## Abstract

Existing methods of well logging interpretation often contain uncertainties in the exploration and evaluation of carbonate reservoirs due to the complex pore types. Based on the time–frequency analysis of array acoustic logging data, the identification of pore types based on a convolutional neural network (CNN) was established. The continuous wavelet transform was first used to transform the 1-D acoustic wave data into 2-D time–frequency spectra as the input data of the CNN based on the pore aspect ratios. According to the acoustic logging data obtained from numerical simulations, a three-type (vug, interparticle-pores, and crack) prediction was established to validate the identification method. The noise-sensitivity analyses demonstrate that our method is stable for noise mixed signals. The CNN-based identification method was used to analyze the field acoustic logging data in a carbonate reservoir. According to the description of the pore structures from the core analysis, the formation pores were divided into two types (cracks and interparticle-pores). The accuracy of the method using field acoustic logging data can reach 90%. This work provides promising means for pore type identification from complex acoustic logging data by applying deep learning technologies, which can be easily extended into other similar neighboring carbonate reservoirs.

## Introduction

In the development of carbonate reservoirs, formations with secondary pores such as cracks and vugs are the key exploration targets [1]. Accurate identification of secondary pores has important guiding significance for oil and gas exploration in the middle and deep formations. The commonly used time average equation [2] describes the relationship between acoustic wave velocity and porosity in the reservoir seismic inversion, AVO analysis, fluid identification, and micro-seismic monitoring. However, complex pore structures cause wide scattering in the velocity-porosity cross-plots of carbonate rocks [3, 4], which often result in considerable uncertainties in predicting the properties (porosity, permeability, and saturation). The low correlation between porosity and permeability has also led to low accuracy of spectrum analysis during seismic exploration.

Carbonate rocks commonly contain a variety of pore types. The best way to characterize the pore types is imaging logging and NMR logging [5], while high financial and time costs deterring applications in most exploration and appraisal wells limit these logging methods in engineering practice [6, 7]. Therefore, it is necessary to construct a method of estimating the pore types using conventional logging data. Array acoustic logging technology provides comprehensive access to the physical property of rocks and formation [8, 9], which is commonly used data to identify formation lithology, fracture distribution, mechanical engineering parameters, and formation anisotropy. Different secondary pores in carbonate reservoirs not only affect the acoustic wave velocity and attenuation but also affect the whole acoustic signal [10], coda wave [11], and reflected wave [12]. At present, researches always focus on the analysis of the velocity or attenuation of the component waves such as P-, S-, and Stoneley waves [8], which is the practical method of identifying the porosity and permeability of the formation. However, the exploitative information on the acoustic logging data is minimal, and the analysis of frequency characteristics still lacks enough attention. On the other hand, the presence of complex pores strongly influences the velocity pickup in the well logging interpretation, especially in low-porosity carbonate formations; thus existing logging methods have high uncertainties in the evaluation of carbonate reservoirs. The porosity measured by the general well logging is often lower than the actual value, although it is proved to be a good gas productive reservoir [13].

The unsupervised learning methods such as k-means, principal components analysis, discriminant function analysis, self-organizing maps, and vector quantization have been used in facies classification [14, 15] and rock type recognition [16], while still have the inherent limitation: unstable, noise sensitive, memory-intensive process, and sometimes difficult to interpret [14]. The application of artificial intelligence (AI) is a new direction for identifying reservoir properties [17,18,19]. The critical issue is to build an extensive database of pattern recognition by extracting characteristic parameters with strong separability and high stability to pore development. Unlike conventional machine learning (ML) techniques, which rely on carefully hand-engineered features, a deep learning model can figure out the feature extraction by itself. In the recent 5 years, as a supervised learning method, artificial neural network (ANN) modeling based on big data analysis has penetrated various fields of oil and gas exploration, significantly improving the accuracy of reservoir identification [20,21,22,23]. However, the traditional ANN has its inherent problems: (1) the number of trainable parameters increases drastically with an increase in the size of the image, which makes it slow in the application of the high-quality images [24], (2) traditional ANN often stucks to an over-fitting problem. Meanwhile, the well logs recorded in pore type formation are essentially the result of complex nonlinear geophysical processes arising primarily due to the variability and interactions of lithology, porosity, fluidity, and other different factors. Further, the well log records are often contaminated by the deplorable borehole conditions. Therefore, pore type identification from well logs constitutes a nonlinear geophysical inverse problem [25, 26]. Deep learning (DL) is a part of a broader family of machine learning and ANN [27]. The advent of backpropagation allows the ANN to update the weight of neurons based on the error rate [28], which makes the DL boom again from the early 2000s. Convolutional neural network (CNN) is a class of feedforward neural networks, most commonly applied to analyze visual imagery, such as video recognition, image classification, and natural language processing [29]. There is also increasing interest in applying CNN technologies to geophysics, such as geophysical feature extraction [30], seismic data denoising [31], as well as reservoir property prediction [32].

Pore aspect ratios (ratio of the minor axis to the major axis of pores, AR) have a remarkable effect on elastic wave propagation [33, 34], which can accurately describe the pore types in carbonate rocks. Existing research shows that the pores with large pore AR, such as vugs and moldic, lead to a higher velocity than the pores with small pore AR (micro-cracks or cracks) [35]. However, it is difficult to retrieve accurate information about the pore types other than porosity via conventional well logs. Therefore, this study raises a prediction method of pore types using CNN based on acoustic logging data. This work aims to build a foundation for pore type identification, which provides a novel idea for the exploration and development of carbonate reservoirs.

## Method

### Input data preprocessing

With the development of the acoustic detection method, time–frequency analysis has been widely used in the field of geological exploration. At present, the time–frequency analysis for processing array acoustic logging data is mainly focused on the qualitative analysis of the relationship between the pore structure parameters and the velocity, attenuation, or dispersion of component waves in the time and frequency domains [12]. Acoustic logging data are closely related to the propagation environment of the formation, which would behave different features in the time or frequency domain. Exploring and accurately extracting these features and applying them to carbonate rocks are the new direction of pore type identification. Robust measurements of statistical characteristics can be achieved by analyzing the nonstationary acoustic signals using the continuous wavelet transform (CWT) [15, 36, 37], which is a formal tool to provide an overcomplete representation of a signal by letting the translation and scale parameters of the wavelets vary continuously [38]. The CWT has the advantages that the width of the window used can be applied flexibly, and the spread of thin reservoirs can be resolved in time and depth domains, which is used to separate the overall waveform for lithology identification [15], seismic data denoising [39], and fracture monitoring [40,41,42]. In comparison with the conventional time domain analysis, the application of CWT enables the network to learn the spectral structures of the signals, which has been found to increase the accuracy of the prediction [43]. In order to fully utilize the information of the acoustic logging waveforms in both the time and frequency domain, we first convert the 1-D signals to the 2-D time–frequency spectrum by the CWT. The graphical spectrum is used as the input data of CNN, and finally, the model is trained to predict the pore type of the “unknown” model. Conceptually, the CWT is a sliding cross-correlation between a signal *x*(*t*) and a family of wavelets as follows:

where *ψ*(*t*) is the mother wavelet, which is a continuous function in both the time and frequency domains:

where *a* is the scale factor and *b* is the time shift of the mother wavelet *ψ*; *ψ*^{*} is the complex conjugate of the mother wavelet, which is a family of functions generated through *b* and *a*. In this way, the wavelet transform is to shift the mother wavelet function and then convolve with the original signal *x*(*t*) on different scales.

### CNN construction

A typical CNN consists of one or more convolutional layers and a fully connected layer at the top (as compared to a traditional ANN) and also includes associated weights and pooling layers [44]. Also, compared with other deep feedforward neural networks, CNN replaces conventional neuron matrix multiplication by convolution [45, 46], which can significantly reduce the number of unknown weights and is more suitable for the image and signal recognition [47]. Here, we construct a CNN structure with 12 layers, including one input layer, three 2-D convolution layers, three ReLU activation layers, three pooling layers, and two fully connected layers (as shown in Fig. 1). The input layer is the image of time–frequency spectra generated by CWT. The filter is separately multiplied and summed with the input layer, and the value of the corresponding convolution feature map is obtained after activating. Convolution refers to the inner product (multiple-by-element multiplication and summation) of an image composed of a plurality of pixels and a constant filter. Convolution with filters is the extraction of feature values (such as color shades and outlines) in different regions of the image. The traditional neural network is only suitable for linear problems, and the activation function introduces nonlinear factors to traditional neurons so that the neural network can approximate any nonlinear function, which further makes the whole neural network more complete and the training results more accurate [48].

The complete feature maps are obtained by using several different kernels. Mathematically, the feature value at location (*i*, *j*) in the *k*th feature map of the *l*th layer \(f_{i,j,k}^{l}\) is calculated by [49]:

where \({\text{w}}_{k}^{l}\) and \(b_{k}^{l}\) are the weight vector and bias term of the *k*th filter of the *l*th layer, respectively, and \({\text{x}}_{i,j}^{l}\) is the input patch centered at the location (*i*, *j*) of the *l*th layer.

In our CNN structure, each convolution layer is activated by the ReLU function, which can transform the whole system into a nonlinear system and improve the versatility of input and output [50]. ReLU is a positive function that guarantees no negative values in the system, and the activation value can be calculated as:

Overfitting is a serious challenge in DL. Therefore, to improve the performance of the network and to avoid overfitting, dropout regularization techniques, also known as pooling, is used in all the hidden layers of the neural network to reduce the number of free parameters [51]. Max pooling layer is used to shrink the size of the image. Both the convolutional layer and pooling layer are used to extract features from the input image, which also belong to the feature engineering layer.

The output of the last fully connected layer is fed to a normalized exponential function (softmax classifier), which calculates a probability distribution over *n* different possible classes [52]:

The objective of the CNN is to optimize the learnable parameters, which minimizes the misfit between the CNN predicted *y* and truth classification *C* of *m* instances using an L2-regularized multinomial logistic loss function:

where *λ* is the regularization parameter controlled the trade-off between the data misfit and model constraints; **Z**_{k} are the model parameters of the *k*th layer.

The weight optimization of the backpropagation process is completed by using the Adam stochastic optimization function [53], which can be used instead of the classical stochastic gradient descent procedure to update network weights iterative based in training data. The following fully connected layer, similar to the model in machine learning, is used to learn the rules in the feature data and then output the predicted results (i.e., pore types). To evaluate the classifier on test data, the classification accuracy is set as the ratio of the number of correct predictions made to the total number of predictions (i.e., number of correct classification/total number of validation set) in each epoch.

### Pore type identification step

The model update has a learning rate of 0.001 with a trade-off parameter *λ *= 10^{−4} and performs 40 iterations of training (epoch). One epoch equals to train all samples once in the training set. Other parameters, such as the number of layers, the number of neurons per layer, the size of the convolution kernel function, and the number of filter steps, are determined according to the training speed and the error range. The flow of the entire method is shown in Fig. 2.

PyTorch is a flexible deep learning framework that allows automatic differentiation via dynamic neural networks [54]. It is a replacement for NumPy, which supports GPU acceleration, distributed training, and various optimization tasks. It also has many more concise features. In this paper, we choose PyTorch to construct a pore type identification in carbonate rocks using a CNN based on time–frequency spectrum images of the acoustic logging signals.

## Method validation by the numerical modeling

A typical process of neural network modeling comprises three basic steps: training, testing, and validating. During training step, the network analyzes the provided data and changes weights between network units to reflect dependencies found in the data. The testing is a process of estimating the quality of the trained neural network. The validating set is a part of the data used to tune the network topology or network parameters other than weights. Since DL is a technology based on big data technology, it usually requires a large amount of input data (tens of thousands to millions) to train the model. If the number of samples is too small, the model will be over-fitting and lead to poor results. Current DL technology does not make full use of the information of the function itself, while DL itself has almost no restrictions on the learning functions. As long as the amount of data is sufficient, only two hidden layers are needed to solve all the problems in theory. Therefore, the current difficulty in DL applications is mainly due to the improvement of the algorithm and how to obtain a large number of representative data [28]. The essential issue of deep learning is to establish a well-classified training set. Here, we use a numerical modeling to produce more training data and validate our identification method.

### Numerical modeling description

The effects of pore types are evident in both time and frequency domains of the acoustic logging data. However, in addition to core analysis, it is difficult to directly obtain accurate pore structures from the actual data [55]. Numerical simulation can not only achieve single-variable control but also significantly reduce the research time and cost. Therefore, numerical simulation of wave propagation has become an essentially auxiliary approach for studying the relationship between elastic wave characteristics and complex pore structures [35, 56, 57]. To quantitatively analyze the effects of pore types on the acoustic logging interpretation, we have developed an acoustic logging scale modeling method using the finite element method (FEM) to describe the actual pores as the randomly distributed ellipses with pore ARs [35]. The model consists of a water-filled borehole, the formation with carbonate properties, and differently shaped pores. The borehole and pores are saturated with fluids, and the governing equations of fluids are expressed as follows [58]:

where **q** is the Dipole Domain Source, N/m^{3}; *Q*_{m} is the Monopole Domain Source, 1/s^{2}; *p*_{t} is fluctuating pressure, Pa; *p*_{b} is background pressure, Pa; *ρ*_{f} is the fluid density, g/cm^{3}.

According to the law of Duhamel-Hooke [59], the governing equations of the carbonate formation under the low-stress condition are expressed as follows:

where **u**(*x*, *y*) is the displacement vector, *σ* is stress tensor, **F**v is the stress vector of the excitation source, *C*(*K*, *G*) is the 4^{th} order elasticity tensor of bulk and shear modulus, “:” stands for the double-dot tensor product, *s*_{0} and *ε*_{0} are the initial stress and strain, *ε* is the total strain, and the superscript T is the transpose of the matrix.

Stress and acceleration are set as continuous at the boundary between the fluids and solid formation:

where **n** is the surface normal vector.

The low reflection boundary is used at the outermost layer to let waves pass out from the model without reflection in the time-dependent analysis [60]:

where *V*_{P} and *V*_{S} are the velocities of the P- and S-waves in the material, respectively.

A dominant frequency of 8 kHz Ricker wavelet is used as the monopole source [9] in borehole acoustic measurements with the uniform and identical intensity in all directions:

where **x**_{0} and *f* are the position and dominant frequency of the monopole source.

According to the geometric model of the borehole with different pore types, FEM is used to solve the problem of acoustic field propagation described by the above formula. A 2-D axisymmetric borehole model is shown in Fig. 3. The basic physical properties of the numerical modeling are given in Table 1.

Due to the high accuracy of the numerical modeling, we obtained a set of acoustic logging data with a single pore type. The typical waveforms received from eight receivers are shown in Fig. 4.

### Data classification

Benefit from the numerical simulation, we can repeatedly run numerical models with randomly distributed pores, and the purpose of increasing the amount of input data can be achieved by calculating a lot of numerical models with consistent pore types. First, the pore is divided into three types according to the pore AR: vugs (AR = 0.8), interparticle-pores (AR = 0.1), and cracks (AR = 0.05). All of the models are established with 10% porosity and 0.06 m pore size, respectively. We performed three sets (set A, set B, and set C) to study the effect of data volume on the stability of the results. The specific data volume is shown in Table 2. Among them, each type of model was repeatedly calculated 10 times in set A, 20 times in set B, and 30 times in set C. Although the amount of data is not large, due to the high stability of the numerical simulation and the controllability of the model variables, the input data structure is relatively simple. For example, set C has 30 models for each pore type (a total of 90 models). We can obtain eight waveforms for each model and 240 waveforms for each pore type of numerical models (a total of 720 waveforms in three pore types). Then, we selected 75% (540 waveforms) of the data as the training set of CNN, and the rest of 25% data (180 waveforms) as the test set. In one epoch, we used 20 waveforms to train the model each time and 10 waveforms to do the test. The prediction accuracy is set as the average of the accuracy of all iterations calculated in the same epoch. The training process is solved by a workstation with 24-thread Intel Xeon E5-2620 CPU and 64 GB RAM. The calculation time is about 0.5–3 h for each training process.

### Pore type identification by numerical modeling

The received waveforms and corresponding wavelet time–frequency spectra of different pore AR models are shown in Fig. 5. Sophisticated graphics are often composed of underlying structures. Local features can represent global features to some extent. The human brain can integrate the multiple local features of an image and then infer the global features. As the pore AR changes, different characteristics can be easily observed in the time–frequency spectrum (Fig. 5b, e, and f). However, it is challenging to quantify the difference rely on manual identification. Computers can solve such problems well through continuous learning and training.

After preprocessing all of the acoustic waveforms, the time–frequency spectra were put into the CNN model. Figure 6 shows the accuracy and loss of our CNN training model with three pore types (vugs, interparticle-pores, and cracks). The training process can converge very well. After 25 epochs, the training accuracy rate (TrainAcc) can reach more than 0.9, and it reaches the maximum value and continues to be stable after 30 epochs, which shows that the training can effectively converge. The loss value in the training process (TrainLost) represents the deviation between the output value and the actual value. It should be noted that TrainLost decreases rapidly to less than 0.001 after 30 epochs. This also indicates that the later training cannot further improve the CNN results. As the amount of data used increases, the speed of reaching the maximum accuracy becomes faster, indicating that increasing the input data is an effective way to improve the capability of CNNs.

Figure 7 shows the results of the three testing sets of A, B, and C. Compared to the training results, the scattering of accuracy curve is apparent, especially when the amount of input data is relatively small (e.g., set A), which also causes to repeatedly oscillate on the function value near the local extreme point. When the amount of data increases, the stability and maximum value of the testing accuracy improve significantly. The accuracy of the testing results is always lower than the training results for the corresponding epoch. However, as to the more massive amount of input data set (set C), the accuracy can still reach above 0.95 after 30 epochs. In addition, it can be inferred that as the amount of data increases further, the prediction will become more accurate, and the number of epochs required to reach the maximum will also decrease. In general, the established CNN training model can better identify the pore types using simulation data, which is also laying a foundation for applying this CNN framework to actual field data.

### Noise-sensitivity analysis

Noise or spikes, such as statistical fluctuations and burr disturbances that are not related to formation properties, typically present in the logging data due to the collision between the logging tool and the wellbore [61], especially in carbonate rocks [62]. Before using our model on field data, we test the stability of the proposed method by adding random white noise to the synthetic data to simulate the noise in the actual observations. A white noise process is one with a mean zero and no correlation between its values at different times. Here, we performed several tests with different white noise level to get complex signals with the signal-to-noise ratio (SNR) from 5 to 25. Typical complex signals are shown in Fig. 8. The plot shows that 2-D time–frequency spectra exhibit better anti-noise performance than 1-D signals.

We trained the CNN method using the power spectra of the white noise models as the inputs. The other setting and parameters are the same as aforementioned description. Each type of noise-level model was repeatedly calculated 20 times and trained by 30 epochs. The noise-sensitivity analyses (Fig. 9) show that all of the accuracies are higher than 0.9, which demonstrates that our method is stable for up to SNR = 5 noise mixed signals.

## Application to field data

### Geologic setting and data classification

At present, the conventional method of pore type identification is to use the core analysis of well drilling and well logs for comparison. Core analysis is an integral part of the drilling process to recognize the geologic characteristics of an oil and gas field, which is also the most intuitive method to get the pore structures. The studied field is located in Shanxi province, in the Ordos Basin, northwest China. The target Majiagou Formation with Ordovician in the Lower Paleozoic in age is the main carbonate reservoir in this structure. Figure 10 shows the well logs and lithology analysis of the S# well in a carbonate reservoir. The total coring footage is 41.5 m in the target depth of 4000 m-4200 m. The statistics and classification of pore structures by core analysis are shown in Table 3.

It is worth noted that core analysis is performed at normal temperature and pressure in the laboratory, while when the core is removed from the ground, the high effective stress initially exerts on the rock attend to be released, which may usually cause the original micro-cracks to widen to a certain extent. On the other hand, the actual logging data are greatly affected by the borehole environment, and other factors can easily cover the influence of the pore structures. Therefore, the pores in the core sections are divided into interparticle-pores (AR > 0.1) and cracks (AR < 0.1) according to the pore ARs. Then, the pore type in the corresponding core section is predicted using the above CNN framework based on field acoustic logging data.

The data sets used in this study consist of 41.5 m classification by core analysis (Table 3) and the corresponding acoustic logging data. The original waveforms of monopole logging are measured every 0.125 m, and there is a total of 322 waveform traces of acoustic data in the core section. The testing sets of the model are 4124.5–4129.5 m for the interparticle-pores and 4132–4137 m for the cracks (approximately 25% of the total data amount), and the remaining 75% of the data are used as the training set. The sections with interparticle-pores and cracks are both 15.75 m (each of 126 traces), which precisely guarantee the same amount of data for each pore type. For each training epoch, 18 traces are randomly selected, and at least seven training processes are needed in an iteration epoch.

### Pore type identification based on core analysis

A CWT was performed on the acoustic logging data with a sampling rate of 50 kHz and a total length of 8 ms to form an image with a resolution of 256 × 256 as the input data of the CNN. Other parameters are consistent with the settings in the previous section.

Figure 11 shows the accuracy and corresponding loss of the CNN model with two classifications (interparticle-pores and cracks) with the field logging data. The initial accuracy is about 0.5, which is consistent with the two-component initial result. At the same time, the training process can converge very well. The TrainAcc can reach more than 0.9 after 15 epochs, and the TrainLost reaches the minimum (less than 0.001) with a certain degree of oscillation after 25 periods. This also indicates that the operations in the later calculating cannot further improve the CNN results. For the testing process, although the TestAcc is less than the TrainAcc, the TestAcc can still reach more than 0.85 after 20 epochs. The accuracy of correct classification is calculated after 30 periods, and the model with the highest accuracy on the validation set is selected as the final classifier. Therefore, the proposed CNN model based on core analysis has a better prediction effect on the pore types.

In addition, increasing the calculation period can improve the accuracy of CNN prediction to a certain extent, but the accuracy of the testing set can only reach 0.9. Core analysis is currently the most accurate means of identifying pore structures. Acoustic logging data are not only affected by pores and lithology, but complex formations and drilling processes can also cause errors in the prediction results. However, the accuracy of DL is greatly affected by the amount of input data. The prediction results can be significantly improved with the continuously increasing amount of trainable data. Therefore, comprehensively utilizing well logging data in the entire oilfield to establish a database of stratum pore structures and combining artificial intelligence technology with big data are the important direction to realize the establishment of digital oilfields or smart oilfields.

## Conclusion

Carbonate reservoirs have more complex pore structures than sandstone reservoirs because of diagenesis. In this study, based on the time–frequency analysis of array acoustic logging data, we established a CNN-based classification method to predict the pore types quantitatively. First, based on the acoustic logging data obtained from numerical simulations, the continuous wavelet transform was used to convert 1-D acoustic wave data into 2-D time–frequency spectra as the inputs of our CNN model, and then, a three-type prediction method of pore aspect ratios (vug, interparticle-pores, and crack) was established to validate the method. Before using our model to field data, we verified the stability of the proposed method by adding random white noise to the synthetic data to simulate the noise in the actual observations. The CNN-based identification method was used to analyze the field acoustic logging data in a carbonate reservoir. Based on the description of pore structures obtained by core analysis, the formation pores were divided into cracks and interparticle-pores according to the pore ARs. The high accuracy (over 90%) indicates that the proposed CNN model based on core analysis has a better prediction effect on the pore types. The presented method not only avoids the errors in velocity picking, but also the application with more information from the logging data can be easily extended to other similar neighboring reservoirs.

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## Acknowledgements

The authors are grateful to the editor and two anonymous reviewers for their insightful reviews. This work was funded by the Future Engergy Systems at the Unversity of Alberta.

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Li, T., Wang, Z., Wang, R. *et al.* Pore type identification in carbonate rocks using convolutional neural network based on acoustic logging data.
*Neural Comput & Applic* (2020). https://doi.org/10.1007/s00521-020-05246-2

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### Keywords

- Pore type
- CNN
- Carbonate rocks
- Acoustic properties
- Time–frequency characteristic