Improved Iterative Methods for NAPL Transport Through Porous Media
Transcript of Improved Iterative Methods for NAPL Transport Through Porous Media
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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