Copula-based dependency modelling of hydraulic properties for non-linear filtration through porous media

Abstract

Predicting hydraulic parameters in porous media, such as Darcy and non-Darcy coefficients, is critical for understanding flow dynamics and designing hydraulic structures. This study develops a stochastic modelling framework using bivariate copula models to estimate these parameters, leveraging easily measurable quantities like packing diameter etc. A synthetic dataset, derived from experimental distributions reported in the literature, was utilized to overcome challenges associated with direct data collection. The analysis revealed that hydraulic radius is strongly correlated with Darcy (Kendall's tau = −0.74) and non-Darcy coefficients (Kendall's tau = −0.76), and an inverse relationship was observed between hydraulic radius and Darcy coefficient. For example, the probability of the Darcy coefficient not exceeding 2 s/m, 20 s/m, and 100 s/m was 10 %, 50 %, and 80 %, respectively, for a hydraulic radius of 0.005 m. Furthermore, the strong correlation between Darcy and non-Darcy coefficients (Kendall's Tau = 0.67) enables the use of the former as a predictor for the latter. For a Darcy coefficient of 50 s/m, the probability of the non-Darcy coefficient not exceeding 300 s²/m² and 750 s²/m² was approximately 50 % and 75 %, respectively. This approach provides designers and engineers with a probabilistic framework for selecting hydraulic parameter values with varying degrees of confidence. It offers practical applications in the design of hydraulic structures, such as rockfill dams, and in estimating discharge through fractures, allowing for more reliable assessments of head loss and stability. The findings underscore the potential of copula models in enhancing the predictive accuracy and practicality of hydraulic analyses in porous media.