Data‑Driven Fracture Morphology Prognosis from High Pressured Modified Proppants Based on Stochastic‑Adam‑RMSprop Optimizers; tf.NNR Study

: Data‑driven models with some evolutionary optimization algorithms, such as particle swarm optimization (PSO) and ant colony optimization (ACO) for hydraulic fracturing of shale reservoirs, have in recent times been validated as one of the best‑performing machine learning al‑ gorithms. Log data from well‑logging tools and physics‑driven models is difficult to collate and model to enhance decision‑making processes. The study sought to train, test, and validate synthetic data emanating from CMG’s numerically propped fracture morphology modeling to support and enhance productive hydrocarbon production and recovery. This data‑driven numerical model was investigated for efficient hydraulic‑induced fracturing by using machine learning, gradient descent, and adaptive optimizers. While satiating research curiosities, the online predictive analysis was conducted using the Google TensorFlow tool with the Tensor Processing Unit (TPU), focusing on lin‑ ear and non‑linear neural network regressions. A multi‑structured dense layer with 1000, 100, and 1 neurons was compiled with mean absolute error (MAE) as loss functions and evaluation metrics concentrating on stochastic gradient descent (SGD), Adam, and RMSprop optimizers at a learning rate of 0.01. However, the emerging algorithm with the best overall optimization process was found to be Adam, whose error margin was 101.22 and whose accuracy was 80.24% for the entire set of 2000 synthetic data it trained and tested. Based on fracture conductivity, the data indicates that there was a higher chance of hydrocarbon production recovery using this method.


Introduction
Hydrocarbon production decline [1,2] poses substantive threats to energy sustainability; hence, the demand for resolving this challenge in both conventional and unconventional wells is on a marathon course.Our contemporary research community has enhanced the practical and technical hydraulic fracturing [3,4] means to intensify the recovery of oil and gas in unconventional reservoirs.This well stimulation [5,6] method could also be termed as fracking [7], which consists of passing high-pressure fluids that are simply made of chemical additives, sand, and water (proppants) for opening and holding up channels for the production of excess stored hydrocarbons.However, under this technique, it is often linear to investigate and initiate the fracturing process, observe the fluid flow from the fractured formation, and determine the fracture propagation.The primary purpose of hydraulic fracturing is to increase the hydrocarbon productive index targeted at lowpermeable formations, for instance, shale formations [8][9][10].Hydrocarbon production decline is mostly attributed to formation damage, which is one of the reasons emanating from poorly designed drilling and completion fluids [11,12].The leak of these fluids seals off the formation of pore throats and void spaces [13,14], preventing the flow of fluids from the formation to the wellbores.
Empirically, it is often prudent to study and design appropriate hydraulic fracturing methods before their inception.Researchers [15,16] have investigated several ways to stimulate non-productive wells coupled with effective predictive analysis by designing numerical models to counter poor flow regimes in the formation.Successful predictive studies are the result of the type and geometric structure of the formation morphology, the type of proppants [17,18], and their mechanical stress capabilities, fracture length, and infinite conductivity.
Based on physics-driven models, Suri-Islam-Hossain (SIH) [19] used an extended finite element method (XFEM) to simulate fluid leak-off effects under proppant transport for fracture propagation.Their hydrodynamic integrated model, as shown in Figure 1, demonstrated an XFEM initial pressure for fracturing set to 7497 psi.The results of their study indicate that the proppants' transport and their relative suspension are largely influenced by an increased rate of injection.production decline is mostly attributed to formation damage, which is one of the reasons emanating from poorly designed drilling and completion fluids [11,12].The leak of these fluids seals off the formation of pore throats and void spaces [13,14], preventing the flow of fluids from the formation to the wellbores.Empirically, it is often prudent to study and design appropriate hydraulic fracturing methods before their inception.Researchers [15,16] have investigated several ways to stimulate non-productive wells coupled with effective predictive analysis by designing numerical models to counter poor flow regimes in the formation.Successful predictive studies are the result of the type and geometric structure of the formation morphology, the type of proppants [17,18], and their mechanical stress capabilities, fracture length, and infinite conductivity.
Based on physics-driven models, Suri-Islam-Hossain (SIH) [19] used an extended finite element method (XFEM) to simulate fluid leak-off effects under proppant transport for fracture propagation.Their hydrodynamic integrated model, as shown in Figure 1, demonstrated an XFEM initial pressure for fracturing set to 7497 psi.The results of their study indicate that the proppants' transport and their relative suspension are largely influenced by an increased rate of injection.However, subsequent physics-driven simulations [20], conducted by Wang et al. [21], explain how permeability testing for coal bed methane deposits can be carried out safely and effectively without blow-ups.The authors further indicated that the direction of fracture can become uncertain, since fracture channels tend to expand in the direction of principal stress.In Figure 2, Wang et al., using PFC2D, modeled and simulated a directional hydraulic fracturing (DHF), whose findings demonstrated that fracture propagation can be regulated using the DHF approach [22], as this overcomes its original or principal stress, and for this reason, it is asserted that fracture propagation extends along and perpendicular to the slotting.
Martyushev et al. [23] expounded the use of machine learning (ML) for the predictive optimization of reservoir pressure in directional hydraulic fracturing (DHF) carbonate reservoirs.Their study considered hydraulically fractured Well 423 on the D3fm oil deposit site, as presented in Figure 3.The focus of ML modeling was based simply on the interactions and influences of the neighboring wells (9070, 430, 424, 427, 433) on Well 423, before and after DHF.The relationship for the model was referred to as the coefficient of correlation (r), as demonstrated in Figure 3.The result of their research indicates that the higher the correlation coefficient, the more accurate the reservoir pressure prediction, and as demonstrated, Well 423, before and after DHF, presents increased pressure levels, indicating a red region, and low reservoir pressures correspond to a lower correlation coefficient, indicating a yellow and blue region.However, subsequent physics-driven simulations [20], conducted by Wang et al. [21], explain how permeability testing for coal bed methane deposits can be carried out safely and effectively without blow-ups.The authors further indicated that the direction of fracture can become uncertain, since fracture channels tend to expand in the direction of principal stress.In Figure 2, Wang et al., using PFC2D, modeled and simulated a directional hydraulic fracturing (DHF), whose findings demonstrated that fracture propagation can be regulated using the DHF approach [22], as this overcomes its original or principal stress, and for this reason, it is asserted that fracture propagation extends along and perpendicular to the slotting.
Martyushev et al. [23] expounded the use of machine learning (ML) for the predictive optimization of reservoir pressure in directional hydraulic fracturing (DHF) carbonate reservoirs.Their study considered hydraulically fractured Well 423 on the D 3 fm oil deposit site, as presented in Figure 3.The focus of ML modeling was based simply on the interactions and influences of the neighboring wells (9070, 430, 424, 427, 433) on Well 423, before and after DHF.The relationship for the model was referred to as the coefficient of correlation (r), as demonstrated in Figure 3.The result of their research indicates that the higher the correlation coefficient, the more accurate the reservoir pressure prediction, and as demonstrated, Well 423, before and after DHF, presents increased pressure levels, indicating a red region, and low reservoir pressures correspond to a lower correlation coefficient, indicating a yellow and blue region.However, reservoir pressures migrating from neighboring wells 429, 427, and 424 to Well 423 before DHF present a case where the tendency of a well blowout is obvious while drilling.The ML predictive analysis presented in the case of Martyushev et al. [23] supports managerial decision-making to optimize drilling operations.
Nonetheless, there has also been abundant research on data-driven models for the prediction of hydraulic fracturing well stimulation.Dong et al. [24] optimized fracture parameters using data-driven algorithms.The authors explain that there is a high cost and driven uncertainty associated with fracture spacing and half-length.For this reason, the research expounded on the use of an evolutionary optimization algorithm (EOA) for  However, reservoir pressures migrating from neighboring wells 429, 427, and 424 to Well 423 before DHF present a case where the tendency of a well blowout is obvious while drilling.The ML predictive analysis presented in the case of Martyushev et al. [23] supports managerial decision-making to optimize drilling operations.
Nonetheless, there has also been abundant research on data-driven models for the prediction of hydraulic fracturing well stimulation.Dong et al. [24] optimized fracture parameters using data-driven algorithms.The authors explain that there is a high cost and driven uncertainty associated with fracture spacing and half-length.For this reason, the research expounded on the use of an evolutionary optimization algorithm (EOA) for However, reservoir pressures migrating from neighboring wells 429, 427, and 424 to Well 423 before DHF present a case where the tendency of a well blowout is obvious while drilling.The ML predictive analysis presented in the case of Martyushev et al. [23] supports managerial decision-making to optimize drilling operations.
Nonetheless, there has also been abundant research on data-driven models for the prediction of hydraulic fracturing well stimulation.Dong et al. [24] optimized fracture parameters using data-driven algorithms.The authors explain that there is a high cost and driven uncertainty associated with fracture spacing and half-length.For this reason, the research expounded on the use of an evolutionary optimization algorithm (EOA) for parametric fracture optimization.Hence, their resulting numerical simulation, based on a gradient-boosted decision tree, random forest, support-vector machine, and multilayer perception (MLP), demonstrated in Figure 4, shows that among all the four productionprediction models, one of the EOA, i.e., particle swarm optimization (PSO), produced the highest net present value.parametric fracture optimization.Hence, their resulting numerical simulation, based on a gradient-boosted decision tree, random forest, support-vector machine, and multilayer perception (MLP), demonstrated in Figure 4, shows that among all the four productionprediction models, one of the EOA, i.e., particle swarm optimization (PSO), produced the highest net present value.In recent times and in this current study, the neural network prognosis architectures have not only looked at the deep neural network Keras architectures, such as sequential, functional, and subclassing API analysis, but there has also been an advance investigation on the use of convolutional neural networks (CNN) and recurrent neural networks (LSTM), with a proposed extension of optimizers.However, the likes of Elbaz and Shen [25][26][27] have proven in their research the possibility of advancing the neural network architecture prognosis.
In other words, while maintaining the TensorFlow Keras Sequential API architecture, a synthetic dataset for training and testing using the most effective neural network optimizers from the current study is essential for reducing predicted errors in the petroleum fracking sector.The stochastic gradient descent [28,29] algorithm used is evaluated for big datasets, with the intention of selecting batches at random from the total dataset for each iteration.In order to roughly obtain a minimum, this optimizer sorts to shuffle the data at random for each iteration.Most importantly, in the case of gradient descent, it is not suitable for large datasets, as the convex algorithm does not randomly shuffle the entire dataset, but instead, for every iteration, the whole data is focused on finding the approximate minimum.For this reason, SGD produces a lot of noise, based on the batches for each iteration, and to reach the desired approximate minimum, a higher number of iterations is needed, which brings the total time for computation to a record high.However, it is purported that SGD with higher iterations can optimize noise cancellation.Nonetheless, another means of countering noise production is by the extension of SGD with momentum, imagine propping a naturally fractured and low permeable formation, where the momentum of the proppants in the natural fracture formation gains maximum convergence.Most of all, while considering the momentum, the likelihood that the desired minimum could be reached is high; hence, careful regulation of the number of iterations is needed for better optimization.
Adaptive moment estimation (Adam) [30] is an extension of SGD [31]; whereas, the weights of the entire network under training are optimized by a single learning rate, Adam, on the other hand, concentrates on upgrading each network's weights.Based on its wide usage, several researchers have indicated it as the benchmark for deep learning and standard optimization approaches, since it does not support overtime computation and requires less memory for computation, thereby reducing the entire cost of In recent times and in this current study, the neural network prognosis architectures have not only looked at the deep neural network Keras architectures, such as sequential, functional, and subclassing API analysis, but there has also been an advance investigation on the use of convolutional neural networks (CNN) and recurrent neural networks (LSTM), with a proposed extension of optimizers.However, the likes of Elbaz and Shen [25][26][27] have proven in their research the possibility of advancing the neural network architecture prognosis.
In other words, while maintaining the TensorFlow Keras Sequential API architecture, a synthetic dataset for training and testing using the most effective neural network optimizers from the current study is essential for reducing predicted errors in the petroleum fracking sector.The stochastic gradient descent [28,29] algorithm used is evaluated for big datasets, with the intention of selecting batches at random from the total dataset for each iteration.In order to roughly obtain a minimum, this optimizer sorts to shuffle the data at random for each iteration.Most importantly, in the case of gradient descent, it is not suitable for large datasets, as the convex algorithm does not randomly shuffle the entire dataset, but instead, for every iteration, the whole data is focused on finding the approximate minimum.For this reason, SGD produces a lot of noise, based on the batches for each iteration, and to reach the desired approximate minimum, a higher number of iterations is needed, which brings the total time for computation to a record high.However, it is purported that SGD with higher iterations can optimize noise cancellation.Nonetheless, another means of countering noise production is by the extension of SGD with momentum, imagine propping a naturally fractured and low permeable formation, where the momentum of the proppants in the natural fracture formation gains maximum convergence.Most of all, while considering the momentum, the likelihood that the desired minimum could be reached is high; hence, careful regulation of the number of iterations is needed for better optimization.
Adaptive moment estimation (Adam) [30] is an extension of SGD [31]; whereas, the weights of the entire network under training are optimized by a single learning rate, Adam, on the other hand, concentrates on upgrading each network's weights.Based on its wide usage, several researchers have indicated it as the benchmark for deep learning and standard optimization approaches, since it does not support overtime computation and requires less memory for computation, thereby reducing the entire cost of computation.While there has been overwhelming research curiosity for better adaptation of deep learn-ing optimizers, previous studies in the research community [32,33], few intelligent applications based on the root mean square propagation (RMSprop) optimizer have been published.This adaptive optimizer takes its roots from RProp, known as resilient backpropagation.Since RProp contradicts the theory behind stochastic gradient descent, RMSprop was developed as an extension of RProp.As a result, just as Adam focuses on each network's weight, so does RMSprop.In this case, a weight's specified learning rate is gradually split by the size of its most recent gradients, averaged over time, and determined using the mean square method.Figure 5 presents an illustrative performance of the current study's choice of gradient and adaptive optimizers for the manipulation and tweaking of the synthetic fracking dataset to optimize the predictive petroleum industry.The previous neural network modeling, as conducted by [34], defined an input shape of 28, 28, 1 and upon successfully splitting the improvised data at a dtype float 32, built its model using Keras Sequential, and with activation functions set to ReLU, and SoftMax introduced a loss function of cross entropy under the following optimizers, seen in Figure 5.It is empirically significant to note that the resulting experiment indicates that the Adam optimizer achieved the best performing algorithm, as it is followed by RMSprop and SGD.
Big Data Cogn.Comput.2023, 7, x FOR PEER REVIEW 5 of 18 computation.While there has been overwhelming research curiosity for better adaptation of deep learning optimizers, previous studies in the research community [32,33], few intelligent applications based on the root mean square propagation (RMSprop) optimizer have been published.This adaptive optimizer takes its roots from RProp, known as resilient backpropagation.Since RProp contradicts the theory behind stochastic gradient descent, RMSprop was developed as an extension of RProp.As a result, just as Adam focuses on each network's weight, so does RMSprop.In this case, a weight's specified learning rate is gradually split by the size of its most recent gradients, averaged over time, and determined using the mean square method.Figure 5 presents an illustrative performance of the current study's choice of gradient and adaptive optimizers for the manipulation and tweaking of the synthetic fracking dataset to optimize the predictive petroleum industry.The previous neural network modeling, as conducted by [34], defined an input shape of 28, 28, 1 and upon successfully splitting the improvised data at a dtype float 32, built its model using Keras Sequential, and with activation functions set to ReLU, and SoftMax introduced a loss function of cross entropy under the following optimizers, seen in Figure 5.It is empirically significant to note that the resulting experiment indicates that the Adam optimizer achieved the best performing algorithm, as it is followed by RMSprop and SGD.Another typical study on various optimizers based on different datasets conducted by Mohapatra et al. [35] demonstrates the efficacy of AdaSwarm compared to SGD, AdaGrad, AdaDelta, RMSprop, AMSGrad, Adam, emulating SGD with PSO parameters.The adaptive gradient-based optimizers under a series of compiled models were used for deep learning comparative mean squared and mean absolute errors (MSE/MAE) loss function analysis.The authors, while focusing on swarm intelligence, thus AdaSwarm and the exponentially weighted momentum particle swarm optimizer (EMPSO), whose various parameters were measured against gradient descent (GD), defined the capabilities of these optimizers to execute precise gradient approximations, which further exposes the novelty of their conducted research.Based on the neural network algorithms (EMPSO/AdaSwarm) and subsequent differential and non-differential models proposed by Mohapatra et al. [35], it resulted that the gradient-free adaptive swarm intelligence algorithm (AdaSwarm) had proven superior over other optimizers, such as RMSprop, SGD, and Adam.
In this current study, stochastic gradient descent (SGD), and Adam and RMSprop optimizers for hydrocarbon production recovery predictive analysis were modeled based on high-pressure hydraulic fracturing.Moreover, the concentration of gradient descent and adaptive optimizers is used to train and test hydraulic fracturing on numerically Another typical study on various optimizers based on different datasets conducted by Mohapatra et al. [35] demonstrates the efficacy of AdaSwarm compared to SGD, Ada-Grad, AdaDelta, RMSprop, AMSGrad, Adam, emulating SGD with PSO parameters.The adaptive gradient-based optimizers under a series of compiled models were used for deep learning comparative mean squared and mean absolute errors (MSE/MAE) loss function analysis.The authors, while focusing on swarm intelligence, thus AdaSwarm and the exponentially weighted momentum particle swarm optimizer (EMPSO), whose various parameters were measured against gradient descent (GD), defined the capabilities of these optimizers to execute precise gradient approximations, which further exposes the novelty of their conducted research.Based on the neural network algorithms (EMPSO/AdaSwarm) and subsequent differential and non-differential models proposed by Mohapatra et al. [35], it resulted that the gradient-free adaptive swarm intelligence algorithm (AdaSwarm) had proven superior over other optimizers, such as RMSprop, SGD, and Adam.
In this current study, stochastic gradient descent (SGD), and Adam and RMSprop optimizers for hydrocarbon production recovery predictive analysis were modeled based on high-pressure hydraulic fracturing.Moreover, the concentration of gradient descent and adaptive optimizers is used to train and test hydraulic fracturing on numerically modeled datasets, based on the Google TensorFlow machine learning algorithms.A linear and non-linear neural network regression (NNR) based on these selected optimizers was used to optimize highly modified proppants [36] for effective fracture propagation and production recovery.

Data-Driven Modeling
Building up models generates data that emanates from intelligent tools.Being aware of the difficulty in reading log data, it is practical to use synthetic data for modeling, making a clear-cut validation with real data.However, in the absence of physics-driven simulations, as demonstrated in Figure 1, data-driven model analysis can be the easiest computationally intelligent tool at hand.Moreover, the wholistic parameters involved in modeling data generation originate from the initial reservoir conditions, hydraulic fracture characteristics, and hydrocarbon production.Figure 6 provides the detailed methods and flow chart for an effective propped fracture prognosis.
used to optimize highly modified proppants [36] for effective fracture propagat production recovery.

Data-Driven Modeling
Building up models generates data that emanates from intelligent tools.Bein of the difficulty in reading log data, it is practical to use synthetic data for mo making a clear-cut validation with real data.However, in the absence of physics simulations, as demonstrated in Figure 1, data-driven model analysis can be the computationally intelligent tool at hand.Moreover, the wholistic parameters invo modeling data generation originate from the initial reservoir conditions, hy fracture characteristics, and hydrocarbon production.Figure 6 provides the d methods and flow chart for an effective propped fracture prognosis.

Numerical Modeling
Based on a commercial black oil simulator, CMG's integrated thir geomechanics-based hydraulic fracturing tools, shown in Figure 7, were u numerically model the data, which generated input and output parameter concentrations on porosity (), height (h), fracture length (Lf), fracture width (wf), permeability, and a productivity index (flowing bottom-hole pressure, Pwf).
The 2000-dataset model was numerically focused on shale formations.The study's 3D design [37][38][39] two-phase flow simulation in assumed vertical reservo saturated with oil and gas.The striated vertical and transverse propped propagation of the simulated reservoir obtained its operation perpendicula minimum principal stress, yet in the direction of its maximum principal stress.Ac to Oritz et al. [40] their study initiated the most appreciable dual-permeability pr for modeling two-phased shale plays and natural fractures [41].Their applicable for simulating naturally induced fractures and hydraulic fractures was made pos CMG-IMEX.Notwithstanding, input parameters modeled with CMG by Kulga e

Numerical Modeling
Based on a commercial black oil simulator, CMG's integrated third-party geomechanics-based hydraulic fracturing tools, shown in Figure 7, were used to numerically model the data, which generated input and output parameters, with concentrations on porosity (ϕ), height (h), fracture length (L f ), fracture width (w f ), fracture permeability, and a productivity index (flowing bottom-hole pressure, P wf ).
The 2000-dataset model was numerically focused on shale formations.The current study's 3D design [37][38][39] two-phase flow simulation in assumed vertical reservoirs was saturated with oil and gas.The striated vertical and transverse propped fracture propagation of the simulated reservoir obtained its operation perpendicular to its minimum principal stress, yet in the direction of its maximum principal stress.According to Oritz et al. [40] their study initiated the most appreciable dual-permeability procedure for modeling two-phased shale plays and natural fractures [41].Their applicable method for simulating naturally induced fractures and hydraulic fractures was made possible by CMG-IMEX.Notwithstanding, input parameters modeled with CMG by Kulga et al. [42] yielded promising hydraulic fracturing [43] parameters for numerically synthesizing the data in Table 1.yielded promising hydraulic fracturing [43] parameters for numerically synthesizin data in Table 1.According to the minimum and maximum synthetic data generated, the initiati propping fractures in vertical shale reservoirs is mostly termed to have initial rese conditions with pressures (Pi) of about 500 psi, thermal conditions of 100 de Fahrenheit (T), and an area (A) of about 1000 acres.Additionally, while considerin efficient predictive analysis, synthetic data for hydraulic fractures obtained a maxim fracture length (Lf) of 1500 ft at a height (h) of 500 ft and 30% porosity (ϕ) and a fract permeability (kf) at 0.1 mD.Nonetheless, the width of the fracture based on the data from 0.1 to 0.4 to conductively expound the pore channels for higher productivity increase flowing bottom-hole pressure.

Fluid-Fracture Equations
Figure 8 schematically demonstrates a one-wing infinite homogenous dimensional formation hydraulic fracturing model that was originally propose Perkins and Kern, also known literally as the P-K equation [44].This fracture flow dia depicts how high-pressure proppants or fluids move in the direction of the x-axis w constant height of h on the y-axis.The diameter of the fracture morphology on the z remains the width.It is interesting to note that the fracture length is exponentially gr than that of a constant height and width.These are mathematically represented base the following P-K assumptions: (a) There is no storage effect nor fluid leak off.According to the minimum and maximum synthetic data generated, the initiation of propping fractures in vertical shale reservoirs is mostly termed to have initial reservoir conditions with pressures (P i ) of about 500 psi, thermal conditions of 100 degrees Fahrenheit (T), and an area (A) of about 1000 acres.Additionally, while considering an efficient predictive analysis, synthetic data for hydraulic fractures obtained a maximum fracture length (L f ) of 1500 ft at a height (h) of 500 ft and 30% porosity (ϕ) and a fractured permeability (k f ) at 0.1 mD.Nonetheless, the width of the fracture based on the data was from 0.1 to 0.4 to conductively expound the pore channels for higher productivity or to increase flowing bottom-hole pressure.

Fluid-Fracture Equations
Figure 8 schematically demonstrates a one-wing infinite homogenous two-dimensional formation hydraulic fracturing model that was originally proposed by Perkins and Kern, also known literally as the P-K equation [44].This fracture flow diagram depicts how highpressure proppants or fluids move in the direction of the x-axis with a constant height of h on the y-axis.The diameter of the fracture morphology on the z-axis remains the width.It is interesting to note that the fracture length is exponentially greater than that of a constant height and width.These are mathematically represented based on the following P-K assumptions:  Young Modulus [45] (vertical plane), E'; Young Modulus (plane strain), E'; Maximum fracture width, wm; Fluid continuity [46,47]; The continuity based on the assumption (a) would be further expressed as; Integrating the Newtonian laminar fluid where Equation (3) is at x = 0; Since one of the P-K assumptions [48] is made on a constant flow rate along the fractured axis, attention is given to the fracture length (L), maximum fracture width at the bottom-hole (w ), and net pressure at downhole (P ), which are represented as; Nordgren's improved model adds storage and leak-off effects to make P-K's equation more convincing and practical, as presented in Equations ( 10)-( 12) [49,50]; Young Modulus [45] (vertical plane), E ′ ; Young Modulus (plane strain), E ′ ; Maximum fracture width, w m ; Fluid continuity [46,47]; The continuity based on the assumption (a) would be further expressed as; Integrating the Newtonian laminar fluid where Equation (3) is at x = 0; w 0 = 0.38 Since one of the P-K assumptions [48] is made on a constant flow rate along the fractured axis, attention is given to the fracture length (L), maximum fracture width at the bottom-hole (w 0 ), and net pressure at downhole (P 0 ), which are represented as; ) Big Data Cogn.Comput.2023, 7, 57 9 of 18 Nordgren's improved model adds storage and leak-off effects to make P-K's equation more convincing and practical, as presented in Equations ( 10)-( 12) [49,50]; (11) For hydrocarbon production recovery in vertical shale wells, there is an inflow of the two-phase process.
Further Fracture Assumptions

1.
The rate of flow is assumed.

2.
Fracture is conducted in vertical wells.

3.
Time for injection is considered.

4.
Existing proppants at high pressure are included.

TensorFlow
TensorFlow is an all-inclusive open-source machine learning platform.Its large, versatile ecosystem of tools, libraries, and community resources enables academics to improve the state-of-the-art of machine learning while simultaneously enabling developers to swiftly construct and deploy ML-powered products.TensorFlow was developed by engineers and researchers on the Google Brain team, a division of Google's Machine Intelligence Research department, for the purpose of conducting machine learning and deep neural network research.The technique is versatile enough to be applied in several other industries.TensorFlow offers non-guaranteed backward compatibility for various languages, in addition to established Python and C++ APIs.

Data Pre-Processing and Splitting
In spite of this, the deep neural network analysis developed for the present study used a linear and non-linear method while focusing on selected optimizers (SGD, Adam, and RMSprop) under the impact of learning rates, activation, and loss functions.
Moreover, there was no need to pre-process or normalize the data because it had already been cleaned before importing, based on the Pandas library.The complete length of the 2000 synthetic data was trained at 80% and tested at 20%.

Deep Neural Network (Non-Linear Regression)
The study generated a model for non-linear regression [51][52][53] to determine the influence and prediction of various fractured input parameters over the production recovery with Keras sequential stable input dense layer of 100, 10 and an output layer shape of 1, shown in Figure 9 and Table 2.Moreover, while maintaining a learning rate of about 0.01, the activation function for the input layer was set to a rectified linear unit (ReLU; thus, a non-linear activation function intended for deep neural networks).
The generated model for training the data was compiled, setting the loss function to the mean absolute error and the optimizers to the stochastic descent gradient (SGD), Adam, and RMSprop, respectively, for each of the models' build-ups.Figures 10-15 demonstrates loss curves for various input parameters emanating from fractured height, width, length, permeability, and porosity of the formation and fractured conductivity raw synthetic data, validated in Figure 9.
The study generated a model for non-linear regression [51][52][53] to determine the influence and prediction of various fractured input parameters over the production recovery with Keras sequential stable input dense layer of 100, 10 and an output layer shape of 1, shown in Figure 9 and Table 2.Moreover, while maintaining a learning rate of about 0.01, the activation function for the input layer was set to a rectified linear unit (ReLU; thus, a non-linear activation function intended for deep neural networks).The generated model for training the data was compiled, setting the loss function to the mean absolute error and the optimizers to the stochastic descent gradient (SGD), Adam, and RMSprop, respectively, for each of the models' build-ups.Figures 10-15 demonstrates loss curves for various input parameters emanating from fractured height, width, length, permeability, and porosity of the formation and fractured conductivity raw synthetic data, validated in Figure 9.

Neural Network (Linear Regression)
Production recovery prediction analysis engaged a linear neural network regressor utilizing the Keras sequential model for the synthetic data.The model was compiled with an input shape of 1 and a linear activation, unlike ReLU, as indicated for the non-linear regressor mentioned earlier.However, while the study maintained its reliability, the model was compiled with the same loss functions (MAE) and selected optimizers for training the model with a 0.01 learning rate and an epoch of 100.

Results and Discussion
The performances of optimizers for the various regressors used to predict hydraulic fracturing and production recovery are compiled in Table 3.For better lay understanding, the average of all optimizers based on the various input parameters was obtained.However, Stochastic descent, Adam, and RMSprop optimizers indicated very good performance optimizers, just as other known optimizers demonstrated the capabilities of molding and shaping the fitted model into an accurate form.

Results and Discussion
The performances of optimizers for the various regressors used to predict hydraulic fracturing and production recovery are compiled in Table 3.For better lay understanding, the average of all optimizers based on the various input parameters was obtained.However, Stochastic descent, Adam, and RMSprop optimizers indicated very good performance optimizers, just as other known optimizers demonstrated the capabilities of molding and shaping the fitted model into an accurate form.

Results and Discussion
The performances of optimizers for the various regressors used to predict hydraulic fracturing and production recovery are compiled in Table 3.For better lay understanding, the average of all optimizers based on the various input parameters was obtained.However, Stochastic descent, Adam, and RMSprop optimizers indicated very good performance optimizers, just as other known optimizers demonstrated the capabilities of molding and shaping the fitted model into an accurate form.

Non-Linear Optimizers
The performances of Keras optimizers based on the neural network ReLU compiled the model into a non-linear form.The input data parameters, while considering SGD, obtained an average loss function or a mean absolute error of 261.83,Adam's optimizer computed as 101.22, and RMSprop was 163.87.The final loss comparative analysis indicates that the lower the loss, the more accurate the prediction would be.

Linear Optimizers
The model was compiled based on the linear activation function, and the input parameters fitted the models to obtain an average Adam optimizer loss function of 209.23.Generally, the linear optimizers for the fracture propagation data were determined to have performed inadequately, in contrast to the same optimizers for non-linear functions.This could be as a result of inadequate neurons used, unfavorable learning rates, and a limited number of iterations.However, Figure 21.draws out the entire computation for this study, demonstrating the best optimizer.As mentioned earlier, the lower the loss, the higher the performance; hence, from the visual bar graph, Adam demonstrated the lowest loss for the synthetic fracture propagation data.

Non-Linear Optimizers
The performances of Keras optimizers based on the neural network ReLU compiled the model into a non-linear form.The input data parameters, while considering SGD, obtained an average loss function or a mean absolute error of 261.83,Adam's optimizer computed as 101.22, and RMSprop was 163.87.The final loss comparative analysis indicates that the lower the loss, the more accurate the prediction would be.

Linear Optimizers
The model was compiled based on the linear activation function, and the input parameters fitted the models to obtain an average Adam optimizer loss function of 209.23.Generally, the linear optimizers for the fracture propagation data were determined to have performed inadequately, in contrast to the same optimizers for non-linear functions.This could be as a result of inadequate neurons used, unfavorable learning rates, and a limited number of iterations.However, Figure 21.draws out the entire computation for this study, demonstrating the best optimizer.As mentioned earlier, the lower the loss, the higher the performance; hence, from the visual bar graph, Adam demonstrated the lowest loss for the synthetic fracture propagation data.

Production Recovery Optimization
The synthetic data emanating from CMG modeling was shuffled and indexed to read production recovery, as demonstrated in Figure 21.However, the model used for the earlier prediction indicated that, out of the entire 2000 synthetic datasets replicated, the Adam optimizer prediction, with a 101.22 loss function, was at best 80.24% accurate.Production recovery was based on the flowing bottom-hole pressure; hence, a plot to determine the fracture conductivity where the propped fracture tends to convey formation fluids into the wellbore is demonstrated in Figure 22.
The measure of permeability and fracture width from the data generated explains that the shale formation under review has a better chance of maximizing hydrocarbon production.

Production Recovery Optimization
The synthetic data emanating from CMG modeling was shuffled and indexed to read production recovery, as demonstrated in Figure 21.However, the model used for the earlier prediction indicated that, out of the entire 2000 synthetic datasets replicated, the Adam optimizer prediction, with a 101.22 loss function, was at best 80.24% accurate.Production recovery was based on the flowing bottom-hole pressure; hence, a plot to determine the fracture conductivity where the propped fracture tends to convey formation fluids into the wellbore is demonstrated in Figure 22.
The measure of permeability and fracture width from the data generated explains that the shale formation under review has a better chance of maximizing hydrocarbon production.

Validation and Limitations
The weight of this study was compared with the work of Dong et al. [24], who in recent times developed a machine learning algorithm coupled with other multilayer perception algorithms with an emphasis on particle swarm optimization (PSO).After several tweaks, the authors noticed the PSO, provided the best results, as indicated in Figure 3. Drawing an intense analysis between PSO and Adam optimizers, the current study did not model using PSO, which is recommended for consideration in subsequent research while considering AdaSwarm [25], which is embedded in the NumPy and TensorFlow libraries.
Neural network optimizers, as researched by Mohapatra et al. [25], explicitly provide a comparative analysis of various optimizers, which throws enough light on the current study.In addition, while considering the efficacy of SGD, Adam, and RMSprop, it is quite fair to validate their performances with other relevant datasets.Furthermore, since some of Mohapatra et al.'s derived models consisted of second-order differential equations, validation of their optimizers is best for the current study's non-linear neural network regressions.In so doing, it is worth noting that, when the average of the previous study's loss functions from SGD, Adam, and RMSprop withholding tolerance was duly compared, Adam still performed much better, even considering other datasets.
However, it is advisable to keep in mind that there was probably not enough model and other hyperparameter tweaking to enhance optimization.In the case of Adam, the algorithm was able to achieve an accuracy of 80.24%.Most importantly, Adam's remaining accuracy of 20% could be further optimized to 100% accuracy if long periods of iterations at an epoch of, i.e., 500 to 1000 for both linear and non-linear neural network training could be considered, with a decreasing learning rate of 0.001.

Conclusions
The data-driven sensitive machine learning algorithms, with an emphasis on SGD, Adam, and RMSprop optimizers, have indeed paved an additional artificially intelligent way to optimize synthetic data and its input parameters for fracturing hydrocarbon wells with low permeability indexes, and their tendency to optimize production recovery predictions.Moreover, based on high-pressure proppants, the novelty of the study was able to identify, using Google TensorFlow libraries, that:

Validation and Limitations
The weight of this study was compared with the work of Dong et al. [24], who in recent times developed a machine learning algorithm coupled with other multilayer perception algorithms with an emphasis on particle swarm optimization (PSO).After several tweaks, the authors noticed the PSO, provided the best results, as indicated in Figure 3. Drawing an intense analysis between PSO and Adam optimizers, the current study did not model using PSO, which is recommended for consideration in subsequent research while considering AdaSwarm [25], which is embedded in the NumPy and TensorFlow libraries.
Neural network optimizers, as researched by Mohapatra et al. [25], explicitly provide a comparative analysis of various optimizers, which throws enough light on the current study.In addition, while considering the efficacy of SGD, Adam, and RMSprop, it is quite fair to validate their performances with other relevant datasets.Furthermore, since some of Mohapatra et al.'s derived models consisted of second-order differential equations, validation of their optimizers is best for the current study's non-linear neural network regressions.In so doing, it is worth noting that, when the average of the previous study's loss functions from SGD, Adam, and RMSprop withholding tolerance was duly compared, Adam still performed much better, even considering other datasets.
However, it is advisable to keep in mind that there was probably not enough model and other hyperparameter tweaking to enhance optimization.In the case of Adam, the algorithm was able to achieve an accuracy of 80.24%.Most importantly, Adam's remaining accuracy of 20% could be further optimized to 100% accuracy if long periods of iterations at an epoch of, i.e., 500 to 1000 for both linear and non-linear neural network training could be considered, with a decreasing learning rate of 0.001.

Conclusions
The data-driven sensitive machine learning algorithms, with an emphasis on SGD, Adam, and RMSprop optimizers, have indeed paved an additional artificially intelligent way to optimize synthetic data and its input parameters for fracturing hydrocarbon wells with low permeability indexes, and their tendency to optimize production recovery predictions.Moreover, based on high-pressure proppants, the novelty of the study was able to identify, using Google TensorFlow libraries, that:

•
The linear function for the trained deep neural network on the synthetic dataset was not fully optimized, and the weakest optimizer among them was stochastic gradient descent (SGD), with a mean absolute error of 295.95.

•
While iterating for a non-linear algorithm, Adam emerged as the best-performing optimizer, with a loss function of 101.22.

•
The proliferation of non-linear neural network algorithms for the prediction and optimization of hydraulic fracture morphology is highly recommended.

•
The synthetic data and other conventional data are both suitable for machine learning algorithms and for decisive decision-making procedures.Google TensorFlow libraries present easy access to coding and validation.

•
The overall novelty of the study is that it automates data-driven prognosis by optimizing the hydraulic fracture parameters, from complex CMG numerical modeling to using Keras Sequential API algorithms and several optimizer compilations for decisionmaking analysis.

•
This study limits the complexity of physics-driven computational fracking analysis and provides an industrial automation means of predicting the expectations and remedies for fracking petroleum shale reservoirs.

Data Availability Statement:
The data used is confidential.
(a) There is no storage effect nor fluid leak off.(b) At the tip, the net pressure remains zero.(c) Fluids are Newtonian and incompressible.(d) Fluid injection is assumed to be in constant volumetric flow rate.(e) Because much less energy is needed to propagate a fracture than to simply allow the fluid to flow along it, the toughness of the formation can be disregarded.

Table 2 .
Output screen; standard model summary for all training.
params: 102,201 Trainable params: 102,201 Non-trainable params: 0The generated model for training the data was compiled, setting the loss function to the mean absolute error and the optimizers to the stochastic descent gradient (SGD), Adam, and RMSprop, respectively, for each of the models' build-ups.Figures10-15demonstrates loss curves for various input parameters emanating from fractured height, width, length, permeability, and porosity of the formation and fractured conductivity raw synthetic data, validated in Figure9.

18 2. 4 . 3 .
illustrate the predictive version of the linear regression.Big Data Cogn.Comput.2023, 7, x FOR PEER REVIEW 12 of Neural Network (Linear Regression)Production recovery prediction analysis engaged a linear neural network regressor utilizing the Keras sequential model for the synthetic data.The model was compiled with an input shape of 1 and a linear activation, unlike ReLU, as indicated for the non-linear regressor mentioned earlier.However, while the study maintained its reliability, the model was compiled with the same loss functions (MAE) and selected optimizers for training the model with a 0.01 learning rate and an epoch of 100.Figures16-20illustrate the predictive version of the linear regression.

AuthorFunding:
Contributions: D.D.K.W. and S.I. designed the numerical models and generated the data.D.D.K.W. computed the machine learning workflow and wrote the manuscript.The methods and results of the manuscripts were reviewed by S.I., A.S. and J.K. Project administration and funding acquisition was performed by A.S., J.K. and S.I.All authors have read and agreed to the published version of the manuscript.This research was funded by [Nazarbayev University] grant number [11022021CRP1512] And the APC was funded by [Nazarbayev University].The authors are grateful for this support.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Nazarbayev University.

Table 1 .
Input Parameters for CMG Numerical Modeling.

Table 1 .
Input Parameters for CMG Numerical Modeling.

Table 2 .
Output screen; standard model summary for all training.

Table 2 .
Output screen; standard model summary for all training.