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Prediction of compressive strength and elastic modulus of carbonate rocks
Measurement 88 (2016) 202–213
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Prediction of compressive strength and elastic modulus of carbonate rocks N. Madhubabu a, P.K. Singh b,⇑, Ashutosh Kainthola b, Bankim Mahanta b, A. Tripathy b, T.N. Singh b a b
Geo Constech Pvt. Ltd., New Delhi, India Department of Earth Sciences, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India
a r t i c l e
i n f o
Article history: Received 8 June 2015 Received in revised form 18 March 2016 Accepted 24 March 2016 Available online 1 April 2016 Keywords: Young’s modulus Porosity Ultrasonic velocity Carbonate rocks Artificial Neural Network
a b s t r a c t Uniaxial Compressive Strength (UCS) and Modulus of elasticity (E) of carbonate rocks are very critical properties in petroleum, mining and civil industries. UCS is the measure of the strength of the rock and E depicts the stiffness, together they control the deformational behavior. But the heterogeneity introduced as a result of fractures, dissolution and dependency on pH and temperature makes them a difficult material to study. Complex diagenesis and resulting pore system makes the job even more daunting. So, an attempt is made to predict these properties using simple index parameters such as Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. Multiple Linear Regression Analysis (MVRA) and Artificial Neural Networking (ANN) have been used for predicting the two properties and the accuracy is tested by root mean square error. The results show that ANN has a better predictive efficiency than MVRA and they can be applied for predicting UCS and Young’s modulus of carbonate rocks with reasonable confidence. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Carbonate rocks are formed in a variety of geological settings and are one of the best reservoir rocks for oil exploration [60]. It is assumed that the proportion of crude oil confined in carbonate rocks is around 50–60% with an estimated production lifetime of more than 50 years [11]. Carbonate rocks are very low strength and are highly susceptible to weathering due to chemically unstable mineralogical and textural combination, and their high reactivity with pH and temperature [41]. As important it is to determine the geomechanical properties with high accuracy, difficulty in sample preparation adhering certain standards along with several inherent heterogeneities makes it a formidable task for rock engineers. In rock ⇑ Corresponding author. Tel.: +91 22 25767271. E-mail address: [email protected] (P.K. Singh). http://dx.doi.org/10.1016/j.measurement.2016.03.050 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
engineering, both uniaxial compressive strength (UCS) and the Modulus of elasticity (E) are widely used parameters as they are important for intact rock classification and rock failure criteria [16]. These parameters have great importance in rock physics applications, viz., onshore and offshore geomechanical engineering, tunneling, dam design, rock drilling and blasting, rock excavation and even for slope stability. There are two methods for assessing the properties of rocks. First is the direct method where tests are conducted on a crafted specimen in the laboratory, the other, known as the indirect method, uses the previously derived empirical equations from the literature [8]. The test procedure for measuring UCS and E have been standardized by both the American Society for Testing and Materials (ASTM) and the International Society for Rock Mechanics (ISRM). Sound rock specimens are required for direct determination of UCS and Young’s modulus of rocks in
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the laboratories. However, high quality core samples in sufficient quantities are not possible to obtain from weak, highly fractured, weathered carbonate rocks. Moreover, the test procedures are expensive because of the requirement to precisely, carefully prepare the specimens to ensure that the ends are smooth and perfectly parallel and several other details as per standards. In addition, careful execution of these tests are difficult, time consuming and requires expensive equipment [20,8,34]. Point load test is an alternative method to determine UCS as it gives reliable values at a lower cost and with ease [50]. In order to overcome these difficulties, indirect methods employing simple index parameters like the density, porosity, P-wave velocity, Point load index, Schmidt hammer were used [29,14,13,47]. These simple index tests require a relatively small number of samples, are quick, less expensive and easy to execute. In recent years, statistical methods used in rock engineering, such as simple and multiple regression techniques have been used for establishing predictive models. In addition to these conventional methods, new techniques like Artificial Neural Networks (ANN), fuzzy interference systems, genetic programming and regression trees have also been tested to estimate the required properties [39,22,40,55,31,8,64,51,62] ANN methods have become popular since the year 1990. It is a form of analysis which is based on the understanding of the brain and human nervous system [23]. Major advantage of ANN is its efficient handling of highly non-linear relationships among data, even when the exact nature of such relationship is not fully unknown [19]. Therefore, neural networks are best suited for UCS and Young’s modulus predictions of carbonate rocks due of the complex nature of inter-relationships among the various quality parameters, composition and processing conditions [16]. The performance of the ANN methods was also compared with other statistical methods in the present study (e.g., Regression analysis). Previous studies have shown that ANN predictive models have better efficiency than the conventional statistical methods [46,56,8,16,62,34]. The objective of this study is to predict UCS and Young’s modulus of carbonate rocks using the data gathered from previous studies (163 sample data) and the data set of carbonate rocks which have been collected from the Miocene section of the Kutch basin, Gujarat, India.
2. Study area The investigated area lies in the Kutch district of Gujarat and is the westernmost part of India. It is a part of Kutch basin which is a peri-cratonic rift basin and has preserved a complete sequence from Triassic to Recent [9]. The area composed of Cenozoic sequences, especially the Miocene sediments, lies in Naliya, Kutch District, Gujarat. The Lower Miocene rocks are exposed in Khari river section (Fig. 1). A graphical log was also prepared showing the lithounits, lithology, sedimentary textures, structures and different lithofacies from the field data (Fig. 1). The first step towards preparation of graphic log was
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identification of different sedimentary units which was almost done in the field itself. In the present work, individual beds in the sequence are given a standard color, the thickness of individual beds, the texture, the sedimentary structures (if any), and fossil contents are marked. As can be seen from the log, clay minerals are ubiquitous throughout the sequence, even the carbonate samples collected from Miocene Formation for laboratory tests contains a maximum clay content ranging up to 20%. The limestone samples are enriched with fossils like Turritella, Bivalves (Oysters and Pecten), etc. (Fig. 2) and also the presence of cross lamination in the limestones were observed in the field (Fig. 3). 3. Methodology: laboratory tests 3.1. Petrography Thin section of two areas (S14 and S18) are presented in this study as the prime focus was to predict strength parameters of carbonate rocks. No correlation was made between textural changes and geomechanical properties, although we don’t neglect the possibility. The sections clearly show the presence of fossils of different assemblage along with micrite (brown) and sparite (off white) matrix (Fig. 4). In case of S14, fossils are cemented together with sparry calcite whereas mixed skeletal remains are surrounded by micritic matrix in S18. Signatures of re-precipitation of micrites on fossils can also be seen. Presence of micrite indicates deposition under calm water condition whereas, presence of sparry calcite in pore spaces suggests deposition under agitated water condition [10]. 3.2. Ultrasonic velocity Ultrasonic techniques for measuring P and S-wave velocities are non-destructive and easy to execute, both in the laboratories and field. Nowadays, pulse generation method is used for the determination of parameters using Pundit testing machine (Fig. 5) which consists of a pulse transmission generator, transducers and electronic tester for measurements of time [55,63]. A coupling gel is applied on the faces to avoid any air gap between the sample and transducers. A number of factors have been known to influence the seismic velocity of rocks which include water, clay, grain size, density, texture, porosity anisotropy and others [30]. Considering the fact that the data obtained by other researchers used in this study is to the standard along with the data obtained in this study, P-wave can be considered as a reliable parameter to relate with other geomechanical properties [43]. 3.3. Strength test A number of block samples of carbonate rocks were collected from the Miocene section of the Kutch basin. In addition, 163 sample data of carbonates were taken from previous research work around the world. Each data set includes Porosity (£), Density (q), P-wave velocity (V p ), Poisson’s ratio (l) Point load index (Is ), uniaxial
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Fig. 1. Geological map of the study area and lithology of the collected samples prepared after Biswas [9].
Turitella
Pecten
Oyster
Fig. 2. Turritella and Bivalve rich limestones in Kutch basin of Miocene formation.
compressive strength (UCS) and Modulus of elasticity (E). From the recent study, thirteen number of block samples, each having an approximate dimension of 0.2 0.2 0.2 m3 were collected in the field. Core samples of 38 mm in diameter were prepared from the blocks and the edges of the samples were cut parallel as per ASTM standards (ASTM D4543) (Fig. 6). The tests were conducted on the finished specimens by the procedures specified in ASTM and ISRM standards (Table 1). Proper care was taken to adhere to the standards
and hence, the material properties thus obtained are of very high accuracy. The destructive tests like UCS and Point load index involves compression of the rock sample until the failure occurs. Some of the failure modes for point load test and UCS are shown in (Fig. 7). Failure modes under compression is very important in rock engineering problems and has attracted many researchers. Basu et al. [7] have particularly studied the failure pattern of three different rock types (granite, schist and sandstone) and observed
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Fig. 3. Cross stratification in a thinly laminated limestones with clear boundaries.
Fig. 4. Thin section of fossiliferous limestone of S14 and S18 samples.
Fig. 5. Experimental set up for determination of P wave velocities in laboratory.
that the degree of fracturing increases as the strength of the rock increases whereas, rocks with predefined structural control tend to fail along the plane of weakness. The test results of all the parameters are presented in Table 2. Together with the data obtained from this study, the total data set of carbonate rocks gathered from all over the world are represented on the scatter plots for Porosity, Young’s modulus, UCS and Point load index against their respective P-wave velocities (Fig. 8). 3.4. Statistical observation: outlier study
Fig. 6. Cylindrical test samples prepared from carbonate rocks as per standards.
Data collected from different geological environment is likely to show variability because of several source of inhomogeneities introduced in their formation. Particularly,
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N. Madhubabu et al. / Measurement 88 (2016) 202–213 Table 1 Standards for determination of different parameters of rocks. Parameter
Standard
Formula
Density
ISRM, 1977
sat fluid qdry ¼ wsat wsub
w
Porosity
ISRM, 1977
P-wave velocity
ASTM D2845 [4], ISRMc [26]
v ¼ Lt
Point load index
ASTM D5731 [5], ISRM [28]
Is ¼ DP2
UCS, Young’s modulus
ASTM D7012 [6], ISRM [27]
rc ¼ AF
£¼
q
wsat wdry wsat wsub
e
Fig. 7. Failure modes observed in broken test samples of Point load index (left) and UCS (right).
Table 2 Laboratory test results of carbonate samples for different properties. Samples
Density (g/cc)
Porosity (%)
P-wave velocity (m/s)
Poisson’s ratio
Point load index in (MPa)
UCS (MPa)
Young’s modulus (GPa)
S-1 S-2 S-3 S-5 S-7 S-9 S-13 S-14 S-15 S-16 S-17 S-18 Aida-1
2.23 2.53 2.34 2.41 2.13 2.34 2.37 2.24 2.23 2.11 2.43 2.00 2.46
15.15 7.85 7.79 2.74 8.13 7.01 8.64 1.09 11.69 14.04 5.82 18.66 1.98
4233 4450 2939 3248 2664 2887 2963 3333 3109 3623 4348 2204 5000
0.25 0.36 0.34 0.35 0.23 0.39 0.36 0.35 0.39 0.37 0.39 0.36 0.36
0.52 1.35 – 0.51 1.04 1.23 0.86 0.83 0.86 – 0.78 0.35 1.36
11.03 29.11 3.53 5.73 11.38 15.58 6.17 6.62 5.29 3.09 14.56 5.73 33.08
5.56 6.89 6.29 6.52 5.23 6.50 6.44 6.06 6.25 5.79 6.78 5.46 6.68
carbonate rocks are most affected by the geological processes owing to the nature of their formation. To observe this variability, statistical theory was applied to predict the outliers (a point distant from other observation points) in the data sets by using box and whisker plot [61]. In a box plot, the upper part of the box is called third quartile (Q3) and the lower part is first quartile (Q1). The middle line (median) dividing the data into two halves and is also termed as second quartile (Q2). Another important term is the interquartile range which is the difference between Q3 and Q1. After obtaining these values, entire box plot is divided into four invisible regions [15]. To determine the outliers, the limits of inner and outer fences are defined and the whiskers extend to most extreme value of the inner fence.
Two possibilities can be defined now based on the position of data. If the data lies within inner fence it is termed as mild outlier and if outside outer fence, extreme outliers. The fences are defined as shown in Eqs. (1) and (2):
Inner fence ¼ Q 1 1:5IQR and Q 1 3IQR
ð1Þ
Outer fence ¼ Q 3 1:5IQR and Q 3 þ 3IQR
ð2Þ
4. Methodology: soft computing analysis UCS and E are a prerequisite for rock mechanics related studies in mining and civil works and also in any geotechnical studies pertinent to petroleum industry. Difficulty in
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Fig. 8. Plot of correlation between different geomechanical parameters against Ultrasonic velocity.
obtaining good quality cores from the carbonates and requirement of high quality samples as per standards restricts a geotechnician to obtain these properties with reasonable confidence. Therefore, along with the data from the present study, a large number of data set was collected to avoid the complexity of the test procedures and present an alternative and easy to use method. Multiple linear regression analysis (MVRA) and Artificial Neural Networks (ANN) were adopted to develop a relationship between the various easily determinable index tests with an aim to estimate corresponding UCS and E with relative confidence. 4.1. Multiple Linear Regression Analysis (MVRA) Multiple linear regression uses several multiple input parameters to predict output parameter. It is generally expressed as the relation of a dependent variable (Y) and a set of independent variables (X 1 ; X 2 ; X 3 ; . . . ::; X n ) where the coefficient of each independent variables are constants [42]. A multiple linear equation can be expressed as
Y 1 ¼ A þ B1 X 1 þ B2 X 2 þ B3 X 3 þ . . . . . . BN X N
ð3Þ
The predicted Y value (Y 1 ) is then compared with the measured Y value for the goodness of the prediction equation. The result is evaluated using the correlation coefficients. 4.2. Artificial Neural Network analysis (ANN) Artificial Neural Networks have been widely used in various branches of science and technology since the 1940s. ANN is generally a soft-computing system which uses parallely arranged processing units called ‘‘neurons”, organized in the form of layers. The interconnection between two consecutive layers is called as ‘‘weights” (Fig. 9). The first and last layer of ANN structure is called the input and the output layers respectively. The input layer only serves to feed the input data to the hidden layer which is between the input and output layers [12]. Generally, there can be many number of hidden layers in the ANN structure, however, from practical purposes, only one or two hidden layers are used [23]. The basic principle behind the working of ANN is same as that of human brain
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Fig. 9. A multiple perception neural network.
which analyzes the data and stores the data for future predictions. The application process of an ANN model design involves the following steps viz. collecting the entire data in one place, converting the data into ANN inputs as it can read only normalized data i.e. between 0 and 1, training, validating and testing the network topology and repeat the above steps as long as it is required to predict the optimum model [58,57,54]. For the study conducted, the type of neural network employed is a Feed- forward back propagation network [59] which is basically a multiple perception network (MLP) (Fig. 9).
The network structure consists of two layers with logarithmic sigmoid transfer function in the hidden layer with 6 neurons and tangent sigmoid transfer function in the output layer. The training function used is Gradient Descent (TRAINGDX). The learning function used is LEARNGDM which is an adaptive learning function. The other training parameters selected are 1000 epochs and the momentum coefficient as 0.9. Corresponding to the five input parameters, five neurons were taken in the input layer and in the same way two neurons corresponding to the output properties in the output layer. The network structure employed in this model is given in Fig. 10. Previously, several researchers have done similar studies
Fig. 10. Network structure for ANN analysis.
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employing ANN which is discussed in detail by Lai and Serra [2]; [38]. In order to get good satisfactory performance of the model, the data should be normalized i.e. the data range should vary between 0 and 1. The formula used for normalizing the data is given by Rafiq et al. [48]
X Norm ¼
X actual X min X max X min
ð4Þ
where X Norm is the normalized value, XActual is the actual value, XMax is the maximum value of the data set and XMin is the minimum value of the data set. 5. Results and discussion From Fig. 8, the relationships of v p versus Porosity and Young’s modulus of carbonate rocks appear to be linear but clearly the data shows two regression trends, especially, the v p versus Porosity plot demonstrates both positive [33,49,37,21] and inverse linear relationships between them. However, the data of v p versus UCS and PLI are scattered implying that there might not be a good relationship between them possibly because of varying mineral composition, presence of secondary porosity and other factors (Fig. 8). On the other hand, the correlations of v p versus E [3,44,63] and v p versus UCS [17,44,63] shows
209
positive relationships. Kainthola et al. Kainthola et al. [35] conducted experiments for different rocks including carbonates and found a linear relationship between point load index and uniaxial compressive strength. Overall, there is not a good relationship between the different observed geomechanical properties with ultrasonic velocity. This could be due to the variations in depositional environments, composition, texture and complex diagenesis processes which alters the internal structures completely, creating a significant effect on properties of the carbonate rocks [1]. Flavio et al. [18] have compared vp versus porosity with varying depositional environments and suggested that the carbonates deposited on the shallow water platform are confined to a narrow range with low porosities and high ultrasonic velocity, whereas, the carbonates from deeper shelf, slope and basin shows large variations of porosity values towards the lower ultrasonic velocity, however at higher ultrasonic velocity values the trend is same as in the platform deposit. Han et al. [24] observed that clay minerals exhibit lower P-wave velocities (1.1–2.8 km/s) relative to other minerals, the reason being the higher compressibility of clay minerals significantly affects the ultrasonic waves travelling through the rocks [53]. In addition [53], found out that high clay content leads to change in mechanical and elastic properties of rocks due to specific mineral changes as a result of changes in water content.
Fig. 11. Box and whisker plot showing the existence of outlier in different parameters.
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Fig. 12. Plot showing (a) predicted UCS versus Measured UCS and (b) predicted Young’s modulus versus measured Young’s modulus from MVRA.
For all the parameters, a normal distribution function is assumed, which also fits well over the scatter plot as can be seen from Fig. 11. Normal distribution is mostly used in geotechnical engineering unless there are reasons valid enough for selecting other types and also because most random variables confirm to this distribution [25]. The mean in all cases is above the median which shows that the data is skewed. Also, there are several data points which lie between inner and outer fence (Eqs. (1) and (2)) showing the existence of mild outlier (outer fence is not shown in Fig. 11) and only few data points lie outside outer fence indicating minor presence of extreme outliers. Outliers are supposed to show discrepancy in data estimation or measurement and is commonly observed while comparing two or more samples. In this case, a prediction model was the desired result by comparing test data from several different locations, although of the same rock type. Outlier in this case is not a result of measurement error, it simply shows that carbonate rocks from different environment may have variable strength, porosity and elastic modulus which is also observed from Fig. 8. In cases of porosity, there are two trends observed in scatter plot (Fig. 8) and mean is much greater than mode which results in more number of outliers unlike other properties (Fig. 11). For prediction of both UCS and E, the input parameters considered here are Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. The stepwise regression procedure is used for analysis. The equations proposed for the prediction of Young’s modulus and UCS from the input parameters using MVRA are
E ¼ 43:214 2:867 £ þ 1:384 Is 127:411 l þ 18:251 q 0:0162 V p
ð5Þ
UCS ¼ 11:813 2:572 £ þ 23:665 Is þ 41:654 l þ 12:197 q 0:001 V p
ð6Þ
where E = Young’s modulus (GPa), UCS = uniaxial compressive strength (MPa), £ = Porosity (%), Is = Point load index
(MPa), l = Poisson’s ratio, q = Density (gm/cc) and V p = Pwave velocity (m/s) The relationship between measured and predicted E and UCS and the correlation coefficient (R2) obtained from multiple linear regression are illustrated in Fig. 12. From MVRA, the data predicted shows high degree of randomness for E as well as in UCS, higher in case of E. A total of 176 data sets were subjected to modeling, out of which, a first set of 163 data points were used to train and validate the network for prediction. Later, the second set of 13 new data points from present work were used to check the efficiency of prediction and accuracy of the trained model. The input parameters consists of Porosity (£), Density (q), P-wave velocity (V p ), Poisson’s ratio (l) and Point load index (Is ) while the output of the model is to predict UCS and E. The validation curve of the neural network generated is given in Fig. 13. The graph shows that the root mean squared error for the training curve decreases gradually with increasing epochs. The validation line is almost close to the best representing line throughout the 1000 epochs (Fig. 13). Regression plot for the training, testing and validation of the model is shown in Fig. 14. In each graph, the higher correlation coefficient shows that the model is accurate in predicting the output. The results of the ANN model can be plotted on a scatter plot showing the measured output versus predicted output. Plotting the data points for the predicted versus measured output against a 1:1 line is a best way of finding out the prediction capacity of the model. Point that lies on the 1:1 line shows the exact prediction of the output by the model and closer a point to the 1:1 line, the better is the prediction. The measured versus predicted plots were constructed for UCS and Young’s modulus. For both the output parameters, the plotted points lie close to the 1:1 line implying a good prediction of ANN model, which is independent of the simplified assumptions like linear behavior or exponential behavior (Fig. 15). The results obtained from ANN and multiple linear regression analysis were compared on the basis of
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Fig. 13. Validation plot of ANN analysis.
Fig. 14. Regression plot for ANN analysis.
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Fig. 15. Plot showing (a) predicted Young’s modulus versus measured Young’s modulus and (b) predicted UCS versus measured UCS from ANN.
Table 3 Comparison of result from different models. Output Parameter
Prediction model
Correlation coefficient (R2)
RMSE
Young’s modulus
Multiple linear regression ANN Multiple linear regression ANN
0.70 0.96 0.91 0.97
9.4 3.8 11.4 6.6
UCS
correlation coefficient (R2) and root mean square error (RMSE). The following Table 3 shows the correlation coefficient and RMSE belonging to each output parameter given by different models. A significant reduction in RMSE, a measure of accuracy, for both the parameters in case of ANN indicates better predictive capacity over multiple linear regression analysis in predicting the data. The observation of ANN’s superiority over MVRA were also made by previous researchers [32]; [54]; [55].
6. Conclusion In the present study, an attempt was made to establish a correlation between several index parameters with ultrasonic velocity. The aim was to predict the uniaxial compressive strength (UCS) and modulus of elasticity (E) of carbonate rocks. ANN and MVRA were employed in order to find suitable equations for the prediction of UCS and E from easily determinable parameters like Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. However, a good correlation was not observed for UCS and Point load index with P-wave velocity indicating complex nature of geological setting of carbonate rocks, while Young’s modulus and Porosity showed linear relationship with two distinct trends. The result of MVRA shows high correlation coefficient for uniaxial compressive strength, while the correlation coefficient was found to be lower for Young’s modulus. RMSE, when compared for
both predictive models, show significant reduction in case of ANN for both the parameters. Out of two prediction models, ANN proves to be a more competent model for estimation of even complex rock properties. ANN has the capability of solving a non-linear problem without even going into the complexity of dealing with high precession sample preparation.
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Measurement journal homepage: www.elsevier.com/locate/measurement
Prediction of compressive strength and elastic modulus of carbonate rocks N. Madhubabu a, P.K. Singh b,⇑, Ashutosh Kainthola b, Bankim Mahanta b, A. Tripathy b, T.N. Singh b a b
Geo Constech Pvt. Ltd., New Delhi, India Department of Earth Sciences, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India
a r t i c l e
i n f o
Article history: Received 8 June 2015 Received in revised form 18 March 2016 Accepted 24 March 2016 Available online 1 April 2016 Keywords: Young’s modulus Porosity Ultrasonic velocity Carbonate rocks Artificial Neural Network
a b s t r a c t Uniaxial Compressive Strength (UCS) and Modulus of elasticity (E) of carbonate rocks are very critical properties in petroleum, mining and civil industries. UCS is the measure of the strength of the rock and E depicts the stiffness, together they control the deformational behavior. But the heterogeneity introduced as a result of fractures, dissolution and dependency on pH and temperature makes them a difficult material to study. Complex diagenesis and resulting pore system makes the job even more daunting. So, an attempt is made to predict these properties using simple index parameters such as Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. Multiple Linear Regression Analysis (MVRA) and Artificial Neural Networking (ANN) have been used for predicting the two properties and the accuracy is tested by root mean square error. The results show that ANN has a better predictive efficiency than MVRA and they can be applied for predicting UCS and Young’s modulus of carbonate rocks with reasonable confidence. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Carbonate rocks are formed in a variety of geological settings and are one of the best reservoir rocks for oil exploration [60]. It is assumed that the proportion of crude oil confined in carbonate rocks is around 50–60% with an estimated production lifetime of more than 50 years [11]. Carbonate rocks are very low strength and are highly susceptible to weathering due to chemically unstable mineralogical and textural combination, and their high reactivity with pH and temperature [41]. As important it is to determine the geomechanical properties with high accuracy, difficulty in sample preparation adhering certain standards along with several inherent heterogeneities makes it a formidable task for rock engineers. In rock ⇑ Corresponding author. Tel.: +91 22 25767271. E-mail address: [email protected] (P.K. Singh). http://dx.doi.org/10.1016/j.measurement.2016.03.050 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
engineering, both uniaxial compressive strength (UCS) and the Modulus of elasticity (E) are widely used parameters as they are important for intact rock classification and rock failure criteria [16]. These parameters have great importance in rock physics applications, viz., onshore and offshore geomechanical engineering, tunneling, dam design, rock drilling and blasting, rock excavation and even for slope stability. There are two methods for assessing the properties of rocks. First is the direct method where tests are conducted on a crafted specimen in the laboratory, the other, known as the indirect method, uses the previously derived empirical equations from the literature [8]. The test procedure for measuring UCS and E have been standardized by both the American Society for Testing and Materials (ASTM) and the International Society for Rock Mechanics (ISRM). Sound rock specimens are required for direct determination of UCS and Young’s modulus of rocks in
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the laboratories. However, high quality core samples in sufficient quantities are not possible to obtain from weak, highly fractured, weathered carbonate rocks. Moreover, the test procedures are expensive because of the requirement to precisely, carefully prepare the specimens to ensure that the ends are smooth and perfectly parallel and several other details as per standards. In addition, careful execution of these tests are difficult, time consuming and requires expensive equipment [20,8,34]. Point load test is an alternative method to determine UCS as it gives reliable values at a lower cost and with ease [50]. In order to overcome these difficulties, indirect methods employing simple index parameters like the density, porosity, P-wave velocity, Point load index, Schmidt hammer were used [29,14,13,47]. These simple index tests require a relatively small number of samples, are quick, less expensive and easy to execute. In recent years, statistical methods used in rock engineering, such as simple and multiple regression techniques have been used for establishing predictive models. In addition to these conventional methods, new techniques like Artificial Neural Networks (ANN), fuzzy interference systems, genetic programming and regression trees have also been tested to estimate the required properties [39,22,40,55,31,8,64,51,62] ANN methods have become popular since the year 1990. It is a form of analysis which is based on the understanding of the brain and human nervous system [23]. Major advantage of ANN is its efficient handling of highly non-linear relationships among data, even when the exact nature of such relationship is not fully unknown [19]. Therefore, neural networks are best suited for UCS and Young’s modulus predictions of carbonate rocks due of the complex nature of inter-relationships among the various quality parameters, composition and processing conditions [16]. The performance of the ANN methods was also compared with other statistical methods in the present study (e.g., Regression analysis). Previous studies have shown that ANN predictive models have better efficiency than the conventional statistical methods [46,56,8,16,62,34]. The objective of this study is to predict UCS and Young’s modulus of carbonate rocks using the data gathered from previous studies (163 sample data) and the data set of carbonate rocks which have been collected from the Miocene section of the Kutch basin, Gujarat, India.
2. Study area The investigated area lies in the Kutch district of Gujarat and is the westernmost part of India. It is a part of Kutch basin which is a peri-cratonic rift basin and has preserved a complete sequence from Triassic to Recent [9]. The area composed of Cenozoic sequences, especially the Miocene sediments, lies in Naliya, Kutch District, Gujarat. The Lower Miocene rocks are exposed in Khari river section (Fig. 1). A graphical log was also prepared showing the lithounits, lithology, sedimentary textures, structures and different lithofacies from the field data (Fig. 1). The first step towards preparation of graphic log was
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identification of different sedimentary units which was almost done in the field itself. In the present work, individual beds in the sequence are given a standard color, the thickness of individual beds, the texture, the sedimentary structures (if any), and fossil contents are marked. As can be seen from the log, clay minerals are ubiquitous throughout the sequence, even the carbonate samples collected from Miocene Formation for laboratory tests contains a maximum clay content ranging up to 20%. The limestone samples are enriched with fossils like Turritella, Bivalves (Oysters and Pecten), etc. (Fig. 2) and also the presence of cross lamination in the limestones were observed in the field (Fig. 3). 3. Methodology: laboratory tests 3.1. Petrography Thin section of two areas (S14 and S18) are presented in this study as the prime focus was to predict strength parameters of carbonate rocks. No correlation was made between textural changes and geomechanical properties, although we don’t neglect the possibility. The sections clearly show the presence of fossils of different assemblage along with micrite (brown) and sparite (off white) matrix (Fig. 4). In case of S14, fossils are cemented together with sparry calcite whereas mixed skeletal remains are surrounded by micritic matrix in S18. Signatures of re-precipitation of micrites on fossils can also be seen. Presence of micrite indicates deposition under calm water condition whereas, presence of sparry calcite in pore spaces suggests deposition under agitated water condition [10]. 3.2. Ultrasonic velocity Ultrasonic techniques for measuring P and S-wave velocities are non-destructive and easy to execute, both in the laboratories and field. Nowadays, pulse generation method is used for the determination of parameters using Pundit testing machine (Fig. 5) which consists of a pulse transmission generator, transducers and electronic tester for measurements of time [55,63]. A coupling gel is applied on the faces to avoid any air gap between the sample and transducers. A number of factors have been known to influence the seismic velocity of rocks which include water, clay, grain size, density, texture, porosity anisotropy and others [30]. Considering the fact that the data obtained by other researchers used in this study is to the standard along with the data obtained in this study, P-wave can be considered as a reliable parameter to relate with other geomechanical properties [43]. 3.3. Strength test A number of block samples of carbonate rocks were collected from the Miocene section of the Kutch basin. In addition, 163 sample data of carbonates were taken from previous research work around the world. Each data set includes Porosity (£), Density (q), P-wave velocity (V p ), Poisson’s ratio (l) Point load index (Is ), uniaxial
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Fig. 1. Geological map of the study area and lithology of the collected samples prepared after Biswas [9].
Turitella
Pecten
Oyster
Fig. 2. Turritella and Bivalve rich limestones in Kutch basin of Miocene formation.
compressive strength (UCS) and Modulus of elasticity (E). From the recent study, thirteen number of block samples, each having an approximate dimension of 0.2 0.2 0.2 m3 were collected in the field. Core samples of 38 mm in diameter were prepared from the blocks and the edges of the samples were cut parallel as per ASTM standards (ASTM D4543) (Fig. 6). The tests were conducted on the finished specimens by the procedures specified in ASTM and ISRM standards (Table 1). Proper care was taken to adhere to the standards
and hence, the material properties thus obtained are of very high accuracy. The destructive tests like UCS and Point load index involves compression of the rock sample until the failure occurs. Some of the failure modes for point load test and UCS are shown in (Fig. 7). Failure modes under compression is very important in rock engineering problems and has attracted many researchers. Basu et al. [7] have particularly studied the failure pattern of three different rock types (granite, schist and sandstone) and observed
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Fig. 3. Cross stratification in a thinly laminated limestones with clear boundaries.
Fig. 4. Thin section of fossiliferous limestone of S14 and S18 samples.
Fig. 5. Experimental set up for determination of P wave velocities in laboratory.
that the degree of fracturing increases as the strength of the rock increases whereas, rocks with predefined structural control tend to fail along the plane of weakness. The test results of all the parameters are presented in Table 2. Together with the data obtained from this study, the total data set of carbonate rocks gathered from all over the world are represented on the scatter plots for Porosity, Young’s modulus, UCS and Point load index against their respective P-wave velocities (Fig. 8). 3.4. Statistical observation: outlier study
Fig. 6. Cylindrical test samples prepared from carbonate rocks as per standards.
Data collected from different geological environment is likely to show variability because of several source of inhomogeneities introduced in their formation. Particularly,
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N. Madhubabu et al. / Measurement 88 (2016) 202–213 Table 1 Standards for determination of different parameters of rocks. Parameter
Standard
Formula
Density
ISRM, 1977
sat fluid qdry ¼ wsat wsub
w
Porosity
ISRM, 1977
P-wave velocity
ASTM D2845 [4], ISRMc [26]
v ¼ Lt
Point load index
ASTM D5731 [5], ISRM [28]
Is ¼ DP2
UCS, Young’s modulus
ASTM D7012 [6], ISRM [27]
rc ¼ AF
£¼
q
wsat wdry wsat wsub
e
Fig. 7. Failure modes observed in broken test samples of Point load index (left) and UCS (right).
Table 2 Laboratory test results of carbonate samples for different properties. Samples
Density (g/cc)
Porosity (%)
P-wave velocity (m/s)
Poisson’s ratio
Point load index in (MPa)
UCS (MPa)
Young’s modulus (GPa)
S-1 S-2 S-3 S-5 S-7 S-9 S-13 S-14 S-15 S-16 S-17 S-18 Aida-1
2.23 2.53 2.34 2.41 2.13 2.34 2.37 2.24 2.23 2.11 2.43 2.00 2.46
15.15 7.85 7.79 2.74 8.13 7.01 8.64 1.09 11.69 14.04 5.82 18.66 1.98
4233 4450 2939 3248 2664 2887 2963 3333 3109 3623 4348 2204 5000
0.25 0.36 0.34 0.35 0.23 0.39 0.36 0.35 0.39 0.37 0.39 0.36 0.36
0.52 1.35 – 0.51 1.04 1.23 0.86 0.83 0.86 – 0.78 0.35 1.36
11.03 29.11 3.53 5.73 11.38 15.58 6.17 6.62 5.29 3.09 14.56 5.73 33.08
5.56 6.89 6.29 6.52 5.23 6.50 6.44 6.06 6.25 5.79 6.78 5.46 6.68
carbonate rocks are most affected by the geological processes owing to the nature of their formation. To observe this variability, statistical theory was applied to predict the outliers (a point distant from other observation points) in the data sets by using box and whisker plot [61]. In a box plot, the upper part of the box is called third quartile (Q3) and the lower part is first quartile (Q1). The middle line (median) dividing the data into two halves and is also termed as second quartile (Q2). Another important term is the interquartile range which is the difference between Q3 and Q1. After obtaining these values, entire box plot is divided into four invisible regions [15]. To determine the outliers, the limits of inner and outer fences are defined and the whiskers extend to most extreme value of the inner fence.
Two possibilities can be defined now based on the position of data. If the data lies within inner fence it is termed as mild outlier and if outside outer fence, extreme outliers. The fences are defined as shown in Eqs. (1) and (2):
Inner fence ¼ Q 1 1:5IQR and Q 1 3IQR
ð1Þ
Outer fence ¼ Q 3 1:5IQR and Q 3 þ 3IQR
ð2Þ
4. Methodology: soft computing analysis UCS and E are a prerequisite for rock mechanics related studies in mining and civil works and also in any geotechnical studies pertinent to petroleum industry. Difficulty in
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Fig. 8. Plot of correlation between different geomechanical parameters against Ultrasonic velocity.
obtaining good quality cores from the carbonates and requirement of high quality samples as per standards restricts a geotechnician to obtain these properties with reasonable confidence. Therefore, along with the data from the present study, a large number of data set was collected to avoid the complexity of the test procedures and present an alternative and easy to use method. Multiple linear regression analysis (MVRA) and Artificial Neural Networks (ANN) were adopted to develop a relationship between the various easily determinable index tests with an aim to estimate corresponding UCS and E with relative confidence. 4.1. Multiple Linear Regression Analysis (MVRA) Multiple linear regression uses several multiple input parameters to predict output parameter. It is generally expressed as the relation of a dependent variable (Y) and a set of independent variables (X 1 ; X 2 ; X 3 ; . . . ::; X n ) where the coefficient of each independent variables are constants [42]. A multiple linear equation can be expressed as
Y 1 ¼ A þ B1 X 1 þ B2 X 2 þ B3 X 3 þ . . . . . . BN X N
ð3Þ
The predicted Y value (Y 1 ) is then compared with the measured Y value for the goodness of the prediction equation. The result is evaluated using the correlation coefficients. 4.2. Artificial Neural Network analysis (ANN) Artificial Neural Networks have been widely used in various branches of science and technology since the 1940s. ANN is generally a soft-computing system which uses parallely arranged processing units called ‘‘neurons”, organized in the form of layers. The interconnection between two consecutive layers is called as ‘‘weights” (Fig. 9). The first and last layer of ANN structure is called the input and the output layers respectively. The input layer only serves to feed the input data to the hidden layer which is between the input and output layers [12]. Generally, there can be many number of hidden layers in the ANN structure, however, from practical purposes, only one or two hidden layers are used [23]. The basic principle behind the working of ANN is same as that of human brain
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Fig. 9. A multiple perception neural network.
which analyzes the data and stores the data for future predictions. The application process of an ANN model design involves the following steps viz. collecting the entire data in one place, converting the data into ANN inputs as it can read only normalized data i.e. between 0 and 1, training, validating and testing the network topology and repeat the above steps as long as it is required to predict the optimum model [58,57,54]. For the study conducted, the type of neural network employed is a Feed- forward back propagation network [59] which is basically a multiple perception network (MLP) (Fig. 9).
The network structure consists of two layers with logarithmic sigmoid transfer function in the hidden layer with 6 neurons and tangent sigmoid transfer function in the output layer. The training function used is Gradient Descent (TRAINGDX). The learning function used is LEARNGDM which is an adaptive learning function. The other training parameters selected are 1000 epochs and the momentum coefficient as 0.9. Corresponding to the five input parameters, five neurons were taken in the input layer and in the same way two neurons corresponding to the output properties in the output layer. The network structure employed in this model is given in Fig. 10. Previously, several researchers have done similar studies
Fig. 10. Network structure for ANN analysis.
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employing ANN which is discussed in detail by Lai and Serra [2]; [38]. In order to get good satisfactory performance of the model, the data should be normalized i.e. the data range should vary between 0 and 1. The formula used for normalizing the data is given by Rafiq et al. [48]
X Norm ¼
X actual X min X max X min
ð4Þ
where X Norm is the normalized value, XActual is the actual value, XMax is the maximum value of the data set and XMin is the minimum value of the data set. 5. Results and discussion From Fig. 8, the relationships of v p versus Porosity and Young’s modulus of carbonate rocks appear to be linear but clearly the data shows two regression trends, especially, the v p versus Porosity plot demonstrates both positive [33,49,37,21] and inverse linear relationships between them. However, the data of v p versus UCS and PLI are scattered implying that there might not be a good relationship between them possibly because of varying mineral composition, presence of secondary porosity and other factors (Fig. 8). On the other hand, the correlations of v p versus E [3,44,63] and v p versus UCS [17,44,63] shows
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positive relationships. Kainthola et al. Kainthola et al. [35] conducted experiments for different rocks including carbonates and found a linear relationship between point load index and uniaxial compressive strength. Overall, there is not a good relationship between the different observed geomechanical properties with ultrasonic velocity. This could be due to the variations in depositional environments, composition, texture and complex diagenesis processes which alters the internal structures completely, creating a significant effect on properties of the carbonate rocks [1]. Flavio et al. [18] have compared vp versus porosity with varying depositional environments and suggested that the carbonates deposited on the shallow water platform are confined to a narrow range with low porosities and high ultrasonic velocity, whereas, the carbonates from deeper shelf, slope and basin shows large variations of porosity values towards the lower ultrasonic velocity, however at higher ultrasonic velocity values the trend is same as in the platform deposit. Han et al. [24] observed that clay minerals exhibit lower P-wave velocities (1.1–2.8 km/s) relative to other minerals, the reason being the higher compressibility of clay minerals significantly affects the ultrasonic waves travelling through the rocks [53]. In addition [53], found out that high clay content leads to change in mechanical and elastic properties of rocks due to specific mineral changes as a result of changes in water content.
Fig. 11. Box and whisker plot showing the existence of outlier in different parameters.
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Fig. 12. Plot showing (a) predicted UCS versus Measured UCS and (b) predicted Young’s modulus versus measured Young’s modulus from MVRA.
For all the parameters, a normal distribution function is assumed, which also fits well over the scatter plot as can be seen from Fig. 11. Normal distribution is mostly used in geotechnical engineering unless there are reasons valid enough for selecting other types and also because most random variables confirm to this distribution [25]. The mean in all cases is above the median which shows that the data is skewed. Also, there are several data points which lie between inner and outer fence (Eqs. (1) and (2)) showing the existence of mild outlier (outer fence is not shown in Fig. 11) and only few data points lie outside outer fence indicating minor presence of extreme outliers. Outliers are supposed to show discrepancy in data estimation or measurement and is commonly observed while comparing two or more samples. In this case, a prediction model was the desired result by comparing test data from several different locations, although of the same rock type. Outlier in this case is not a result of measurement error, it simply shows that carbonate rocks from different environment may have variable strength, porosity and elastic modulus which is also observed from Fig. 8. In cases of porosity, there are two trends observed in scatter plot (Fig. 8) and mean is much greater than mode which results in more number of outliers unlike other properties (Fig. 11). For prediction of both UCS and E, the input parameters considered here are Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. The stepwise regression procedure is used for analysis. The equations proposed for the prediction of Young’s modulus and UCS from the input parameters using MVRA are
E ¼ 43:214 2:867 £ þ 1:384 Is 127:411 l þ 18:251 q 0:0162 V p
ð5Þ
UCS ¼ 11:813 2:572 £ þ 23:665 Is þ 41:654 l þ 12:197 q 0:001 V p
ð6Þ
where E = Young’s modulus (GPa), UCS = uniaxial compressive strength (MPa), £ = Porosity (%), Is = Point load index
(MPa), l = Poisson’s ratio, q = Density (gm/cc) and V p = Pwave velocity (m/s) The relationship between measured and predicted E and UCS and the correlation coefficient (R2) obtained from multiple linear regression are illustrated in Fig. 12. From MVRA, the data predicted shows high degree of randomness for E as well as in UCS, higher in case of E. A total of 176 data sets were subjected to modeling, out of which, a first set of 163 data points were used to train and validate the network for prediction. Later, the second set of 13 new data points from present work were used to check the efficiency of prediction and accuracy of the trained model. The input parameters consists of Porosity (£), Density (q), P-wave velocity (V p ), Poisson’s ratio (l) and Point load index (Is ) while the output of the model is to predict UCS and E. The validation curve of the neural network generated is given in Fig. 13. The graph shows that the root mean squared error for the training curve decreases gradually with increasing epochs. The validation line is almost close to the best representing line throughout the 1000 epochs (Fig. 13). Regression plot for the training, testing and validation of the model is shown in Fig. 14. In each graph, the higher correlation coefficient shows that the model is accurate in predicting the output. The results of the ANN model can be plotted on a scatter plot showing the measured output versus predicted output. Plotting the data points for the predicted versus measured output against a 1:1 line is a best way of finding out the prediction capacity of the model. Point that lies on the 1:1 line shows the exact prediction of the output by the model and closer a point to the 1:1 line, the better is the prediction. The measured versus predicted plots were constructed for UCS and Young’s modulus. For both the output parameters, the plotted points lie close to the 1:1 line implying a good prediction of ANN model, which is independent of the simplified assumptions like linear behavior or exponential behavior (Fig. 15). The results obtained from ANN and multiple linear regression analysis were compared on the basis of
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Fig. 13. Validation plot of ANN analysis.
Fig. 14. Regression plot for ANN analysis.
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Fig. 15. Plot showing (a) predicted Young’s modulus versus measured Young’s modulus and (b) predicted UCS versus measured UCS from ANN.
Table 3 Comparison of result from different models. Output Parameter
Prediction model
Correlation coefficient (R2)
RMSE
Young’s modulus
Multiple linear regression ANN Multiple linear regression ANN
0.70 0.96 0.91 0.97
9.4 3.8 11.4 6.6
UCS
correlation coefficient (R2) and root mean square error (RMSE). The following Table 3 shows the correlation coefficient and RMSE belonging to each output parameter given by different models. A significant reduction in RMSE, a measure of accuracy, for both the parameters in case of ANN indicates better predictive capacity over multiple linear regression analysis in predicting the data. The observation of ANN’s superiority over MVRA were also made by previous researchers [32]; [54]; [55].
6. Conclusion In the present study, an attempt was made to establish a correlation between several index parameters with ultrasonic velocity. The aim was to predict the uniaxial compressive strength (UCS) and modulus of elasticity (E) of carbonate rocks. ANN and MVRA were employed in order to find suitable equations for the prediction of UCS and E from easily determinable parameters like Porosity, Density, P-wave velocity, Poisson’s ratio and Point load index. However, a good correlation was not observed for UCS and Point load index with P-wave velocity indicating complex nature of geological setting of carbonate rocks, while Young’s modulus and Porosity showed linear relationship with two distinct trends. The result of MVRA shows high correlation coefficient for uniaxial compressive strength, while the correlation coefficient was found to be lower for Young’s modulus. RMSE, when compared for
both predictive models, show significant reduction in case of ANN for both the parameters. Out of two prediction models, ANN proves to be a more competent model for estimation of even complex rock properties. ANN has the capability of solving a non-linear problem without even going into the complexity of dealing with high precession sample preparation.
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