident and reflected waves.

This equation helps us understand how incident and reflected waves interact constructively, providing insights into the PWP peak at the fissure end [19].

In the context of the Bayesian belief network (BBN) model for hydraulic fracturing, nodes are interconnected using probability distributions to express the scenario likelihood. Continuous probability distributions are used for non-root nodes, while conditional probabilities are applied to other nodes. Since the model often deals with qualitative data, expert elicitation is preferred when no quantitative data is available [20,27,28].

Hazard identification, fault analysis, and expert opinions contribute to constructing the Conditional Probability Table (CPT), ensuring the model’s credibility and realism [21]. Integrating these mathematical equations into our BBN model enhances our ability to predict and effectively manage the hydraulic fracturing process.

Furthermore, the relative weight values of each node play a vital role. Root formations are compatible with each state of nodes, such as reservoir contamination, fluid efficiency, exceeding equipment loading capacity, error in the hydraulic system, near-wellbore friction, damage in the reservoir, and shortening the well’s life. The relative weight of the node varies between one and zero, expressing the value of the weight in the node and the mutual effects between the root node and the consequence node [22,28,29].

To develop the hybrid Bayesian network (HBN), we utilize the fault tree model from the second section. The CPT is defined according to the location and function of nodes and their influence on the states of consequence nodes. In the BBN, nodes are categorized into pivot nodes with respect to sample sizes of root nodes, safety nodes, and consequence nodes [22,27,30].

Table 2. Scenario analysis of the BBN obtained from the three-method framework.

Nodes						Probability Measurement
CN1	SN2	RN2	RN8	RN12	CN10	Likelihood	EBBN	WSA
High	No	No	High	Low	Yes	0.978	0.227	0.375
High	No	No	Low	High	No	0.981	0.341	0.412
Low	No	Yes	Low	High	No	0.991	0.369	0.488
Low	No	Yes	Low	Low	No	1	0.382	0.497

summarizes the scenario analysis for hydraulic fracturing failure using three methods: likelihood, EBBN, and Weighted Sum Algorithm. These methods calculate the likelihood of failure scenarios related to the propagation of Pressure Wave Peaks (PWPs), providing valuable insights for risk management and decision-making [19,20,21,22]. These methods are essential for risk assessment in hydraulic fracturing.

3. The Main Results of Bayesian Belief Networks

The primary outcomes of Bayesian belief networks (BBNs) hinge on a judicious assignment of a priori probabilities, expert input, and the application of artificial intelligence algorithms, particularly in scenarios with limited data. Sensitivity analysis becomes crucial in contexts of restricted data to pinpoint influential input parameters that significantly affect the output results. The BBN model relies heavily on nodes, shaping its structure through conditional probabilities in both inputs and outputs. The strategic setting of probabilities serves the dual purpose of exploring sensitivity and facilitating analysis-assisted optimization through model calibration. To enhance the integrity of the BBN model, which deals with both discrete and continuous values of input parameters, a variance reduction method is employed. This method allows for an in-depth assessment of the BBN model’s sensitivity to variations in specific input parameters, contributing to the model’s overall optimization [6,8,22,24,31].

3.1. Finding Sensitivity Parameters

The Sobol indices are utilized for global sensitivity analysis, providing a robust method to assess the relative significance of input parameters on model outputs. This analysis reveals the extent to which interactions between specific inputs influence the overall output. For instance, it might indicate that 90% of the output is due to the interaction between two inputs, with one contributing 8% of the variance and the other 2% (see Figure 9). This approach is applied to characterize the dynamic system of the environment impacted by hydraulic fracturing, incorporating changes and developments expressed within the BBN model for shale gas reservoir hydraulic fracturing [21,25,26].

The variance-based Sobol sensitivity method explores the multidimensional space of the unknown input parameters X with a certain number of Monte Carlo samples. The sensitivity indices, both first order and the higher interactions between the unknown input parameters, are generated by a decomposition of the BBN model function $Y=f\left(X\right)$ in a d-dimensional factor space into summands of increasing dimensionality [30,32,33]:

$$\gamma =f_0+{\sum }_{i=1}^d{f}_{ij}\left(X_i\right)+{\sum }_{i<j}^d{f}_{ij}\left({\mathsf{{\rm X}}}_i,{\mathsf{{\rm X}}}_j\right)\dots \dots \dots +f_{1,2},\dots .,d\left({\mathsf{{\rm X}}}_1,{\mathsf{{\rm X}}}_2,\dots \dots ,{\mathsf{{\rm X}}}_d\right)$$

(6)

where the constant f_i is a function of X_i and f_ij is a function $(X_i,X_J,\dots ..)d{x}_{ik}=0.1\le k\le S_i,S_{ij}=1,\dots d,$ etc. A condition of this decomposition is the total number of summands in Equation (2).

$${\int }_0^1{f}_{i_1{i}_2\dots \dots i_s\left({\mathsf{{\rm X}}}_{i_1}{\mathsf{{\rm X}}}_{i_2},\dots \dots .{\mathsf{{\rm X}}}_{i_s}\right)d{X}_K=0, forK=i_1\dots \dots ..i_s}$$

(7)

Functional elements are expressed by conditional probability values,

$$f_0=E\left(Y\right)$$

(8)

$$f_i\left(X_i\right)=E\left(\frac{Y}{X_i}\right)-f_0$$

(9)

$$f_{ij}\left(X_i,X_j\right)=E\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{$X_i{X}_j$}\right.\right)-f_0-f_i-f_j$$

(10)

We develop the function $f\left(X_i\right)$ to become of the second degree so that the function $f_{ij}$ can be defined by the terms ij and the main variables $X_i$ and $X_j$. This assumes that the factorization results from the integral of the function $\left(X\right)$ are squared as follows:

$$f^2\left(X\right)dx-f_0^2{\sum }_{s=0}^d{\sum }_{i_1<\cdots .<i_s}^d\int f_{i=1}^2\dots ..i_{s}d\left(X_{i_1}\right)\dots \dots .d{\left(X\right)}_{i_s}$$

(11)

The following properties hold for variance expression indices:

$$Var\left(Y\right)={\sum }_{i_1}^d{V}_i+{\sum }_{i<j}^d{V}_{ij}+\cdots ..V_{12\dots \dots ..d}$$

(12)

where

$$V_i=Var{X}_i(E_{X_{~i}}\left(\frac{Y}{X_i}\right)$$

(13)

$$V_{ij}=Var{X}_{ij}\left(E_{X_{~ij}}\left(\frac{Y}{X_i{X}_j}\right)\right)-V_i-V_j$$

(14)

Based on the total conditions of variance, the outputs are explained through the analysis of variance according to the condition of each input. $X~i$ stands for variables excluding the value of $X_i$, which is the main sensitivity indicator.

$S_i$ is an indicator of sensitivity called the first order of sensitivity and is given as follows:

$$S_i=\raisebox{1ex}{$V_i$}\!\left/ \!\raisebox{-1ex}{$Var\left(Y\right)$}\right.=\raisebox{1ex}{$d_i$}\!\left/ \!\raisebox{-1ex}{$D\left(Y\right)$}\right.$$

(15)

It is considered that the main outputs of the variables $X_i$ express the state of variable $X_i$ alone in order to calculate the mean value of the various inputs through the total variance, which in turn enables the formation of additional indicators, $S_{ij}$ and $S_{ijk}$. Depending on $\left(Y\right)$, the outputs are expressed as follows:

$${\sum }_{i=1}^d{S}_i+{\sum }_{i<j}^d{S}_{ij}+\cdots \dots \dots S_{12\dots .d}=1$$

(16)

Since the model contains a large number of variables, the indicators are computationally expensive, so the total order index $S_{T_i}$ that is used to measure all the variance outputs for $X_i$ is not given here.

3.2. Calculation of Indices

In examining the statistical dependencies and their implications for the performance of the Bayesian belief network (BBN) in hydraulic fracturing, we employ Monte Carlo integrations and Sobol’s estimation. This methodology unveils crucial parameters within the BBN framework, facilitating a comprehensive assessment of system behavior. By integrating Sobol’s variance-based method with Monte Carlo simulations, primary order and total sensitivity indices are evaluated. Utilizing the BN Toolbox© in Matlab^®, these analyses reinforce the robustness of the model for risk assessment in hydraulic fracturing processes, contributing to improved risk diagnosis [20,22,34].

Table 3. The sensitivity analysis results for the BBN by Sobol’s estimation based on Monte Carlo integrations.

Input Parameter BBN Model	Primary Order Index	Output Parameter BBN Model	Finally Order Index
RN1	0.0095	RN1	0.0090
RN2	0.1242	RN2	0.1441
RN3	0.0188	RN3	0.0184
RN4	0.0156	RN4	0.0154
RN5	0.0781	RN5	0.0190
CN1	0.1271	CN1	0.1582
CN2	0.0063	CN2	0.0078
CN3	0.0094	CN3	0.0049
CN4	0.0087	CN4	0.0065
RN6	0.0071	RN6	0.0064
CN5	0.0072	CN5	0.0083
CN6	0.0072	CN6	0.0079
RN7	0.0081	RN7	0.0061
RN8	0.1155	RN8	0.1201
CN7	0.0069	CN7	0.0058
RN9	0.0089	RN9	0.0077
CN8	0.0081	CN8	0.0087
SN1	0.0004	SN1	0.0003
CN9	0.0048	CN9	0.0041
RN10	0.0067	RN10	0.0046
SN2	0.1011	SN2	0.1109
RN11	0.0009	RN11	0.0009
RN12	0.1470	RN12	0.1605
RN13	0.0099	RN13	0.0069
CN10	0.1380	CN10	0.1480
RN14	0.0081	RN14	0.0094
CN11	0.0035	CN11	0.0023
RN15	0.0074	RN15	0.0069
RN16	0.0007	RN16	0.0009

illustrates the results of a simulation with over 5000 iterations using Matlab, displaying calculated sensitivity indicators for the BBN model. These outcomes underscore the model’s reliability and its potential to optimize decision-making in hydraulic fracturing scenarios, aligning with the objective of ensuring safer and more efficient operations. To quantitatively compare the prediction accuracy of the order index, two evaluation indexes—root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²)—are introduced. RMSE and MAE reflect the predictive error, with lower values indicating better algorithm performance. R², assessing goodness of fit, ranges from 0 to 1, with a score closer to 1 indicating a superior model fit [35,36,37].

Root mean square error (RMSE):

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n(\widehat{y_i}-y_i{)}^2}$$

(17)

Mean absolute error (MAE):

$$MAE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left|\widehat{y_i}-y_i\right|}^2$$

(18)

Goodness of fit (R²):

$$R^2=1-\frac{{\sum }_{i=1}^n(\widehat{y_i}-y_i{)}^2}{{\sum }_{i=1}^n{(y_i-{\overline{Y}}_i)}^2}$$

(19)

4. Discussion

4.1. Analysis of the Most Important Scenarios Proposed from the BBN Framework

During the well hydraulic fracturing, Bayesian network-driven intelligent artificial analysis explores scenarios, identifying components causing process failure or undesired outcomes. Utilizing a purpose function measures the risk score (RS), enabling precise risk assessment. The function is combined with Bayesian network probabilities to evaluate failure consequences through a sorting probability calculation, as depicted in the Equation below [37,38,39].

$$RS\left(Se\right)=\frac{E\left[\frac{U}{Se}\right]P\left(Se\right)}{E\left[U\right]}$$

(20)

where

$Se$ is the scenario exclusion;

$P\left(Se\right)$ is the probability of scenario Se;

$E\left[U\right]$ is the expected disutility where $E\left[U\right]=\sum _{s\in S^{+}}P\left(S\right)U\left[X_u\left(S_{I\left(u\right)}\right)\right]$;

$X_u\left(S_{I\left(u\right)}\right)$ is the function of combinations of parent states of the value for 1, which belongs to components that fail in the system, and 0, which presents them otherwise;

$RS\left(Se\right)$ is the risk share; and

$E\left[\frac{U}{Se}\right]$ is the conditional expected disutility function for scenario $Se$.

Table 4. The four riskiest scenarios based on the consequence framework.

Important Risks Scenarios from BBN	Nodes Leading Directly to Top Event						The Result Obtained of Probability Failure Hydraulic Fracturing Process, Probability of Scenario and Risks Share
Scenarios	CN1	SN2	RN2	RN8	RN12	CN10	${\mathit{P}}_{\mathit{f}\mathit{a}\mathit{i}\mathit{l}}$	$\mathit{P}\left(\mathit{S}\mathit{e}\right)$	$\mathit{R}\mathit{S}\left(\mathit{S}\mathit{e}\right)$
S1	High	No	No	High	Low	Yes	0.979	0.148	0.204
S2	High	No	No	Low	High	No	0.971	0.082	0.103
S3	Med	No	Yes	Low	Med	No	0.994	0.071	0.088
S4	Low	No	Yes	Low	Low	No	1	0.043	0.079

outlines the outcomes of the sensitivity analysis, emphasizing the most critical scenarios in hydraulic fracturing that pose a potential risk of process failure. Significantly higher probability rates are observed for CN1, CN10, and RN8, signaling heightened risks to the well and reservoir. In contrast, lower rates are attributed to RN12. These results underscore the substantial influence of specific crucial variables, especially inherent factors that are challenging to control. Effectively addressing factors like rock–gel friction in diverse reservoirs becomes essential for enhancing fracture conductivity and projecting fracture pressure to achieve desired operational results [19,26,40].

The findings reveal substantial variations between scenarios, notably indicating a heightened failure rate in the fourth scenario. This emphasizes the critical need to prevent non-homogeneous tank pressure surges as they can significantly jeopardize well performance and overall results. Industry experts in oil and gas, particularly those focused on engineering and production, are urged to reassess the most perilous scenarios, particularly involving CN1, RN8, and factors influencing the RN12 node, such as temperature and unfavorable physical properties [29,40,41].

4.2. The Results of Sensitivity Analysis and Partial Validation

This section consolidates previous efforts and outcomes in constructing an integrated model aimed at averting failures in hydraulic fracturing processes within non-conventional reservoirs. Failures in such processes can result in diminished production, shortened well lifespan, and increased financial and environmental losses. The fault tree was initially developed, followed by the creation of the Bayesian belief network (BBN) model (Figure 10). The MATLAB simulations in Figure 11 provide a real-time, quantitative comparison of prediction accuracy within the BBN modeling framework. This crucial analysis focuses on three specific scenarios: RN1, RN8, and RN10. Two evaluation indices—root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²)—were employed to quantitatively compare predictive accuracy. These results, validated through expert consultation, offer insights into mitigating the most perilous scenarios and conducting simulations to minimize failure risks based on artificial intelligence technology [18,30,35]. The network, depicted in Figure 12 and Figure 13, calculates probabilities based on expert opinions from the shale gas industry, integrating the predictive capabilities of artificial intelligence technology [18,35].

This study contributes to a realistic model, leveraging Bayesian belief network (BBN) and intelligent artificial technology as shown in Figure 14 and Figure 15. The figures illustrate the calculation of sensitivity indices using MATLAB software Version information for MathWorks R2022b with a number of runs between 5000 and 10,000, ensuring the accuracy and robustness of the model. This is crucial for preventing failures in real-time hydraulic fracturing processes in heterogeneous reservoirs with poor permeability. Through sensitivity analysis repetitions and Sobol method utilization, in coordination with expert insights, the approach ensures effective results, aiding in identifying critical failure scenarios and developing control models for operators and engineers to avert failures and achieve desired outcomes [8,30,42].

4.3. The Cost–Benefit Analysis

To assess the economic feasibility of the proposed risk management approach using the BBN model, a cost–benefit analysis was conducted [7,10,31,33]. This study has presented a comprehensive analysis of the hydraulic fracturing process in hydrocarbon reservoirs, specifically focusing on identifying potential failure scenarios and the factors contributing to these failures. This study adopted a Bayesian belief network (BBN) model to effectively predict risks and enable informed decision-making to avoid failures in advance. It also involved experts to help identify the most influential scenarios and calculate the probability of failure using sensitivity analysis techniques. A critical aspect of the present study was the cost–benefit analysis, which allowed the evaluation of the effectiveness of various mitigation plans in reducing the probability of failure and their associated costs. This analysis aimed to provide a comprehensive understanding of the balance between the costs of implementing mitigation measures and the benefits of reduced failure probabilities [7,8]. The cost–benefit analysis, based on the mitigation plans and associated costs of the hydraulic fracturing process, is outlined in

Table 5. Mitigation plans (based on nodes), associated costs, and reduction of probability of failure in the hydraulic fracturing process.

Node	Mitigation Plan	Associated Cost (USD)	Reduction in Probability of Failure
CN1	Improved wellbore stability	50,000	20%
SN2	Enhanced fluid control	30,000	15%
RN2	Optimized pressure management	20,000	10%
RN8	Better reservoir characterization	60,000	25%
RN12	Advanced real-time monitoring	10,000	12%
CN10	Rigorous training of personnel	5000	18%
Total		175,000	100%

and can be summarized as follows.

The total cost of implementing all the proposed mitigation plans is USD 175,000.

Implementing these mitigation plans can significantly reduce the risks associated with hydraulic fracturing operations by leading to more efficient and reliable processes, increased production, and extended lifespan of the wells. The cost–benefit analysis highlights the importance of investing in mitigation measures to minimize the risks of failure in hydraulic fracturing processes, which can ultimately result in significant economic and environmental benefits.

5. Conclusions

The hydraulic fracturing process plays a pivotal role in enhancing production and prolonging well life in shale gas reservoirs. Adopting a dynamic Bayesian belief network (BBN) model becomes imperative to comprehend the multifaceted factors influencing well cracking processes and reservoir dynamics. This model facilitates risk prediction and informed decision-making, steering clear of potential failures. Field studies and expert insights contribute to scenario identification, and a resulting event tree model transforms into a BBN model, depicting causal relationships between factors.

Conducting sensitivity analysis techniques on this dynamic BBN model achieves an impressive 85% accuracy in predicting failure scenarios. Beyond cost savings, the model’s risk management capabilities potentially extend well life by up to 20%, translating into substantial yearly revenue. The BBN model emerges as a financial boon while significantly enhancing operational sustainability and longevity. Its precision in predictions and risk mitigation strategies serves as a linchpin for the success and economic viability of hydraulic fracturing operations.

To enhance the BBN model’s effectiveness in mitigating hydraulic fracturing failures, crucial strategic measures are imperative. Collaboration with industry stakeholders is essential for comprehensive data collection, reinforcing the model’s accuracy with precise datasets. Advanced machine learning and additional data refinement can ensure high accuracy, even in scenarios with limited information like S1 and S4, preventing the risk prevalence probability rate from exceeding 0.01 after model adoption. This strategic enhancement has the potential to elevate prediction accuracy beyond 96%, transforming decision-making. Regular assessment of mitigation plans prioritizes cost-effectiveness, minimizing operational costs while maintaining safety and fostering environmental and financial sustainability in the shale gas sector. This comprehensive approach underscores the pivotal role of the BBN model in revolutionizing hydraulic fracturing operations, providing a pathway to economic success, safety, and environmental responsibility.