Application of numerical modeling and genetic programming to estimate rock mass modulus of deformation

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Abstract

Estimation of the rock mass modulus of deformation (Em) is one of the most important design parameters in designing many structures in and on rock. This parameter can be obtained by in situ tests, empirical relations between deformation modulus and rock mass classification, and estimating from laboratory tests results. In this paper, a back analysis calculation is performed to present an equation for estimation of the rock mass modulus of deformation using genetic programming (GP) and numerical modeling. A database of 40,960 datasets, including vertical stress (σz), horizontal to vertical stresses ratio (k), Poisson’s ratio (ν), radius of circular tunnel (r) and wall displacement of circular tunnel on the horizontal diameter (d) for input parameters and modulus of deformation for output, was established. The selected parameters are easy to determine and rock mass modulus of deformation can be obtained from instrumentation data of any size circular galleries. The resulting RMSE of 0.86 and correlation coefficient of 97% of the proposed equation demonstrated the capability of the computer program (CP) generated by GP.

Introduction

Rock mass modulus of deformation (Em) is one of the most important design parameters in a rock engineering practice and is used in designing many structures in and on rock. It refers to the ratio of stress to the corresponding strain during loading of a rock mass, including elastic and inelastic behavior. Several methods are available to estimate rock mass modulus of deformation including in situ tests, empirical relations between deformation modulus and rock mass classification, geophysical (usually seismic) methods and estimating from laboratory tests results. Despite their great features, the stated methods have some restrictions in the application. In situ tests are time-consuming and expensive, and require large scale galleries and difficult procedures. In situ tests are usually ignored in small sized projects but since laboratory tests on small specimens cannot predict the deformability of rock mass, in situ tests which provide direct information on deformability are preferred for site investigation studies. Empirical relations are indirect methods that relate the rock mass deformation modulus to index properties such as RQD or to rock mass classifications systems such as RMR and Q. These empirical relations were developed by different authors due to difficulties encountered during the in situ tests. As shown in Table 1, the most widely known empirical equations were studied by Nicholson et al. [1], [2], [3], [4], [5]. Although the empirical equations for the indirect estimation of the deformation modulus of rock mass are simple and cost-effective, the equations include some uncertainties relating to the limited data availability, variability of rock type and the heterogeneous nature of the rock masses [6]. Another method for obtaining rock mass deformation modulus is back analysis. Back analysis is an indirect technique that has been used to determine the mechanical properties of rock masses by using field measurements of displacements. Since displacements of rock masses induced by excavation can be measured easily and reliably, the displacement based on back analysis techniques have been always a research topic since 1970s [7], [8], [9], [10], [11], [12].

Numerical modeling has been used to investigate a variety of problems in geotechnics and geoengineering. If extensive geotechnical and geological data are available, then comprehensive predictions of deformations and stability can be made by numerical stress analyses. If not, the model can still be used to perform parametric studies, providing insight into the possible range of responses of a system, given the likely ranges for the various parameters. This understanding of the key parameters can then help set priorities for site investigation and material testing, which in turn will produce data that are used in design. FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions) is a three-dimensional numerical modeling program that uses finite difference for engineering mechanics computation. FLAC3D simulates the behavior of three-dimensional structures built of rock, soil or other materials that undergo plastic flow when their yield limits are reached. Different authors have used numerical modeling for different purposes [11], [12], [13], [14], [15].

Genetic programming (GP) and other computational techniques such as genetic algorithm (GA), artificial neural network (ANN) and fuzzy logic (FL) are applied as artificial intelligence (AI) tools for data analysis, modeling and optimization. These tools are becoming more significant due to their ability to extract information from data and transforming information into knowledge [15]. They have the ability to learn and generalize, whereby they can produce reasonable results for inputs not seen during training [16]. The application of the capabilities of artificial intelligence (AI) has been widely appreciated in geotechnics and geoengineering, as well as in other fields but the applications of GP has been limited to a few researches in the recent years [5], [17], [18], [19].

According to the successful employment of numerical modeling, genetic programming and back analysis method in previous works, in this study a back analysis calculation is presented for the determination of rock mass modulus of deformation using a genetic programming approach based on the data gathered by numerical modeling (FLAC3D) and the effectiveness of the combination of the stated methods in providing an estimation of rock mass modulus of deformation is demonstrated.

Section snippets

Genetic programming

Genetic programming as an artificial intelligence technique has recently been used successfully to extract knowledge in the form of IF-THEN rules and has been utilized in various fields particularly in finance and technical analysis [20], [21]. GP was first introduced by Koza as an extension of the conventional genetic algorithms (GA) [22]. The main difference between genetic programming and genetic algorithms is the representation of the solution. Genetic programming creates computer programs

Methodology

In this paper, a back analysis method is presented for estimating rock mass modulus of deformation using numerical modeling (FLAC3D) and genetic programming (GP). For this purpose 40,960 hypothetical different cases (different geometrical and mechanical properties) were simulated by FLAC3D program version 3, all in the form of unlined circular tunnels and the horizontal diameter wall displacements of these tunnels were calculated. The hypothetical rock mass properties were obtained using RocLab

Rock mass modulus of deformation estimation using genetic programming

The main aim of this study is to employ genetic programming techniques to generate new equation for the purpose of rock mass modulus of deformation estimation. To reach the stated goal, first of all, the whole data set is separated into two different sets: namely the training set and the testing one. The training set which contains 75% of the whole data set is used to evaluate the final or optimum computer programs (CPs) generated by the GP system while the testing set (the remaining 25% of the

Validation of GP equation by numerical modeling (FLAC3D) results

Returned values of deformation modulus by Eq. (2) for three circular tunnels are compared with deformation modulus values used in the FLAC3D simulations. Table 5 shows the information of the tunnels as well as percent error of our approach results in back calculation of deformation modulus of rock mass (Em).

Comparison between GP model and real case studies

Predicted and real deformation modulus of rock masses of two water conveyance tunnel both in Iran are compared in this study [23], [24]. The necessary information on these tunnels and error analysis of our approach results compared with real deformation modulus values of rock masses are summarized in Table 6. As can be seen from Table 6, the calculated modulus of deformation by proposed equation is close to real rock mass deformation modulus value of the stated tunnels, and the percentage error

Conclusions

This study provides an equation representative of numerical modeling (FLAC3D) for determination of rock mass modulus of deformation using the genetic programming (GP). The database used for GP modeling was gathered by numerical modeling (FLAC3D). Input parameters for the GP model are easy to determine (σz, r, ν, k and d). There is no need to excavate large scale galleries to estimate rock mass modulus of deformation and it can be obtained from instrumentation data of any size circular

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