- 4093 -

A New Pore Pressure Prediction

Method—Back Propagation Artificial Neural Network

Lianbo Hua,*, Jingen Denga, Haiyan Zhub, Hai Lina, Zijian Chena, Fucheng Denga, Chuanliang Yana

a State Key Laboratory of Petroleum Resource & Prospecting, China University of Petroleum (Beijing), Beijing, 102249,China

b School of Petroleum Engineering, Southwest Petroleum University, Chengdu, Sichuan, 610500, China

* e-mail: hlb1988cupb@163.com

ABSTRACT

Accurate pore pressure prediction is a necessary requirement to well structure optimizing,

drilling difficulty minimizing, drilling accidents preventing. It plays a very important role in

the economically and efficiently well operating. Previous methods for pore prediction have

their own hypothesizes and cannot take into consideration factors that indicate or influence

the pore pressure, so their applicability is limited to one or several regions and overpressure

causes. This paper presents a new kind of pore pressure prediction BP neural network making

use of the powerful learning capability of the neural network. Our model has three layers: two

hidden layers and the output layer. The inputs of the first hidden layer include gamma ray and

formation density. The result of the first hidden layer is 4 * input to the second hidden layer

which additionally has depth, interval transit time and formation density as inputs. The

optimized numbers of the neurons of each layer are 2, 5, and 1 respectively. The activation

function of the first layer is hard-limit function, and the second and the third layer both has

hyperbolic function as activation function. The feasibility and accuracy of our network is

verified by the successful applications in a normal pressure field and an overpressure field.

The average error is 4.61%. Compared with normal-structure pore pressure prediction neural

network, our model doubled the accuracy of the prediction result.

KEYWORDS: pore pressure prediction; artificial neural network; overpressure field;

high accuracy

Vol. 18 [2013], Bund. S 4094

INTRODUCTION

Pore pressure, also known as formation pressure, is the pressure which the fluid in the pore is

subjected to. Pore pressure is one of the most important parameters for well structure optimizing,

economically and efficiently well operating, wellbore stability analyzing, etc. wellbore stability

problems, kicks, or other serious problems such as blow-outs can be avoided with an accurate

estimate of pore pressure, which means a good pore pressure prediction can not only save time

and money but also guarantee the safety of drilling process. Having releasing the significant

importance of pore pressure, researchers and engineers have done extensive work on the

prediction of pore pressure since the classic paper written by Hottman and Johnson [1]. Hottman &

Johnson’s method for pore pressure prediction uses a crossplot to relate departures of a pore

pressure indicator from the normal trend line to the pore pressure gradient at that depth. Using

this method needs the establishment of the crossplot which means a lot of statistical work.

Moreover, the crossplot developed at one region maybe improper for another region. Hottman &

Johnson’s method products accurate result only for prediction of pore pressure generated by

under-compaction mechanism and may overestimate pore pressure in same region such as the

deepwater Gulf of Mexico [2].

The Equivalent Depth method [3, 4] is another frequently mentioned pore pressure estimation

techniques in the literature. If normal-pressured and over-pressured formations follow the same,

unique relation for compaction as a function of effective stress, the Equivalent Depth method can

provide a good estimation of pore pressure. Equivalent Depth method is valid only over limited

depth ranges because of the fact that fluid properties and lithology change with depth [5] and it

may underestimate the pore pressure of unloading formation [6]. Eaton [7, 8] published a new

method in 1972 and enriched it in 1975. Eaton’s method predicts pore pressure with parameters:

interval transit time, resistivity, conductivity, and dxc-exponent. However, Eaton’s method uses

these parameters individually without taking into consideration the combined influence of these

parameters. Moreover, Eaton’s method is inapplicable for pore pressure prediction of unloading

formation as it underestimates the overpressure generated by fluid expansion [6]. Bowers [6, 9]

developed a novel method in 1995 and complemented is in 2001. This method provides good

estimation for unloading formation, but it requires a proper understanding of the stress state of the

formation. These methods are used extensively and efficaciously, and more methods were listed

in literature [2]. Summarizing these methods, we will acknowledge that they just utilize part of the

factors which indicate or influence the pore pressure without an overall consideration of these

factors and that their applicability is limited to one or several regions and overpressure causes.

Artificial Neural Networks (ANNs) are powerful and efficient tools for dealing with complicated

problems and can make overall generalization. It is very tempting to use ANNs for predicting

pore pressure.

Artificial Neural Network (ANN) [10] is a method which can simulate the capability of

receiving signals and making proper responses of biological neural network. It was introduced

Vol. 18 [2013], Bund. S 4095

half a century ago, and is utilized in numerous fields including petroleum industry. T. Sadiq and I.

S. Nashawi [11] established a Back Propagation Neural Network (BPNN) and Generalized

Regression Neural Networks (GRNN) for predicting the formation fracture gradient and

compared their performance. It is revealed in the study that the GRNN provides an adequate

approximation of fracture gradient as a function of depth, overburden stress gradient and

Poisson’s ratio. Manabu Doi [12] successfully developed a BPNN to predict the uniaxial tensile

strength formation. C. Siruvuri [13] et al established a neural network for understanding the causes

of stuck pipe and verified its feasibility with actual data from Gulf of Mexico.

Rahman Ashena [14] developed an Artificial Neural Network for evaluating bottom hole

pressure in the inclined annulus and applied it successfully in two major Iranian Oil Fields.

According to his literature, his ANN model shows a much better performance than Naseri et al

mechanistic model. Keshavarzi, R. et al. [15] developed a feed-forward with back-propagation

neural network (FFBPNN) to predict the fracture gradient as a function of pore pressure

gradient, depth and rock density in one of southern oilfields in Iran, and the results indicate

that the neural network approach is not only feasible but also yield quite accurate outcome.

Jahanbakhshi, R. and Keshavarzi, R. [16] successfully applied the ANN approach to prediction

the wellbore stability risk. The ANN was developed in such a way that in-situ stresses, drilling

string properties, drilling operation, geological conditions and mud properties were the inputs and

wellbore stability/instability was the output. ANN was also used for pore pressure estimation

before. Liang changbao et al. [17] use a common three-layer BPANN for predicting of pore

pressure and fracture pressure. The inputs of the BPANN include interval transit time, formation

density, and the depth. However, the literature did not provide any application case of the

established model.

Husam AlMustafa [18] combined an unsupervised vector quantization (UVQ) analysis and a

supervised vector quantization (SVQ) to identify the acoustic impedance along with its Fourier

transform as potential attributes indicative of geopressure and select attributes to generate a map

consisting of 2 distinct target classes, one reflecting a zone of geopressure, and the other a normal

formation-pressure zone. Successful application verified the model, but its process is complex.

James W. Bridges [19] mentioned using ANN to predict pore pressure. However, as pointed out in

the paper, the results were inconclusive. Xue Yadong, Gao Deli [20] applied ANN to the recognize

the deep formation pore pressure. In this paper, the formation depth is compared to a time axis,

and then the pore pressure can be assimilated to time series. Last, the ANN was use for

recognizing the pore pressure. The ANN used in this paper also falls into the common and wildly-

used category. What should be pointed out is that they utilized the normal pore pressure data of a

well to train the ANN and then applied the trained ANN to recognize the normal pore pressure of

the lower formation in the same well and that they utilized the overpressure data of the well to

train the ANN and then applied it to recognize the overpressure of the lower formation in the

same well. The successful case verified the feasibility of the approach, however, according to the

literature, the number of iterations that the training process needed to converge reached up to

21000. David N. Dewhurst [21] utilized ANN to calculate the pore pressure with seismic data.

Vol. 18 [2013], Bund. S 4096

However the ANN was used for prediction the net stress. Hu Quanming and Ji Haipeng [22]

applied ANN to predict the pore pressure the Saertu oil field and Xingshugang oil field in

Daqing. According to the literature, the inputs of their ANN model included log data such as

interval transit time, spontaneous potential, gamma ray, and 3-resistivity record and the average

error of the ANN is approximate 8.9%. Without depth as one of the inputs, the provided result is

questionable. The structure or the topology of the ANNs mentioned above is common and wildly-

used. The goal of this paper is to present a new pore pressure prediction ANN model, to verify its

feasibility and accuracy and to discuss the influence of the parameters of the ANN.

MODEL ESTABLISHMENT

Artificial neuron model

Artificial neural network [10] can loosely simulate the structure and capability of biological

neural network: network node or artificial neuron works as biological neuron and connection

weights of the neural network function as chemical transmitter and electric transmitter. An ANN

has the capability of intelligent analyzing with simple mathematical method and dealing with

non-linear, fuzzy, and complex relationship.

Artificial neuron is the basic element of an ANN, whose model is showed in Figure 1. The

function of artificial neuron, as the name implies, is to simulate the biological neuron.

j jS jy( )jf S

1 j

Threshold

Sum Activation Function1x

2x

px pj

2 j

Weights

Output

j jS j jS jy( )jf S

1 j

Threshold

Sum Activation Function1x

2x

px pj

2 j

Weights

Output

Figure 1: The basic artificial neuron model

The artificial neuron showed in Figure 1 has P inputs and one output. The inputs are xi (

i=1，2,…,P) and the output is yj. The relationship of inputs and output can be formulated as

following:

Vol. 18 [2013], Bund. S 4097

)(

p

1ij

jj

ji

ij

sfy

xws

(1)

Where, j is the threshold; wij is the connection weight or weight from signal i to the neuron

j; js is the net activation; f ( ) is called activation function. There are a lot of activation function

including: linear function, ramp function, threshold function, squashing function etc.

FFBP artificial neural network

The neurons are arranged and organized in different forms depending on the type of the

network. These neurons are organized in the form of layers. Then layers are linked by the

connection weight, and an ANN is formed. After half of a century’s research, researches

established numerous types of ANN. Feed-forward with back-propagation artificial neural

network (FFBPANN) is well-known and widely used in engineering applications. The structure

or topology of a typical feed forward neural network is showed is Figure 2.

Input nodes

Hidden layers

Output layers

Model in

puts

Mod

el outputs

Figure 2: The structure or topology of a typical feed forward neural network

The neural network showed in Figure 2 has three layers (the inputs nodes are not counted as a

layer of the neural network): two hidden layers and the output layer. Each neuron in a certain

layer receives the signals coming from the upper layer and produces an output using the

algorithm described in formula (1). The outputs of all the neurons of a certain layer are the output

of that layer.

Before an ANN can be used to perform its task, it should be trained to do so, that’s to say, it

need a training or learning process. The training process is a procedure to determine the weights

and thresholds using an appropriate learning algorithm. FFBPANN which this paper also uses

applies the error back propagation approach to realize the goal. The training process needs a

Vol. 18 [2013], Bund. S 4098

sample database. Generally, the data in the database is randomly divided into 3 subsets: the

training set, the validation set, and the test set. The training set is used for calculating weights and

thresholds. The validation set is used for stopping training process to prevent the network from

overfitting the data. The error which is defined as the difference between the predicted output and

the desired output (target) on the validation set is monitored during the training process. The

validation error will normally decreases in the initial stage of the training process, so does the

training set error. However, when the network begins to overfit the data, the validation error

begins to rise. When the validation error increases for a specified number of iterations, the

training is completed and the weights and thresholds are determined. The test error is used for

compare different models.

As mentioned in the introduction part of this paper, ANNs are used for predicting the bottom

hole pressure, formation fracture pressure, pore pressure, and the wellbore stability. According to

the literatures, the structure or the topology of these ANNs belongs to the common and wildly-

used type. However, an ANN with this type of structure cannot perfectly simulate the process we

calculate the pore pressure with log data. This paper presents a new kind of pore pressure

prediction ANN with a novel structure.

New kind of pore pressure prediction FFBPANN

It takes several steps to establish an ANN model. Firstly, we should determine the type of

ANN and the output parameters. After decades of research, researches established numerous

types of ANN. However, only several kinds of them are wildly used. Some kinds of them such as

sensation network and linear network are only capable for problem with linear separability or

linear relationship. FFBPANN is the most wildly used ANN due to its powerful adaptability for

different problem and excellent performance when dealing with complex relationship. This paper

also applies FFBPANN to pore pressure prediction. The output of our ANN is pore pressure.

Then the next important job is to determine the number of the network layers, the link condition

of the layers, the number of neurons on each layers and the activation function. Previous research

reveals that a three-layer BPANN can almost approximate any nonlinear function, so for a

comparatively simple problem such as the pore pressure prediction in this paper, a three-layer

BPANN is competent [23]. If an ANN has too many layers than the problem really need, the

ANN will overfit the training data. The determination of the linking condition of the layers

depends on the problem to solve and the principle is the ANN with optimized linking relationship

can better simulate the process we human beings solve the same problem. The number of neurons

on each layer is greatly influenced by the complexity of the problem and the structure of the

ANN. An ANN with underestimated neurons cannot solve the problem and its capacity is limited

while an ANN with too many neurons also can overfit the training data. So the number of neurons

on each layer is determined by some empirical methods or try-and-error method. The selection of

activation function depends on the rough relationship between the inputs and the outputs. A

higher degree of linear relationship between the inputs and the outputs requires the ANN has

Vol. 18 [2013], Bund. S 4099

more linear activation functions, or the ANN needs more nonlinear activation functions. The

inputs parameters, in some situation, are definite, but in some cases such as the pore pressure in

this paper, inputs parameters are optional. It is well known that we can use different logging data

to predict the pore pressure. In this paper the performance of ANNs with different combination of

logging data will be compared and the optimal combination will be recommended. What should

be pointed out is that all the data in the database and the data used for predicting pore pressure

should be normalized, and the output of the ANN should be reversely normalized to get the real

pore pressure.

After optimizing, this paper established a new pore pressure prediction FFBPANN, whose

structure is showed as Figure 3.

Input nodes

Hidden layers

Output layer

The first

set ofin

puts

Pore p

ressure

The secon

d set of in

puts

Figure 3: The structure of our FFBPANN for pore pressure prediction

Our optimized FFBPANN has three layers: two hidden layers and the output layer. The first

set of inputs which includes gamma ray, formation density is the inputs of the first layer. The

second set of inputs, which includes interval transit time, formation density and depth, and the

result of the first layer are the inputs of the second layer. Compared with the ANNs used in the

literatures mentioned before, the novel design of this ANN is the fact that it has two sets of inputs

and a different structure. It is this novel design that enables the ANN to generate a more accurate

pore pressure prediction with logging data because it can simulate the process we predict the pore

pressure more precisely. In the discussion section, the author will explain why this structure and

combination of logging data is chosen.

Vol. 18 [2013], Bund. S 4100

FIELD APPLICATION

To verify the feasibility, the reliability and the accuracy of our ANN model, five wells

selected from two different oil fields are used for training and verifying the ANN. Two of these

five wells including well A and well B come from LiuHua oil field located in the Pearl River

Mouth Basin of South China Sea and have normal pore pressure distribution. The rest of these

wells including well C, well D and well E come from WeiZhou oil field located in the Beibuwan

Basin of South China Sea and have abnormal pressure. The logging data and pore pressure data

(measured or/and predicted) of these five wells is presented from Figure 4 to Figure 8. the DT

stands for the interval transit time, Gamma ray is the gamma logging of the formation, the

Density represents the formation density, and the Pressure is the equivalent mud density of the

formation pore pressure (measured or/and predicted).

950

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Depth(m)

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DT(us/ft)

Depth(m)

Figure 4: The profile map of logging data and pore pressure (known) of well A

Vol. 18 [2013], Bund. S 4101

2100

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Measured

data

Our model

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Figure 5: The profile map of logging data and pore pressure of well B

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pressure(g/cm^3)

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Figure 6: The profile map of logging data and pore pressure (known) of well C

Vol. 18 [2013], Bund. S 4102

1850

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Depth(m)

Eaton's method

Our method

Measured data1850

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Figure 7: The profile map of logging data and pore pressure of well D

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Pressue(g/cm^3)Depth(m)

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Eaton's methodth d

Measured data

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Depth

Figure 8: The profile map of logging data and pore pressure (known) of well E

1) The logging data and pore pressure of well A acts as training database to training our ANN

and then the trained ANN is applied to predict the pore pressure of well B with its logging data

showed in Figure 5. Both the result and measured data are showed in Figure 5. From the graph, it

is obvious that the predicting result is excellent and the trained ANN does well. Because pore

pressure prediction dose not focus on the normal pressure formation, the following discussion

Vol. 18 [2013], Bund. S 4103

will not contain the normal pressure prediction. This example is used for verifying the capability

and feasibility of our model for normal pore pressure prediction.

2) The logging data and pore pressure of well C acts as training database to training our ANN

and then the trained ANN is applied to predict the pore pressure of well D and well E with their

logging data showed in Figure 7 and Figure 8 respectively. The result and measured data of well

D and well E are also showed in Figure 7 and Figure 8 respectively. The result showed in Figure

7 and Figure 8 verifies the feasibility and the capability of our model. In the following section,

the influence of the ANN parameters on the prediction result will be discussed with the data of

wells coming from WeiZhou oil field.

DISCUSSION

The influence of the structure of the ANN

As mentioned before, ANNs used for predicting the bottom hole pressure, formation fracture

pressure, pore pressure, and the wellbore stability have the structure or the topology showed in

Figure 2. However, the author found that the ANNs with the structure cannot perform well in

pore pressure prediction. To solve the problem, we should analyze the process we calculate the

pore pressure using conventional methods. Because most of these methods are based on the

theory of shale compaction, before we use the logging data to predict calculate the pore pressure

we need to exclude the logging data of non-shale formation with the criterions such as the gamma

ray [24]. It is obvious that a reasonable and competent pore pressure prediction ANN should have

this ability, and that the ANN with structure showed in Figure 2 is lack of this ability.

Considering the shortage of common pore pressure prediction ANN, the author develops a new-

structured pore pressure prediction ANN. Our model can partly realize the process of excluding

the logging data of non-shale formation since the result of the first layer whose inputs are the

indicators of formation lithology can reveals the influence of formation lithology. The result of

common ANN and our model is presented and compared. Moreover, the result of Eaton’s method

is also presented. The comparative criterions include the maximum, minimum and average value

of the prediction error. The subsequent comparative analysis also applies these criterions. The

comparative result is listed in Table.1. The data of the Table.1 reveals that the performance our

model is much better that the conventional ANN and the accuracy is doubled.

Table 1: The error of different methods

Method maximum error minimum error average error

well D well E well D well E well D well E average

conventional

ANN 26.08% 35.41% 0.04% 0.13% 5.95% 14.13% 10.4%

our ANN 10.76% 14.81% 0.00% 0.50% 2.07% 7.15% 4.61%

Eaton’s 31.71% 33.09% 0.07% 0.39% 6.55% 8.05% 7.03%

Vol. 18 [2013], Bund. S 4104

The prediction result of well D and well E using Eaton’s method is showed in Figure 7 and

Figure 8 respectively. From Figure 7 and Figure 8, it is revealed that the result of our model has

the same trend with the Eaton’s method, but the result of Eaton’s method changes roughly and

has a lower smoothness than that of our model. The error of Eaton’s method also listed in

Table.1, from which we find that the maximum error of Eaton’s method is much bigger than our

model while the difference of the average error of these two methods is small. Eaton’s method is

more sensitive to the formation lithology change than our model and the comparison also verifies

the feasibility and accuracy of our model.

The influence of the neurons’ number

There are numerous methods to determine the number of neurons on each layer, but most of

them are empirical or semi-empirical approaches [23]. Try-and-error method is also used for

determining the number of neurons. This paper compares the results of several our ANN models

with different number of neurons and the errors of these ANNs are listed in Table.2. Table.2

reveals that the number of neurons has a great influence on the prediction result and that the best

combination of the neurons is 2-5-2 which means the first layer has two neurons and the second

layer has five neurons and the output layer has one neuron.

Table 2: the errors of ANNs with different number of neurons

number of neurons maximum error minimum error average error

first layer

second layer

output layer well D well E well D well E well D well E average

1 5 1 9.54% 17.30% 0.08% 0.01% 4.00% 9.70% 6.85%

1 10 1 15.79% 15.42% 0.02% 0.02% 3.39% 7.30% 5.35%

2 5 1 10.76% 14.81% 0.00% 0.50% 2.07% 7.15% 4.61%

2 10 1 13.19% 16.41% 0.00% 0.00% 3.80% 8.93% 6.37%

3 5 1 16.17% 17.29% 0.59% 0.01% 4.37% 9.71% 7.04%

3 10 1 25.60% 17.53% 0.04% 0.00% 5.71% 7.97% 6.84%

The influence of the activation function

As introduced before, the selection of activation function depends on the rough relationship

between the inputs and the outputs and it is difficult to determine the activation functions [23] and

try-and-error method is also effective one to do the job. It has mentioned that activation function

includes linear function, ramp function, threshold function, squashing function et al. This paper

compares the results of several our ANN models with different activation function combination

and the errors of these ANNs are listed in Table.3. In Table.3, h stands for hardlim function

which belongs to threshold function, t represents hyperbolic function which belongs to squashing

function and s is short of saturating linear function which falls into ramp function. Table.3 reveals

Vol. 18 [2013], Bund. S 4105

that the selection of activation function has a great influence on the prediction result and that the

best combination of activation functions is h-t-t. The error of the ANN’s output increases with the

increasing of its linear activation function. The comparison result is not a surprise, considering

the fact that relationship of the logging data and the formation pore pressure highly non-linear.

The activation functions of our optimized ANN for each layer are h-t-t which means

activation function of the first layer is hardlim function and the activation of the second layer and

the output layer is the hyperbolic function.

Table 3: the errors of ANNs with different activation functions activation function maximum error minimum error average error

first layer

second layer

output layer

well D well E well D well E well D well E average

h t t 10.76% 14.81% 0.00% 0.50% 2.07% 7.15% 4.61%

s t t 8.73% 16.75% 0.39% 0.19% 4.83% 8.66% 6.75%

h t s 38.07% 34.47% 13.51% 7.47% 24.36% 16.67% 20.52%

h s t 11.55% 15.18% 0.47% 0.80% 5.20% 8.90% 7.05%

h s s 38.07% 34.47% 13.51% 7.47% 24.36% 16.67% 20.52%

The influence of the inputs’ parameters

The formation parameters that conventional methods use for pore pressure prediction are part

of logging data including interval transit time, resistivity, conductivity, dxc-exponent and gamma

ray el at [1-9]. Because of the character of the ANN that it can have several inputs and does not

need a clear formulated relationship between these inputs and the outputs. We can take into

consideration as more factors that influence or indicate the overpressure as possible. This paper

compares the results of several our ANN models with different combinations of the logging data

and the errors of these ANNs are listed in Table.4. Table.4 reveals that the error of ANN with

inputs of the third combination is the smallest. The first set of inputs of the third combination

includes interval transit time, formation density, and the depth. The second set of the inputs

includes gamma ray and the formation density.

Table 4: The errors of ANNs with different inputs combination inputs combination maximum error minimum error average error

The first set of inputs

The second set of inputs

well D well E well D well E well D well E average

interval transit time, density

gamma ray 51.89% 34.54% 0.02% 0.09% 15.39% 19.41% 17.40%

interval transit time, density,

depth gamma ray 15.79% 15.42% 0.02% 0.02% 3.39% 7.30% 5.35%

interval transit time, density,

depth

gamma ray,

density 10.76% 14.81% 0.00% 0.50% 2.07% 7.15% 4.61%

Vol. 18 [2013], Bund. S 4106

CONCLUSIONS

The paper presents a new pore pressure prediction FFBPANN. The FFBPANN model has

three layers: two hidden layers and one output layer. The inputs of the first layer include gamma

ray and formation density and the inputs of the second layer include depth, interval transit time,

formation density and the result of the first layer.

The feasibility and accuracy of our model is verified by the successful application in five

wells coming from two fields. The maximum average errors of our model is 7.15%, and the

average value of average errors is 4.61%，The results indicate that our model is not only feasible

but also yield quite accurate outcome.

The numbers of the neurons of each layer are 2, 5, and 1 respectively. The activation function

of the first layer is hard-limit function, and the second and the third layer both has hyperbolic

function as activation function.

With powerful learning ability and excellent adaptability, the trained ANN can predict

accurate pore pressure which can save a lot of time and parameters our ANN use are vary

available. If new data is available, our ANN model is able to be further trained.

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