Abstract
Reactive transport in saturated/unsaturated porous media is numerically upscaled to the spacetime scale of a hypothetical measurement through coarse grained space-time (CGST) averages. The one-dimensional reactive transport is modeled at the fine-grained Darcy scale by the actual number of molecules involved in reactions which undergo advective and diffusive movements described by global random walk (GRW) simulations. The CGST averages verify identities similar to a local scale balance equation which allow us to derive expressions for the flow velocity and the intrinsic diffusion coefficient in terms of averaged microscopic quantities. The latter are further used to verify the CGST-GRW numerical approach. The upscaling approach is applied to biodegradation processes in saturated aquifers and variably saturated soils and the CGST averages are compared to classical volume averages. One finds that if the process is characterized by slow variations in time, as in homogeneous systems or in case of observations of reactive transport in heterogeneous aquifers made at large times or far away from the contaminant source, the differences between the two averages are negligible. Instead, the differences are significant if the averages are computed close to thesource at early times, in case of aquifer simulations, and can be extremely large in simulations of biodegradation in soils. In the latter case, the volume average is totally inappropriate as model for experimental measurements, leading for instance to overestimations by 100% of the CGST average.
Authors
Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Florin A. Radu
Department of Mathematics, University of Bergen, Allegaten 41, 5007 Bergen, Norway
Keywords
Space-time upscaling, Global random walk, Reactive transport, Richards equation MSC: 76S05, 35K57, 86A05, 65C35
Cite this paper as:
N. Suciu and F.A. Radu, Space-time upscaling of reactive transport in porous media, Arxiv: https://doi.org/10.48550/arXiv.2112.10692
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[1] Alecsa, C.D., Boros, I., Frank, F., Knabner, P., Nechita, M., Prechtel, A., Rupp, A., Suciu, N., 2019. Numerical benchmark study for fow in heterogeneous aquifers. Adv. Water Resour., 138, 103558. https://doi.org/10.1016/j.advwatres.2020.103558
[2] Andricevic R., 1998. Effects of local dispersion and sampling volume on the evolution of concentration fluctuations in aquifers. Water Resour. Res., 34(5):1115–29. http://dx.doi.org/10.1029/98WR00260
[3] de Barros, F.P.J., Fiori, A., 2014. First-order based cumulative distribution function for solute concentration in heterogeneous aquifers: Theoretical analysis and implications for human health risk assessment. Water Resour. Res., 50(5), 4018–4037. http://dx.doi.org/10.1002/2013WR015024
[4] Battiato, I. Multiscale dynamics of reactive fronts in heterogeneous media with fluctuating forcings. SIAM Conference on Mathematical and Computational Issues in the Geosciences September 11–14, 2017, Erlangen, Germany. https://archive.siam.org/meetings/gs17/program/invited-talks-prize-presentations/.
[5] Bayer-Raich, M., Jarsjo, J., Liedl, R., Ptak, T., Teutsch, G.2004. Average contaminant concentration and mass flow in aquifers from time-dependent pumping well data: Analytical framework. Water Resour. Res., 40(8), W08303. http://dx.doi.org/10.1029/2004WR003095
[6] Bause, M., Knabner, P., 2004. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput. Visual. Sci., 7(2), 61–78. https://doi.org/10.1007/s00791-004-0139-y
[7] Berkowitz, B., 2021. HESS Opinions: Chemical transport modeling in subsurface hydrological systems – Space, time, and the holy grail of “upscaling”. Hydrol. Earth. Syst. Sci. Discuss., [preprint]. https://doi.org/10.5194/hess-2021-537
[8] Brunner, F., Radu, F.A., Bause, M., Knabner, P., 2012. Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media. Adv. Water Resour., 35, 163–171. http://dx.doi.org/10.1016/j.advwatres.2011.10.001
[9] Bringedal, C., Berre, I., Pop, I.S., Radu, F.A., 2015. A model for non-isothermal flow and mineral precipitation and dissolution in a thin strip. J. Comput. Appl. Math., 289, 346–355. https://doi.org/10.1016/j.cam.2014.12.009
[10] Bringedal, C., Berre, I., Pop, I. S., Radu, F. A., 2016. Upscaling of nonisothermal reactive porous media flow under dominant Peclet number: the effect of changing porosity. Multiscale Model. Simul. 14(1), 502–533. https://doi.org/10.1137/15M1022781
[11] Caroni, E., Fiorotto, V., 2005. Analysis of concentration as sampled in natural aquifers. Transport Porous Media, 59(1),19–45. http://dx.doi.org/10.1007/s11242-004-1119-x
[12] Cirpka, O.A., Frind, E.O., Helmig, R., 1999. Numerical simulation of biodegradation controlled by transverse mixing. J. Contam. Hydrol., 40(2), 159-182. https://doi.org/10.1016/S0169-7722(99)00044-3
[13] Cushman, J.H., 1983. Multiphase transport equations: I-general equation for macroscopic statistical, local space-time homogeneity. Transport Theor. Stat. Phys., 12(1), 35–71. http://dx.doi.org/10.1080/00411458308212731
[14] Destouni, G., Graham, W., 1997. The influence of observation method on local concentration statistics in the subsurface. Water Resour. Res., 33(4), 663–676. http://dx.doi.org/10.1029/96WR03955
[15] Eberhard, J.P., Suciu, N., Vamos, C., 2007. On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. Math. Theor., 40(4), 597–610. https://doi.org/10.1088/1751-8113/40/4/002
[16] Gray, W.G., Miller, C.T., 2013. A generalization of averaging theorems for porous medium analysis. Adv. Water Resour. 62, 227–37. https://doi.org/10.1016/j.advwatres.2013.06.006
[17] Gray, W.G., Dye, A.L., McClure, J.E., Pyrak-Nolte, L.J., Miller, C.T., 2015. On the dynamics and kinematics of two-fluid-phase flow in porous media. Water Resour. Res., 51(7), 5365-81. https://doi.org/10.1016/10.1002/2015WR016921
[18] He, Y., Sykes, J.F., 1996. On the spatial-temporal averaging method for modeling transport in porous media. Transport Porous Media, 22(1), 1–51. https://doi.org/10.1007/BF00974310
[19] Hornung, U., 1996. Homogenization and porous media (Vol. 6). Springer Science & Business Media. https://link.springer.com/book/10.1007/978-1-4612-1920-0
[20] Illiano, D., Pop, I.S., Radu, F.A., 2020. Iterative schemes for surfactant transport in porous media. Comput. Geosci. https://doi.org/10.1007/s10596-020-09949-2
[21] Klofkorn, R., Kr¨oner, D., Ohlberger, M., 2002. Local adaptive methods for convection dominated problems. Int. J. Numer. Meth. Fluid., 40(1-2), 79–91. https://doi.org/10.1002/fld.268
[22] Kumar, K., Van Noorden, T.L., Pop, I.S., 2011. Effective dispersion equations for reactive flows involving free boundaries at the microscale. Multiscale Model. Simul., 9(1), 29–58. https://doi.org/10.1137/100804553
[23] Kumar, K., Neuss-Radu, M., Pop, I.S., 2016. Homogenization of a pore scale model for precipitation and dissolution in porous media. IMA J. Appl. Math., 81(5), 877–897. https://doi.org/10.1093/imamat/hxw039
[24] List, F. and Radu, F.A., 2016. A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20 (2), 341–353. https://doi.org/10.1007/s10596-016-9566-3
[25] Mailloux B.J., Fuller, M.E., Rose, G.F., Onstott, T.C., DeFlaun, M.F., Alvarez, E., Hemingway, C., Hallet, R.B., Phelps, T.J., Griffin, T., 2003. A modular injection system, multilevel sampler, and manifold for tracer tests. Ground Water, 41(6), 816–27. http://dx.doi.org/10.1111/j.1745-6584.2003.tb0242
[26] McClure, J.E., Dye, A.L., Miller, C.T., Gray, W.G., 2017. On the consistency of scale among experiments, theory, and simulation. Hydrol. Earth Syst. Sci., 21(2), https://doi.org/10.5194/hess-21-1063-2017
[27] Nolen, J., Papanicolaou, G., Pironneau, O., 2008. A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul., 7, 171–196. http://dx.doi.org/10.1137/070693230
[28] Nordbotten, J.M., Celia, M.A., Dahle, H.K., Hassanizadeh, S.M., 2007. Interpretation of macroscale variables in Darcy’s law. Water Resour. Res., 43(8), W08430. http://dx.doi.org/10.1029/2006WR005018
[29] Pop, I.S., Radu, F.A., Knabner, P., 2004. Mixed finite elements for the Richards’ equation: linearization procedure. Journal of computational and applied mathematics 168 (1-2), 365–373. https://doi.org/10.1016/j.cam.2003.04.008
[30] Porta, G.M., Riva, M., Guadagnini, A., 2012. Upscaling solute transport in porous media in the presence of an irreversible bimolecular reaction. Adv. Water Resour., 35, 151–162. https://doi.org/10.1016/j.advwatres.2011.09.004
[31] Schuler, L., Suciu, N., Knabner, P., Attinger, S. 2016. A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers. Adv. Water Resour., 96:55–67. http://dx.doi.org/10.1016/j.advwatres.2016.06.012
[32] Schwarze, H., Jaekel, U., Vereecken, H., 2001. Estimation of macrodispersion by different approximation methods for flow and transport in randomly heterogeneous media. Transport Porous Media, 43(2), 265-287. https://doi.org/10.1023/A:1010771123844
[33] Schwede, R.L., Cirpka, O.A., Nowak, W., Neuweiler, I., 2008. Impact of sampling volume on the probability density function of steady state concentration. Water Resour. Res., 44(12), W12433. http://dx.doi.org/10.1029/2007WR006668
[34] Skøien, J.O., Bloschl, G., Western, A.W., 2003. Characteristic space scales and timescales in hydrology. Water Resour. Res., 39(10). http://dx.doi.org/10.1029/2002WR001736
[35] Srzic, V., Cvetkovic, V., Andricevic, R., Gotovac, H., 2013. Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty. Water Resour. Res., 49(6), 3712–3728. http://dx.doi.org/10.1002/wrcr.20314
[36] Suciu N., 2001. On the connection between the microscopic and macoscopic modeling of the thermodynamic processes. Ed. Univ. Pitesti (in Romanian). English abstract: http://www1.am.uni-erlangen.de/suciu/Rabsth.pdf.
[37] Suciu, .N, Radu, F.A., Attinger, S., Sch¨uler, L., Knabner, P., 2015. A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media. J. Comput. Appl. Math., 289, 241–252. http://dx.doi.org/10.1016/j.cam.2015.01.030
[38] Suciu, N., Schuler, L., Attinger, S., Knabner, P., 2016. Towards a filtered density function approach for reactive transport in groundwater. Adv. Water Resour., 90, 83–98. http://dx.doi.org/10.1016/j.advwatres.2016.02.016
[39] Suciu, N., 2019. Diffusion in Random Fields. Applications to Transport in Groundwater. Birkhauser, Cham. https://doi.org/10.1007/978-3-030-15081-5
[40] Suciu, N., Illiano, D., Prechtel, A., Radu, F.A., 2021. Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media. Adv. Water Resour., 152, 103935. https://doi.org/10.1016/j.advwatres.2021.103935
[41] Suciu, N., Radu, F.A., 2021. Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media. arXiv: 2107.08745
[42] Suciu, N., Radu, F.A., 2021. https://github.com/PMFlow/MonodReactions. Git repository. https://doi.org/10.5281/zenodo.5104076
[43] Suciu, N., Radu, F.A., 2021. https://github.com/PMFlow/SpaceTimeUpscaling. Git repository.
[44] Teutsch, G., Ptak, T., Schwarz, R., Holder, T., 2000. Ein neues Verfahren zur Quantifizierung der Grundwasserimmission: I. Theoretische Grundlagen. Grundwasser, 4(5):170–175. http://dx.doi.org/10.1007/s767-000-8368-7
[45] Tonina, D., Bellin, A., 2008. Effects of pore-scale dispersion, degree of heterogeneity, sampling size, and source volume on the concentration moments of conservative solutes in heterogeneous formations. Adv. Water Resour., 31(2), 339–354. http://dx.doi.org/doi:10.1016/j.advwatres.2007.08.009
[46] Vamos C., Georgescu A., Suciu N., Turcu I., 1996. Balance equations for physical systems with corpuscular structure. Phys. Stat. Mech. Appl., 227(1), 81–92. http://dx.doi.org/10.1016/0378-4371(95)00373-8
[47] Vamos C., Suciu N,, Georgescu, A., 1997. Hydrodynamic equations for one-dimensional systems of inelastic particles. Phys. Rev. E, 55(5), 6277–6280. http://doi.org/10.1103/PhysRevE.55.6277
[48] Vamos, C., Suciu, N., Peculea, M., 1997. Numerical modelling of the one-dimensional diffusion by random walkers. Rev. Anal. Numer. Theor. Approx., 26(1), 237–247. http://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art32.
[49] Vamos, C., Suciu, N., Blaj, W., 2000. Derivation of one-dimensional hydrodynamic model for stock price evolution. Phys. Stat. Mech. Appl., 287(3), 461–467. http://dx.doi.org/10.1016/S0378-4371(00)00385-X
[50] Vamos, C., Suciu, N., Vereecken, H., 2003. Generalized random walk algorithm for the numerical modeling of complex diffusion processes. J. Comput. Phys., 186, 527–544. https://doi.org/10.1016/S0021-9991(03)00073-1
[51] Vamos C., 2007. Continuous fields associated to corpuscular systems. Risoprint, Cluj-Napoca (in Romanian). English version: http://ictp.acad.ro/vamos/cv-thesis/
[52] Wiedemeier, T.H., Rifai, H.S., Newell, C.J., Wilson, J.T., 1999. Natural attenuation of fuels and chlorinated solvents in the subsurface. John Wiley & Sons.
[53] Wright, E. E., Sund, N. L., Richter, D. H., Porta, G. M., Bolster, D., 2021. Upscaling bimolecular reactive transport in highly heterogeneous porous media with the LAgrangian Transport Eulerian Reaction Spatial (LATERS) Markov model. Stoch. Environ. Res. Risk. Assess., 35, 1529–1547. https://doi.org/10.1007/s00477-021-02006-z