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Hybrid Numerical Methods for Multiscale Simulation of Subsurface Biogeochemical Processes 1 Tim Scheibe Pacific Northwest National Laboratory tim.scheibe@pnl.gov Interagency Steering Committee on Multimedia Environmental Modeling (ISCMEM) Annual Public Meeting November 28, 2011

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Acknowledgments Work presented here was mostly funded by the Office of Science, Biological and Environmental Research, Subsurface Biogeochemistry Research (SBR) Program: PNNL-led SciDAC project (Scientific Discovery through Advanced Computing) PNNL’s SBR Subsurface SFA (Scientific Focus Area) MAP flowchart was developed during planning stages of a PNNL initiative in Energy Systems Simulation (LDRD – Laboratory Directed Research and Development)

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What is the appropriate level of complexity in reactive transport modeling? “The secret to successful solute-transport modeling may simply be to lower expectations.” Leonard Konikow (2011 Ground Water) “Entities should not be multiplied beyond necessity.” William of Ockham (c. 1320) “The variety of beings should not rashly be diminished.” Immanuel Kant (c. 1800) (Thanks to Chris Green, USGS) “A model should be as simple as possible, but no simpler.” Albert Einstein (c. 1940)

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What is the appropriate level of complexity in transport modeling? Complex Simple Predictive Fundamental (first-principles) physics/chemistry Descriptive Phenomenological (empirical) physics/chemistry Two Questions: 1.Where do you need/want to be? (level of predictive power) 2.Where can you be in practical terms? Satisfaction occurs when (1) and (2) coincide Quantum Mechanics (electron scale) Molecular Dynamics (molecular scale) Solid/Fluid Dynamics (pore scale) Darcy’s Law / ADE (porous medium scale) Calibrated aquifer model (field scale) 2 Technology-driven Problem-driven 1

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Problems… We often don’t know how to answer question 1 (and may not even give it any thought) If we do think carefully about question 1, we are often limited by characterization, fundamental understanding, and/or computation such that we cannot achieve satisfaction Solutions… Lower our expectations (and communicate clearly) Develop rigorous tools for analyzing multiscale problems, and apply ongoing technology advances to bring questions 1 and 2 into harmony 5

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Problems… We often don’t know how to answer question 1 (and may not even give it any thought) If we do think carefully about question 1, we are often limited by characterization, fundamental understanding, and/or computation such that we cannot achieve satisfaction Solutions… Lower our expectations (and communicate clearly) Develop rigorous tools for analyzing multiscale problems, and apply ongoing technology advances to bring questions 1 and 2 into harmony 6

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Multiscale Analysis Platform - MAP Motif C: Numerical Upscaling/Parameterization Motif C: Numerical Upscaling/Parameterization Motif B: Formal Upscaling with Closure Motif B: Formal Upscaling with Closure Motif A: Multiresolution Solvers Motif A: Multiresolution Solvers Loosely Coupled What is the first principle model for your problem? Q0 Q1 Complete Fine Scale Solution? Q2 Degree of Coupling? Q3 Spatial Scale Separation? Yes No Fully Decoupled Tightly Coupled Q5 Temporal Scale Separation? (relaxation times) Q4 Self Similar? Q8 Macroscopic Model Known? Q6 Small % of Domain? Q7 Macro- scopic Model Known? Sufficient Insufficient/None Motif D: Fractal Methods Motif D: Fractal Methods ? ? No Yes No Long Relaxation Time at Microscale Yes Motif E: Concurrent Hybrid Multiscale Motif E: Concurrent Hybrid Multiscale Yes Motif F: Hierarchical Hybrid Multiscale – Time Bursts Motif F: Hierarchical Hybrid Multiscale – Time Bursts Short Relaxation Time at Microscale F 1 : Top Down F 2 :Bottom Up Yes No Q8 Macro- scopic Model Known? Motif G: Hierarchical Hybrid Multiscale – Gap Tooth Motif G: Hierarchical Hybrid Multiscale – Gap Tooth Yes No Motif H: Time-Parallel Hierarchical Hybrid Multiscale Motif H: Time-Parallel Hierarchical Hybrid Multiscale

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Mixing-Controlled Mineral Precipitation Mixing-controlled calcium carbonate precipitation (Zhang et al., ES&T 44(20), 2010). 8

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Mixing Controlled Mineral Precipitation 9 Battiato et al. Hybrid models of reactive transport in porous and fractured media. Adv Water Resour (2011), doi:10.1016/j.advwatres.2011.01.012 Appropriate level of complexity: Darcy-scale simulation with effective process models and parameters Feasible level of complexity: Darcy-scale simulation with effective process models and parameters

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Pore-Scale Modeling and Upscaling Why Pore-Scale Modeling? “…It is important to have a reliable physically based tool that can provide plausible estimates of macroscopic properties. Any theoretical or numerical approach to this problem not only needs a detailed understanding of… mechanisms at the pore level but also an accurate and realistic characterization of the structure of the porous medium.” (emphasis added) 10 Piri and Blunt, Phys. Rev. E, 026310, 2005

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Pore-Scale Modeling and Upscaling 11 Computationally intensive Multiple numerical approaches Makes use of advanced characterization techniques (X-ray CT Geometry Courtesy John Zachara, PNNL) (MRI Data Courtesy Joe Seymour, Montana St. U.) (Pore-scale Visualization by Chad Jones, UC Davis)

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Pore-Scale Dispersion Nature of pore-scale dispersion 12 1.Particles from inlet face (“..zselect_points”)zselect_points Simulations by Bruce Palmer, PNNL Animations by Chad Jones and Kwan-Liu Ma, UC Davis

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Direct Numerical Upscaling Explicit pore-scale simulation using SPH method Two time snapshots With and without intragranular diffusion Effective measure: Breakthrough curve Fit with 1) ADE and 2) MRM 13 t = 300 t = 600 With IGD No IGD Example: Intragranular Diffusion Simulations by Zhangshuan Hou, PNNL

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3D Pore-Scale Velocity Benchmark Based on MRI experiments by Joe Seymour, Montana State University 14

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Mixing Controlled Mineral Precipitation 15 Battiato et al. Hybrid models of reactive transport in porous and fractured media. Adv Water Resour (2011), doi:10.1016/j.advwatres.2011.01.012 Appropriate level of complexity: Pore-scale resolution of flow and reactive transport Feasible level of complexity: Darcy-scale simulation with effective process models and parameters

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Hybrid Multiscale Simulation 16 Reaction at each discretization point i Averaged reaction over all i Define mixing coefficient (equals unity if fully mixed) Macroscopic equation – must be “closed” by computing m from pore-scale information

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Hybrid Multiscale Simulation Multiscale dimension reduction approach Reduce degrees of freedom (number of time steps) solved in microscale simulation by iterating between microscale and macroscale Perform numerical closure on microscale with short bursts of pore- scale simulation where insufficient general closure exists 17 (figure after Kevrekidis et al. 2003) Tartakovsky and Scheibe, “Dimension reduction method for advection-diffusion-reaction systems,” Advances in Water Resources, 34(12): 1616-1626, doi:10.1016/j.advwatres.2011.07.011, 2011.

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Hybrid Multiscale Simulation Multiscale dimension reduction approach 18 Complete Pore-Scale Solution Dimension Reduction Solution

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Conclusions “…as simple as possible” depends on the problem at hand Rigorous analysis approaches are needed to define the necessary level of complexity for specific problems Where the necessary level of complexity exceeds technological capabilities, new methods are needed to close this gap Hybrid multiscale simulation methods offer a means for solving complex problems (in which pore- and Darcy- scale processes are tightly coupled) in a computationally efficient manner Multiphase flow (e.g., CO 2 sequestration) Microbial dynamics (biofilms, chemotaxis, transport) 19

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Questions? 20 http://subsurface.pnnl.gov/

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