# Modelling Geotechnical Heterogeneities Using Geostatistical Simulation and Finite Differences Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

#### 2.1. General

#### 2.2. Geostatistical Simulation

- Primarily, the data are analysed and basic statistics are calculated;
- For each variable of interest, the data are transformed into data with a standard Gaussian distribution (normal scores);
- The normal scores’ experimental variograms are computed and the spatial behaviour, including the anisotropy, of the transformed parameters is determined. This stage is followed by model fitting;
- Using the zone centroids as a grid, Gaussian random fields, with covariance functions given by the models fitted in step 3, are simulated at the target points using the turning bands algorithm [25]. This algorithm is chosen because of its ability to accurately reproduce the distribution and spatial correlation structure of the Gaussian random field to be simulated and because of its low central processing unit (CPU) time and memory storage requirements, which outperform other simulation algorithms. In addition, a post-processing stage based on kriging is used to condition the simulation to the normal scores;
- The Gaussian values of each realisation are back-transformed to their original scale.

#### 2.3. Scenario Reduction

- Calculate the Euclidean distance of any pair of realisations.
- Define a dissimilarity matrix between the n realisations using the Euclidean distances resulting from step 1.
- Apply Multi-Dimensional Scaling (MDS) to represent the dissimilarity matrix in a 2D space, and herewith a spatial representation of the n realisations is accomplished. The scaling chosen was classical multidimensional scaling [26].
- Apply a kernel transform to obtain a featured space. In detail, a scale factor is chosen so that the nonlinearity of the Euclidean distance values in the 2D space is reduced.
- Study and find the optimal number of clusters by performing for a first time the kernel k-medoid clustering technique for different numbers of clusters (from 1 until a predefined maximum). To choose the optimal number of clusters, an evaluation criterion is used, such as the average silhouette method.
- Perform the kernel k-medoid clustering for a previously defined optimum number of clusters.
- Plot (in 2D) the optimum number of clusters and their respective medoids (centres).
- Perform the post-processing analysis to understand the validity of each cluster.

## 3. Case Study

#### 3.1. Available Data

_{50mm}). The remaining parameters (P2 to P5) were estimated directly from borehole logging. According to the results of the rock mechanics laboratory tests and the interpreted RMR values from the borehole samples, the rock mass is classified as fair to good (mostly in the range of 50 to 60) in terms of geotechnical quality.

#### 3.2. Three-Dimensional Numerical Model

^{3}and a Poisson ratio of 0.20 were assumed for the rock mass.

#### 3.3. Exploratory Data Analysis

#### 3.4. Model Spatial Continuity

#### 3.5. Conditional Simulation Results

#### 3.6. From Geotechnical Data to Geomechanical Parameters

#### 3.7. Scenario Reduction

## 4. Numerical Modelling Results

## 5. Discussion

## 6. Conclusions

_{m}) through the use of empirical formulas. If detailed in situ data based on large-scale tests existed, this step could be avoided.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Finite difference grid of the tunnel model: (

**a**) $\mathrm{xyz}$ perspective; (

**b**) $\mathrm{xz}$ plan view of the tunnel zone.

**Figure 4.**Experimental (crosses) and theoretical (solid lines) variograms along the main anisotropy directions: horizontal plane (black) and vertical (blue): (

**a**) RMR; and (

**b**) UCS.

**Figure 5.**Three-dimensional (3D) maps of the RMR simulated on the target grid mesh, for: (

**a**) realisation n. 1; and (

**b**) variance of the 100 realisations.

**Figure 6.**Histograms of the ${\mathrm{E}}_{\mathrm{m}}$ values (in GPa) obtained for all of the 100 realisations (each colour represents one realisation).

**Figure 7.**Scenario reduction analysis for the ${\mathrm{E}}_{\mathrm{m}}$ 100 realisations resulting in: (

**a**) a two-dimensional (2D) spatial representation of the ${\mathrm{E}}_{\mathrm{m}}$ 100 realisations (black points) using the Euclidean distance computed (X and Y represent the distances of the Dissimilarity matrix); (

**b**) a 2D Feature space representation for the ${\mathrm{E}}_{\mathrm{m}}$ 100 realisations (black points); and (

**c**) the clusters’ final configuration (coloured points) with the matching medoids (coloured squares) for four clusters.

**Figure 8.**Validation of the four-cluster configuration in comparison with the 100 realisations using: (

**a**) the average values of percentiles 10, 50, and 90; (

**b**) a point-by-point percentile sum.

**Figure 9.**Workflow applied to build the different models to represent the rock mass characterisation of the rock mass using as an input the RMR system.

**Figure 10.**Representation of the displacements (in mm) obtained for all of the analysed models (102), the deterministic model 1, cluster 1 (C1), 2 (C2), 3 (C3), and 4 (C4) of model 2, and the 100 realisations, in: (

**a**) the centre of the tunnel arch; and (

**b**) the centre of the tunnel invert.

**Figure 11.**Representation of the displacements (in mm) obtained for all of the analysed models (102), the deterministic model 1, cluster 1 (C1), 2 (C2), 3 (C3), and 4 (C4) of model 2, and the 100 realisations, in: (

**a**) the centre of the left sidewall; and (

**b**) the centre of the right sidewall.

**Figure 12.**$\mathrm{XZ}$ plane at y = 0 with the contour of displacement (colour scale in m) of the rock mass after the excavation for: (

**a**) model 1; (

**b**) model 2-cluster 1; (

**c**) model 2-cluster 2; (

**d**) model 2-cluster 3; and (

**e**) model 2-cluster 4.

**Figure 13.**Histograms and distribution fitting curve for the 100 realisations and values (lines) for models 1 to 5 of displacements in: (

**a**) the centre of the tunnel arch; (

**b**) the centre of the tunnel invert.

**Figure 14.**Histograms and distribution fitting curve for the 100 realisations and values (lines) for models 1 to 5 values (lines) of displacements in: (

**a**) the centre of the left sidewall; (

**b**) the centre of the right sidewall.

Statistical Parameter | RMR | UCS (in MPa) |
---|---|---|

Number of samples | 3969 | 3969 |

Minimum | 48 | 138 |

Maximum | 78 | 208 |

Mean | 66.7 | 176.9 |

Variance | 14.2 | 1054.7 |

Statistical Parameter | Realisation n. 1 of the RMR | Realisation n. 1 of the UCS |
---|---|---|

Number of grid points | 100,800 | 100,800 |

Minimum | 64 | 137 |

Maximum | 77 | 207 |

Mean | 71.1 | 159.5 |

Variance | 2.5 | 391.5 |

**Table 3.**Empirical expressions used to obtain ${\mathrm{E}}_{\mathrm{m}}$ and the corresponding authors.

Number | Authors | Required Parameters | Limitations | Equation (${\mathit{E}}_{\mathit{m}}$ in GPa) |
---|---|---|---|---|

1 | [30] | RMR | RMR > 50 | ${\mathrm{E}}_{\mathrm{m}}=2\times \mathrm{RMR}-100$ |

2 | [31] | RMR | - | ${\mathrm{E}}_{\mathrm{m}}=0.0003\times {\mathrm{RMR}}^{3}-0.0193\times {\mathrm{RMR}}^{2}+0.315\times \mathrm{RMR}+3.4064$ |

3 | [32] | UCS, GSI | UCS > 100 MPa | ${\mathrm{E}}_{\mathrm{m}}=\left(1-\frac{\mathrm{D}}{2}\right)\times {10}^{\left(\mathrm{GSI}-10\right)/40}$ |

4 | [33] | RMR | - | ${\mathrm{E}}_{\mathrm{m}}=0.1\times {\left(\frac{\mathrm{RMR}}{10}\right)}^{3}$ |

**Table 4.**Statistical analysis of the ${\mathrm{E}}_{\mathrm{m}}$ (in GPa) values obtained through the application of empirical formulae using the geotechnical information.

Statistical Parameter | ${\mathit{E}}_{\mathit{m}}$ (in GPa) |
---|---|

Total of points with information | 100,800 |

Mean | 39.50 |

Variance | 7.30 |

Standard deviation | 2.70 |

Minimum | 19.74 |

Maximum | 71.53 |

Statistical Parameter | Vertical Displacement (mm) | Horizontal Displacement (mm) | Max. Stress (MPa) | Min. Stress (MPa) | ||
---|---|---|---|---|---|---|

Arch | Invert | Left Wall | Right Wall | |||

Minimum | 5.20 | 6.82 | 1.50 | 1.42 | 6.67 | 1.77 |

Maximum | 8.34 | 11.31 | 2.75 | 2.63 | 7.91 | 1.84 |

Mean | 6.60 | 8.62 | 2.00 | 2.00 | 6.97 | 1.80 |

P10 | 5.92 | 7.76 | 1.73 | 1.71 | 6.81 | 1.77 |

P50 | 6.60 | 8.62 | 2.00 | 2.00 | 6.97 | 1.79 |

P90 | 7.49 | 9.80 | 2.28 | 2.33 | 7.21 | 1.82 |

Model | Corresponding Realisation | Vertical Displacement (mm) | Horizontal Displacement (mm) | Max. Stress (MPa) | |||
---|---|---|---|---|---|---|---|

Arch | Invert | Left Wall | Right Wall | Arch | Invert | ||

Model 1 | - | 5.86 | 9.36 | 1.79 | 1.79 | 6.91 | 1.77 |

Model 2-Cluster 1 | Realisation 28 | 6.31 | 9.25 | 2.00 | 2.14 | 6.85 | 1.80 |

Model 2-Cluster 2 | Realisation 46 | 6.07 | 8.24 | 1.82 | 1.87 | 7.05 | 1.80 |

Model 2-Cluster 3 | Realisation 53 | 6.95 | 10.1 | 2.11 | 2.22 | 6.99 | 1.80 |

Model 2-Cluster 4 | Realisation 74 | 6.02 | 7.03 | 1.56 | 1.84 | 7.91 | 1.80 |

**Table 7.**Distribution curve fitting details (first three moments) regarding the 100 realisations’ obtained values for displacements.

Displacement | Mean | Standard Deviation | Skewness |
---|---|---|---|

Maximum horizontal displacement (mm): left sidewall | 2.01 | 0.67 | 0.091 |

Maximum horizontal displacement (mm): right sidewall | 2.00 | 0.24 | 0.100 |

Maximum vertical displacement (mm): centre of the arch | 6.63 | 0.67 | 0.006 |

Maximum vertical displacement (mm): centre of the invert | 8.71 | 0.83 | 0.003 |

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**MDPI and ACS Style**

Pinheiro, M.; Emery, X.; Miranda, T.; Lamas, L.; Espada, M.
Modelling Geotechnical Heterogeneities Using Geostatistical Simulation and Finite Differences Analysis. *Minerals* **2018**, *8*, 52.
https://doi.org/10.3390/min8020052

**AMA Style**

Pinheiro M, Emery X, Miranda T, Lamas L, Espada M.
Modelling Geotechnical Heterogeneities Using Geostatistical Simulation and Finite Differences Analysis. *Minerals*. 2018; 8(2):52.
https://doi.org/10.3390/min8020052

**Chicago/Turabian Style**

Pinheiro, Marisa, Xavier Emery, Tiago Miranda, Luís Lamas, and Margarida Espada.
2018. "Modelling Geotechnical Heterogeneities Using Geostatistical Simulation and Finite Differences Analysis" *Minerals* 8, no. 2: 52.
https://doi.org/10.3390/min8020052