Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Pietrzyk, Kyle; Battiato, Ilenia

doi:10.1029/2022wr033600

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Journal Article · 01 July 2023 · Water Resources Research

DOI:https://doi.org/10.1029/2022wr033600· OSTI ID:2420935

^[1]; ^[2]

Department of Energy Resources Engineering Stanford University Stanford CA USA; OSTI
Department of Energy Resources Engineering Stanford University Stanford CA USA

Abstract

In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number ofad hocapproaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Research Organization:: Stanford Univ., CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

DOE Contract Number:: SC0019075

OSTI ID:: 2420935

Journal Information:: Water Resources Research, Journal Name: Water Resources Research Journal Issue: 7 Vol. 59; ISSN 0043-1397

Publisher:: American Geophysical Union (AGU)

Country of Publication:: United States

Language:: English

References (57)

Macroscopic models for filtration and heterogeneous reactions in porous media Municchi, Federico; Icardi, Matteo Advances in Water Resources, Vol. 141 https://doi.org/10.1016/j.advwatres.2020.103605	journal	July 2020
On Veracity of Macroscopic Lithium-Ion Battery Models Arunachalam, Harikesh; Onori, Simona; Battiato, Ilenia Journal of The Electrochemical Society, Vol. 162, Issue 10 https://doi.org/10.1149/2.0771509jes	journal	January 2015
Taylor dispersion in porous media. Determination of the dispersion tensor Salles, J.; Thovert, J. ‐F.; Delannay, R. Physics of Fluids A: Fluid Dynamics, Vol. 5, Issue 10 https://doi.org/10.1063/1.858751	journal	October 1993
Dispersion and Convection in Periodic Porous Media Rubinstein, Jacob; Mauri, Roberto SIAM Journal on Applied Mathematics, Vol. 46, Issue 6 https://doi.org/10.1137/0146060	journal	December 1986
Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst Iliev, Oleg; Mikelić, Andro; Prill, Torben Advances in Water Resources, Vol. 146 https://doi.org/10.1016/j.advwatres.2020.103779	journal	December 2020
Homogenization of a Nonlinear Convection-Diffusion Equation with Rapidly Oscillating Coefficients and Strong Convection Marušić-Paloka, Eduard; Piatnitski, Andrey L. Journal of the London Mathematical Society, Vol. 72, Issue 2 https://doi.org/10.1112/s0024610705006824	journal	October 2005
On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media Battiato, I.; Tartakovsky, D. M.; Tartakovsky, A. M. Advances in Water Resources, Vol. 32, Issue 11 https://doi.org/10.1016/j.advwatres.2009.08.008	journal	November 2009
Multiscale Investigation on Biofilm Distribution and Its Impact on Macroscopic Biogeochemical Reaction Rates: BIOFILM DISTRIBUTION AND RATE SCALING Yan, Zhifeng; Liu, Chongxuan; Liu, Yuanyuan Water Resources Research, Vol. 53, Issue 11 https://doi.org/10.1002/2017wr020570	journal	November 2017
Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media Allaire, Grégoire; Brizzi, Robert; Mikelić, Andro Chemical Engineering Science, Vol. 65, Issue 7 https://doi.org/10.1016/j.ces.2009.09.010	journal	April 2010
Nonhysteretic Capillary Pressure in Two‐Fluid Porous Medium Systems: Definition, Evaluation, Validation, and Dynamics Miller, C. T.; Bruning, K.; Talbot, C. L. Water Resources Research, Vol. 55, Issue 8 https://doi.org/10.1029/2018wr024586	journal	August 2019
Nonlocal dispersion in media with continuously evolving scales of heterogeneity Cushman, John H.; Ginn, T. R. Transport in Porous Media, Vol. 13, Issue 1 https://doi.org/10.1007/bf00613273	journal	October 1993
The Method of Volume Averaging Whitaker, Stephen Theory and Applications of Transport in Porous Media https://doi.org/10.1007/978-94-017-3389-2	book	January 1999
Upscaling and Automation: Pushing the Boundaries of Multiscale Modeling through Symbolic Computing: An Introduction to Symbolica Pietrzyk, Kyle; Korneev, Svyatoslav; Behandish, Morad Transport in Porous Media, Vol. 140, Issue 1 https://doi.org/10.1007/s11242-021-01628-9	journal	August 2021
Theory and Applications of Macroscale Models in Porous Media Battiato, Ilenia; Ferrero V., Peter T.; O’ Malley, Daniel Transport in Porous Media, Vol. 130, Issue 1 https://doi.org/10.1007/s11242-019-01282-2	journal	April 2019
Upscaling of Diffusion–Reaction Phenomena by Homogenisation Technique: Possible Appearance of Morphogenesis Bloch, Jean-Francis; Auriault, Jean-Louis Transport in Porous Media, Vol. 127, Issue 1 https://doi.org/10.1007/s11242-018-1187-y	journal	November 2018
Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media Allaire, Grégoire; Mikelić, Andro; Piatnitski, Andrey SIAM Journal on Mathematical Analysis, Vol. 42, Issue 1 https://doi.org/10.1137/090754935	journal	January 2010
Multiscale Approaches in Reactive Transport Modeling Molins, Sergi; Knabner, Peter Reviews in Mineralogy and Geochemistry, Vol. 85, Issue 1 https://doi.org/10.2138/rmg.2019.85.2	journal	September 2019
Rigorous Upscaling of the Reactive Flow through a Pore, under Dominant Peclet and Damkohler Numbers Mikelić, Andro; Devigne, Vincent; van Duijn, C. J. SIAM Journal on Mathematical Analysis, Vol. 38, Issue 4 https://doi.org/10.1137/050633573	journal	January 2006
Multiscale formulation of frictional contact mechanics at the pore scale Mehmani, Yashar; Castelletto, Nicola; Tchelepi, Hamdi A. Journal of Computational Physics, Vol. 430 https://doi.org/10.1016/j.jcp.2020.110092	journal	April 2021
The dissolution kinetics of major sedimentary carbonate minerals Morse, John W.; Arvidson, Rolf S. Earth-Science Reviews, Vol. 58, Issue 1-2 https://doi.org/10.1016/s0012-8252(01)00083-6	journal	July 2002
Homogenizability conditions for multicomponent reactive transport Boso, Francesca; Battiato, Ilenia Advances in Water Resources, Vol. 62 https://doi.org/10.1016/j.advwatres.2013.07.014	journal	December 2013
A hybrid pore-scale and continuum-scale model for solute diffusion, reaction, and biofilm development in porous media Tang, Youneng; Valocchi, Albert J.; Werth, Charles J. Water Resources Research, Vol. 51, Issue 3 https://doi.org/10.1002/2014wr016322	journal	March 2015
Solute diffusion in fractured porous media with memory effects due to adsorption Lewandowska, Jolanta; Auriault, Jean-Louis; Empereur, Sabrina Comptes Rendus Mécanique, Vol. 330, Issue 12 https://doi.org/10.1016/s1631-0721(02)01546-2	journal	December 2002
Multi-scale Model of Reactive Transport in Fractured Media: Diffusion Limitations on Rates Molins, Sergi; Trebotich, David; Arora, Bhavna Transport in Porous Media, Vol. 128, Issue 2 https://doi.org/10.1007/s11242-019-01266-2	journal	March 2019
An Adaptive Hybrid Vertical Equilibrium/Full‐Dimensional Model for Compositional Multiphase Flow Becker, Beatrix; Guo, Bo; Buntic, Ivan Water Resources Research, Vol. 58, Issue 1 https://doi.org/10.1029/2021wr030990	journal	January 2022
Dispersion, convection, and reaction in porous media Mauri, Roberto Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 5 https://doi.org/10.1063/1.858007	journal	May 1991
Modeling non-Fickian transport in geological formations as a continuous time random walk Berkowitz, Brian; Cortis, Andrea; Dentz, Marco Reviews of Geophysics, Vol. 44, Issue 2 https://doi.org/10.1029/2005RG000178	journal	January 2006
Upscaling diffusion–reaction in porous media Bourbatache, M. K.; Millet, O.; Moyne, C. Acta Mechanica, Vol. 231, Issue 5 https://doi.org/10.1007/s00707-020-02631-9	journal	February 2020
Homogenization in a Periodic and Time-Dependent Potential Garnier, Josselin SIAM Journal on Applied Mathematics, Vol. 57, Issue 1 https://doi.org/10.1137/s0036139995282001	journal	February 1997
A spectral approach for homogenization of diffusion and heterogeneous reaction in porous media Le, Tien Dung; Moyne, Christian; Bourbatache, Khaled Applied Mathematical Modelling, Vol. 104 https://doi.org/10.1016/j.apm.2021.12.017	journal	April 2022
General conservation equations for multi-phase systems: 1. Averaging procedure Hassanizadeh, Majid; Gray, William G. Advances in Water Resources, Vol. 2 https://doi.org/10.1016/0309-1708(79)90025-3	journal	March 1979
A primer on upscaling tools for porous media Cushman, John H.; Bennethum, Lynn S.; Hu, Bill X. Advances in Water Resources, Vol. 25, Issue 8-12 https://doi.org/10.1016/s0309-1708(02)00047-7	journal	August 2002
Eulerian-Lagrangian Theory of transport in space-time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation Neuman, Shlomo P. Water Resources Research, Vol. 29, Issue 3 https://doi.org/10.1029/92wr02306	journal	March 1993
General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations Hassanizadeh, Majid; Gray, William G. Advances in Water Resources, Vol. 2 https://doi.org/10.1016/0309-1708(79)90035-6	journal	March 1979
The role of scaling laws in upscaling Wood, Brian D. Advances in Water Resources, Vol. 32, Issue 5 https://doi.org/10.1016/j.advwatres.2008.08.015	journal	May 2009
Striving to translate shale physics across ten orders of magnitude: What have we learned? Mehmani, Yashar; Anderson, Timothy; Wang, Yuhang Earth-Science Reviews, Vol. 223 https://doi.org/10.1016/j.earscirev.2021.103848	journal	December 2021
On the late-time behavior of tracer test breakthrough curves Haggerty, Roy; McKenna, Sean A.; Meigs, Lucy C. Water Resources Research, Vol. 36, Issue 12 https://doi.org/10.1029/2000WR900214	journal	December 2000
Homogenization of a convection–diffusion model with reaction in a porous medium Allaire, Grégoire; Raphael, Anne-Lise Comptes Rendus Mathematique, Vol. 344, Issue 8 https://doi.org/10.1016/j.crma.2007.03.008	journal	April 2007
Applicability regimes for macroscopic models of reactive transport in porous media Battiato, I.; Tartakovsky, D. M. Journal of Contaminant Hydrology, Vol. 120-121 https://doi.org/10.1016/j.jconhyd.2010.05.005	journal	March 2011
Method of homogenization applied to dispersion in porous media Mei, Chiang C. Transport in Porous Media, Vol. 9, Issue 3 https://doi.org/10.1007/bf00611970	journal	November 1992
A homogenization approach for buoyancy-induced flows over micro-textured vertical surfaces Ahmed, Essam Nabil; Bottaro, Alessandro; Tanda, Giovanni Journal of Fluid Mechanics, Vol. 941 https://doi.org/10.1017/jfm.2022.320	journal	May 2022
Patch‐Based Multiscale Algorithm for Flow and Reactive Transport in Fracture‐Microcrack Systems in Shales Wang, Ziyan; Battiato, Ilenia Water Resources Research, Vol. 56, Issue 2 https://doi.org/10.1029/2019wr025960	journal	February 2020
Homogenization and Porous Media Hornung, Ulrich Interdisciplinary Applied Mathematics https://doi.org/10.1007/978-1-4612-1920-0	book	January 1997
Physics-based hybrid method for multiscale transport in porous media Yousefzadeh, Mehrdad; Battiato, Ilenia Journal of Computational Physics, Vol. 344 https://doi.org/10.1016/j.jcp.2017.04.055	journal	September 2017
An Analysis Platform for Multiscale Hydrogeologic Modeling with Emphasis on Hybrid Multiscale Methods Scheibe, Timothy D.; Murphy, Ellyn M.; Chen, Xingyuan Groundwater, Vol. 53, Issue 1 https://doi.org/10.1111/gwat.12179	journal	March 2014
Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems Gray, William G.; Miller, Cass T. Advances in Geophysical and Environmental Mechanics and Mathematics https://doi.org/10.1007/978-3-319-04010-3	book	January 2014
Taylor dispersion in porous media: Analysis by multiple scale expansions Auriault, J. L.; Adler, P. M. Advances in Water Resources, Vol. 18, Issue 4 https://doi.org/10.1016/0309-1708(95)00011-7	journal	January 1995
Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach Bachmat, Yehuda; Bear, Jacob Transport in Porous Media, Vol. 1, Issue 3 https://doi.org/10.1007/BF00238181	journal	September 1986
Multiscale simulation of transport phenomena in porous media: from toy models to materials models Schiller, Ulf D.; Wang, Fang MRS Communications, Vol. 8, Issue 02 https://doi.org/10.1557/mrc.2018.29	journal	March 2018
Three-dimensional stochastic analysis of macrodispersion in aquifers Gelhar, Lynn W.; Axness, Carl L. Water Resources Research, Vol. 19, Issue 1 https://doi.org/10.1029/WR019i001p00161	journal	February 1983
Hybrid models of reactive transport in porous and fractured media Battiato, Ilenia; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M. Advances in Water Resources, Vol. 34, Issue 9 https://doi.org/10.1016/j.advwatres.2011.01.012	journal	September 2011
A Priori Parameter Estimation for the Thermodynamically Constrained Averaging Theory: Species Transport in a Saturated Porous Medium Miller, Cass T.; Valdés-Parada, Francisco J.; Ostvar, Sassan Transport in Porous Media, Vol. 122, Issue 3 https://doi.org/10.1007/s11242-018-1010-9	journal	February 2018
Limits of Classical Homogenization Procedure for Coupled Diffusion-Heterogeneous Reaction Processes in Porous Media Bourbatache, Mohamed Khaled; Le, Tien Dung; Millet, Olivier Transport in Porous Media, Vol. 140, Issue 2 https://doi.org/10.1007/s11242-021-01683-2	journal	October 2021
Modeling and scale-bridging using machine learning: nanoconfinement effects in porous media Lubbers, Nicholas; Agarwal, Animesh; Chen, Yu Scientific Reports, Vol. 10, Issue 1 https://doi.org/10.1038/s41598-020-69661-0	journal	August 2020
Automated Solution of Differential Equations by the Finite Element Method Logg, Anders; Mardal, Kent-Andre; Wells, Garth Lecture Notes in Computational Science and Engineering https://doi.org/10.1007/978-3-642-23099-8	book	January 2012
Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions Golfier, Fabrice; Wood, Brian D.; Orgogozo, Laurent Advances in Water Resources, Vol. 32, Issue 3 https://doi.org/10.1016/j.advwatres.2008.11.012	journal	March 2009
Upscaling Reactive Transport and Clogging in Shale Microcracks by Deep Learning Wang, Ziyan; Battiato, Ilenia Water Resources Research, Vol. 57, Issue 4 https://doi.org/10.1029/2020wr029125	journal	April 2021

Similar Records

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Journal Article · 2023 · Water Resources Research · OSTI ID:1989525

Automated Symbolic Upscaling: 2. Model Generation for Extended Applicability Regimes

Journal Article · 2023 · Water Resources Research · OSTI ID:2420936

Automated Symbolic Upscaling: 2. Model Generation for Extended Applicability Regimes

Journal Article · 2023 · Water Resources Research · OSTI ID:1994234

Related Subjects

Environmental Sciences & Ecology
Marine & Freshwater Biology
Water Resources

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Citation Formats

References (57)

Similar Records

Related Subjects