^{*}

Edited by: Zbigniew R. Struzik, University of Tokyo, Japan

Reviewed by: Gunnar Pruessner, Imperial College London, UK; Michael Ronayne, Colorado State University, USA; Franci Gabrovšek, Karst Research Institute, Slovenia

*Correspondence: Martin Hendrick

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

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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 [

In a previous paper [_{f} ≈ 1.5, conductivity exponent _{w} ≈ 2.4. We also observed that these exponents are related by the Einstein relation.

Here we build on this work and propose a new model allowing the description of those systems, and study how their structures may emerge from general energy dissipation principles.

Previous authors have proposed several models for describing the geometry of karstic networks. The most comprehensive are based on a full mathematical description of the physical and chemical processes leading to the dissolution of carbonates (for example

Many natural systems exhibit fractal properties and can be modeled as critical phenomena [

Here we adopt the framework of percolation theory that is recognized as particularly well-suited to study connected structures [

For clarity, we provide here a brief summary of previous findings [

The Ox Bel Ha and Sac Actun coastal karstic systems (Figure ^{2} with a limited vertical extension (average conduit depth is approximately 12 m). They formed within a horizontal layer of a relatively young carbonate platform and is relatively homogeneous in comparison to the network extension. These two remarkable properties (large extension and homogeneous geology) make these networks ideal for study. Due to the flat topography of the area, the hydraulic head gradient is small and ranges from 1 to 10 cm/km [

In Hendrick and Renard [

Both Ox Bel Ha and Sac Actun have fractal dimension _{f} ≈ 1.5, and exhibit the same conductivity scaling behavior. For each network, the conductivity σ between two nodes scales with the Euclidean distance _{w} ≈ 2.4. It was also found that this set of exponents is in agreement with the (2-dimensional) Einstein relation

In the following, we study the fractal dimension, conductivity exponent, and walk exponent of subnetworks of backbones using a large number of numerical simulations. The fractal dimension is computed using the Maximum-Excluded-Mass-Burning algorithm, described in Song et al. [

The conductivity exponent is obtained evaluating the relation

The model is based on percolation theory that provides a standard framework for studying connected structures. The percolation model is that a fraction _{c} (_{c} ≈ 0.593 for a square lattice), a cluster spans the lattice (and would also span an infinite lattice). The correlation length ξ (the characteristic size of clusters) depends on _{c} as

In our numerical simulations, the lattices are finite and the infinite cluster is defined as the largest cluster that connect the lattice's boundaries. We call this the the percolation cluster.

At criticality, i.e., at _{c}, the percolation cluster is self-similar on all length scales (for an infinite lattice). Here we focus on the backbone of the percolation cluster, defined as the conducting part (i.e., links of the percolation cluster carrying non-zero flow rates) of the percolation cluster. The fractal dimension, walk exponent and conductivity exponent of the critical infinite percolation cluster and critical backbone are known [

_{f}, conductivity exponent _{w} of critical percolation cluster (cPC), critical backbone percolation cluster (cBB) and Tulum karst networks

_{f} |
_{w} |
||
---|---|---|---|

cPC | 91/48 ≈ 1.896 | 0.9826 ± 0.0008 | 2.878 ± 0.001 |

cBB | 1.6432 ± 0.0008 | 0.9826 ± 0.0008 | 2.62 ± 0.03 |

Ox Bel Ha | 1.51 ± 0.03 | 0.917 ± 0.037 | 2.39 ± 0.03 |

Sac Actun | 1.49 ± 0.03 | 0.920 ± 0.036 | 2.40 ± 0.03 |

_{c}, percolation clusters are described by the same set of exponents up to ξ

The probability _{∞} that a node of a percolation cluster belongs to the infinite spanning cluster is zero for _{c} and is given by a power law _{c}, with a universal exponent β = 5/36 in 2 dimensions. The percolation model is an archetype model for continuous phase transition. The probability _{∞} is the order parameter that distinguishes the disconnected phase to the connected phase. At the critical point _{c}, a collective behavior emerges at all length scales since the correlation length ξ diverges.

Karst networks are structures connecting inlets to outlets to allow the transport of water in heterogeneous media. Therefore, percolation theory is a natural starting point to model such of systems.

We consider networks in a busbar configuration where a fixed flux is prescribed between two parallel edges (Figure

In the remainder of this work, we always consider the (largest) percolation cluster that connects the two edges of the busbar configuration.

The proposed modeling procedure consists of numerically generating a percolation cluster for a given value of _{c}). A prescribed flux is imposed on this network as a boundary condition between the edges of the busbar configuration. Water flows in the percolation cluster from the upstream edge to the downstream edge. The magnitude of the flow is computed by solving numerically Kirchhoff equation (using the Hagen-Poiseuille equation and a resistance of one for each link). It is then assumed that the karst network grows along the links of the percolation cluster that carry the strongest flow rates and that form a connected network that spans the two edges. Thus, we study karst networks as subnetworks of backbones of the percolation clusters.

We take as parameter θ = _{t}/_{max} with _{t} the imposed threshold flow rate (i.e., minimal flow rate allowed for a link to be part of the karst network). _{ij} the flow rate on the link 〈^{1}

^{−8} to θ = 0.06 and then reduces.

We investigate, through numerical simulations, the properties of θ-subnetworks for different site occupation probabilities at the critical point and above. We consider ^{2}

_{f}, _{w}, and _{c} = 0.593, _{w} is represented by a line.

Dimensionally speaking, a karst network can be a θ-subnetwork of a critical backbone, as the fractal dimension of the backbone is larger than the fractal dimension of observed karst network (Table

Figure _{f}, decreases with the percolation probability _{w} can be understood intuitively by considering that, as _{w} decreases as _{w} → 2).

We notice that for _{f} ≈ 1.5 and also have conductivity exponent close to the one observed for the networks of Tulum (Figure

There is no guarantee that the generated structures respect the Einstein relation. We examine the validity of this relation as a hallmark since this one is satisfied by the karst networks of Tulum and for homogeneous fractal structures such as percolation cluster or the Sierpinski gasket. It is noteworthy that, even if the relation is not fulfilled exactly with the proposed model, the behavior of the walk exponent computed via the Einstein relation _{f}, _{w}, and _{c} and for small θ (i.e., large _{f}). This is not surprising because critical percolation clusters and their backbones satisfy the Einstein relation up to the scale of the lattice size.

_{f}, _{w}, and

If we consider θ-subnetworks of percolation clusters above _{c}, we lose self-similarity at the large scale. The typical size of percolation clusters is given by the correlation length _{c}, clusters are self similar up to the correlation length (ξ correspond to the typical finite size of finite clusters). Above _{c}, ξ corresponds also to the size of largest holes on the infinite spanning cluster. If we consider _{c}, ξ is reduced and percolation looks more and more homogeneous (the size of largest holes of infinite spanning cluster are reduced). However, here we consider only θ-subnetworks which are sparser structures than backbones. The holes (loops) sizes vary non-monotonically with θ. Holes size grows with θ until the θ-subnetwork collapses into a small structure only containing small loops (Figure _{θ} of its θ-subnetworks. Nonetheless, ξ and ξ_{θ} correspond for small θ, i.e., close to the backbone. This observation aims to clarify why proper scaling behavior is expected for θ-subnetwork even above criticality.

Karst networks are natural systems. Therefore, it is reasonable to assume that they grow in a manner such that the resulting structures minimize the dissipation of energy related to the transport of underground water.

To investigate whether this assumption holds, we compute the (rate of) energy _{ij} be the resistance associated to the link 〈_{ij} its flow rate, the dissipated energy (assuming a laminar flow described by the Hagen-Poiseuille equation) is given by

The networks generated by our model, whose structures tend to approach the characteristics of real karst networks, are rather different than backbones. However, the backbone of a percolation cluster is the subnetwork that, by maximizing the number of conducting links, minimizes

Our model is static, meaning we work with a steady state flow and do not allow conduit radii to change over time. Consequently, we cannot study how a karst evolves from its early stage to its maturity. What we can do is to compare the θ-subnetworks with each other to try to understand the structure of observed karst networks.

Our very first hypothesis is that karst systems develop along links carrying the strongest flow rates. One can argue that a strong flow rate is needed to create a link and keep it open. A limiting process of the physics of dissolution is the mass transport. When water dissolves limestone, it saturates in calcite and hence, if there is not a sufficient flux to flush out dissolved calcite, the dissolution process stops. In this way, to guarantee a sufficient water pressure gradient along each links, the network has to concentrate the water flow by limiting the number of conduits. Hence, we are led to consider θ-subnetworks that optimize the dissipated energy for a limited size. Therefore, we make the assumption that karst networks are structures that minimizes

with _{max}/_{max}, with

Figure ^{−2}. When these results are plotted as a function of _{f} (Figure _{f} ≈ 1.52. The minimum of _{f} ≈ 1.52.

_{f}

To refine the understanding of the links between the structure and the overall energy dissipated in the system, the minimum of

_{f} |
_{w} |
||||
---|---|---|---|---|---|

0.593 | 0.050 | 1.50 ± 0.04 | 1.040 ± 0.029 | 2.56 ± 0.02 | 2.54 ± 0.05 |

0.60 | 0.040 | 1.53 ± 0.04 | 0.992 ± 0.030 | 2.53 ± 0.02 | 2.52 ± 0.05 |

0.61 | 0.038 | 1.56 ± 0.05 | 0.919 ± 0.035 | 2.52 ± 0.02 | 2.48 ± 0.06 |

0.62 | 0.035 | 1.61 ± 0.05 | 0.794 ± 0.039 | 2.49 ± 0.02 | 2.40 ± 0.07 |

In Rodríguez-Iturbe and Rinaldo [

It is temping to assume an analog dynamical scenario for the description of the development of karst networks: during the short freezing time period, the structure of a karst system quickly takes on the observed structure (described by the minimum of

It results from this discussion that the study of the structure of young karst systems, in comparison with mature networks, may support or deny the freezing time hypothesis. However, young karst networks are generally characterized by conduits of small apertures and thus are not directly observable. Therefore, their properties should be studied through dynamical models of dissolution.

Siemers and Dreybrodt [_{c}. A busbar configuration is assumed and a prescribed water head difference is imposed between the edges. The percolation networks were built on a 30 × 30 square lattice. This model shows that the early development stage (that occurs under laminar flow), corresponding to widening of the fractures, determines the structure of the mature karst network. This initial period is brief compared to the entire evolution of the network toward its maturity. The authors show that preferentially dissolved fractures correspond to those forming the pathways offering the least resistance to the flow. However, for percolation probabilities ^{3}_{c}, there is a small number of pathways offering small resistances to the flow, and these pathways are decisive for the resulting karst network structure and the details of the under saturation process is not important.

The model of Siemers and Dreybrodt [

To conclude, in the work of Siemers and Dreybrodt [

The model presented here aimed to reproduce the observed exponents (fractal dimension, conductivity exponent, and walk exponent) of karst networks of the area of Tulum. We studied the dissipated energy due to viscosity

We showed that for ranges of site occupation probability close to _{c}, the generated structures minimizing

A study of statistical properties of the structure of younger karst networks, and the set up of a dynamical model of dissolution adapted to Tulum's karst networks, could provide interesting additional information to assess the freezing time hypothesis, and may ultimately reinforce our approach.

Although, our model reproduced the fractal dimension and conductivity exponent of karst networks quite well, it failed to reproduce the walk exponent accurately. Improvement of the measure of _{w}, using for example the probability of first-passage time of random walk, should be considered. The results may confirm or deny the violation of the Einstein relation by the generated structures.

Others percolation models such as directed percolation [

The multifractal properties of moments of flow rate (current) distribution is an important research topic for characterizing random resistor networks [

Finally, our model relies on the fine tuning of the percolation probability

MH conducted this work. MH and PR wrote the paper. PR initiated the study and participated in the discussion.

This work was funded by the Swiss National Science Foundation under the contract nb. 200021L_141298.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors are grateful to James G. Coke IV, Bil Phillips, and Robbie Schmittner of Quintana Roo Speleological survey for having provided the data at the basis of this study, as well as James Thornton and Stephen A. Miller for having revised the final version of this manuscript.

^{*}Application to a synthetic case

^{1}Indices

^{2}The largest θ at which there exists a θ-subnetwork varies between percolation clusters. Therefore, the interval of θ in which we sample the 6 θ-subnetworks differs from one percolation cluster to another. We do not sample at fixed θ's to avoid biased results or empty samples. Therefore, we extract 6 values of θ equally distributed on the interval [0.01, θ_{max}], with θ_{max} being the maximum value of θ for which a θ-subnetwork exists.

^{3}In Siemers and Dreybrodt [^{b} the bond percolation probability. The authors show that the details about the dissolution of calcite is determining for final network structure when ^{b} > 0.6 which correspond to