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Mathematical Geosciences

, Volume 49, Issue 5, pp 657–675 | Cite as

A Numerical Framework for Wall Dissolution Modeling

Analysis of Flute Formation
  • Aleksander GrmEmail author
  • Tomaž Šuštar
  • Tomaž Rodič
  • Franci Gabrovšek
Article

Abstract

Scallops and flutes are common dissolution rock forms encountered in karst caves and surface streams. Their evolution is only partially understood and no numerical model that simulates their formation has been presented. This work at least partially fills the gap by introducing a numerical approach to simulate the evolution of different initial forms of soluble surfaces embedded in a turbulent fluid. The aim is to analyze wall dissolution phenomena from basic principles and to identify stable profiles. The analysis is based on a finite volume moving boundary method. The underlying mathematical model is a \(k-\epsilon \) turbulent model for fluid flow coupled with turbulent scalar transport. The rock wall is treated as a moving boundary, where the normal wall retreat velocity is proportional to the under-saturation of the boundary fluid cells with respect to the mineral comprising the wall. As the flow time scale is several orders of magnitude smaller than the dissolution time scale, stationary flow field, concentration field and wall propagation velocity are calculated for each iteration. The boundary at all points is then moved by distracting minimal velocity along the entire boundary from the actual velocity at a certain location, and then normalized to the maximum allowed shift, which is equal to half the height of the boundary cell. In this way only deformation of the initial wall is calculated. The method was applied to several different initial profiles. During the evolution, the profiles progressively converged towards stable forms. In this work, a framework is proposed for a computation of the moving boundary problem related to slow dissolution of a soluble surface.

Keywords

Moving boundary problem Quasi-stationary turbulent flow and turbulent transport Finite volume Soluble surface 

Notes

Acknowledgments

This research has been supported by the Grant J2-4093 of Slovenian Research Agency (ARRS).

References

  1. Bird A, Springer G, Bosch R, Curl R (2009) Effects of surface morphologies on flow behavior in karst conduits. In: 15th International Congress of Speleology. National Speleological Society, Kerrville, Texas, pp 1417–1421Google Scholar
  2. Blumberg P, Curl RL (1974) Experimental and theoretical studies of dissolution roughness. J Fluid Mech 65(4):735–751CrossRefGoogle Scholar
  3. Curl RL (1966) Scallops and flutes. Trans Cave Res Group G B 7(2):121–160Google Scholar
  4. Curl RL (1974) Deducing flow velocity in cave conduits from scallops. Bull Natl Speleol Soc 36(2):1–5Google Scholar
  5. Dreybrodt W, Buhmann D (1991) A mass-transfer model for dissolution and precipitation of calcite from solutions in turbulent motion. Chem Geol 90(1–2):107–122CrossRefGoogle Scholar
  6. Ferziger J, Perić M (2002) Computational methods for fluid dynamics. Springer, BerlinCrossRefGoogle Scholar
  7. Geuzaine C, Remacle J (2009) Gmsh a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331CrossRefGoogle Scholar
  8. Hammer Ø, Lauritzen S, Jamtveit B (2011) Stability of dissolution flutes under turbulent flow. J Cave Karst Stud 73(3):181–186CrossRefGoogle Scholar
  9. Jasak H, Tuković Z (2007) Automatic mesh motion for the unstructured finite volume method. Trans FAMENA 30(2):1–18Google Scholar
  10. Jeschke AA, Vosbeck K, Dreybrodt W (2001) Surface controlled dissolution rates of gypsum in aqueous solutions exhibit nonlinear dissolution kinetics. Geochim Cosmochim Acta 65(1):27–34CrossRefGoogle Scholar
  11. Kaufmann G, Dreybrodt W (2007) Calcite dissolution kinetics in the system CaCO\(_3\)-H\(_2\)O-CaCO\(_3\) at high undersaturation. Geochim Et Cosmochim Acta 71(6):1398–1410CrossRefGoogle Scholar
  12. Pope SS (2013) Turbulent flows. Cambridge University Press, New YorkGoogle Scholar
  13. Schlichting H, Gersten K (2000) Boundary layer theory. Springer, BerlinCrossRefGoogle Scholar
  14. van Noorden TL, Pop IS, Röger M (2007) Crystal dissolution and precipitation in porous media: L \(^1\)-contraction and uniqueness. In: Discrete and continuous dynamical systems (Dynamical systems and differential equations. Proceedings of the 6th AIMS international conference, suppl.) Poitiers, France, pp 1013–1020Google Scholar
  15. Weller HG, Tabor G, Jasak H, Fureby C (1998) A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput Phys 12(6):620–631CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  1. 1.C3MLjubljanaSlovenia
  2. 2.NTFUniversity of LjubljanaLjubljanaSlovenia
  3. 3.FPPUniversity of LjubljanaPortorožSlovenia
  4. 4.Karst Research Institute, ZRC-SAZUPostojnaSlovenia

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