ANN-based estimation of pore pressure of hydrocarbon reservoirs—a case study

In seismic methods, pore pressure is estimated by converting seismic velocity into pore pressure and calibrating it with pressure results during the well-testing program. This study has been carried out using post-stack seismic data and sonic and density log data of 6 wells in one of the fields in SW Iran. While an optimum number of attributes is selected, the general regression (GRNN) provides higher accuracy than back propagation (BPNN) at the initial prediction stages. However, acoustic impedance (AI) is the most applicable seismic attribute used as root and reverses AI for estimating P-wave and density. Using a set of attributes can train the system to estimate the property. The correlation coefficient of actual and predicted P-wave using an AI seismic attribute has been calculated as 0.74 and the multi-attribute technique as 0.79. Also, density and three attributes reach from 0.57 to 0.60, which shows a better relationship between seismic attributes and density. After determining optimum layers with the principal component analysis (PCA), formation pressure was modeled with the feed forward–back propagation (FFBP-ANN) method. Five information layers, including gamma, VP, AI, density, and overburden pressure, have the most linear convergence with the initial pressure model and are used to modify the ANN model of effective pressure.


Introduction
In order to determine the pore pressure gradient in a hydrocarbon field, well logs and drilling data are required. Based on screening the existing data and preparing its database, the required well logs are prepared using estimation models (Haris et al. 2017;Jindal and Biswal 2016;; Ostad-Ali-Askari and Shayan 2021). In Fig. 1, of the total number of wells in field A, several wells located in its central part have a sound passing time (sonic log). However, in the side parts of the field, this well log does not exist, and to calculate the pore pressure gradient in the whole field, this diagram should be estimated. For this purpose, using artificial neural networks, an estimation of the sonic log is performed for the number of wells without data. Due to the division of the hydrocarbon reservoir of this field into different zones, estimation of the sonic log for each zone should be performed separately. After designing and training neural networks and estimating time-lapse diagrams in each zone, the generalization of the networks and the convergence between the actual and estimated values in each zone should be investigated and analyzed (Hu et al. 2013;Ostad-Ali-Askari and Shayannejad 2021;Ostad-Ali-Askari et al. 2017;Poursiami 2013;Sadiq and Nashawi 2000). The ANN, first introduced half a century ago, is a method that can simulate the ability to receive signals and appropriate responses from the biological neural network and is used in countless fields, including the oil industry. In recent years, a back propagation neural network (BPNN) and a general regression neural network (GRNN) have been used to predict the formation gradient and compare their performance. The study showed that GRNN provides a sufficient approximation of the fracture gradient as a function of depth, overburden pressure gradient, and Poisson's ratio, but the values of the fracture gradient predicted by the Eaton method (1969;1975) differed from the actual data (Fooshee 2009;Ouadfeul and Aliouane 2012;Sadiq and Nashawi 2000). Badri et al. (2000) also studied the feasibility of formation pressure estimation in Nile Delta offshore wells with seismic data, which required integrating surface survey data and well data to better predict pore pressure. Also, the pore pressure from a Fig. 1 Stacking (red) and sonic log (blue) compared in 3 wells (Chatterjee et al. 2012) single and repetitive seismic data set was estimated, which suggested more authentic relationships between seismic velocity and pore pressure (Kvam and Landrø 2005). Thus, pressure data became increasingly crucial for drilling planning and field development in high-risk areas for abnormal ground pressure (Bowers 2002;Ramdhan and Goulty 2011). Also, ANN was used to detect formation pressures. They use a well's natural pore pressure data to train ANN and then use the trained ANN to detect the natural pore pressure of the lower formation in the same well. The ANN is then used to detect the excessive pressure of the lower formation in the same well. Dewhurst et al. (2004) used seismic trained ANN to calculate pore pressure and considered the overpressure data to detect the abnormal pressure.
Attention to pore fluid and rock stresses in sedimentary sequences in continuation of drilling mud characteristics (such as density and shale factor) to estimate and evaluate the formation pressure was considered very valuable (Dutta and Khazanehdari 2006;Dutta 2002;Ostad-Ali-Askari et al. 2017;Ostad-Ali-Askari and Shayannejad 2021;Zoback 2007). The structure or topology of the ANNs mentioned above is common and widely used, while seismic data was considered the only data to predict pore pressure in the predrilling stage (Esmersoy et al. 2013;Golian et al. 2019). Chatterjee et al. (2012) predict the formation pressure using seismic velocity for deep-water high-pressure and hightemperature wells (HTHP). Indirect estimation of the formation pressure is provided using the velocities obtained from seismic data. Poursiami (2013) proposed reservoir pressure modeling using well data. The sonic log was estimated using an artificial neural network, and the reservoir pressure model was performed by applying geostatistics. However, neural network modeling was performed only with the back propagation (BP) method. Hu et al. (2013) proposed using a feed forward-back propagation artificial neural network (FFBP-ANN) to determine pore pressure. However, network training was not performed correctly due to lack of data. Kumar Singha et al. (2013) applied a probabilistic neural network (PNN)-based approach to predict subsurface pore pressure from 2D seismic data. Nouri et al. (2013) presented the pore pressure estimation of the Sefid Zakhour field in the south of Iran using seismic data and the pore pressure model with 2D seismic data and comparison with well test data. Abidin (2014) successfully developed a BPNN to predict the uniaxial tensile strength.
However, most pressure formation methods are qualitative, so their effect in reducing drilling risk is less than that of the exploration methods such as seismic data; the results are only suitable around the well, and those in the interwell space are less accurate. Keshavarzi and Jahanbakhshi (2013) presented a real-time prediction of pore pressure gradient through an artificial intelligence approach as a case study from one of the Middle East oil fields. Aliouane and Amar (2015) present two neural network and fuzzy logic applications to Barnett Shale in a gas reservoir in Algeria. Zhang et al. (2016) present BP neural network with genetic algorithm for prediction of geo-stress state from wellbore pressures. Hadi et al. (2019) used regression analysis and artificial neural networks for real-time pore pressure prediction in depleted reservoirs. Hutomo et al. (2019) showed with reasonable accuracy that attributes of seismic amplitude, seismic frequency, acoustic impedance (AI), and shear impedance can be used as parameters to create a model using a neural network. Moreover, Ahmed et al. (2019), Tanko and Bello (2020), and Abdelaal et al. (2022) presented a new ANN for pore pressure prediction while drilling. In addition, Mahmoud et al. (2020) used machine learning techniques such as ANN and functional neural networks (FNN) to develop models for estimating static Young's modulus for sandstone formations. Khatibi and Aghajanpour (2020) proposed ANN approaches for shear sonic log prediction, comparing with the empirical Greenberg-Castagna method. Gowida et al. (2022) explained a new approach to develop a new ANN data-driven model to estimate the safe mud weight range in no time and without additional cost. Finally, Beheshtian et al. (2022) developed an ANN method to predict a safe mud window from ten well-log input combined with machine learning algorithm hybridized with optimizers.
In this study, different neural network models used in estimating formation pressures were evaluated and the validation of the four main ANN methods was discussed. Also, as a case study in a field in Abadan Plain in SW Iran, the estimation of compressional velocity (V P ) and density with seismic attributes has been done. Finally, for completing the cube of formation pressure model, the feed forward-back propagation (FFBP-ANN) method using principal component analysis (PCA) to select neural network layers has been done successfully.

Methodology
Calculating the quantity amount of pore pressure by Eaton's (1975) method is the most widely used relation as below (Eq. (1)): In this relation, OB is the overburden pressure, P n is the pore pressure on the natural trend line, Δ t n is the sonic log on the natural trend line, Δt is the sonic log, and x is the exponential coefficient of the Eaton relationship (Azadpour and Shad Manaman 2015;Poursiami 2013). Also, pore (1) P p = OB − OB − P n Δt n Δt x 302 Page 4 of 17 pressure could be obtained by subtracting effective stress from overburden pressure. For determining the amount of pore pressure to the overburden pressure, hydrostatic pressure and velocity obtained from the well log and velocity on the diagram's natural trend line are required to determine the amount of pore pressure to the overburden pressure. Overburden pressure at the location of the wells is obtained according to the density log. In order to calculate the hydrostatic pressure, the hydrostatic pressure gradient introduced for the Middle East region, especially Iran, was used. The hydrostatic pressure gradient is assumed to be 0.464 (psi/ft; Poursiami 2013; Xie et al. 2010).
Attributes are mathematical functions related to the fundamental information in seismic data: time, amplitude, frequency, and attenuation. Most of the attributes we use are post-stack from the stacked and migrated data volume loaded on our workstations. The pre-stack attributes are principally derived from amplitude variations with offset (AVO) measurements (Brown 2001(Brown , 2011Carcione et al. 2003;Riahi and Fakhari 2022).
Seismic attributes and log data predict reservoir properties throughout the field. One of these important attributes is the AI, which represents the layer's lithology and partially represents the reservoir fluid in the layers. The AI model is an important characteristic that enables the interpolation of seismic data and other features individually or in groups. Correspondingly, density estimation is calculated from seismic data with the help of seismic value, and then the gradient and overburden pressure will be calculated (Brown 2001;2011;Fazli 2015;Fazli et al. 2017;Russell 1988Russell , 2017. The network node or artificial neuron acts as the biological neuron, and the connection weight of the neural network functions as the chemical transmitter and the electrical transmitter. ANN can intelligently analyze with simple mathematical methods and deal with nonlinear, fuzzy, and complex relationships. A neuron is the main element of an ANN, the model. The function of artificial neurons, as the name implies, is to simulate biological neurons. The artificial neuron has a P input and an output (Haris et al. 2017;Liu et al. 2013;Tong et al. 2014).
The inputs are x i (i = 1, …, p), and the output is y j . The relationship between inputs and outputs can be set as follows (Eq. (2)) (Hu et al. 2013): Here, is the threshold. W ij is the weight or weight of the connection from signal i to neuron j. S j is pure activation, and f(S j ) is called the activation function (Hu et al. 2013). There are many activation functions, including linear function, ramp function, threshold function, crushing function, etc. Neurons are arranged in different ways depending on the type of network. The feed forward-back propagation artificial neural network (FFBP-ANN) is well known and widely used in engineering applications. The structure or topology of the feed-forward neural network is shown in Fig. 2a. ANN needs a training or learning process. The training process is an excellent way to determine the weight and threshold using a learning algorithm (Hu et al. 2013;Pirnazar et al. 2018). The importance of validation data is that it prevents overtraining. When the training data perform the training process, the validation data is carried out such that the system is not too dependent on the training data. The following is a sample of the training process and the amount of training, validation, and test errors in each "epoch." The higher the number of the epoch results, the lower the training error, but to a point where the validation error gradually increases; this is where the overtraining may occur, which is why the training process stops (Haris et al. 2017;Hu et al. 2013; Ostad-Ali-Askari and Shayannejad 2021).
The validation method used will be compared and evaluated using test data, including root mean square error (RMSE), mean bias error (MBE), and coefficient of determination (R 2 ) model. The basis of neural network training is based on the trial-and-error method to provide the best-hidden layer arrangement by changing the number of hidden layers and their neurons, type of stimulus function, training algorithm, and several epochs of the training step to estimate the desired output parameter. Also, the transfer functions and the number of neurons in the hidden layer are tested, and the best network structure for each of the variables will be obtained.
Statistics used: mean bias error (MBE), evaluation of deviation of estimated values from actual values (Eq. (4)); root mean square error (RMSE), model accuracy based on the difference between actual and estimated values (Eq. (3)); coefficient of determination (R 2 ), determines the fit of the linear regression model to the pair of estimated and observational data (Eq. (5)). The value of this coefficient is always between zero and one; a value closer to one indicates better performance of the model (Hu et al. 2013; Ostad-Ali-Askari and Shayan 2021; Ostad-Ali-Askari and Shayannejad 2021; Ostad-Ali-Askari et al. 2017).
where n is the number of observation points, ẑ x i is the estimated value for point i, z x i is the observation value for point i, and z x i is the average observation value.
The criterion of the suitability of the hidden layer arrangement can be expressed by applying linear regression between the measured values and the estimated values and calculating the coefficient of determination. The high coefficient of determination (R 2 ) indicates the achievement of a suitable arrangement for estimating the output parameter.
The closer the RMSE and MBE criteria are to zero, the more accurate and minor error the method indicates. Therefore, selecting the best network will be based on the lowest RMSE and the highest R 2 (Haris et al. 2017; Ostad-Ali-Askari and Shayannejad 2021; Pirnazar et al. 2018).
The FFBP-ANN structure optimized to predict pore pressure has three layers: two hidden layers and an outlet layer. The first input set, which includes gamma rays and the formation density, is the first layer inputs. The second category of inputs includes distance transfer time, formation density, and depth, and the result of the first layer is the second layer inputs. Thus, compared to the ANNs used in the literature mentioned earlier, the new design of this ANN has two different input sets and structures.
The back propagation neural network (BPNN) is the most widely used type of artificial neural network. This typically consists of many simple processing elements called neurons in layers and are connected by connections called synapses. The Levenberg-Marquard (LM) training algorithm was adopted for the pore pressure gradient. The most important advantage of the LM training algorithm is its rapid convergence. However, its storage rate is higher than many other training algorithms (Sadiq and Nashawi 2000).
For linear regression analysis of sonic logs, generally good results will not be obtained. The diagram below shows a V P log in front of a single seismic attribute ( Fig. 3a; mathematical functions derived from seismic data). A higherdegree curve can be better adapted to these points. There are several options for calculating this curve. One possible option is to apply a nonlinear transition to both variables and draw a straight line between the points. The second option is to draw a polynomial of higher degrees, and the third option is to use a neural network to find the existing relationship (Amirzadeh et al. 2013;Fazli 2015;Fazli et al. 2017).
The general regression neural network (GRNN) is a feed-forward neural network based on nonlinear regression theory consisting of four layers: input layer, pattern layer, Fig. 2 a Structure or topology of a typical forward neural network, b FFBP-ANN structure made by "Lianbo Hu" to predict pore pressure (Hu et al. 2013) summation layer, and output layer. While the nerve cells in the first three layers are fully connected, each output neuron connects only to specific processing units in the summation layer.
As the distance between the input vector and the weight vector decreases, the efficiency increases; thus, the radial base neuron acts as a detector that produces one whenever the input is of the same weight vector. The summation layer has two types of processing: addition units and division units. The number of summation units always equals the number of the GRNN output units. The division units only sum the weighted activation of the pattern units without using any activation function (Abdelaal et al. 2022;Sadiq and Nashawi 2000).
The GRNN training is entirely different from the training used for the BPNN. Each input-output vector pair from the training set to the GRNN input layer is completed only once. Pattern unit training is unsupervised but uses a special clustering algorithm that prevents specifying the number of pattern units in advance. Instead, the radius of the clusters (or expansion) must be determined before training begins (Khatibi and Aghajanpour 2020;Veeken et al. 2020).
The suggested neural network model uses depth, overburden stress gradient, and Poisson's ratio to predict the fracture gradient. Overburden stress gradient and Poisson's ratio were approximated from curves given by Eaton's equation. Poisson's ratio (υ) has been calculated by having the data of the final shear velocity as well as the completed compressional velocity. Then it has modeled based on the ratio of compressional to shear velocity by Eq. (6) in the form of a log and finally as a cube. Poisson's ratio values are generally between 0.2 and 0.1, which is acceptable. Finally, by overburden pressure, pore pressure, and Poisson's ratio and using Eaton's Eq. (1969), the formation fracture pressure is calculated according to Eq. (7) as follows (Baouche et al. 2020;Eaton 1969Eaton , 1975; Sen and Ganguli 2019): As a sample, input data including depth, overburden pressure gradient, and Poisson's coefficient used in the training phase is shown in Table 1.
Among the various neural networks, multi-layer feedforward neural network (MLFN) is one of the most effective types. The multi-layer feed-forward neural network consists of a layer of input points (or nerve cells), a layer of hidden points, and a layer of output points. These layers are generally called input, hidden, and output layers. Signals are transmitted from the input layer only in the forward direction to the hidden layer and output. Thus, every single point in each layer (except the input layer) receives only the output signals from the previous layer. These points compute the input signals by the weights connecting the different points in the two adjacent layers (Fazli 2015;Fazli et al. 2017).
The threshold function manages the output signal of each point. The hidden layer contains three points, and due to the use of an input attribute, the input layer has only one point (Fig. 3b). The advantage of this method is that the obtained regression curve is consistent with the data for many seismic attribute values, and its accuracy is more than linear regression. Correspondingly, instability at low attribute values (values less than 9000), as the 7) P fraction = P overburden − P pore * 1 − + P pore  MLFN tries to model the data as close as possible, could be a disadvantage. In other words, the network has suffered from overtraining at this point (Ahmed et al. 2019;Haris et al. 2017;Mahmoud et al. 2020). Another form of a neural network includes probabilitybased neural networks. These networks are mathematical interpolation programs that use the structure of a neural network. Their application has a potential advantage because by studying the mathematical formulation, the understanding of the performance of this network is more tangible than the feed-forward network. Here, the data used to train the network is similar to the forward feed network and includes training examples for each instance in the time window. By providing data to the network, the probabilistic neural network (PNN) assumes that any new output value for the log can be written as a linear combination of the log values in the data. The measurement of each value is the minimum validity error.
The PNN also has the same desired characteristic as the MLFN, i.e., good correlation with the data (Fig. 3c). In addition, it does not have the instability mentioned in the previous case for the range of the lower part of the attribute. The problem with the PNN is that it has a high execution time, as it receives all the training data and compares each output sample with each input sample (Golian et al. 2019;Hu et al. 2013;Sadiq and Nashawi 2000).

Applications
Artificial neural networks (ANNs) are a powerful and effective tool for dealing with complex problems and can provide a general overview. For example, using ANN is very useful for predicting pore pressure.

Impact of ANN structure and activation function
Structured neural networks cannot perform well in predicting formation pressures. To solve the problem, we must analyze the process of calculating the formation pressures using conventional methods.
Since most of these methods are based on shale density theory, before using the log data to predict the pore pressure calculation, it is necessary to exclude non-shale formation data with some criteria such as gamma rays. In addition, ANN must have the ability to predict pore pressure reasonably.
In Table 2, the results of the standard ANN model and the "Lianbo Hu" model for two hypothetical wells D and E are presented and compared. In addition, the result of Eaton's method is also presented. Comparative criteria include the maximum, minimum, and average amount of prediction error (Adim et al. 2018;Hu et al. 2013), and as a result, the average error of the suggested ANN method is less than that of the conventional ANN and traditional Eaton's method.
The number of neurons has a significant effect on the prediction results (Table 3); for example, in the Lianbo Hu ANN model, the best combination of neurons is the 2-5-1 format.
The choice of activation function depends on the relationship between inputs and outputs; it is difficult to determine the activation functions of the work, and the trial-and-error method is effective for doing the job.
The h stands for the Hardlim function, which belongs to the threshold function; t stands for the hyperbolic function, which belongs to the squashing function; and s represents the saturated linear function (Table 4). The best combination is the h-t-t activation functions. The comparison shows that the relationship between well log data and pore pressure of the formation is very nonlinear. Our optimized ANN activation functions for each layer are h-t-t, which means that the activation function of the first layer is the Hardlim function and the activation of the second layer and the output layer has a hyperbolic function.
The ANN model can have multiple inputs according to its characteristics and does not require a precise relationship between inputs (such as distance transmission time and gamma rays) and outputs, and it can affect or show abnormal pressure as much as possible (Hu et al. 2013).

Data transfer validation
One of the main steps for transferring data from wells to seismic lines is the validation step of this transfer. Results shows that increasing the number of attributes always improves the fit between the actual and estimated values. In the case of new data that in total training data (previous wells) have not been applied, it can be ineffective and sometimes destructive. Overtraining causes this phenomenon. Using more attributes is like using a higher-degree curve to match the points of a crossover diagram. A highdegree curve may provide better estimation for existing data; however, it may increase the error if a new point is added to previous points (Fig. 4).
Although the higher-grade curve is more consistent with the training data, its estimation error for the validation data (white squares) is greater than that of the lower-grade curve (Fig. 5a).
Most of these methods are calculated for linear regression and cannot be used for nonlinear predictions such as neural networks. Instead, the cross-validation method can be used in all cases of prediction. Cross-validation includes dividing the total training data into two subsets; the first is a set of training data, and the second is a set of validation data.
The training data obtains the relationship between the log and the attributes, and the validation data estimates the final prediction error. The test data set includes all existing wells except one well not included in the calculations and is used as validation data. In the cross-validation process, the validity of the mentioned analysis is repeated to the number of wells, and each time one of the wells is separated from the rest (Aliouane and Amar 2015;Beheshtian et al. 2022;Fazli et al. 2017).
Validation error for any number of attributes is always more significant than the training error because deleting a well from the training data always reduces the predictive power. Validation error does not have a uniform descending form of training error but has a minimum point of four attributes and increases. The interpretation is that any attribute added to the set of attributes after the fourth attribute only causes the set to be overtrained, and the error increases. Therefore, if the validity curve of the minimum value is known, the same point is selected as the optimal number of attributes. For example, Fig. 5b has several minimum points, but after the first two attributes, the downward trend of the curve is interrupted. Therefore, the same number of attributes is selected as the optimal number (Abdelaal et al. 2022;Ahmed et al. 2019;Fazli et al. 2017;Riahi and Fakhari 2022).

Estimating compressional velocity (VP) and density with seismic attributes
The Emerge module of HRS.8 software was used to estimate the well parameters, including compressional velocity  (Fazli et al. 2017) and density, and finally, their multiple products as AI. After integrating well logs and seismic data, the Emerge module, using seismic attributes, determines a suitable set of attributes to determine the seismic velocity in the well area. The next step determines the target log using linear regressions or ANN. The present studies have been carried out using post-stack seismic data and sonic and density log data of 6 wells in one of the fields in southwestern Iran, located in Abadan plain. It was necessary to select the optimal number of seismic attributes. For this purpose, the validation diagram was used, and some suitable attributes were selected. Seven attributes were suitable due to the reduction of the estimation error by increasing each attribute to the previous attributes (Table 5).
The degree of correlation with the target P-wave (sound speed) log (m/s) and seismic attributes are shown in Fig. 6. Furthermore, the correlation coefficient of 0.71 shows a good relationship between the root acoustic impedance (Sqrd (AI)) seismic attribute and the P-wave log (target diagram). The correlation coefficient of the cross-sectional diagram of the actual and predicted P-wave (target diagram) using an AI seismic attribute has been calculated as 0.74 and using all seven attributes as 0.79 (Fig. 6a,b). Hence, the increase of the correlation coefficient of + 0.05 shows the improvement Fig. 5 a Diagram with total validation error (red curve). The second attributes onwards have only a tiny contribution to improving the validity error, and after the fourth attribute, they also increase the validity error, b V P log prediction error diagram against the number of attributes used in the stepwise regression method (Fazli et al. 2017)  of the relationship. It means that the estimated values of P-wave velocity are in good agreement with the original values (well log). According to Table 6 and Fig. 7, by evaluating the training and validation curve, it was found that the first three seismic attributes of the selected attributes are suitable for density (g/cm 3 ) estimation. Therefore, the density volume was estimated using these three seismic attributes and density on the seismic data cube. The estimation was done based on single-attribute analysis and then multi-attribute analysis.
By adding the number of attributes, the correlation coefficient between density and attributes including three attributes of "1/AI," "Quadrature Trace," and "Integrated Absolute Amplitude" are 0.60 (Fig. 8b). The density volume is estimated by applying this relationship between seismic data (seismic attributes) and density on the seismic data cube.

Using principal component analysis (PCA) to select neural network layers
Before using a neural network, a primary model should be constructed using traditional modeling methods like inverse distance weighted (IDW). For example, a primary effective pressure cube with formation pressure test data by the IDW method.
To model the effective pressure cube using the initial ANN in the entire study area, all information layers need to be completed as scaled up or cubes. The best method for modeling the neural network in this study is to complete the formation pressure model after determining the principal component analysis (PCA), which is the feed forward-back propagation (FFBP-ANN) method. All the information layers, including the completed data cubes of the Abadan Plain of the SW Iran field, have been used according to Table 7. According to the analysis of Petrel 2016 software, the correlation between 0.2 and 0.3 (green values) is suitable for generating a neural network layer, and values below 0.2 (blue values) have a low correlation. In the initial neural network model, the very low correlation of many information layers, generally less than 0.2 and in some cases above 0.6, has caused disturbances and affected the effective pressure values in the neural network model. Therefore, to modify the model, five layers of information, including formation gamma, compressional velocity, seismic acoustic impedance inversion (AI), density, and overburden pressure, have the most linear convergence with the initial effective pressure model and have been used. Among the selected layers of gamma cubes, compressional velocity and acoustic impedance have a correlation coefficient in the acceptable range, and density and overburden pressure have also been selected due to their direct impact on other formation pressures (Table 8 and Table 9).
The final pressure cube obtained from the initial IDW model is presented below. The maximum effective pressure is in the range of 1000-9000 psi, which correlates well with the DST pressure test data while drilling the wells (Fig. 9).
In the effective pressure cube made with the neural network based on the initial model using 30 epochs, the value of the training error is 1083.5, the test error is 1083.64, and the relative error is calculated as 0.536. ANN errors decrease with increasing epochs (Fig. 10).

Samples of validating ANN for pore pressure modeling accuracy
To validate the feasibility and accuracy of the ANN model, selected well logs and pressure data from one oil field location were used. These data are used to train and verify the ANN. BPNN was created using MATLAB toolbox, nn-tool, with the optimized model with three layers consisting of two hidden layers and one output layer. The inputs are gamma ray, density, sonic log, and depth. This combination of input enables the BPNN to generate a more accurate result. Also, the GRNN was created with the nntool and tested with the data. As a result (Table 10), the GRNN tends to predict that pore pressure is higher at the initial stages. The most suitable model architecture is back propagation neural network (BPNN), with an average error of 1.0048%. As another sample, the tables below present the results obtained from pore pressure modeling using the artificial neural network. The artificial neural network model showed a high degree of correlation with the actual pore pressures, as evidenced by the low errors recorded at individual pressure points (Table 11). The results of the Fig. 8 Cross plot of actual and predicted density using a a single seismic attribute of reverse acoustic impedance (1/AI) with a correlation coefficient of 0.57, b multiple attributes (1/AI, quadrature trace, and integrated absolute amplitude) with a correlation coefficient of 0.60 statistical error analysis (Table 12) further underscore the model's accuracy, with the model yielding a coefficient of correlation of 0.9927, a root mean square error of 1.6628, and a mean bias error of 10.95%. These results imply a substantial degree of correlation and, by extension, the high accuracy of the artificial neural network model in predicting pore pressure.

Conclusion
To conclude all the results: • Before using the log data to predict the pore pressure calculation, it is necessary to exclude non-shale formation data with some criteria such as gamma rays based   on shale density theory. In addition, ANN must have the ability to predict pore pressure reasonably. • In this study, The FFBP-ANN structure and LM algorithm have been optimized to predict pore pressure. Also, the proposed GRNN and MLFN models are used to predict the fracture gradient and acoustic impedance (AI). • After integrating well logs and seismic data by the Emerge module of HSR.8 software, using seismic attributes, determine a suitable set of attributes to determine   1984% the seismic velocity in the well area. The next step determines the target log using linear regressions or artificial neural networks (ANN). • Using a set of attributes is more beneficial than using raw seismic data. After that, these data can train the system to estimate the P-wave velocity property. Acoustic impedance is the most important and applicable seismic attribute that is used as root acoustic impedance (Sqrd (AI)) and reverse (1/AI) for estimating P-wave and density. • The correlation coefficient of the actual and predicted P-wave (target diagram) using an AI seismic attribute has been calculated as 0.74 and all seven indicators as 0.79. Hence, the increase of the correlation coefficient of + 0.05 means that the estimated values of the P-wave velocity are in good agreement with the original values (well log). • By using the multi-attribute technique, the correlation coefficient between density and attributes (three attributes of "1/AI," "Quadrature Trace," and "Integrated Absolute Amplitude") reaches from 0.57 to 0.60 which shows a better relationship between seismic data (seismic attributes) and density on the seismic data cube. • The best method for modeling the neural network in this study is to complete the formation pressure model after determining the PCA, which is the FFBP-ANN method.
As the results, the correlation between 0.2 and 0.3 (green values) is suitable for generating a neural network layer, and values below 0.2 (blue values) have a low correlation. • To modify the ANN model of effective pressure, five information layers, including formation gamma, compressional velocity, seismic acoustic impedance inversion (AI), density, and overburden pressure, have the most linear convergence with the initial effective pressure model and have been used. In the ANN model based on the effective pressure cube using 30 epochs, the value of the training error is 1083.5, the test error is 1083.64, and the relative error is calculated as 0.536, generally showing ANN errors decrease with increasing epochs, but, after 30 epochs, may be overtrained.
It is suggested to use the results of this research for many applications especially determining formation pressure in similar hydrocarbon reservoirs.