Document Type : Original Article

Authors

• Amir Malekpour ¹
• Nima Sadeghian ²
• Mohammad Javad Farrokhi ³

¹ Assistant Prof., Department of Water Engineering, Faculty of Agricultural Sciences, University of Guilan

² Master's degree in water engineering department, majoring in water structures.Department of Water Engineering, Faculty of Agricultural Sciences, University of Guilan

³ Student of the water engineering department, Faculty of Agricultural Sciences, University of Guilan

Abstract

Extended Abstract
Introduction
The consolidation and the subsequent settlement can lead to the land subsidence, building destruction, pipeline ruptures in water supply networks, and damage to the asphalt pavement. In soil consolidation analyses and many other geotechnical problems, the uncertainty of geotechnical variables and their spatial variations is of significant importance. As a result, the uncertainty-based approaches are currently employed to consider these problems rather than deterministic analyses. In this regard, some researches have demonstrated the considerable influence of the random variables of hydraulic conductivity and volume compressibility on the soil consolidation phenomenon. However, the studies have rarely addressed the correlation between these random variables and its effect on the probabilistic consolidation analysis. The current research aims at investigating the impact of correlation between two random variables of hydraulic conductivity and volume compressibility using copula functions via the development of a computer program in MATLAB. The performance of different copula functions in the bivariate probabilistic analysis of consolidation and the temporal variations of pore water pressure distributions are studied in a case study in Guilan province of Iran.
Methodology
In this research, a computer program was developed in MATLAB to estimate the marginal distributions of two random variables of hydraulic conductivity and soil volume compressibility. Then the bivariate probability distributions of two random variables were obtained using two copula groups of Archimedean (Clayton, Gumbel and Frank) and elliptic (Gaussian and t-student). The bivariate distributions of random variables were applied to estimate the temporal pore water pressure distributions during consolidation in soil depths of 2 and 4 meters. The best joint probability distribution and the corresponding copula function was determined on MvCAT software based on the correlation of random variables and using certain criteria such as AIC, BIC, RMSE, and NSE. As a feature of developed computer program in this research,1000 pair values of hydraulic conductivity and volumetric compressibility were generated by copula functions (from the primary 24 field data) in order to create more accurate results. Then, after numerically solving the governing differential equation of consolidation using the implicit central finite difference method, the probability distributions of pore water pressure over time including the probability density functions (PDFs) and cumulative distribution functions (CDFs) were calculated using different copulas and compared with each other.
Results and Discussion
The results showed that the inverse Gaussian distribution properly fits to the marginal distributions of each single random variable, according to BIC criterion. In this research, Kendall’s correlartion coefficient showed a positive correlation between the random variables of hydraulic conductivity and soil volume compressibility. After 15 days from the beginning of consolidation with an initial loading of 400 kPa, the pore water pressures in the depth of 2 meters were estimated equal to 398.75, 398.9 and 398.95 kPa for Clayton, Gumbel and Clayton copulas, respectively. Whereas the pore pressure in the same depth were obtained equal to 399 and 399.05 kPa for Gaussian and t-student copulas, respectively. In the depth of 4 meters, Clayton, Gumbel and Clayton copulas, estimated the pore pressures equal to 399.54, 399.55, 399.54 kPa, respectively. It shows that Archimedean copulas create almost similar results in deeper regions within a soil layer. For elliptical copulas in the depth of 4 meters, the pore water pressures were calculated equal to 399.8 and 399.75 kPa for Gaussian and t-student copulas, respectively.
Conclusions
Considering the correlation of random variables, it is concluded that Archimedean copulas are more accurate in extreme values than elliptic copulas but elliptic copulas according to AIC, BIC and other evaluation criteria provide better balance between the number of parameters, the accuracy and the complexity of model. Generally, for both Archimedean and elliptic copulas, the temporal variations of pore water distributions show an increase in uncertainty with time via changing from sharp and narrow curves to flat and wide curves. Moreover, the consolidation rate (pore pressure dissipation rate) is slower for elliptic copulas than Archimedean copulas. Gaussian copula was found to be the best copula among all investigated copulas. The error of neglecting the correlation of random variables is bigger when a shallow foundation is to be designed by an engineer. Meanwhile, the consolidation rate is overestimated when the correlation of random variables is ignored.
 Acknowledgement
The authors express their gratitude to the experts working in technical and soil mechanics laboratory of Guilan province for their collaborations and carrying out the experiments on the soil samples of this research.

Keywords

• Copula functions
• Fine-grained soil
• Joint probability analysis
• Monte-Carlo method

Main Subjects

• Geotechnic