Studying petrophysical properties of micritic limestones using machine learning methods
Tuan Nguyen-Sy, Minh-Ngoc Vu, Anh-Dung Tran-Le, Bao-Viet Tran, Thi-Thu-Nga Nguyen, Thoi-Trung Nguyen
PII: S0926-9851(20)30609-1
DOI: https://doi.org/10.1016/j.jappgeo.2020.104226
Reference: APPGEO 104226
To appear in: Journal of Applied Geophysics Received date: 9 August 2020
Revised date: 15 November 2020 Accepted date: 20 November 2020
Please cite this article as: T. Nguyen-Sy, M.-N. Vu, A.-D. Tran-Le, et al., Studying petrophysical properties of micritic limestones using machine learning methods, Journal of Applied Geophysics(2020),https://doi.org/10.1016/j.jappgeo.2020.104226
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© 2020 Published by Elsevier.
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Studying petrophysical properties of micritic limestones using machine learning methods
Tuan Nguyen-Sya,b, Minh-Ngoc Vuc,d, Anh-Dung Tran-Lee, Bao-Viet Tranf, Thi-Thu-Nga Nguyeng, Thoi-Trung Nguyena,b
(a) Division of Construction Computation, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
(b) Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam (c) Institute of Research and Development, Duy Tan University, Danang 550000, Vietnam
(d) Faculty of Natural Sciences, Duy Tan University, Danang 550000, Vietnam (e) Laboratoire des Technologies Innovantes, Université de Picardie Jules Verne Avenue
des Facultés, 80025 Amiens Cedex 1, France
(f) Research and Application Center for Technology in Civil Engineering, University of Transport and Communications, 3 Cau Giay, Dong Da Hanoi, Vietnam
(g) Le Quy Don Technical University, Institute of Special Engineering, 236 Hoang Quoc Viet, Hanoi, Vietnam
Corresponding author: nguyensytuan@tdtu.edu.vn
Abstract: It is important in geophysical applications to relate the compressional and shear ultrasonic wave velocities of micritic limestone to its porosity, volume fraction and density of micrite grains as well as the effective confining pressure. In this paper, this difficulty task is successfully realized by using the most relevant machine learning methods: the artificial neural network method, the support vector machine method and the extreme gradient boosting (XGB). A relevant dataset available in literature is considered to train and test the models. It is observed that the XGB method significantly outperform the other methods in term of
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accuracy and training time. It allow obtaining a very high R-squared value of 0.96 and a very small relative root mean squared error of 3% while predicting the sonic velocities from other petrophysical properties. The robustness of the models is also confirmed by studying the sensitivity of the random splittings between the training and the testing sets.
Keywords: micritic limestone; carbonate rock; machine learning; XGBoost
1. Introduction
Carbonate rocks are the host of numerous petroleum reservoir around the world (Xu and Payne, 2009). Therefore, predicting mechanical and physical properties of carbonate rock is an important problem in geophysics and geomechanics. Such prediction is very complex because of the complex diagenesis processes including compaction, dissolution, precipitation and cementation, etc. (Assefa et al., 2003; Vanorio et al., 2008; Verwer et al., 2008). This rock type is mainly composed of macro grains, micro grains (micrite), cement (filling in the space between the grains) and pore (Leighton and Pendexter, 1962; Regnet et al., 2019). Such heterogeneous medium can be regarded as a mixture between micritic limestone matrix and macro grains (and/or macro pore). Different types of macro grains including organic grains such as corals can be found in carbonate rock. Therefore, it is not possible to characterize grains dominated carbonate rock without specify the grain types. Besides, observations using scanning electron microscopy (SEM) have shown complex microtextures with different pore types in micritic limestone depending on the volume fraction of micrite grains and cement (Moshier, 1989a; Lambert et al., 2006). That is why the ultrasonic and wave velocities measured on carbonate rock are extremely scattered (Eberli et al., 2003; Regnet et al., 2015;
Kittridge, 2015). This is also the reason of a quite limited success of the classical homogenization theories while dealing with carbonate rocks (Baechle et al., 2008; Fournier et al., 2011; Adelinet et al., 2019).
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In numerous carbonate reservoir, microporous micritic limestone is the dominant component. Hence, studying the geophysical and geomechanical properties of micritic limestone is an attractive subject in literature (Moshier, 1989a, 1989b; Munnecke and Samtleben, 1996; Kazmierczak et al., 1996; Pandey et al., 2019). Fournier et al. (2011) measured the compressional and shear ultrasonic wave velocities of micritic limestone at different stress conditions. They observed a strong dependency of both P- and S-wave velocities to the porosity, the volume fraction of micrite and stress condition. However, data are highly scattered. Indeed, they tried to model their data by the homogenization theories (including the self-consistent and the differential effective medium schemes) but the predictive capacity of these models is very limited in this case.
Machine learning (ML) is now known as an appropriate choice to face such a complex problem (see e.g. Zhang et al., 2020 for a review). For example, Madhubabu et al. (2016) considered the artificial neural networks method to predict the Young modulus and the uniaxial compressive strength of carbonate rock and obtained a high accuracy. Ghasemi et al.
(2018) considered a tree model and reviewed other ML models in literature to predict those mechanical properties of carbonate rock. Nguyen-Sy et al. (2020a, 2020b) employed the three most relevant machine learning methods for predicting the compressive strength of concrete and the anisotropic elastic stiffnesses of shale. However, no model exists yet to relate the ultrasonic wave velocities of micritic limestone to its petrophysical properties such as the porosity, the volume fraction of micrite and stress condition.
In this paper, the extreme gradient boosting (XGB) method will be considered to predict the ultrasonic P- and S-wave velocities of micritic limestone from its basic features: the porosity, the volume fraction and the density of micrite and the effective confining pressure.
This method is considered as the most relevant ML method nowadays because it wins or contributes to the winning of almost every machine learning competitions on the famous
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online platform Kaggle (Chen and Guestrin, 2016). Two other relevant ML methods that are the artificial neural networks (ANN) method (Hassoun, 1995) and the support vector machine (SVM) method (Cortes and Vapnik, 1995) are also considered to compare with the XGB method. The relevant dataset provided by Founier et al. (2011) will be considered to train and test the considered ML models.
The paper is organized as following: first, the statistical analysis of the considered dataset is presented. Second, the performance of the XGB method is demonstrated with detailed sensitivity analysis. Third, the XGB method is compared with the ANN and SVM methods to confirm the exceptional high performance of the XGB method for the present application.
Concluding remarks are given at the end of the paper.
2. Statistical analysis of the dataset
A relevant dataset measured on micritic limestone by Fournier et al. (2011) is used in this paper for training and testing the machine learning methods. This dataset includes 378 data measured on 80 samples at different stress conditions. The samples are taken from a Lower Cretaceous formation in France. Only the samples that have a grain-stone texture with absence of macro-porosity were selected. They samples were first dried and then equilibrated in ambient temperature and humidity conditions. The porosity was measured by the weighting method. The P- and S- wave velocities of each sample are measured at different confining effect stress conditions, up to 40 (MPa). The S-wave velocity is measured on two perpendicular directions. However, the measured values for those two directions are quite close because of the isotropy of the medium. Therefore, an average value for each sample at each stress condition will be considered. The statistical parameters of the dataset are shown on Table 1. The porosity varies from almost zero to 25.5 (%) and the volume fraction of micrite varies in a large range from 40.9 to 85.3 (%). The grain density is in the range from 2.66 to 2.75. The samples are measured in a large confining stress range up to 40 (MPa). Both
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P- and S- wave velocities vary in very large ranges up to 6.223 (km/s) for and 3.318 (km/s) for .
Statistical distributions of the porosity, the volume fraction and the density of micrite grains as well as the P- and S-wave velocities are shown on Figure 1. The distribution of both wave velocities and the porosity are quite close to the normal distribution. The correlation between those features are also shown on that figure. It can be observed that the two velocities correlate very well. Also, it can be observed clearly that both wave velocities decrease with increasing porosity. However, no correlation between the velocities versus the volume fraction and the density of the micrite grains can be found.
Machine learning methods are considered to relate the ultrasonic wave velocities to the other petrophysical properties. So the input features are: the micrite volume, the porosity, the effective confining pressure and the grain density. The output of the model are the sonic wave velocities. They are predicted separately from the input features.
Table 1 : Statistical parameters of the dataset.
Micrite volume (%)
Porosity (%)
Effective confining pressure (MPa)
Grain density (g/cm3)
Vp (km/s) Vs (km/s)
Mean 65.216 14.753 15.470 2.705 4.440 2.489
Standard deviation
10.813 5.313 13.445 0.015 0.643 0.291
Min 40.900 0.400 2.500 2.660 2.803 1.751
Max 85.300 25.500 40.000 2.750 6.223 3.318
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Figure 1 : Statistical distribution and correlation between the features of the dataset.
3. XGB model
3.1. Theoretical basis of XGB model
The XGB model is an advanced tree boosting system (Chen and Guestrin, 2016). It is an improvement of the gradient boosting method that was developed by Friedman et al. (2000).
It use many additive functions to predict the result as
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∑
(1)
where is the predicted result for ith sample of which the vector of the features is ; is the number of estimators and each estimator (with in the range from 1 to ) corresponds to an independent tree structure; is the initial guess that is the mean of the measured value in the training set; is the learning rate (also called shrinkage parameter) that help to improve smoothly the model while adding new tree and avoid over-fitting. It is of interest to remark that over-fitting is the main issue of all ML models.
The training process is realized in an additive manner. With respect to Eq. (1), at kth step, a kth estimator is added to the model and the kth predicted result is calculated from the predicted value at the previous step and the estimation of the additional kth estimator as
(2)
where is defined by the leaves weights that are found by minimizing the objective function of the kth tree. The latter is constructed by splitting the leaves starting from a single leaf. Such procedure is realized by maximizing the gain parameter. The splitting is accepted if the gain parameter is superior to 0. Thus, increasing the regularization parameters allows reducing the gain parameter and then allow avoiding the complexity of the leaf splitting, i.e. keep the tree structure simple. But it will reduce the capacity of the model to fit with the training data. The readers who are interested in the details of the XGB method could be refered to the paper of Chen and Guestrin (2016).
3.2. Sensitivity analysis
The whole dataset is randomly splitted into a training and testing sets with a ratio 70:30, i.e. 70% of the initial dataset is considered for the training process and 30% of the dataset is to
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test the trained model. The XGB model is trained and tested with the number of estimators varies in a large range from to 1000. For the other parameters, the default values in the XGBoost package are considered: , and . It is observed that the accuracy of the model increases with increasing number of estimators up to 500 (Figure 2).
Above this value, the model becomes more complicated but its accuracy is not improved.
Similarly, the value yields minimum relative root mean squared error (RMSE) (the relative error is the absolute error normalized by the corresponding measured velocities) (see Figure 2, right side): 2.5% for and 3 (%) for . The RMSE is considered instead of the absolute mean squared error because it is better to evaluate the accuracy of the model on the considered large ranges of the compressional and shear wave velocities. So, the optimal number of estimators is .
Using this optimal value of , the sensitivity of the learning rate is tested in a large range from to 1. It is observed that the default value yields optimal results (Figure 3). Similarly, the optimal regularization parameters and are 1 and 0, respectively (see Figure 4 and Figure 5). The optimal parameters of the XGB model are listed in Table 2.
Figure 2 : R-squared values and relative RMSE for and evaluated on the testing set by the XGB model: effect of number of estimator .
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Figure 3 : XGB model with sensitivity of the learning rate parameter .
Figure 4 : XGB model with sensitivity of the regularization parameter .
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Figure 5 : XGB model with sensitivity of the regularization parameter .
Table 2 : Optimal model parameters.
500 0.3 1 0
3.3. Effect of splitting the training and testing datasets
The random splitting of the whole dataset into training and testing datasets may influence the results of the model. In this section, such sensitivity will be clarified. The optimal model parameters listed in Table 2 are considered for the simulation. While randomly changing the split between the training and testing datasets, with the ratio 70:30, it is observed that the R- squared values that are evaluated on the testing set for and vary slightly around 0.96.
The corresponding relative RMSE is about 2.5 to 3 (%). The high RMSE of 5% for observed on the 9th random splitting between the training and testing set may due to a localization effect. More clearly, this random splitting does not allow the training set to appropriately represent the whole dataset. This may happen while considering machine
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learning for a dataset of small to medium size. But the average R2 score and RMSE over all the random splitting between the training and testing datasets are very good.
Figure 6 : R-squared values for and evaluated on the testing set by the XGB model:
effect of the random selection of the training and testing sets.
3.4. Modeling the stress dependence
The trained XGB model is used to predict the S- and P-wave velocities with a continuous range of the effective confining pressure up to 50 (MPa). This stress corresponds to an in-situ effective stress at about 5 km in sedimentary basins. The optimal parameters listed in Table 2 are considered for the simulation. Figure 7 shows the results obtained for the sample with the highest porosity of 25.5%. The volume fraction and the density of the micrite grains are 67 (%) and 2.7 (g/cm3), respectively. The results given by the XGB model are not smooth curves but they are in form of stepwise constants for different effective confining pressure ranges.
For this case with high porosity, the effective confining pressure dependence of the velocities is significant. It can be observed also that the predicted curves fit perfectly with the
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experimental data. The results obtained for the case with almost zero porosity are shown on Figure 8. For this highly compacted sample, the effective confining pressure dependence of the velocities is negligible. Indeed, the predicted velocities are almost constant for a large effective confining pressure range up to 50 (MPa). The prediction for the cases with medium porosity (12% and 16.8 %) are shown on Figure 9 and Figure 10. The predicted curves fit with data for all the samples.
The stress effect on the wave velocities is quite complicated: for samples with low velocities, the velocities increase with increasing stress because of the compaction process (Figure 8). Inversely, the velocities of cemented samples decrease with increasing stress because of pore collapse and cracking/damage effects as shown in Figure 7, Figure 9 and Figure 10.
Figure 7 : Prediction of and of the sample with highest porosity (25.5%): a comparison with data.
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Figure 8 : Prediction of and of the sample with lowest porosity (0.4%): a comparison with data.
Figure 9 : Prediction of and of the sample a porosity of 12 (%): a comparison with data.
Figure 10 : Prediction of and of the sample a porosity of 16.8 (%): a comparison with data.
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Let us consider an ANN with a single hidden layer that has neurons. The ANN model relates the vector of the features to a target of each sample by the function
(3) where the notation stands for a vector and is for a dot product between two vectors or between a matrix and a vector. The function is the activation function. is the matrix of weights linking the input layer (that has input nodes for features) and the hidden layer (that has neurons). The bias vector contains bias parameters of hidden neurons. The vector contains the weights associated to outputs of the hidden neurons and is the bias of the output nosude. In this example, a single output node is considered. The ANN model is trained by modifying the weights and biases to minimize the objective function by the gradient descent techniques. It should be noted that the while using the ANN method, the features need to be normalized to a mean of 0 and a standard deviation of 1.
The Keras package (see e.g. Chollet, 2018) is used to develop the ANN model. The sensitivity of the ANN method to the number of hidden neurons (with a single hidden layer) is shown on Figure 11. A large range of hidden neurons from to was considered for sensitivity purpose. The rectified linear unit activation function (RELU) and the root mean square propagation optimizer (RMSProp) were considered. With a large number of training epochs (3000 epochs) R-squared values around 0.90 are obtained for the testing dataset. The relative errors of P- and S-wave velocities are around 4 (%) for different random splitting between the training and testing subsets (Figure 12).
The sensitivities of the optimizer and the activation function are considered. It is observed that the RELU activation function and the RMSProp optimizer provided the best result. Deep neural networks with two or four hidden layers are also considered but the accuracy could not
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be significantly improved. In any case, the accuracy of the ANN model is far below the one of the XGB model.
Figure 11 : ANN method with a single hidden layer: sensitivity of the number of hidden neurons.
Figure 12 : R-squared values for and evaluated on the testing set by the ANN model with a single hidden layer: effect of the random selection of the training and testing sets.
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The SVM model was also considered to compare with the XGB model. The well- known Scikit-learn package (Pedregosa et al., 2011) was considered to build the SVM model.
Optimal parameters of the SVM model were obtained by sensitivity analysis. It is observed that the SVM model with optimized parameters is far less accurate than the XGB and the ANN models for both S- and P-wave velocities (Figure 13, Figure 14). The R2 score obtained by the SVM model on the testing set is only 0.82 for and 0.64 for . The XGB model outperforms the two other methods. It provides a value for both velocities and is extremely fast to train comparing to the ANN model.
Figure 13 : Prediction of P-wave velocity: a comparison between the XGB, ANN and SVM
models.
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Figure 14 : Prediction of S-wave velocity: a comparison between the XGB, ANN and SVM models.
5. Conclusion
The XGB model was considered to predict the S- and P- ultrasonic wave velocities of micritic limestone from the basic features including the porosity, the volume fraction and the density of micrite grains as well as the confining effective stress. A relevant dataset available in literature was considered to train and test the model. It is randomly splitted in training and testing subsets. The training of the model on the training subset was extremely fast, less than a minute by using personal computer. Optimal model parameters were obtained by a sensitivity analysis: , , and . With those optimized parameters, the XGB model predict (on the testing subset) a high R2 score of 0.96 for both S- and P-wave velocities.
Two other relevant ML models that are the ANN and SVM models were also considered to compare with the XGB model. The optimal parameters of these model were also obtained by sensitivity analysis. The effect of the random splitting between the training and testing sets was also studied. It is observed that, the XGB model outperforms the ANN and SVM
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methods for this case. From this study, it should be suggested that the XGB model is an appropriate model to study the relationship between the ultrasonic wave velocities and other petrophysical properties of micritic limestone.
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22 Dear Editors,
The submitted manuscript tackles the problem of “Studying petrophysical properties of micritic limestones using machine learning methods”. This is an original paper which has neither previously, nor simultaneously, in whole or in part been submitted anywhere else. The developed results should have significant contribution to the domain of Applied Geophysics.
Sincerely, The authors
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23 Highlight
Compressional and shear ultrasonic wave velocities of micritic limestone are modeled
The most relevant machine learning methods are considered
XGB method significantly outperform the other methods
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24 Conflict of interest
We wish to confirm that there are no known conflicts of interest associated with this
publication and there has been no significant financial support for this work that could have influenced its outcome.
Sincerely, The authors
