Filtered density function uncertainty assessments for reactivetransport in groundwater
Final Report Abstract
Monitoring groundwater contamination and remediation scenarios requires mathematical models and numerical solutions for reactive transport in groundwater systems. In most relevant cases, the spatial heterogeneity of such systems at the field scale cannot be completely described and parameterized to allow deterministic descriptions. Stochastic approaches are therefore intensively used. One way to consider randomness in modeling transport in hydro-geological systems is through stochastic parameterizations of incompletely known hydraulic conductivity fields. This parameterization induces the randomness of the Darcy velocity field which in turn implies the randomness of the observables of the transport process. Beyond mean and variance, traditionally used in stochastic approaches, the full one-point onetime concentration probability density (PDF) is needed to estimate exceedance probabilities in assessments of groundwater contamination. The PDF approach is mainly useful in case of reactive transport: because reaction terms are in a closed form there is no need to upscale them, as in case of modeling the mean behavior of species concentrations. Solving such evolution equations also avoids the cumbersome Monte Carlo simulations often used to assess concentration PDFs. We have shown that evolution equations for concentration PDFs weighted by a conserved scalar (e.g. the density or the sum of the reactant species concentrations) take the form of a Fokker-Planck equation. Consequently, the associated systems of Itô equations can be used to estimate the weighted concentration PDF. We used a global random walk (GRW) algorithm consisting of a superposition of large numbers of solutions for the system of Itô equations on regular lattices embedded in Cartesian products of physical and concentration spaces. This procedure results in a huge reduction of computational costs. Using a spatial filtering operation to infer the statistics of the transport process, we further developed a filtered density function (FDF) approach. The FDF equations are formally identical to the PDF equations and can be solved by the same GRW algorithm. The PDF/FDF equations contain unclosed terms, depending on multi-point PDFs, which have to be modeled. We used upscaling methods to model the terms describing the transport in the physical space. To close the terms which govern the transport in the concentration space, we proposed a mixing model with a variable time scale, appropriate for groundwater systems. The modeled PDF/FDF equations were validated by comparisons with mean values and PDFs estimated from Monte Carlo simulations. The convergence with increasing filter width of the FDF solution to the PDF solution demonstrates the consistency of the FDF approach.
Publications
- , Global random walk algorithm for transport in media with discontinuous dispersion coefficients, EGU General Assembly, Geophysical Research Abstracts, Vol. 15, EGU2013-12751-1, 2013
uciu, N., and C. Vamoş
- Evolution of the concentration PDF in random environments modeled by global random walk, EGU General Assembly, Geophysical Research Abstracts, Vol. 15, EGU2013-12709-1, 2013
Suciu, N., C. Vamoş, S. Attinger, and P. Knabner
- (2014), Diffusion in random velocity fields with applications to contaminant transport in groundwater, Adv. Water Resour., 69, 114-133
Suciu, N.
(See online at https://doi.org/10.1016/j.advwatres.2014.04.002) - Concentration PDF modeled by global random walk, XX. International Conference on Computational Methods in Water Resources, Stuttgart University, June, 10–13, 2014
Suciu, N., S. Attinger, L. Schüler, C. Vamos and P. Knabner
- Filtered Density Function Methods Applied to Solute Transport in Heterogenous Media, XX. International Conference on Computational Methods in Water Resources, Stuttgart University, June, 10–13, 2014
Schüler, L., S. Attinger
- (2015), A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media, J. Comput. Appl. Math. 289, 241-252
Suciu, N., F.A. Radu, S. Attinger, L. Schüler, P. Knabner
(See online at https://doi.org/10.1016/j.cam.2015.01.030) - (2015), Automatic algorithm to decompose discrete paths of fractional Brownian motion into self-similar intrinsic components, Eur. Phys. J. B, 88, 250
Vamoş, C., M. Crăciun, and N. Suciu
(See online at https://doi.org/10.1140/epjb/e2015-60515-5) - (2015), Consistency issues in PDF methods, Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica, 23(3), 187-208
Suciu, N., L. Schüler, S. Attinger, C. Vamoş, and P. Knabner
(See online at https://doi.org/10.1515/auom-2015-0055) - (2015), Solute transport in aquifers with evolving scale heterogeneity, Analele Stiintifice ale Universitatii Ovidius Constanta- Seria Matematica, 23(3), 167-186
Suciu, N., S. Attinger, F. A. Radu, C. Vamos, J. Vanderborght, H. Vereecken, and P. Knabner
(See online at https://doi.org/10.1515/auom-2015-0054) - A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media, The 6th International Conference on Advanced Computational Methods in engineering, ACOMEN 2014, 23–28 June 2014; J. Comput. Apl. Math, 289, 241-252, 2015
Suciu, N., F. A. Radu, S. Attinger, L. Schüler, C. Vamos and P. Knabner
(See online at https://doi.org/10.1016/j.cam.2015.01.030) - Consistency issues in PDF methods, 10th Workshop on Mathematical Modelling of Environmental and Life Sciences Problems, October 16-19, Constanţa, Romania; Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica, 23(3), 187-208, 2015
Suciu, N., F. A. Radu, S. Attinger, L. Schüler, C. Vamos and P. Knabner
(See online at https://dx.doi.org/10.1515/auom-2015-0055) - Filtered density functions for solutes transported in heterogeneous aquifers, EGU General Assembly 2015, Vienna, April, 12– 17, 2015
Schüler, L., N. Suciu, S. Attinger, and P. Knabner
- Parameterization and Monte Carlo solutions to PDF evolution equations, EGU General Assembly 2015, Vienna, April, 12–17, 2015, Poster Y35
Suciu, N., L. Schüler, S. Attinger, and P. Knabner
- (2016), Building a Bridge from Moments to PDF's: A new approach to finding PDF mixing models
Schüler, L., N. Suciu, P. Knabner, and S. Attinger
- (2016), Towards a filtered density function approach for reactive transport in groundwater, Adv. Water Resour
Suciu, N., L. Schüler, S. Attinger, C. Vamoş, and P. Knabner
(See online at https://doi.org/10.1016/j.advwatres.2016.02.016)
