# Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results and Discussion

**B**, located in the center at the bottom of the pit, the rate of evaporation, q, increases sharply with the increase in H, reaching q = 0.99 at H = 4 mm (Figure 5f). Additionally, at H ≥ 6 mm, the evaporation front is almost absent in the point B area (Supplementary Figure S2); i.e., the values of moisture content exceed ${\theta}_{m}$, indicating that only a capillary zone is present in this area. In the next simulations, we add two symmetrically positioned 15 mm-wide and 4 mm-deep pits (H = 4.0 mm). The distance between pits, L, is varied while keeping ${\theta}_{0}$ constant at ${\theta}_{0}=0.11$ (Figure 5i, Supplementary Figure S3). The reduction in L from 25 mm to 15 mm results in the decrease in the rate of evaporation, q, at the vertex A located between two pits, from 0.20 to 0.03, while the minimum value of q = 0.03 is observed in the case of immediately adjacent pits with L = 15.0 mm (Figure 5j). With a further reduction in L, a sharp increase in the rate of evaporation, q, can be observed at the vertex A (Figure 5j).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Honeycombs: (

**a**) Apolena Rock City, CZ; (

**b**) Bukhta Desantnaya, Ussuri Bay, Primorsky Krai, RU.

**Figure 2.**Relationship between the relative humidity, ${h}_{m}\left(\theta \right)$, of the surface material and moisture content $\theta $.

**Figure 5.**Influence of boundary conditions on the hydraulic field. (

**a**–

**d**) Smooth surface of rock at various levels of saturation, ${\theta}_{0}$. (

**a**) Computational model. (

**b**) Rate of evaporation, q, as the function of ${\theta}_{0}$. (

**c**) Moisture content distribution at ${\theta}_{0}=0.11$. (

**d**) Distribution of evaporation rate, q, at ${\theta}_{0}=0.11$. (

**e**–

**h**) Single 15 mm-wide pit located at the center of the right edge, at various values of H, with constant ${\theta}_{0}=0.11$. I Computational model. (

**f**) Rate of evaporation, q, as the function of ${\theta}_{0}$. (

**g**) Moisture content distribution at H = 4.0 mm. (

**h**) Distribution of evaporation rate, q, at H = 4.0 mm. (

**i**–

**l**) Two adjacent 15 mm-wide and 4.0 mm-deep pits with various center-to-center distances,

**L**, at constant ${\theta}_{0}=0.11$. (

**i**) Computational model. (

**j**) Rate of evaporation, q, as the function of L. (

**g**) Moisture content distribution at L = 17.0 mm. (

**h**) Distribution of evaporation rate, q, at L = 17.0 mm.

**Figure 6.**Erosion process simulation. The value of constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.115$. Moisture content over the whole volume exceeds ${\theta}_{m}$, indicating that the model is fully wet, and no erosion takes place (3).

**Figure 7.**Erosion process simulation. The value of constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.110$. The virtually flat surface can be observed. This can be explained that only the protruding parts dry out below ${\theta}_{m}$, and the erosion occurs only in protruding parts.

**Figure 8.**Erosion process simulation. The value of the constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.109$. The formation of the single isolated lip can be observed.

**Figure 9.**Erosion process simulation. The value of the constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.108$. Formation of several isolated lips can be observed.

**Figure 10.**Erosion process simulation. The constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.106$. The increase in number and thickness of lips can be observed.

**Figure 11.**Erosion process simulation. The constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.100$. The further increase in lips thickness and merging of several adjacent lips can be observed.

**Figure 12.**Erosion process simulation. The constant distribution of moisture content at the bottom side is set to ${\theta}_{0}$ = 0.060. Formation of the deep isolated pit (tafoni) from the initial pit can be observed. In other regions, the erosion process is less pronounced because the evaporation front is removed from the surface.

**Figure 13.**Erosion process simulation. The constant distribution of moisture content at the bottom side is set to ${\theta}_{0}=0.030$. Evaporation front is far away from the surface, and no erosion occurs.

**Figure 14.**The example of various shapes formed at the same rock under the same environmental conditions. It is assumed that formation of various shapes is related to height variations in moisture content inside the rock. The object is located in Apolena Rock City, CZ.

**Table 1.**Typical parameters values [37].

Symbol | Value | Unit |
---|---|---|

${\theta}_{m}$ | 0.01 | - |

${\theta}_{\mathrm{\infty}}$ | 9.14 × 10^{−4} | - |

$\beta $ | 6.36 × 10^{−4} | mm s^{−1} |

${h}_{a}$ | 0.61 | - |

${h}_{m}\left({\theta}_{m}\right)$ | 1.0 | - |

${h}_{m}\left({\theta}_{\mathrm{\infty}}\right)$ | 0.61 | - |

${h}_{m}\left(0\right)$ | 0 | |

$D\left(0.25\right)$ | 5.41 | mm s^{−2} |

$D\left(0.1\right)$ | 0.1 | mm s^{−2} |

$D\left({\theta}_{m}\right)$ | 0.0003 | mm s^{−2} |

$D\left({\theta}_{\mathrm{\infty}}\right)$ | 0.01 | mm s^{−2} |

$\mathbf{Moisture}\mathbf{Content}{\mathit{\theta}}_{0}$ | Landform Shapes |
---|---|

0.115 | No erosion |

0.110 | Flat surface |

0.109 | Formation of single isolated lip (honeycombs) |

0.108 | Formation of several isolated lips (honeycombs) |

0.106 | Increase in number and thickness of lips (honeycombs) |

0.100 | Further increase in lips thickness merging of several adjacent lips (honeycombs) |

0.060 | Formation of deep isolated pit (tafoni) |

0.030 | No erosion |

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**MDPI and ACS Style**

Safonov, A.; Minchenkov, K.
Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition. *Geosciences* **2023**, *13*, 161.
https://doi.org/10.3390/geosciences13060161

**AMA Style**

Safonov A, Minchenkov K.
Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition. *Geosciences*. 2023; 13(6):161.
https://doi.org/10.3390/geosciences13060161

**Chicago/Turabian Style**

Safonov, Alexander, and Kirill Minchenkov.
2023. "Mathematical Simulation of Honeycomb Weathering via Moisture Transport and Salt Deposition" *Geosciences* 13, no. 6: 161.
https://doi.org/10.3390/geosciences13060161