A New Empirical Correlation for Pore Pressure Prediction Based on Artificial Neural Networks Applied to a Real Case Study

Abstract

Pore pressure prediction is a critical parameter in petroleum engineering and is essential for safe drilling operations and wellbore stability. However, traditional methods for pore pressure prediction, such as empirical correlations, require selecting appropriate input parameters and may not capture the complex relationships between these parameters and the pore pressure. In contrast, artificial neural networks (ANNs) can learn complex relationships between inputs and outputs from data. This paper presents a new empirical correlation for predicting pore pressure using ANNs. The proposed method uses 42 datasets of well log data, including temperature, porosity, and water saturation, to train ANNs for pore pressure prediction. The trained model, with the Bayesian regularization backpropagation function, predicts the pore pressure with an average absolute percentage error (AAPE) and correlation coefficient (R) of 4.22% and 0.875, respectively. The trained ANN is then used to develop a new empirical correlation that relates pore pressure to the input parameters considering the weights and biases of the optimized ANN model. To validate the proposed correlation, it is applied to a blind dataset, where the model successfully predicts the pore pressure with an AAPE of 5.44% and R of 0.957. The results show that the proposed correlation provides accurate and reliable predictions of pore pressure. The proposed method provides a robust and accurate approach for predicting pore pressure in petroleum engineering applications, which can be used to improve drilling safety and wellbore stability.

1. Introduction

Formation pressure, also known as pore pressure, denotes the pressure exerted by fluids within the pore spaces of rock formations. This pressure plays a crucial role in various aspects of oil and gas well drilling and completion. It influences the choice of drilling rig, design of drilling mud, casing and cementing programs, and overall completion scenarios. Understanding and accurately assessing formation pressure are vital for ensuring the safety, efficiency, and integrity of drilling operations, as well as for mitigating the risks associated with wellbore stability and integrity. Thus, it serves as a fundamental parameter guiding decision making throughout the drilling and completion processes [1,2].

The pore pressure is classified as normal, subnormal, or over-pressure (i.e., geopressure). The normal pore pressure differs from region to region but usually ranges from 0.433 psi/ft to 0.465 psi/ft. Subnormal pressure is less than normal pressure and might result in differential pipe sticking and mud circulation loss. On the other hand, over-pressure, also known as geopressure, occurs when pore pressure exceeds normal pressure. This pressure results from an additional source of pressure being added to the system, which can result in pressure kicks and blowouts during drilling operations. An excess of pressure can be caused by geological, mechanical, geochemical, or geothermal factors, or a combination of these factors. Real-time pore pressure prediction may improve well trajectory, casing, and mud program designs, and provide better wellbore stability analyses, saving drilling time and expense [3,4,5,6].

Typically, drilling through normally pressurized formations poses minimal challenges, particularly when drilling parameters are meticulously planned. Conversely, navigating through subnormally pressurized or over-pressured formations presents substantial risks. These risks encompass a range of issues such as wellbore instability, lost circulation, stuck pipe, kicks, and blowouts [7,8]. Such complications not only disrupt project execution but also incur additional costs and, in the worst-case scenario, pose severe health and safety hazards. Consequently, the precise prediction of pore pressure becomes indispensable for ensuring the safety, efficiency, and success of drilling operations.

Accurate pore pressure prediction aids drilling engineers in making informed decisions for well construction and operation. Anticipating pressure variations allows for tailored drilling strategies, fluid selection, casing programs, and parameters adjustment, mitigating hazards and risks to personnel, equipment, and the environment [3,4]. Accurate pore pressure prediction optimizes operational efficiency, reducing downtime and unexpected issues during drilling. Integrating reliable prediction methodologies enhances project outcomes, lowers costs, and ensures safety standards, aligning with the overarching goals of safety, efficiency, and success in drilling operations [7,8].

Several methods have been developed for predicting pore pressure, including empirical, analytical, and numerical methods [2,9,10,11]. Empirical methods are based on correlations between pore pressure and other well log data, such as sonic velocity, porosity, and resistivity logs. These methods are widely used in the industry due to their simplicity and ease of use. However, empirical methods have limitations, particularly in cases where the correlation is based on a limited dataset, and when the geological setting is different from that used to develop the correlation, which requires modifying the original model parameters to be used in this new geological formation [12].

In recent times, artificial neural networks (ANNs) have surfaced as a viable substitute for empirical approaches in predicting pore pressure. Modeled on the human brain, ANNs are computational frameworks adept at discerning intricate associations between input data and output variables. Their capacity to manage extensive datasets and non-linear connections renders them invaluable in petroleum engineering and various geological disciplines. ANNs offer a promising avenue for pore pressure forecasting, capitalizing on their capability to grasp nuanced patterns within datasets. This marks a significant advancement in predictive techniques within the field, providing a more robust and versatile approach to pore pressure estimation compared to traditional empirical methods [13].

The objective of this study is to employ ANNs in forecasting pore pressure by leveraging well log data such as resistivity, Gamma Ray, bulk formation density, sonic porosity, and neutron porosity. In the first part of this paper, the data were partitioned, with a portion utilized for training the ANNs model. In the second part of this paper, the steps for developing an imperial correlation for pore pressure estimation were explained. The remaining data (after training) served to validate the developed correlation performance.

This proposed methodology for pore pressure prediction holds significant importance, particularly in cases where alternative tools for pore pressure estimation are not feasible. By harnessing ANNs, this approach offers a novel and potentially more effective means of predicting pore pressure, capitalizing on the network’s capacity to discern intricate patterns and relationships within complex datasets. This study presents a valuable contribution to the field of pore pressure forecasting, presenting a viable alternative that may prove instrumental in enhancing accuracy and reliability, especially when conventional methods fall short in certain geological contexts.

2. Field Background

Epsilon is the most recent oilfield to be developed in the Prinos basin, which is currently Greece’s only operational basin for the production of hydrocarbons [14]. The Epsilon oil field is located in the northern part of the Aegean Sea, approximately 11 km south-southeast of Kavala, and between the Greek mainland and Thassos island (Figure 1). The year 1976 marked the beginning of the company’s production activities. However, the first exploration well, drilled in 1971 and situated 20 km east of the island of Thassos, only produced oil with a very low gravity. The subsequent drilling operations, which took place west of the island in 1972 and 1973, led to the discovery of the South Kavala gas field, which was located at a sea depth of 52 m. Finally, the fourth attempt at drilling, which took place in the middle of the Prinos basin toward the end of 1973, was ultimately successful in locating the Prinos oil field, which was situated in water approximately 30 m deep.

Figure 2 depicts a stratigraphy model of the taphrogenetic basin as presented by Pollak [17], Proedrou [18], and Proedrou and Sidiropoulos [19]. On the basis of this, the stratigraphic column can be divided into three series: the Pre-Evaporitic Series is composed of conglomerates, sandstones, siltstone/shales, and limestones, the Evaporitic Series consists of salt and anhydrite, and the Post-Evaporitic Series is primarily made up of siltstone/shales and sandstones. The general structure of the taphrogenetic basin consists of an anticline resembling a dome, which is formed by northwest-oriented and southwest-dipping synkinematic faults. According to Proedrou and Papaconstantinou [15] and Kiomourtzi [14], these faults are interconnected with the trapping mechanism. The reservoirs are primarily composed of sandstones and siltstones that were deposited during the upper Miocene in various environments, including deltaic, marine, and turbiditic settings. These depositional particulars are documented in Kiomourtzi et al. [16,20] and Choustoulakis’ [21] works. Evaporites encompass the entire basin, effectively trapping hydrocarbons beneath them. However, in the South Kavala and Ammodhis fields, the upward migration of hydrocarbons is possible due to the activation of faults. Porosity and permeability typically decrease with increasing depth due to the weight of overlying layers, clay content, and dolomitization [15].

3. Methodology

3.1. Data Preparation

In order to achieve accurate pore pressure predictions, data from 48 measurements, including formation temperature, porosity, water saturation, and pore pressure, were collected from Well-1. The formation temperature, porosity, and water saturation were considered in this work because of their physical relationship with the pore pressure. Porosity, the volume of pore space in a rock formation, directly influences pore pressure by determining the fluid storage capacity within the reservoir. Temperature affects pore pressure through its impact on fluid density and viscosity, altering the fluid flow behavior within the pores. Water saturation, the fraction of pore space occupied by water, plays a significant role in pore pressure prediction, as it dictates the fluid pressures exerted within the reservoir. By leveraging these physical relationships, ANN models can effectively learn and capture the complex nonlinear interactions between these variables to accurately forecast pore pressure in subsurface formations.

Before learning the ANN model, the collected data underwent a preprocessing stage aimed at enhancing data quality. Initially, this involved filtering out irregular values, such as water saturation readings below 0 or exceeding 1.0, which were considered invalid. Subsequently, a secondary preprocessing step targeted outlier removal. Outliers were identified based on the statistical distribution of the dataset, with values lying beyond ±3.0 standard deviations deemed as outliers. These outlier values, potentially indicative of errors or anomalies, were eliminated from the dataset to ensure the reliability and integrity of the input data. Through these preprocessing measures, the dataset was refined, thus enhancing the robustness and accuracy of subsequent pore pressure predictions derived from the processed data. All input parameters were predicted with errors of less than 5%.

The preprocessing procedure plays a critical role in maintaining the integrity and reliability of data utilized for pore pressure prediction. Eliminating erroneous or invalid values ensures that only meaningful data contribute to the accuracy of the predictive model. Filtering out outliers reduces the impact of extreme observations, enhancing the model’s robustness and effectiveness in forecasting pore pressure. By ensuring the quality of input data, preprocessing minimizes the risk of inaccuracies and distortions, thus bolstering the model’s capability to generate the reliable predictions essential for informed decision making in drilling and reservoir engineering applications.

3.2. ANN Model Learning

The ANN used in this work is a computational model inspired by the structure and function of the human brain, designed to process complex data and learn patterns. These models consist of interconnected nodes arranged in layers: input, hidden, and output. Each node applies a transformation to its inputs using weighted connections and activation functions. Through iterative training, ANNs adjust these weights to minimize errors and optimize performance in tasks such as classification, regression, and pattern recognition [22,23,24].

Following the completion of the data preprocessing phase, a total of 46 measurements were selected for the subsequent model-learning process. The objective was to develop an ANN model capable of accurately predicting pore pressure based on the gathered dataset. Throughout this learning phase, the careful optimization of both the percentage of training data points and the design parameters of the ANN model was conducted.

The optimization process involved fine-tuning several key design parameters critical to the effectiveness and efficiency of the ANN model. These parameters encompassed the selection of an appropriate learning function, determining the optimal number of learning layers within the network architecture, specifying the ideal number of learning neurons allocated to each layer, and defining the most suitable transferring function to facilitate information flow within the network.

Optimizing these design parameters is fundamental to ensuring the ANNs model’s ability to capture complex relationships within the data and make accurate predictions of pore pressure. By meticulously adjusting these parameters, the model can effectively learn from the training data and generalize its predictions to unseen data points. This optimization process enhances the model’s predictive performance and robustness, ultimately yielding more reliable estimations of pore pressure in the target well.

The optimization of the training data percentage, ranging from 40% to 95% with a step size of 5%, was executed using a loop embedded within MATLAB. This iterative process aimed to determine the most suitable proportion of data to be utilized for training the model. Subsequently, the performance of various training functions was evaluated to identify the most effective approach for training the ANN model. This assessment encompassed testing the Levenberg–Marquardt backpropagation function [25], resilient backpropagation function [26], and Bayesian regularization backpropagation function [27], each known for its unique characteristics and suitability in different scenarios.

Furthermore, the effectiveness of different transferring functions in predicting pore pressure was investigated. This analysis involved testing the tangential sigmoidal function, logarithmic sigmoidal function, and linear function [28,29,30], each offering distinct mathematical properties that can impact the model’s predictive capabilities. Additionally, the influence of the number of layers in the ANN model architecture was explored, ranging from single-layer configurations to models with two or three layers. Within each layer, varying neuron counts, ranging from 4 to 30 neurons per layer, were examined to assess their impact on the model’s performance [31].

The extensive assessment undertaken in this study sought to pinpoint the most effective amalgamation of training data percentage, training and transferring functions, and network architecture crucial for precise pore pressure prediction. Through a methodical examination of these parameters, the objective was to refine the predictive accuracy and resilience of the ANN model, thereby furnishing dependable estimations of pore pressure within the designated well. By systematically probing these variables, this study aimed to bolster the model’s capability to generate reliable predictions, thus enhancing its applicability and reliability in real-world scenarios.

This comprehensive evaluation process underscores the meticulous approach taken to optimize the ANN model for pore pressure prediction, ensuring that it is finely tuned to capture the intricate relationships within the data. Ultimately, the goal is to equip stakeholders with a robust tool that delivers accurate and consistent pore pressure estimations, facilitating informed decision making and enhancing operational efficiency in drilling and reservoir engineering endeavors. Through this rigorous evaluation, this study endeavors to advance the field’s understanding of pore pressure dynamics and contribute to the development of more effective predictive models tailored to the unique challenges of geological formations.

The outcomes of the learning phase indicated that employing 91% of the available data for training purposes, coupled with a single learning layer comprising five neurons, yielded optimal results for predicting pore pressure. The choice of the Bayesian regularization backpropagation function for training the model, along with the Logarithmic sigmoidal transferring function, further enhanced predictive accuracy. These configurations were determined through rigorous testing and an evaluation of various combinations of the ANN design parameters.

The Bayesian regularization backpropagation function used in this study integrates Bayesian principles into the traditional neural network training algorithm to tackle overfitting. It introduces prior distributions over model parameters, encouraging simpler models and accounting for parameter uncertainty. During training, these priors are combined with the likelihood of the data to compute posterior distributions over parameters, resulting in more robust models capable of better generalization to unseen data.

Figure 3 illustrates the schematic representation of the optimized ANN model designed specifically for pore pressure estimation. Meanwhile,

Table 1. The optimized parameters for pore pressure prediction.

Parameter	Optimum Value
Training data points	42
Training/testing data ratio	91/9
Training layers	Single
Number of neurons	23
Training function	Bayesian regularization backpropagation
Transferring function	Logarithmic sigmoidal function

provides a concise summary of the key properties and characteristics of this optimized model, offering insights into its architecture and functionality.

Additionally,

Table 2. The statistical features of the 42 datasets used to train the ANNs for pore pressure estimation. This reference table represents the applicable range of the training data and the recommended range to apply the developed ANN model and the extracted equation.

	Temp. (°C)	Porosity (frac.)	Sw (frac.)	Pore Pressure (psi)
Minimum	137.3	0.037	0.001	4254
Maximum	149.2	0.182	0.999	6496
Average	143.0	0.104	0.510	5473
Mean	142.9	0.098	0.352	5440
Median	142.8	0.106	0.473	5165
Standard deviation	3.139	0.036	0.289	615.5

presents the statistical features and ranges of the datasets used to train the ANN model. This table serves as a reference point for understanding the distribution and variability of the data utilized in the training process, providing valuable context for interpreting the model’s performance. This data range in Table 2 (i.e., minimum and maximum values) is also important when the model optimized in this work is to be used to predict the pore pressure for other new data, so it is important to ensure that these new data fall in the range of the training data shown in Table 2.

3.3. Extracting Equation from the Optimized ANN Model

After evaluating the ANNsmodel depicted in Figure 3 and examining the properties outlined in Table 1, the foundation was laid for formulating a novel equation to predict pore pressure. This equation, represented as Equation (1), encapsulates the essence of the ANN model, incorporating its key features and parameters. The development of empirical correlations from optimized models is common practice nowadays in many fields, not only for the prediction of ROP or the use of an ANN [32].

By leveraging the insights gained from the ANN model and the properties detailed in Table 1, this new equation offers a comprehensive approach to pore pressure prediction, potentially enhancing accuracy and reliability in forecasting this critical parameter in geological settings.

$$P_{p,n}=\sum _{i=1}^I{w}_{o_i}logsig\left(\sum _{j=1}^J{w}_{t_{i,j}}{Y}_j+b_{t_i}\right)+b_o=\sum _{i=1}^I{w}_{o_i}\frac{1}{1+e^{-\left(\sum _{j=1}^J{w}_{t_{i,j}}{Y}_j+b_{t_i}\right)}}+b_o$$

(1)

where P_p,n is the normalized pore pressure and I and J denote the total number of neurons and input parameters, which are five and four, respectively, in this case. w and b represent the weights and biases, respectively. t denotes the training layer, while o represents the output layer.

Data normalization is essential for ensuring the optimal performance of the ANN model. Normalization involves scaling the input data to a standard range, typically between 0 and 1 or −1 and 1, to eliminate variations in magnitude and units among different features [33,34]. This process prevents certain input features from dominating the learning process due to their larger scale, ensuring fair treatment for all features during training. Failure to normalize data may lead to slow convergence during training, instability in the learning process, and suboptimal model performance. Additionally, normalization helps the ANN generalize better to unseen data by reducing the risk of overfitting to the training dataset [35,36].

It is important to mention that the parameters introduced into the ANN model are usually automatically normalized by the two-point slope, explained by Equations (2) and (3), to have a value between the 1 and −1 range.

$$\frac{{\mathrm{Y}}_n-Y_{n,min}}{Y_{n,max}-Y_{n,min}}=\frac{\mathrm{Y}-Y_{min}}{Y_{max}-Y_{min}}$$

(2)

$${\mathrm{Y}}_n=\left(\frac{\mathrm{Y}-Y_{min}}{Y_{max}-Y_{min}}\right)\left(Y_{n,max}-Y_{n,min}\right)+Y_{n,min}$$

(3)

where Y denotes the input parameter, which could be the formation temperature, porosity, water saturation, or the pore pressure; the subscript n indicates the normalized parameter; and min and max represent the minimum and maximum, respectively. For example, since the input parameters are normalized between −1 and 1, Y_n,min will be −1 and Y_n,max will be 1; therefore, Equation (3) could be written as in Equation (4).

$${\mathrm{Y}}_n=2\left(\frac{\mathrm{Y}-Y_{min}}{Y_{max}-Y_{min}}\right)+1$$

(4)

General expressions for normalizing the four inputs considered in this study could be derived by substituting for each parameter’s minimum and maximum values, as listed in Table 2, to obtain the expressions in Equation (5) to Equation (7).

$${Temp}_n=0.168\left(Temp-137.3\right)+1$$

(5)

$${\varnothing }_n=13.76\left(\varnothing -0.037\right)+1$$

(6)

$$S_{w,n}=2.005\left(S_w-0.001\right)+1$$

(7)

where Temp_n, Φ_n, and S_w,n are the normalized formation temperature, porosity, and water saturation, respectively. Now, Equation (1) could be expanded and written as a function of these normalized parameters, as in Equation (8).

$$P_{p,n}=\sum _{i=1}^I{w}_{o_i}\frac{1}{1+e^{-\left(w_{t_{i,1}}{Temp}_n+w_{t_{i,2}}{\varnothing }_n+w_{t_{i,3}}{S}_{w,n}+b_{t_i}\right)}}+b_{\mathrm{o}}$$

(8)

The normalized parameters in Equation (8) could be calculated using Equation (5) to Equation (7), while the weights and biases needed in this equation were extracted from the optimized ANN model and listed in

Table 3. The extracted weights and biases for pore pressure estimation, extracted from the optimized ANN model, to be substituted into Equation (10).

Number of Neurons		Input Layer				Output Layer
		Weights (w1)			Biases (b1)	Weights (w2)	Bias (b2)
		j = 1	j = 2	j = 3
Number of neutrons	i = 1	−1.825	2.375	1.390	4.202	−1.042	−0.517
	i = 2	−1.431	3.561	0.507	3.376	−1.318
	i = 3	1.006	−3.680	0.470	−2.797	−0.071
	i = 4	−1.945	2.549	2.255	2.814	1.344
	i = 5	2.297	−1.653	2.230	−2.424	0.362
	i = 6	−2.276	2.811	0.491	1.865	0.698
	i = 7	2.451	0.371	−3.713	−1.987	3.041
	i = 8	1.078	−3.616	1.589	−0.991	1.077
	i = 9	−0.665	−3.647	−2.651	0.932	−1.268
	i = 10	3.202	−2.151	−0.666	−0.737	−1.491
	i = 11	0.972	3.684	−3.220	−0.105	−1.330
	i = 12	−0.597	−0.094	4.154	0.021	0.340
	i = 13	−3.951	−0.706	−1.273	−0.564	−1.303
	i = 14	−2.755	0.112	−2.463	−0.358	0.044
	i = 15	−1.428	0.139	4.810	−1.523	−2.601
	i = 16	−0.660	1.727	−3.608	0.118	−0.939
	i = 17	−1.077	−1.853	−2.772	−2.021	−0.915
	i = 18	−3.461	1.505	−0.247	−1.509	0.546
	i = 19	−1.757	2.869	−1.241	−2.478	−0.020
	i = 20	1.473	−3.009	−1.478	2.658	0.130
	i = 21	0.104	−4.950	0.983	3.967	3.653
	i = 22	1.288	3.016	1.916	2.977	0.020
	i = 23	−3.257	2.202	−1.787	−3.718	2.052

.

As calculated from Equation (8), the pore pressure is in the normalized form, which requires denormalization to obtain the actual pore pressure values. The denormalized pore pressure could be calculated using Equation (9), which is derived by rearranging Equation (4) and substituting for pore pressure instead of Y.

$$P_p=\frac{\left(P_{p,n}+1\right)}{2}\left(P_{p,max}-P_{p,min}\right)+P_{p,min}$$

(9)

After substituting the minimum and maximum values of pore pressure from Table 2, and the normalized pore pressure from Equation (8) into Equation (9), we can obtain Equation (10).

$$P_p=1121\left(\sum _{i=1}^I\left[w_{o_i}\frac{1}{1+e^{-\left(w_{t_{i,1}}{Temp}_n+w_{t_{i,2}}{\varnothing }_n+w_{t_{i,3}}{S}_{w,n}+b_{t_i}\right)}}\right]+0.483\right)+4254$$

(10)

3.4. Testing the Suggested Equation for Pore Pressure

The validation process of the optimized ANNs model, along with the equations derived from it, are integral for evaluating its generalization performance, averting overfitting, fine-tuning hyperparameters, comparing models, identifying data inconsistencies, and instilling trust and confidence in the model’s efficacy [37,38]. This validation stage serves as a critical checkpoint to ensure that the model effectively addresses the designated problem and performs well on unseen data.

Subsequently, the derived Equation (10) correlation underwent testing using the remaining datasets obtained from Well-1. The accuracy of Equation (10) was assessed by examining the average absolute percentage error (AAPE) and correlation coefficient between the actual and predicted pore pressure values. This evaluation process serves to validate the predictive capability of the developed equation and ascertain its reliability in forecasting pore pressure across different datasets. By comparing the actual and predicted values, analysts can gauge the degree of agreement between the model’s estimations and real-world observations, providing insights into its effectiveness and precision.

Overall, the validation process acts as a vital quality assurance step, ensuring that the ANN model and derived equations are robust, dependable, and capable of delivering accurate predictions. Through rigorous validation, stakeholders can gain confidence in the model’s performance and make informed decisions based on its output, thereby enhancing the efficacy and trustworthiness of predictive modeling in geoscience applications.

4. Results and Discussion

4.1. Learning the Artificial Neural Network Model

Figure 4 presents a comparison between the actual and estimated pore pressure values for the training dataset sourced from Well-1, comprising 42 data points. The analysis reveals a notably high correlation coefficient of 0.875 and an AAPE of 4.22%. These metrics signify the high level of accuracy achieved by the trained ANN model in predicting pore pressure.

Moreover, a visual inspection of the actual versus estimated pore pressure plots further underscores the remarkable concordance between the two, affirming the precision and reliability of the ANN model’s predictions, as indicated in Figure 5. Although the alignment is less for data with pore pressures greater than 6000 psi, which is attributed to the low amount of data present to train the model at that range, the error in the predicted pore pressure is still less than 12%, which is still a small error. The close alignment observed between the actual and estimated pore pressure values reinforces the robustness and effectiveness of the ANN model in capturing the underlying patterns and relationships within the data.

Overall, the findings depicted in Figure 4 underscore the utility of the ANN model as a valuable predictive tool for pore pressure estimation in geoscience applications.

4.2. Testing the Suggested Equation for Pressure

Developing an accurate empirical correlation for pore pressure prediction holds immense practical significance in engineering tasks related to well drilling and well design. Pore pressure prediction is crucial for ensuring drilling safety, optimizing drilling parameters, and minimizing costly drilling hazards. Accurate predictions help engineers anticipate pressure changes within subsurface formations, enabling them to implement appropriate drilling strategies, select suitable drilling fluids, and design effective casing programs. By utilizing ANNs to predict pore pressure, engineers can streamline drilling operations, reduce non-productive time, and enhance wellbore stability. Furthermore, precise predictions aid in optimizing well designs, including the placement of casing strings and perforation intervals, thereby maximizing hydrocarbon recovery while minimizing the risks associated with fluid influxes, formation collapses, and wellbore instability.

Following the development of the empirical correlation represented by Equation (10), its efficacy was scrutinized through testing, utilizing pore pressure data dependent on formation temperature, porosity, and water saturation. The assessment was conducted using the remaining dataset from Well-1, consisting of four data points. The results of this evaluation, depicted in Figure 6, demonstrate the robust predictive capabilities of Equation (10) in estimating pore pressure accurately. Notably, the correlation exhibited a high correlation coefficient of 0.957, indicating a strong linear relationship between the predicted and actual pore pressure values. Additionally, the AAPE was found to be 5.44%, signifying a relatively low level of deviation between the predicted and observed pore pressure values.

This validation process is pivotal in affirming the reliability and accuracy of the developed empirical correlation. By comparing the predicted pore pressure values with the actual measurements, analysts can assess the performance of the correlation equation across different datasets. The high correlation coefficient and low average absolute percentage error observed in Figure 6 reinforce the effectiveness of Equation (10) in accurately estimating pore pressure based on formation temperature, porosity, and water saturation data. These findings instill confidence in the predictive capabilities of the correlation equation, highlighting its utility as a valuable tool for pore pressure estimation in geological settings. Moreover, the successful validation of Equation (10) underscores its potential to enhance decision-making processes in drilling and reservoir engineering applications, ultimately contributing to the optimization of well planning and operational strategies.

Despite the high accuracy of the optimized ANN model and the correlation of Equation (10), it is important here to mention that one of the limitations for these models and correlations is that both of them are developed based on data collected from a single well with only 42 training data points; therefore, it is recommended to consider the outcomes of this study as a guide for using ANNs for pore pressure prediction and improve this prediction by incorporating more data collected from various sites.

5. Conclusions

This study highlights the potential of artificial neural networks (ANNs) in pore pressure prediction and encourages further research in this area. The use of ANNs in predicting pore pressure is a relatively new approach, but our results show that it has significant advantages. The ability of ANNs to handle complex relationships between input variables and output variables is beneficial in pore pressure prediction, where multiple factors can influence the pore pressure of a formation.

This paper proposes a new empirical correlation for pore pressure prediction based on ANNs; this correlation was validated using real data from wells from the Epsilon oil field in Greece. The proposed model and empirical correlation achieved high accuracy in predicting pore pressure, which is important for optimizing the planning of drilling operations.

Based on the study findings, the ANN model trained on the provided data accurately predicted pore pressure, yielding an AAPE of 4.22% and an R of 0.875. The extracted equation from the model was further tested on data obtained from the same training well, resulting in an estimated pore pressure with an AAPE of 5.44% and an R of 0.957.

The proposed model has several advantages in terms of accuracy, robustness, and versatility. Firstly, it can handle complex relationships between input and output variables, which is essential for accurate pore pressure prediction. Secondly, it can be easily integrated into existing drilling software, providing real-time pore pressure predictions during drilling operations. Thirdly, the pore pressure values predicted through this model are also necessary to optimize the drilling parameters, such as the mud weight, which can significantly reduce drilling costs.