Abstract
Understanding the complex structure of karst networks is a challenge. In this work, we characterize the fractal properties of some of the largest coastal karst network systems in the world. They are located near the town of Tulum (Quintana Roo, Mexico). Their fractal dimension $d_f$, conductivity exponent $\tilde{\mu}$ and walk dimension $d_w$ are estimated using real space renormalization and numerical simulations. We obtain the following values for these exponents: $d_f\approx 1.5$, $d_w\approx 2.4$, $\tilde{\mu}\approx 0.9$. We observe that the Einstein relation holds for these structures $\tilde{\mu} \approx -d_f + d_w$. These results indicate that coastal karst networks can be considered as critical systems and this provides some foundations to model them within this framework.
Highlights
Karst network structures are still poorly understood because of the lack of a general framework to study them [1], even if modeling and characterizing karst networks has been a long standing research topic [2,3,4,5,6].It is estimated that karst features cover 20% of the globe’s land surface [7]
We focus on the analysis of karst networks located around the town of Tulum (Quintana Roo, Mexico)
The Einstein relation μ = −2 + d − df + dw holds in the 95% confidence intervals, and notice that μis quite robust through renormalization
Summary
Karst network structures are still poorly understood because of the lack of a general framework to study them [1], even if modeling and characterizing karst networks has been a long standing research topic [2,3,4,5,6]. The area of Tulum hosts two of the largest water filled networks in the world: Ox Bel Ha and Sac Actun (above 200 km of connected conduits for each one [9]) Due to their large sizes and the relatively simple and homogeneous geology (horizontal carbonate platform) of the underground, we expect that these networks exhibit a well marked statistical signature characterizing the physical processes of their formation. The Einstein relation μ = −2 + d − df + dw (we work in 2 dimensions, d = 2) holds in the 95% confidence intervals, and notice that μis quite robust through renormalization These results highlight the deep fractal nature of karst network around Tulum. We finish with a discussion and perspectives for future work
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