1. Hybrid and Multiscale Modeling of Subsurface Flow and Transport Processes Mesa C Organizers: Timothy Scheibe (PNNL) Daniel Tartakovsky (UCSD) 1

2. Hybrid Models for Multiscale Simulation of Subsurface Biogeochemical Processes
Tim Scheibe1
Daniel Tartakovsky2
Alexander Tartakovsky1
George Redden3
1Pacific Northwest National Laboratory
2University of California, San Diego
3Idaho National Laboratory 2

3. Motivation
Continuum reactive transport models may not adequately describe localized coupled precipitation / transport / reaction processes in the presence of high concentration gradients.
Z = 40.5 cm
This slide gives an example that illustrates the need for multi-physics, multi-scale modeling. The continuum-scale models commonly used in reactive transport models are not adequate to represent the important physics and chemistry in this case, and pore-scale models (with different representation of physics/chemistry) must be incorporated.
First image: Shows the mesoscale experimental flow cell at Idaho National Laboratory. This is instrumented to allow control of two different solutions flowing in from the bottom and imaging of the resulting transport
Second image: Shows the result of an experiment in which sodium carbonate and calcium chloride solutions were injected, forming a calcium carbonate preciptate (white band) in the mixing zone between the two solutions.
Third image: Left panel shows a dye tracer experiment (blue and red dyes) indicating the mixing zone (dark area). When simulated using a continuum model with dispersive mixing (parameters based on phenomenological observations), the predicted mixing zone is quite a bit larger than observed (model results shown in right panel).
Fourth image: Plot of model prediction of saturation index along a cross-section at height 40.5 cm. SI > 1.0 indicates that precipitation should theoretically occur. However, obviously it only occurs in a very thin band (< 1 cm thick) – click to re-enlarge experimental picture showing calcite band. This disparity is because of 1) wrong conceptualization of mixing as dispersive, when in fact diffusion is the control for actual mixing, and 2) failure to account for the effect of the precipitate in changing the flow/transport/mixing of the two solutes (coupling between reaction and transport).
Both of these can be accurately simulated at the pore scale using a discrete pore geometry model. However, it is not computationally feasible to simulate the entire experimental domain at this scale. Therefore, there is a clear need to link between the pore-scale model (in regions and times where the reaction is occurring) and the continuum-scale model (for the rest of the domain). 3

4. A+B=C reactive system (Chopard et al., 1994).
Surface reaction
Surface growth
Continuity equation
Momentum conservation 4

5. Uncoupled Precipitation Simulation
Concentration of “A”
Concentration of “Caq”
(Concentration of “B” is mirror image)
A and B are 1.0 at the inlet; Ceq = Last equation is constrained such that rate is zero if C <= Ceq (precipitation only)
Continuum simulation with no change in flow or transport properties. Grid refined to 1 cm in center of domain.
Concentration of “Cs” 5

6. Fully Coupled Continuum Simulation
Continuum simulation with modification of permeability and transverse dispersivity as a function of changes in porosity associated with precipitation.
Porosity reduction only from 0.37 to 0.23 6

7. Smoothed Particle Hydrodynamics
Smoothed Particle Hydrodynamics Lucy (1977), Gingold and Monaghan (1977)
SPH does not require structured computational mesh for calculation of special derivatives. Fluids and solids are replaced with a set of particles. The particles serve as interpolation points from which properties of the fluid can be calculated.
Spatial derivatives can be found by analytical differentiation of the kernel:
The SPH particles are material particles which can be treated like any other particle system. 7

8. SPH Extension to Precipitation/Dissolution Reactions:
Rate of gain/loss of mass in the solid phase is approximated by
The processes of precipitation and dissolution are modeled by tracking the masses, mi, of the solid particles. Once the mass of a solid particle reaches twice the original mass of the solid particle the nearest fluid particle ‘precipitates’, becoming a new solid particle
Similarly, if the mass of a solid particle reaches zero, the solid particle becomes a new fluid particle. Since the new fluid particles will be very close to the solid boundary, where the fluid velocity is very small, the initial velocity of a new fluid particle is set to zero. 8

9. SPH Extension to Precipitation/Dissolution Reactions:
Comparison of SPH and one-dimensional analytical solutions of the diffusion equation with reaction (adsorption) at a fixed solid boundary. 9

10. SPH Extension to Precipitation/Dissolution Reactions:
Position of a solid boundary (in units of h) as a function of time (in dimensionless SPH model time units) obtained by SPH simulations and analytically.
Tartakovsky A. M., P. Meakin, T. D. Scheibe, and R. M. Eichler West, "Simulations of reactive transport and precipitation with smoothed particle hydrodynamics."  Journal of Computational Physics, 222(2): , 2007. 10

11. Side-by-side injection of reacting solutions into two halves of a two-dimensional granular porous medium.
Na2CO3
CaCl2 11

12. t = 1000
t = 3000
t = 6000
Changes in solubility product <CA><CB> as result of precipitation. Reactive flow, Pe=2.8
Steady state solubility product <CA><CB>. Non-reactive flow, Pe=2.8
t = 1000 12

13. Effective reaction rates
Generation of “C”
Precipitation of “C”
Rate of change of <CC> due to reaction between solutes A and B versus the product of the average concentrations <CA><CB> for two Peclet numbers.
Rate of change of <CC> due to precipitation as a function of <CC> - Ceq for two Peclet numbers. 13

14. New Direction: Hybrid Modeling
Conclusion: Pore-scale modeling provides a more fundamental description of these strongly-coupled processes (mixing and reaction) that are not well described at the continuum scale.
Problem: Pore-scale modeling is extremely computationally intensive. Simulation at application-relevant scales is impractical.
Potential Solution: Hybrid multiscale modeling – directly couple simulations at two scales (pore and continuum). 14

15. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade. 15

16. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Materials Science:
“Modeling and simulation on various length and time scales have become a major field of Materials Science and Engineering in academia as well as in industrial research and development.”
(Multiscale Materials Modeling conference; 16

17. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Materials Science:
Bridging domain model of a nanotube
Figures from Belytschko and Xiao, 2003 and Xiao and Belytschko, 2004.
See also reviews by Csanyi et al., 2005; Wang and Zhang, 2006. 17

18. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Chemical Engineering (catalysis and reactor processes):
Comparison of predictions for NH3 decomposition on Ru with and without adsorbate-adsorbate interactions.
From Vlachos et al. 2006 18

19. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Chemical Engineering (catalysis and reactor processes):
Thin film deposition processes – multiple length scales encountered.
From Vlachos 1999 19

20. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Life sciences:
Multiscale model of lac repressor – DNA complex. Grey box is MD domain; DNA loop outside the box is modeled by elasticity theory (see below).
From Villa et al. 2005 20

21. Hybrid Modeling – Background
Hybrid multiscale models can be traced back to the early 1970’s (Gehlen et al. 1972) and have become widely used in some disciplines during the last decade.
Hydrodynamics:
Self-diffusion of Argon gas (two colors). Simulated using hybrid Direct Simulation Monte Carlo / Euler method.
From Wijesinghe et al. 2004 21

22. Hybrid Modeling – Background
From Balhoff et al. 2007
What about porous media?
Two porous media of contrasting permeability (e.g., sand-filled fracture and relatively impermeable matrix). Network pore model in pore-scale region; continuum model in matrix region. 22

23. Current Research
Hybrid Numerical Methods for Multiscale Simulations of Subsurface Biogeochemical Processes
SciDAC Science Application (with two Science Application Partnerships) funded in FY 23

24. Current Research
Hybrid Numerical Methods for Multiscale Simulations of Subsurface Biogeochemical Processes
SciDAC Science Application (with two Science Application Partnerships) funded in FY
Objective: Develop a component-based parallel model of groundwater flow and multicomponent reactive transport by directly coupling sub-pore-, pore-, and continuum-scale models. 24

25. Current Research
Hybrid Numerical Methods for Multiscale Simulations of Subsurface Biogeochemical Processes
SciDAC Science Application (with two Science Application Partnerships) funded in FY
Objective: Develop a component-based parallel model of groundwater flow and multicomponent reactive transport by directly coupling sub-pore-, pore-, and continuum-scale models.
Current focus: Hybrid two-scale model of calcite precipitation problem 25

26. Continuum Model Subsurface Transport Over Multiple Phases (STOMP)
Multiphase flow (sat/unsat) with multicomponent reactions
Common Component Architecture framework
Weak scaling studies
Smoothed Particle Hydrodynamics
Can be used at continuum scale also
Paper in preparation 26

27. Pore-Scale Model Smoothed Particle Hydrodynamics
2D code being used to test model coupling
Parallel 3D code is being debugged and validated
Front-Tracking Method (FronTier)
Collaboration with Xiaolin Li (SUNY Stony Brook) and Harold Trease (PNNL) 27

28. First Hybrid (Continuum-Pore) Model
(Tartakovsky et al., in preparation)
SPH at both scales
No advection; mixing by diffusion only and heterogeneous reaction
Also working on a coupled SPH / FE model using “compatibility coupling” (Rabczuk et al 2006). 28

29. Conclusions
Hybrid multiscale methods represent a potentially powerful approach to subsurface reactive transport simulations in which highly coupled, non-linear, localized processes predominate.
There has been little work undertaken to date in this area but we can build from developments in other disciplines.
We have initiated a project that is developing a component-based hybrid modeling framework for subsurface simulations. 29

30. Other Contributors Bruce Palmer (PNNL): CCA SAP PI
Karen Schuchardt (PNNL): Workflow / data SAP PI
Yilin Fang, Glenn Hammond, Mark White (PNNL): Continuum-scale reactive transport modeling
Vidhya Gurumoorthi (PNNL): CCA implementation 30

31. Acknowledgment
Funding for this research was provided by the U. S. Department of Energy through the following programs:
Laboratory-Directed Research and Development (administered by PNNL through the Computational Science Initiative)
Office of Science, Biological and Environmental Research, Environmental Remediation Sciences Program (ERSP).
Office of Science, Biological and Environmental Research and Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program. 31