1. Deflated Conjugate Gradient Method
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Deflated Conjugate Gradient Method
for modeling Groundwater Flow
Near Faults
Lennart Ros
Deltares & TU Delft
Delft January :
Supervisors:
Prof. Dr. Ir. C. Vuik (TU Delft)
Dr. M. Genseberger (Deltares)
Ir. J. Verkaik (Deltares)

2. 16/11/2018
Outline
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3. Outline Introduction Deltares Subsurface, Geohydrology & Faults
MODFLOW
IBRAHYM & problem
Equation, Discretization & Method
Testcase & Observations
Deflation Techniques & First Results
Further Research & Goals
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4. 16/11/2018
Introduction
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5. Introduction
Deltares
January 1st 2008
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6. Introduction Subsurface Subsurface is schematized in layers .
Successive sand and clay
(aquifers and aquitards)
Assumption:
Horizontal flow in aquifer
Vertical flow in aquitard
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7. Introduction Geohydrology Connected pores give a rock permeability.
The driving force for groundwater flow is the difference in height and pressure.
To represent this difference we introduce the concept of hydraulic heads, h [L].
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8. Introduction
Faults
Medium Faults are vertical barriers inside aquifers.
Faults do not usually consist of a single, clean
fracture  fault zone.
Different types of faults.
Main property: low permeability.
Large contrasts in parameters.
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9. Introduction All Faults in the IBRAHYM model Deflated CG method
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10. Introduction MODFLOW:
MODFLOW is a software package which calculates hydraulic heads.
Developed by the U.S. Geological Survey.
Open-source code: everyone can use and improve this program
Rectangular grid and uses cell- centered variables.
Quasi-3D model.
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11. Introduction IBRAHYM:
groundwater model developed for several waterboards in Limburg.
large variety of faults in subsoil.
faults cause model to suffer from bad convergence behavior of solver.
uses at most 19 layers to model groundwater flow area.
uses grid cells of 25 times 25 meter to get detailed information.
most famous fault is ”de Peelrandbreuk” in Limburg.
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12. Equation, Discretization & Method
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Equation,
Discretization
& Method
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13. Equation, Discretization & Method
Governing Equation:
hydraulic conductivities along x,y, and z coordinate axes [LT-1], h
potentiometric head [L], W
volumetric flux per unit volume representing sources and sinks of water [T-1], Ss
specific storage of porous material [L-1], t
Time [T]
Where:
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14. Equation, Discretization & Method
Finite Volume Discretization:
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15. Equation, Discretization & Method
Finite Volume Discretization:
External Sources:
Time Discretization:
Euler Backwards
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16. Equation, Discretization & Method
Discretized Equation Using Finite Volume Method:
Where:
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17. Equation, Discretization & Method
Faults in MODFLOW :
When we model a fault in the subsoil we update the hydraulic conductance.
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18. Equation, Discretization & Method
Solution Method:
MODFLOW use stress, time and inner iteration loops
We look at inner iteration loop:
solves a linear system of equations
matrix is symmertic negative definite
Preconditioned Conjugate Gradient Method:
Incomplete Cholesky Decomposition
also: SOR
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19. Testcase & Observations
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Testcase &
Observations
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20. Testcase & Observations
Simple Testcase:
15 rows, 15 colums, 1 layer
1 fault on 1/3th of the domain
Cells represent an area of
25 x 25 meters
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21. Testcase & Observations
Observations for simple testcase in Matlab:
Preconditioning:
Incomplete Cholesky
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22. Testcase & Observations
Observations for simple testcase in Matlab:
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23. Testcase & Observations
Observations for simple testcase in Matlab:
Smallest eigenvalue:
Next eigenvalue:
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24. Testcase & Observations
Due to the small eigenvalue we have a
slow converging model.
Want to get rid of this eigenvalue
IDEA: USE DEFLATION
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25. 16/11/2018
Deflation
Techniques
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26. Basic Idea of Deflation:
Deflation Techniques
Basic Idea of Deflation:
General linear system of equations:
Define: ,
where: and assume A to be SPD
So:
and
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27. Basic Idea of Deflation:
Deflation Techniques
Basic Idea of Deflation:
Note we can write:
But since:
we only need to compute
Since we solve the deflated system:
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28. Deflation using Eigenvectors:
Deflation Techniques
Deflation using Eigenvectors:
Assume that A has eigenvalues:
and we choose the corresponding eigenvectors such that
If we now define
Then:
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29. Alternative Deflation Techniques:
Random Subdomain Deflation
Deflation based on Physics:
Use faults as boundary of domain
Define vectors such that an element next to a
fault has value 1 and otherwise 0.
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30. Results for the test problem:
Deflation Techniques
Results for the test problem:
Deflation using subdomain deflation
1 domain left of fault
1 domain right of fault
The eigenvector corresponding to the smallest eigenvalue is in the span of these two vectors.
Eigenvalues of and are almost the same, but the smallest is cancelled now.
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31. Results for the test problem:
Deflation Techniques
Results for the test problem:
Less iterates are needed
Result looks positive
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32. 16/11/2018
Further
Research
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33. Further Research & Goals
Future Research:
How representive is the Matlab model?
Can faults in IBRAHYM be seen as the sum of local faults?
Is deflation always faster, even if we do not have faults?
Future Goals:
Implementing deflation in MODFLOW.
Choose suitable deflation vectors such that:
vectors are easy to construct,
a priori information is used to construct vectors,
choice of vectors is generetic and not problem dependent.
Reduce number of iterations in PCG solver and gain wall-clock times.
Find criterion for when to use deflation for a general problem.
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