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 11 2008 13.00 www.deltares.com
Supervisors Prof. Dr. Ir. C. Vuik (TU
Delft) Dr. M. Genseberger (Deltares) Ir. J.
Verkaik (Deltares)
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Outline
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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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Introduction
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Introduction
Deltares
January 1st 2008
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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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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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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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Introduction
All Faults in the IBRAHYM model
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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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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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Equation, Discretization Method
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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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Equation, Discretization Method
Finite Volume Discretization
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Equation, Discretization Method
Finite Volume Discretization
External Sources
Time Discretization
Euler Backwards
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Equation, Discretization Method
Discretized Equation Using Finite Volume Method
Where
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Equation, Discretization Method
Faults in MODFLOW

• When we model a fault in the subsoil we update
  the hydraulic conductance.

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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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Testcase Observations
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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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Testcase Observations
Observations for simple testcase in Matlab
Preconditioning Incomplete Cholesky
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Testcase Observations
Observations for simple testcase in Matlab
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Testcase Observations
Observations for simple testcase in Matlab
Smallest eigenvalue 0.00010283296716 Next
eigenvalue 0.04870854847951
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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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Deflation Techniques
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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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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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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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Deflation Techniques
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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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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Deflation Techniques
Results for the test problem

• Less iterates are needed
• Result looks positive

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Further Research
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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.