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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