WOA (Whale Optimization Algorithm) Optimizes Elman Neural Network Model to Predict Porosity Value in Well Logging Curve

Abstract

Porosity is a vital parameter in reservoir research. In the process of oil exploration, reservoir research is very important for oil and gas exploration. Because it is necessary to take cores for indoor test in order to accurately obtain the porosity value of cores, this process consumes significant manpower and material resources. Therefore, this paper introduces the method of machine learning to predict the porosity by using logging curves. This paper creatively develops a WOA (whale optimization algorithm) optimized Elman neural network model to predict porosity through logging parameters PE, DEN, M2R1, AC, GR, R25, R4 and CNL. Porosity measurement is constructed by taking cores for indoor experiments. It contains a total of 328 sample points. The data is divided into training set and test set. The logging parameters are used as the input parameters of the prediction model, and the porosity measured in the laboratory are used as the output parameter. In order to evaluate the performance of the model, RMSE, R², MAE and VAF evaluation indexes are introduced to evaluate. This paper also introduces the non-optimized Elman neural network and BP neural network to compare with this optimization model. The research shows that the WOA algorithm optimizes the super parameters of the Elman neural network, so that the performance of the WOA–Elman model is better than the Elman neural network model and the BP neural network model.

1. Introduction

At present, in the process of oil and gas field exploration and development, the direct measurement method (core analysis [1], sidewall coring [2], cuttings analysis [3]) and the indirect interpretation method (seismic data [4] and well logging data [5]) are the main methods to determine conventional reservoir parameters (porosity, permeability, saturation, etc.). One of the most important methods for reservoir parameter prediction is logging interpretation [6], which has the advantages of rich data, high resolution and low cost. Reservoir physical parameters are the major parameters for reservoir prediction and evaluation. Porosity, as one of the most basic reservoir physical parameters, can provide a reliable reference for reservoir interpretation. The methods to obtain reservoir porosity are mainly divided into the direct measurement method and the indirect interpretation method. The direct measurement method obtains porosity through core measurement, and the latter is obtained through logging data interpretation. The porosity obtained by rock physical analysis after drilling coring is more accurate than other methods [7], but the cost of sampling and testing is high. Moreover, with the increase of drilling depth, the difficulty of obtaining cores also increases. Therefore, the direct measurement method is limited and it is difficult to obtain the complete porosity of the whole area. The indirect interpretation method is used to determine reservoir porosity according to the relationship between geological information and logging data [8], which becomes more and more important in oil and gas resource exploration. However, there are many factors affecting reservoir porosity, and the relationship is complex, which brings great difficulties to the accurate prediction of reservoir porosity. Predecessors have established empirical formulas or simplified geological models to calculate and predict the reservoir porosity [9,10,11]. These methods can achieve good results for solving general geological reservoir problems. However, these methods cannot effectively obtain the complex nonlinear relationship and spatial continuity between porosity and logging data. Therefore, it is of great significance to explore and develop new porosity prediction methods to realize accurate reservoir prediction and evaluation.

S. J. Rogers [12] (1995) used the artificial neural network with back propagation architecture to predict the permeability value according to the porosity value at the same depth. The results showed that the permeability prediction effect of neural network is much better than that of regression prediction. P. M. Wong [13] (1995) used discriminant analysis and the back propagation neural network to predict lithofacies. The results showed that the effect of the neural network method was better than discriminant analysis. Patrick M. Wong [14] (1995) used the back propagation neural network method to predict porosity value, and proposed a weight visualization curve to optimize the neural network, so as to greatly reduce the training time. Fatai Anifowse [15] (2010) made use of fuzzy logic and function network to predict porosity and permeability. The results showed that under the same data set, the prediction effect of the mixed model (fuzzy logic- function network) was better than that of the single model. Wafaa El Shahat Afify [16] (2010) proposed an intelligent technology to determine reservoir physical properties from logging data by using fuzzy logic and neural network technology. The experimental results showed that this method could predict reservoir physical properties more accurately and reliably than the traditional reservoir physical property calculation method. A. Mohebbi [17] (2012) compared the predicted results of artificial neural network with the measured permeability in core analysis experiment, and the results showed that the predicted results of artificial neural network are reliable. A. F. Al Anazi [18] (2012) used the support vector regression method based on empirical risk minimization (ERM) to predict porosity and permeability, and compared it with multilayer perceptron. The results showed that the ERM-SVR method is better than the multilayer perceptron method in predicting porosity and permeability under the condition of small samples. Syed Shujath Ali [19] (2013) used the adaptive neuro fuzzy inference system (ANFIS), a functional network and a support vector machine to predict permeability and porosity. The results showed that the three machine learning models could achieve good results in prediction and had a positive impact on oilfield development and field guidance. Ahmed Ali Zerrouki [20] (2014) made use of the combination of fuzzy sorting and artificial neural network to predict porosity. The results showed that the correlation coefficient R² between the porosity predicted by artificial neural network and the actual porosity value in logging data is 0.878, which also reflects the feasibility of applying artificial neural network to porosity prediction. Tahar Aïfa [21] (2014) proposed a hybrid neuro fuzzy model. The fuzzy logic method was used to correct the calculated permeability and the actual core permeability, and the neural network was used as the nonlinear regression method to predict the permeability. The results show that this hybrid method can be used as a tool for estimating reservoir physical properties. Ahmed Amara Konate [22] (2015) compared the generalized regression neural network (GRNN) and the feedforward back propagation neural network (FFBP) in porosity prediction model. The research results show that the GRNN network can estimate porosity parameters more accurately and reliably than the FFBP network. Oki Dwi Saputro [23] (2016) used the artificial neural network based on Levenberg Marquardt back propagation algorithm to predict the porosity in logging data. Through many tests, it was concluded that the optimal network structure is 10 hidden layers. Rafik [24] (2017) used the multiple regression method to predict permeability based on logging data, and also tested several other statistical regression techniques. The results showed that using the statistical regression method to predict permeability was very promising. Yufeng Gu [25] (2018) used a continuous constrained Boltzmann machine and a particle swarm optimization algorithm to optimize SVR to predict permeability. The experimental results showed that the new method was effective in permeability prediction. Salaheldin Elkatatny [26] (2018) used adaptive differential evolution optimization methods to optimize the artificial neural network to predict the total organic carbon. The input data are gamma ray, compression time, deep resistivity and bulk density. The adaptive algorithms were used to determine the best combination of the network structure of the artificial neural network (number of hidden layers, number of neuron layers, training function, transfer function), The experimental results showed that the neural network optimized by the adaptive difference algorithm had higher accuracy than the traditional artificial neural network algorithm. Navid Kardani [27] (2021) combined the equivalent optimization and improved equivalent optimization with two traditional machine learning algorithms, extreme neural network (ELM) and artificial neural network (ANN) to develop a new hybrid model. The experimental results showed that the combination of ANN, ELM and meta heuristic search algorithm can better predict the properties of carbonate reservoir. Jinwoo Lee [28] (2021) used a probabilistic neural network to predict properties in reservoir, and made use of multi-task learning to improve prediction accuracy. The experimental results showed that the model is superior to other supervised machine learning algorithms in prediction accuracy. Alvin K. Mulashani [29] (2021) proposed an improved GMDH based on Levenberg Marquardt (LM) to predict permeability. The results showed that this method could reasonably reduce the prediction time and had high accuracy. Compared with the traditional GMDH and back propagation neural network, it performed well in the training data, and also obtained good results in the test process, which could be used as an improved scheme for predicting reservoir parameters.

The above examples are the application of machine learning modelsused by scholars in the field of logging curve property prediction, but the task of improving the prediction accuracy still needs to continue. Therefore, many multivariate heuristic algorithms have been applied to the field of reservoir prediction, such as genetic algorithm, particle swarm optimization algorithm, gray wolf algorithm, sine cosine algorithm and multiverse algorithm. These meta heuristic algorithms play an important role in improving the accuracy of the original prediction models. Thus, the whale algorithm is proposed to improve the accuracy of Elman neural network prediction model.

2. Methodology

2.1. Elman

Elman neural network [30] is a local recurrent neural network. Its structure includes input layer, hidden layer, undertaking layer and output layer, as shown in Figure 1. The input layer unit plays the role of data input; the output layer unit performs linear weighting operation; the transfer function of the hidden layer element can be linear or nonlinear; the receiving layer unit is used to remember and store the output value of the hidden layer unit at the previous time. The special connection between the undertaking layer and the hidden layer makes the network model sensitive to time-series dynamic data, so the ability of Elman neural network to process complex time-varying data has been improved.

The mathematical model of Elman neural network is as follows:

$$q_{\mathrm{c}}(k)=q(k-1)$$

$$q(k)=g\left[w_1{q}_{\mathrm{c}}(k)+w_{2}u(k)\right]$$

$$h(k)=f\left[q(k)\cdot w_3\right]$$

where $g(*)$ is the transfer function of hidden layer neurons, and sigmoid function is often used; $f(*)$ is the excitation function of the output neuron. The learning index function of the Elman neural network adopts the sum of squares function of error, and the expression is as follows:

$$E(k)={\displaystyle \sum _{k=1}^n{\left[h(k)-h_{\mathrm{c}}(k)\right]}^2}$$

where $h_{\mathrm{c}}(k)$ is target output value.

2.2. WOA

The whale optimization algorithm [31] is mainly divided into three steps: surrounding the prey, attacking by spiral bubble net, and looking for prey randomly. (1) Surrounding the prey. Because the target prey position is unknown, the WOA algorithm regards the best candidate individual position in the current whale group as the target prey position, and other individuals in the whale group update the position according to the position of the best candidate individual. Namely:

$$D=\left|C\cdot X^{*}(t)-X(t)\right|$$

$$X(t+1)=X^{*}(t)-A\cdot D$$

where $X$ is the position vector of the current solution; $t$ is the number of iterations; $A$ and $C$ are coefficient vectors; $X^{*}$ is the position vector of the optimal solution in the current whale population. The calculation methods of A and C are:

$$A=2ar-a$$

$$C=2r$$

where: a is with the increase of the number of iterations, and it decreases linearly from 2 to 0; r is any vector between 0 and 1.

(2) Attacking by spiral bubble net. The WOA algorithm first calculates the distance between the individual whale and the target prey, and then simulates the spiral movement of the humpback whale for hunting behavior:

$$D^{\prime }=\left|X^{*}(t)-X(t)\right|$$

$$X(t+1)=D^{\prime }\cdot e^{bl}\cdot \cos (2\pi l)+X^{*}(t)$$

where: $b$ is constant coefficient of spiral shape, $l$ is a random number in $[-1,1]$.

(3) Looking for prey randomly. When $A$ is greater than 1 or less than −1, in order to improve the global search ability of the algorithm, individuals in the whale population randomly select prey by referring to each other’s positions.

$$D=\left|C\cdot X_{rand }-X\right|$$

$$X(t+1)=X_{rand }-A\cdot D$$

where: $X_{rand }$ is randomly selected position vector of the current whale group.

3. Dataset Preparation

The sample data is the real data of oil wells in the exploration area (Western China). The logging data includes logging parameters such as PE, DEN, M2R1, AC, GR, R25, R4 and CNL. The full names corresponding to the abbreviations written in this article are shown in

Table 1. The full name of the abbreviation mentioned in the text.

Abbreviation	Full Name
PE	Photoelectric absorption cross section index
DEN	Density
M2R1	High resolution array induced resistivity
AC	Acoustic
GR	Gamma ray
R25	2.5 m bottom gradient resistivity
R4	4 m bottom gradient resistivity
CNL	Neutron
POR	Porosity
WOA	Whale optimization algorithm
RMSE	Root mean square error
MAE	Mean absolute error
BP	back propagation

. In order to obtain accurate porosity values, several laboratory tests are carried out to test the porosity value. According to Boyle’s law, at a constant temperature, the volume of core chamber is certain, and the smaller the solid volume of rock sample is, the larger the volume of gas in rock ventricle is, and the lower the equilibrium pressure is after connecting with the standard chamber. On the contrary, the larger the solid volume of the rock sample placed in the core chamber is, the higher the equilibrium pressure is. Standard curve of solid volume and equilibrium pressure are drawn, and the equilibrium pressure of rock sample to be tested is measured, and the solid volume of rock sample is checked according to the standard curve. According to the following formula, the porosity of rock sample can be calculated:

$$\varphi =\frac{V_p}{V_f}=\frac{V_f-V_s}{V_f}$$

where: $\varphi$ is porosity, $V_p$ is pore volume, $V_s$ is skeleton volume.

The measured porosity data corresponds to the logging data in the depth direction, and the prediction model is used to train and test the model, which is the Elman neural network based on the WOA. In order to verify the performance and feasibility of the algorithm, small sample data are selected for algorithm testing, 328 sample points are selected, of which 200 sample points are trained and 128 sample points are tested. The overall analysis and modeling process implemented in this study are shown in Figure 2; Figure 3 is a logging curve, which shows the value of logging parameters.

Table 2. Data statistics and interpretation.

Parameter	Min	Max	Median	Std	Average	Skew
PE	3.042	6.46	4.602	0.7273	4.582238	0.144289
DEN	2.519	2.625	2.58	0.019115	2.579354	−0.25568
M2R1	143	705	382.5	117.3329	390.5579	0.374388
AC	55	63	57	1.425923	57.625	0.713462
GR	39	68	47	5.906085	48.8811	0.87834
R25	111	143	126	6.855572	125.2835	0.126538
R4	237	301	256	15.22972	257.6128	0.62556
CNL	5	12	8	1.549364	8.009146	0.123645
POR	1.272	6.182	2.9075	1.039905	3.161107	0.595966

shows the statistical analysis of various logging data parameters.

The research method is mainly divided into four steps: (1) Data set preparation (2) Model establishment (3) Model verification and evaluation (4) Result analysis. An Elman neural network model based on the WOA is constructed, and then the best model is selected through the modeling results.

In this paper, Pearson and Spearman correlation coefficients are selected as the correlation evaluation criteria. The correlation coefficient is used to characterize the linear correlation between each logging parameter and porosity. Through the comparison, the logging parameters that are sensitive to porosity can be quickly identified. The calculation formula is:

$$r=\frac{\mathrm{cov}(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}$$

where, $r$ indicates the correlation coefficient between two variables; $\mathrm{cov}(X,Y)$ represents the covariance of two variables to describe the linear relationship between two variables; $D(X)$ and $D(Y)$ are the variance of two variables respectively.

The closer the absolute value of the correlation coefficient of the two variables is to 1, the better the correlation is, and the closer it is to 0, the worse the correlation is. The form of correlation heat map is used in Figure 4 and Figure 5 to show the correlation between data parameters. The values in the drawing represent the correlation between the two parameters. It can be observed from Figure 4 that the algorithm correlation between porosity and DEN (based on Pearson algorithm) is the best, and the absolute value reaches 0.70, but the correlation between porosity and PE is the worst, and the absolute value is only 0.03. It can be observed from Figure 5 that porosity also has the best correlation with DEN (based on Spearman algorithm), with an absolute value of 0.65, and the correlation between porosity and PE is the worst, and the absolute value is only 0

4. Model Inspection and Evaluation

It is very necessary to verify and evaluate the developed models. After the models are established, whether the accuracy is excellent or not is related to the implementation of subsequent projects. Evaluating the accuracy and quality of the model can use RMSE, R², MAE and VAF, so this paper introduces these four evaluation indicators.

RMSE is the square root of the ratio of the square of the deviation between the predicted value and the real value to the experimental sample size N; MAE is the average of the absolute value of the error between the predicted value and the real value. The smaller the RMSE and MAE are, the higher the accuracy of the model is. R² And VAF are usually used to evaluate the linear fit of the model. If the value of R² is closer to 1 or the value of VAF is closer to 100, the higher the quality of the model is.

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left({\stackrel{\wedge }{POR}}_i-PO{R}_i\right)}^2}}$$

$$R^2=1-\frac{{\displaystyle \sum _{i=1}^N{\left(PO{R}_i-{\stackrel{\wedge }{POR}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^N{\left(PO{R}_i-{\overline{POR}}_i\right)}^2}}$$

$$MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|{\stackrel{\wedge }{POR}}_i-PO{R}_i\right|}$$

$$VAF=\left[1-\frac{\mathrm{var}\left(PO{R}_i-{\stackrel{\wedge }{POR}}_i\right)}{\mathrm{var}\left(PO{R}_i\right)}\right]\times 100$$

5. Analysis of Prediction Results

Models are developed according to the process in Figure 2. The initialization operation of relevant parameters of Elman neural network model is executed. The hidden layer of the model is set to 10 layers; the training times are 1000; the learning rate is 0.01; the minimum error of training target is 0.0001; the display frequency is 25; the momentum factor is 0.01; the minimum performance gradient is 0.000001; the maximum number of failures is 10. The population number of MFO algorithm is set to 20 and the maximum number of iterations is set to 50.

Next, the training set is used to train the Elman neural network optimized based on the WOA algorithm, the non-optimized Elman neural network and the most widely used BP neural network. As shown in Figure 6 and Figure 7, the correlation between the predicted value and the actual value of the training data set and the testing data set can be seen, and the effect of these models is relatively good. The sampling points are basically distributed near the perfect line (actual porosity = predicted porosity). The WOA–Elman model has better training effect, with RMSE value of 0.1118, R² value of 0.9904, MAE value of 0.0893 and VAF index of 99.10. After the model training, the test data set is used to verify and evaluate the three prediction models. By analyzing the correlation and error between the predicted porosity value and the actual porosity value, it can be seen from the figure that the sampling points are also basically distributed near the perfect line (actual porosity = predicted porosity). The performance of the three prediction models from high to low is WOA–Elman (RMSE = 0.1457; R² = 0.9696; MAE = 0.1182; VAF = 97.01), Elman (RMSE = 0.3066; R² = 0.8749; MAE = 0.2545; VAF = 87.50), and BP (RMSE = 0.7438; R²: 0.8608; MAE = 0.5265; VAF = 86.12). The results show that the prediction performance of the WOA–Elman model can achieve relatively high prediction accuracy. The RMSE of the model proposed in this paper is 0.1457, which is smaller than the back propagation neural network model proposed by P. M. Wong [13] in 1995 (RMSE = 2.700), fuzzy logic and neural network technology proposed by Wafaa El Shahat Afify [16] in 2010 (RMSE = 0.3896), feedforward directional propagation neural network proposed by Majid Jamshidian [32] in 2015 (RMSE = 0.1621) and deep learning model by Peng An [33] in 2018 (RMSE = 1.2688).

In order to more intuitively display the logging curve results, Figure 8 shows the results of porosity prediction by these three models, which can more intuitively display the prediction results. Figure 9 is a line chart comparing the real value with the predicted value, which can also show the prediction accuracy more truly and intuitively. The prediction effects of these three models are relatively good, which can reflect the general trend of porosity curve. It also shows the feasibility of applying machine learning to porosity prediction of logging curve. From Figure 10, we can see that the accuracy of the WOA–Elman neural network model is higher than that of the non-optimized Elman neural network model. Therefore, it can also be concluded that the WOA algorithm has greatly improved the accuracy of the Elman neural network model.

Table 3. Evaluation index values of three machine learning models.

	R2	RMSE	MAE	VAF
Elman	0.8749	0.3066	0.2545	87.50
WOA–Elman	0.9696	0.1457	0.1182	97.01
BP	0.8608	0.7438	0.5265	86.12

shows the values of each evaluation index of the three machine learning models

6. Conclusions

The WOA (whale optimization algorithm) has better optimization search ability, which is more suitable for the search of less parameters such as the Elman neural network weight and threshold. The performance of three models was as follows: WOA–Elman (RMSE: 0.1457; R²: 0.9696; MAE = 0.1182; VAF = 97.01), Elman (RMSE:0.3066; R²: 0.8749; MAE = 0.2545; VAF = 87.50), BP (RMSE:0.7438; R²: 0.8608; MAE = 0.5265; VAF = 86.12). Hence, the hybrid optimization model has good popularization and application value.

To conclude, the WOA–Elman neural network model proposed in this paper has good ability to predict porosity, and can effectively improve the performance of Elman neural network in super parameter adjustment.

In this paper, real oilfield data are used and good prediction results are achieved in the model test stage. The WOA–Elman neural network can be deployed in the prediction of oilfield data porosity curve in the future stage, which can save a great deal of manpower, material and financial resources.