improved iterative methods for napl transport through porous media
TRANSCRIPT
Improved Iterative Methods for NAPL Transport Through
Porous Media
Victor Minden May 2, 2012
Background
• Non-aqueous phase liquid (NAPL) – Gasoline – Pesticides – Mercury
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• Aquifer – Water – Pores
Background (cont.)
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Image source: Geoff Ruth (File:High School Earth Science 1-13.pdf (page 493)) [CC-BY-SA-3.0 (www.creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
• Spills • Leakage • Agricultural run-off
Can we model and simulate the flow of these contaminants?
The Problem
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• Abriola et al. – VALOR • Version 1.0: 2D model, 3-phase flow • Now: 3D model, 2-phase flow
• Well-established theory
Past Work
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L. Abriola, K. Rathfelder, M. Maiza, and S. Yadav, VALOR Code Version 1.0: A PC Code for SimulaGng Immiscible Contaminant Transport in Subsurface Systems, Tech. Rep. EPRI TR-‐101018, The University of Michigan, Ann Arbor,
Michigan, September 1992.
• Explore changes to VALOR solution algorithm
This Work
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Goal: Improve runtime without sacrificing accuracy
• Model • Discretization • IMPES • Solution Scheme • Simulation Results • Conclusion
Outline
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• Conservation of mass
• Effective Mass Density
Model
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Effective mass density Volumetric flux
• Modified Darcy’s Law
• Transmissibility
Model (cont.)
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Volumetric flux
J. Bear, Dynamics of fluids in porous media, Dover, 1988.
• Final Equation
Model (cont.)
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• Model • Discretization • IMPES • Solution Scheme • Simulation Results • Conclusion
Outline
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• Sample space and time on discrete grid
• Round values to nearest floating-point number
Finite problem
Discretization
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• Finite Differences – Approximate derivatives
Discretization (cont.)
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• Model • Discretization • IMPES • Solution Scheme • Simulation Results • Conclusion
Outline
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• Explicit schemes – – CFL conditions
• Implicit schemes – – Expensive
IMPES
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R. LeVeque, Finite Difference Methods for Ordinary and ParGal DifferenGal EquaGons: Steady-‐State and Time-‐Dependent Problems, Classics in Applied MathemaGcs, Society for Industrial and Applied MathemaGcs, 2007.
• Implicit-Pressure, Explicit-Saturation (IMPES)
• Expand time-derivative with product-rule
• Express NAPL pressure in terms of water pressure and capillary pressure
IMPES (cont.)
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• Saturation Equations
IMPES (cont.)
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• Pressure Equation
IMPES (cont.)
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• Problems – Density is pressure-dependent – Capillary pressures and relative
permeabilities have saturation-dependent functional form
IMPES (cont.)
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Nonlinear system
J. C. Parker, R. J. Lenhard, and T. Kuppusamy, A parametric model for consGtuGve properGes governing mulGphase flow in porous media, Water Resour. Res., 23 (1987), pp. 618–624.
• Model • Discretization • IMPES • Solution Scheme • Simulation Results • Conclusion
Outline
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• Picard Linearization – Solve a fixed point equation
– Iteration
Solution Scheme
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• Parallel Picard
Solution Scheme (cont.)
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• Predictor-Corrector Picard
Solution Scheme (cont.)
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• Generalized minimal residual method (GMRES) – Solve – Each step:
Solution Scheme (cont.)
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Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on ScienGfic and StaGsGcal CompuGng, 7 (1986), pp. 856–869
• GMRES(k) – Restart GMRES – Minimize over only k vectors – Choice of k important
Solution Scheme (cont.)
25 Y. Saad, IteraGve Methods for Sparse Linear Systems, Society for Industrial and Applied MathemaGcs, 2003.
• Preconditioning – New matrix, new spectrum – Right preconditioning:
Solution Scheme (cont.)
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• Jacobi
• Algebraic Multigrid (AMG)
Solution Scheme (cont.)
27 U. Troeenberg, C. Oosterlee, and A. Schüller, MulGgrid, Academic Press, 2001.
• Terminating GMRES – Iterations – Absolute residual norm – Relative residual norm – Stagnation
Solution Scheme (cont.)
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• Model • Discretization • IMPES • Solution Scheme • Simulation Results • Conclusion
Outline
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• Test Problem Parameters – 2.8GHz Intel® Xeon, 16GB RAM – – – Medium properties generated in
T-PROGS – 5 days of simulation
Simulation Results
30 S. F. Carle, T-‐PROGS: TransiGon Probability GeostaGsGcal Sogware and User’s Guide, University of California, 1999.
• Original (Fortran) vs PETSc (C) – Left Jacobi – 30 GMRES iterations
Simulation Results (cont.)
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S. Balay, J. Brown, , K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc users manual, Tech. Rep. ANL-‐ 95/11 -‐ Revision 3.2, Argonne NaGonal Laboratory, 2011.
Simulation Results (cont.)
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• Preliminary changes – Switch to right Jacobi – Enforce check of nonlinear
residual
Simulation Results (cont.)
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Simulation Results (cont.)
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• Varying GMRES Tolerances
Simulation Results (cont.)
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Simulation Results (cont.)
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• Varying Restart Parameter
Simulation Results (cont.)
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• Jacobi vs AMG – “Amortize” construction – Experiment with strength
threshold
Simulation Results (cont.)
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• AMG convergence
Simulation Results (cont.)
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Simulation Results (cont.)
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[20] V. E. Henson and U. M. Yang, Boomeramg: a parallel algebraic mulGgrid solver and precondiGoner, Applied Numerical MathemaGcs, 41 (2000), pp. 155–177.
• Alternative Linearization
Simulation Results (cont.)
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• Model • IMPES • Discretization • Solution Scheme • Simulation Results • Conclusion
Outline
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• Summary of Results – C, right Jacobi, stagnation, restart • 4.75x speed-up on one test
problem, 6.25x on another • Stagnation tolerance single biggest
improvement – AMG, restructure • Increased runtime • Restructure: difference norm O(1e-2)
Conclusion
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• Future work – Validation – Jacobi vs AMG • Streamline saturation update?
Conclusion (cont.)
45 U. Troeenberg, C. Oosterlee, and A. Schüller, MulGgrid, Academic Press, 2001.
Thank You!
• Acknowledgements • Questions?
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Extra Slides
• Assumptions • Variables
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Assumptions
• Isothermal system • Viscosity independent of
pressure • Pore space does not change
with time • Fluid saturations account totally
for pore volume • Liquid compressibility constant
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Variables
• Pressure – • Saturation – • Porosity - • Density – • Velocity – • Source/Sink – • Intrinsic soil permeability – • Relative permeability - • Viscosity -
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