# Numerical properties of a model problem for evaluation of natural tracer transport in groundwater

## Main Article Content

## Abstract

We solve a model problem of natural tracer transport in groundwater between the surface and the tunnel, based on field measured data. The problem with a simplified geometry represents the main features of flow inhomogeneity, namely the presence of fractures and matrix, and an influence of the stagnant zones on the tracer breakthrough. From the fictitious pulse tracer input, we calculate the mean residence time. The problem is solved by the mixed-hybrid finite element method for the flow equation and the discontinuous Galerkin method for the advection-diffusion transport, both implemented in Flow123d open-source software. We check a convergence by the time step refinement and find the limit of the mean residence time with rising time interval. The effect of dispersion parameters can explain some of the differences between results obtained by different numerical software in a separate study [5]. We also show how both the flow and the transport problem have a simple and efficient procedure to solve their inverse problems.

## Article Details

How to Cite

HOKR, Milan; BALVÍN, Aleš.
Numerical properties of a model problem for evaluation of natural tracer transport in groundwater.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 292-301, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/418>. Date accessed: 20 sep. 2017.
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Articles

## References

[1] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer, New York, 1991

[2] J. Březina, M. Hokr, Mixed-Hybrid Formulation of Multidimensional Fracture Flow In: Numerical Methods and Applications, Lecture Notes in Computer Science Volume 6046, 2011, pp 125-132.

[3] J. Březina, Mortar-Like Mixed-Hybrid Methods for Elliptic Problems on Complex Geometries, In: D. Sevčovič, A. Handlovičová, Z. Minarechová, editor, ALGORITMY 2012 19th Conference on Scientific Computing, Slovak University of Technology in Bratislava, Publishing House of STU, 2012.

[4] A. Ern, A. F. Stephansen, and P. Zunino, A discontinuous Galerkin method with weighted averages for advectiondiffusion equations with locally small and anisotropic diffusivity, IMA Journal of Numerical Analysis, 29(2) (2009), 235-256.

[5] M. Hokr, H. Shao, W.P. Gardner, A. Balvín, H. Kunz, Y. Wang, M. Vencl, Real-case benchmark for flow and tracer transport in the fractured rock, In preparation, to be submitted to Envir. Earth Sci., 2015

[6] M. Hokr, I. Škarydová, and D. Frydrych, Modelling of tunnel inflow with combination of discrete fractures and continuum. Computing and Visualization in Science, 15(1) (2013), pp.21-28.

[7] M. Hokr, A. Balvín, D. Frydrych, I. Škarydová, Meshing issues in the numerical solution of the tunnel inflow problem, In: Mathematical Models in Engineering and Computer Science (Marascu-Klein, ed.), NAUN, 2013, pp. 162-168

[8] H.A. Loáiciga, Residence time, groundwater age, and solute output in steady-state groundwater systems, Adv. Water Res. 27 (2004), 681–688.

[9] P. Maloszewski and A. Zuber, On the Theory of Tracer Experiments in Fissured Rocks with a Porous Matrix, Journal of Hydrology, 79 (1985), 333–358.

[10] A. Suckow, The age of groundwater – Definitions, models and why we do not need this term, Applied Geochemistry, 50 (2014), 222–230.

[11] TUL, Flow123d version 1.8.2, Documentation of file formats and brief user manual, Technical University of Liberec, 2015, Online: http://flow123d.github.io/.

[2] J. Březina, M. Hokr, Mixed-Hybrid Formulation of Multidimensional Fracture Flow In: Numerical Methods and Applications, Lecture Notes in Computer Science Volume 6046, 2011, pp 125-132.

[3] J. Březina, Mortar-Like Mixed-Hybrid Methods for Elliptic Problems on Complex Geometries, In: D. Sevčovič, A. Handlovičová, Z. Minarechová, editor, ALGORITMY 2012 19th Conference on Scientific Computing, Slovak University of Technology in Bratislava, Publishing House of STU, 2012.

[4] A. Ern, A. F. Stephansen, and P. Zunino, A discontinuous Galerkin method with weighted averages for advectiondiffusion equations with locally small and anisotropic diffusivity, IMA Journal of Numerical Analysis, 29(2) (2009), 235-256.

[5] M. Hokr, H. Shao, W.P. Gardner, A. Balvín, H. Kunz, Y. Wang, M. Vencl, Real-case benchmark for flow and tracer transport in the fractured rock, In preparation, to be submitted to Envir. Earth Sci., 2015

[6] M. Hokr, I. Škarydová, and D. Frydrych, Modelling of tunnel inflow with combination of discrete fractures and continuum. Computing and Visualization in Science, 15(1) (2013), pp.21-28.

[7] M. Hokr, A. Balvín, D. Frydrych, I. Škarydová, Meshing issues in the numerical solution of the tunnel inflow problem, In: Mathematical Models in Engineering and Computer Science (Marascu-Klein, ed.), NAUN, 2013, pp. 162-168

[8] H.A. Loáiciga, Residence time, groundwater age, and solute output in steady-state groundwater systems, Adv. Water Res. 27 (2004), 681–688.

[9] P. Maloszewski and A. Zuber, On the Theory of Tracer Experiments in Fissured Rocks with a Porous Matrix, Journal of Hydrology, 79 (1985), 333–358.

[10] A. Suckow, The age of groundwater – Definitions, models and why we do not need this term, Applied Geochemistry, 50 (2014), 222–230.

[11] TUL, Flow123d version 1.8.2, Documentation of file formats and brief user manual, Technical University of Liberec, 2015, Online: http://flow123d.github.io/.