# Modulus of Elasticity for Grain-Supported Carbonates—Determination and Estimation for Preliminary Engineering Purposes

^{1}

^{2}

^{3}

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## Abstract

**:**

## Featured Application

**Modulus of elasticity for grain-supported carbonate rocks is successfully estimated by using simple estimation models. The methodology and models described in this paper can help engineers in the early design stages when they need a quick and easy estimate.**

## Abstract

^{2}) used to evaluate the success of the estimation, simple regression tree models were found to perform well for the preliminary estimation, while more complex models based on Bagging performed very well.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Grain-Supported Carbonates

#### 2.2. Methods for Determining Young’s Modulus and Other Properties

_{a}is change in axial strain. It is emphasised that the test interval for the average modulus of elasticity is not fixed by ISRM recommendations and should be adjusted for common rock, depending on how large the linear part of the stress-strain curve is.

_{c}) was determined according to the peak stress required for failure of the specimen. The stress load was 0.75 MPa per second and was performed using an ELE ADR 2000 laboratory press.

#### 2.3. Use of R Statistical Environment for Estimating Youngs Moduli

^{2}, tree for the random tree analysis, and randomForest for bagging and the random forest model.

_{0}and β

_{1}are constants.

_{1}, X

_{2}, …, X

_{p}and a response variable Y. The model is given by the Formula (3)

_{0}, β

_{1}, …, β

_{p}are regression coefficients and $\u03f5$ is a random error term with zero mean. The coefficient β

_{0}is also known as the intercept term. The interpretation of the coefficients is as follows. If X

_{i}increase for one unit, holding all other predictors fixed, the regression coefficient β

_{i}equals the average change of Y. In practice, the coefficients are unknown and estimated using the least squares method [14]. In the case of one predictor (k = 1), the data are organised in a set of points (x

_{i}, y

_{i}) where x

_{i}is a measurement of X and y

_{i}is a measurement of Y that corresponds to x

_{i}. The least squares method gives “the best” line that fits the data—so that y

_{i}≈ β

_{0}+ β

_{1}x

_{i}, for i = 1, …, n. In the case of two predictors (k = 2), the method gives “the best” plane that fits the data. In the case of more than two predictors, there is a lack of intuition and imagination, but the model is still correct and interpretable in the described way.

^{2}and given by (4)

_{i}using x

_{i}and a statistical method, and $\overline{y}$ is the overall sample mean.

^{2}statistic takes the form of the proportion of the variability in the response variable that is explained by the model. If R

^{2}takes the value close to 1, that indicates that a large amount of variability in the response variable has been explained by the fitted model. However, it is clear that R

^{2}will increase when more variables are added to the model. If adding some new predictor to the model leads to just a tiny increase in R

^{2}, then it can be dropped from the model [14]. Adding such predictors to the model also leads to overfitting. Therefore, it is useful to have more measures of goodness of model fit.

^{2}, called adjusted R

^{2}, which adjusts thevpreviously described R

^{2}by taking into account the size of the dataset and number of parameters in the model. It is given by (7).

_{(n)}(9) [14]. RMSECV is given by (8).

_{i}is the observed ith value and ${\widehat{y}}_{i}$ is the predicted value for y

_{i}using x

_{i}and a statistical method fitted on the remaining n − 1 observations. The main advantage of this model accuracy measure is its unbiasedness.

_{C}and RSS

_{R}are the residual sum of squares for the complete and reduced model, respectively. Further, ${p}_{C}$ and ${p}_{R}$ are the numbers of predictors in the complete and reduced model and n is the length of the dataset. If the null hypothesis is true, the test statistic has an F distribution with ($n-{p}_{R}$) and ($n-{p}_{C}$) degrees of freedom.

_{i}equals 0.

_{k}and split value s that partitions the data into two groups S

_{1}= {X|X

_{k}< s} and S

_{2}= {X|X

_{k}≥ s} such that the overall sums of squares are minimised (15)

_{1}and S

_{2}, respectively [15].

_{1}, S

_{2}, …, S

_{N}called terminal nodes or leaves. Each of the terminal nodes is presented by the average of the response variable of observations settled in that node.

## 3. Results

#### 3.1. Petrographic Characteristics and Physical-Mechanical Properties

^{3}) and P-velocity (average 5710.12 m/s).

#### 3.2. Models for Estimating Modulus of Elasticity

^{3}), I

_{S}

_{(50)}is the point load strength index (MPa), v

_{p}is the P-wave velocity (m/s), σ

_{c}uniaxial compressive strength (MPa). The constructed regression tree is shown in Figure 8.

_{p}), the point load strength index (I

_{s}

_{(50)}), Schmidt rebound hardness (SRH), and uniaxial compressive strength (σ

_{c}). They stand as the most important predictors for the prediction of modulus of elasticity. In function “prune.tree” the parameter “best” equals the number of terminal nodes. If there is no tree of a certain size, the next largest is constructed. Since the tree in Figure 8 has five leaves, the pruned tree could have three or four leaves. Of specific interest may be the one with four leaves (and three most important predictors) shown in Figure 9.

_{p}) and the least important is uniaxial compressive strength (σ

_{c}).

## 4. Discussion

^{2}and RMSECV (Table 3). It is obtained using a leave-one-out cross-validation. The best result according to RMSECV is achieved by the models (17) and (18).

^{2}.

^{2}than every linear model. The models based on regression trees give significantly better results according to R

^{2}and adjusted R

^{2}. Except for bagging, they give slightly worse results due to the value of RMSECV.

^{2}models in this paper was compared with those in other papers [30,31,43]. One obstacle to the comparison is that the same success coefficients were not used, so the comparison can only be made using the size R

^{2}that all models had. This is most evident when comparing the R

^{2}model which was created using artificial neural networks which have a black box in their structure and therefore cannot explain their outputs, and it has undesirable negative properties such as complexity in its multilayer structure, possibility of overlearning [31]. Taking this into consideration, it can only be argued that the models developed in this paper have lower R

^{2}than the neural network models [43].

^{2}(0.8934) to the model created earlier, whose coefficient of determination is 0.908 [31], and it was created using the Least Square Support Vector Machine method (LS-SVM). This is an advanced version that is superior to the artificial neural networks method in terms of predictive performance, generality, and robustness. This is consistent with the research of [29] who investigated the advantages and disadvantages of fuzzy inference systems, artificial neural networks and LS -SVM methods and found that the performance of the LS -SVM model was the best among the other methods. However, the small difference in R

^{2}between the models in this paper and [31] shows the similarity of the success of both models and does not imply that the Bagging model is less useful or less accurate. More so, if the emphasis is on simplicity and great attention is paid to avoiding overfitting, then it is quite justified to use the models of the authors of this paper, as they have used other very rigorous measures (RMSECV and adjusted R

^{2}) to evaluate model performance.

## 5. Conclusions

- Petrographic characteristics of rocks have proven to be extremely important since they directly affect the physical and mechanical properties and modulus of elasticity. According to the obtained results, the packstone and grainstone carbonates from Croatia show similar properties and can be mutually combined when estimating the modulus of elasticity of carbonates. Still, it is important to point out that presented results consider low porosity carbonates without or with minor secondary defects. Other types of carbonate rock materials may have considerably different properties, therefore better results, it is suggested that each group of carbonates is considered and estimated separately.
- It is useful in engineering terms to create simple estimation models based on multiple regression and regression trees, and in this sense, the R statistical environment has proven to be a viable and accessible platform.
- Regression tree models can be used to easily estimate the modulus of elasticity for carbonates with packstone and grainstone textures with input parameters of P-velocity, uniaxial compressive strength, and point load test index.
- The bagging tree model showed the best evaluation results. Its performance parameters are 0.8934 for R
^{2}and 0.8687 for adjusted R^{2}. This model can be used with input parameters listed in the previous point, as well as density, porosity and Schmidt rebound hardness.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Determination of the average modulus of elasticity: (

**a**) stress-strain curve; (

**b**) LVDT based displacement transducer.

**Figure 4.**Grainstone depositional textures from different locations: (

**a**) dolomite from Brušane; (

**b**) dolomite from Škrobotnik; (

**c**) limestone from Salakovci; (

**d**) limestone from Korenići.

**Figure 5.**Packstone depositional texture in limestones from different locations: (

**a**) Međurače; (

**b**) Debelo brdo; (

**c**) Trget; (

**d**) Kanfanar.

**Figure 7.**Correlation between physical and mechanical characteristics, which were later used as input variables in modelling.

General Statistics | $\mathit{\rho}$ | n | I_{s(50)} | SRH | v_{p} | σ_{c} | E |
---|---|---|---|---|---|---|---|

min | 2376 | 0.14 | 0.5 | 38.2 | 4944 | 51.44 | 38.38 |

max | 2823 | 11.25 | 6.8 | 72.5 | 6378 | 181.36 | 93.26 |

average | 2634.64 | 3.22 | 4.30 | 59.05 | 5710.12 | 126.81 | 57.11 |

standard deviation | 122.10 | 3.01 | 1.43 | 8.98 | 389.80 | 32.45 | 12.99 |

Coefficient of variation (%) | 4.63 | 93.48 | 33.26 | 15.21 | 6.83 | 25.59 | 22.75 |

^{3}); n—porosity (%); I

_{s(50)}—point load test index (MPa); v

_{p}—P wave velocity (m/s); SRH—Schmidt rebound hardness; σ

_{c}—uniaxial compressive strength (MPa); E—Young’s modulus of elasticity (GPa).

Location | Lithology | Depositional Texture | Number of In Situ Samples |
---|---|---|---|

Škrobotnik, Brušane | dolomite | dolograinstone | 7 |

Korenići, Salakovci | limestone | grainstone | 7 |

Međurače, Debelo brdo, Trget, Kanfanar | limestone | packstone | 19 |

**Table 3.**RMSECV, adjusted R

^{2}values and their ranking numbers for multiple linear regression models.

Linear Model | RMSECV Rank (Value) | Adjusted R^{2} Rank (Value) |
---|---|---|

(17) | 1. (10.0025) | 4. (0.5129) |

(18) | 2. (10.0313) | 7. (0.4890) |

(19) | 3. (10.0798) | 8. (0.4837) |

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**MDPI and ACS Style**

Briševac, Z.; Pollak, D.; Maričić, A.; Vlahek, A.
Modulus of Elasticity for Grain-Supported Carbonates—Determination and Estimation for Preliminary Engineering Purposes. *Appl. Sci.* **2021**, *11*, 6148.
https://doi.org/10.3390/app11136148

**AMA Style**

Briševac Z, Pollak D, Maričić A, Vlahek A.
Modulus of Elasticity for Grain-Supported Carbonates—Determination and Estimation for Preliminary Engineering Purposes. *Applied Sciences*. 2021; 11(13):6148.
https://doi.org/10.3390/app11136148

**Chicago/Turabian Style**

Briševac, Zlatko, Davor Pollak, Ana Maričić, and Andreja Vlahek.
2021. "Modulus of Elasticity for Grain-Supported Carbonates—Determination and Estimation for Preliminary Engineering Purposes" *Applied Sciences* 11, no. 13: 6148.
https://doi.org/10.3390/app11136148