# Voronoi implicit interface method for geometry evolution of two minerals with applications in reactive porous media

## Main Article Content

## Abstract

We present a numerical method to describe the precipitation and dissolution processes of two interacting mineral phases and one fluid phase in porous media. We use the Voronoi implicit interface method [9] to track an interface evolving in normal direction that can contain triple points. To represent the interface implicitly, one uses the $\epsilon$ set of an evolving level set function that is given as the (unsigned) distance function initially or after reinitialization. To obtain the speed of evolution, we combine the Voronoi implicit interface method with a constant extrapolation of the normal speed which is prescribed only at the interface.

## Article Details

How to Cite

Frolkovič, P., Gajdošová, N., Gärttner, S., & Ray, N.
(2020).
Voronoi implicit interface method for geometry evolution of two minerals with applications in reactive porous media.

*Proceedings Of The Conference Algoritmy,*, 121 - 130. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1566/825
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Articles

## References

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[2] T. D. Aslam. A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys., 193:349–355, 2003.

[3] L. Chen, Q. Kang, B. Carey, W.-Q. Tao Pore-scale study of diffusion–reaction processes involving dissolution and precipitation using the lattice Boltzmann method. International Journal of Heat and Mass Transfer , 75: 483-496 (2014).

[4] J. Gao, H. Xing, Z. Tian, J. K. Pearce, M. Sedek, S. D. Golding, V. Rudolph Reactive transport in porous media for CO2 sequestration: Pore scale modeling using the lattice Boltzmann method. Computers & Geosciences, 98, 9-20, 2017.

[5] P. Frolkovič, K. Mikula, and J. Urbán Distance function and extension in normal direction for implicitly defined interfaces. Discrete and continuous dynamical systems-series S, 8(5), 871-880.

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[8] R.I. Saye and J.A. Sethian The Voronoi Implicit Interface Method for computing multiphase physics. Proc. Natl. Acad. Sci. U. S. A., 108:19498–19503, 2011.

[9] R.I. Saye and J.A. Sethian Analysis and applications of the Voronoi Implicit Interface Method. J. Comp. Phys., 231:6051–6085, 2012.

[10] J. Sethian. Level Set Methods and Fast Marching Methods. Cambridge University Press, 1999.

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[12] H. Zhao, T. Chan, B. Merriman, and S. Osher. A Variational Level Set Approach to Multiphase Motion. J. Comput. Phys., 127:179–195, 1996.

[2] T. D. Aslam. A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys., 193:349–355, 2003.

[3] L. Chen, Q. Kang, B. Carey, W.-Q. Tao Pore-scale study of diffusion–reaction processes involving dissolution and precipitation using the lattice Boltzmann method. International Journal of Heat and Mass Transfer , 75: 483-496 (2014).

[4] J. Gao, H. Xing, Z. Tian, J. K. Pearce, M. Sedek, S. D. Golding, V. Rudolph Reactive transport in porous media for CO2 sequestration: Pore scale modeling using the lattice Boltzmann method. Computers & Geosciences, 98, 9-20, 2017.

[5] P. Frolkovič, K. Mikula, and J. Urbán Distance function and extension in normal direction for implicitly defined interfaces. Discrete and continuous dynamical systems-series S, 8(5), 871-880.

[6] S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.

[7] E. Rouy and A. Tourin. A viscosity solutions approach to shape-from-shading. SIAM J. Num. Anal., 29:867–884, 1992.

[8] R.I. Saye and J.A. Sethian The Voronoi Implicit Interface Method for computing multiphase physics. Proc. Natl. Acad. Sci. U. S. A., 108:19498–19503, 2011.

[9] R.I. Saye and J.A. Sethian Analysis and applications of the Voronoi Implicit Interface Method. J. Comp. Phys., 231:6051–6085, 2012.

[10] J. Sethian. Level Set Methods and Fast Marching Methods. Cambridge University Press, 1999.

[11] J.A. Sethian Fast Marching Methods. SIAM Review, 41:199–235, 1999.

[12] H. Zhao, T. Chan, B. Merriman, and S. Osher. A Variational Level Set Approach to Multiphase Motion. J. Comput. Phys., 127:179–195, 1996.