Subnetworks of Percolation Backbones to Model Karst Systems Around Tulum, Mexico

Martin Hendrick,Philippe Renard

doi:10.3389/fphy.2016.00043

Abstract

Karstic caves, which play a key role in groundwater transport, are often organized as complex connected networks resulting from the dissolution of carbonate rocks. In this work, we propose a new model to describe and study the structures of the two largest submersed karst networks in the world. Both of these networks are located in the area of Tulum (Quintana Roo, Mexico). In a previous work \cite{hendrick2016fractal} we showed that these networks behave as self-similar structures exhibiting well-defined scaling behaviours. In this paper, we suggest that these networks can be modeled using substructures of percolation clusters ($\theta$-subnetworks) having similar structural behaviour (in terms of fractal dimension and conductivity exponent) to those observed in Tulum's karst networks. We show in addition that these $\theta$-subnetworks correspond to structures that minimise a global function, where this global function includes energy dissipation by the viscous forces when water flows through the network, and the cost of network formation itself.

Highlights

In a previous paper [1], we studied the fractal properties of the Ox Bel Ha and Sac Actun karstic networks in the region of Tulum, Mexico using real space renormalization and numerical simulations. We found that both networks have similar structures with well defined fractal dimension df ≈ 1.5, conductivity exponent μ ≈ 0.9, and walk dimension dw ≈ 2.4
We study the fractal dimension, conductivity exponent, and walk exponent of subnetworks of backbones using a large number of numerical simulations
Analogy with Rivers Networks In Rodríguez-Iturbe and Rinaldo [14], the authors make a static model of river networks, the Optimal Channel Network (OCN)

Summary

Introduction

In a previous paper [1], we studied the fractal properties of the Ox Bel Ha and Sac Actun karstic networks in the region of Tulum, Mexico using real space renormalization and numerical simulations. Even simpler models are based on the statistical resampling of existing data sets and allow stochastic networks to be generated which reproduce the main statistical characteristics of the training networks [8]. Amongst these studies, Ronayne and Gorelick [9] and Ronayne [10] used the invasion percolation model of Stark [11] and investigated its large scale flow and transport properties.

Methods

Results

Discussion

Conclusion