Deflated Conjugate Gradient Method for modeling Groundwater Flow Near Faults

datum

Deflated Conjugate Gradient Method

for modeling Groundwater FlowNear Faults

Lennart RosDeltares & TU Delft

Delft January 11 2008: 13.00www.deltares.com

Supervisors:

Prof. Dr. Ir. C. Vuik (TU Delft)Dr. M. Genseberger (Deltares)Ir. J. Verkaik (Deltares)

2Deflated CG method 04/22/23

Outline

04/22/23Deflated CG method 3

Outline

Introduction Deltares Subsurface, Geohydrology & Faults MODFLOW IBRAHYM & problem

Equation, Discretization & Method

Testcase & Observations

Deflation Techniques & First Results

Further Research & Goals


Introduction


Introduction

Deltares

January 1st 2008


Introduction

Subsurface is schematized in layers . Successive sand and clay

(aquifers and aquitards) Assumption:

• Horizontal flow in aquifer• Vertical flow in aquitard

Subsurface


Introduction

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].

Geohydrology


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.


Introduction

All Faults in theIBRAHYM

model


Introduction

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.

MODFLOW:


Introduction

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.

IBRAHYM:


Equation,Discretization

& Method



xx yy zz sh h h hK K K W S

x x y y z z t

Where: 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]

, ,xx yy zzK K K

Governing Equation:



Finite Volume Discretization:

ihQ SS Vt

, 1/ 2, , 1/ 2, , 1, , ,i j k i j k i j k i j kq CR h h



Finite Volume Discretization:

External Sources:

Time Discretization:

Euler Backwards

, , , , , , , , , , , , , , , , ,1 1 1

N N N

i j k n i j k n i j k i j k n i j k i j k i j kn n n

a p h q P h Q

1, , , , , ,

1

m mi j k i j k i j k

m m

h h ht t t



, 1/ 2, , 1, , 1/ 2, , 1, 1/ 2, , 1, ,

1/ 2, , 1, , , , 1/ 2 , , 1 , , 1/ 2 , , 1

, 1/ 2, , 1/ 2, 1/ 2, , 1/ 2, ,

, , 1/ 2 , , 1/ 2

m m mi j k i j k i j k i j k i j k i j k

m m mi j k i j k i j k i j k i j k i j k

i j k i j k i j k i j k

i j k i j k

CR h CR h CC h

CC h CV h CV h

CR CR CC CC

CV CV H

, , , , , ,m

i j k i j k i j kCOF h RHS

, ,, , , , 1

1, ,

, , , , , , 1

,

.

i j k j i ki j k i j k m m

mi j k

i j k i j k i j k j i k m m

SS r c vHCOF P

t th

RHS Q SS r c vt t

Where:

Discretized Equation Using Finite Volume Method:



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

, 1/ 2,, 1/ 2,

, 1/ 2,

originali j k barrier

i j k originali j k barrier

CR CCR

CR C

Faults in MODFLOW :

1/ 2, ,1/ 2, ,

1/ 2, ,

originali j k barrier

i j k originali j k barrier

CV CCV

CV C



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

Solution Method:



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



Observations for simple testcase in Matlab:

Preconditioning:

Incomplete Cholesky




Smallest eigenvalue: 0.00010283296716

Next eigenvalue: 0.04870854847951



Due to the small eigenvalue we have a slow converging model.

Want to get rid of this eigenvalue

IDEA: USE DEFLATION


DeflationTechniques


Deflation Techniques

Basic Idea of Deflation:

General linear system of equations:

Define: ,

where: and assume A to be SPD

So:

and

Ay b

1 TP I AZE Z TE Z AZ

1

1

T

T T

E E

P I ZE Z A



Basic Idea of Deflation:

Note we can write:

But since:

we only need to compute

Since we solve the deflated system:

,T Ty I P y P y

1 1 ,T T T TI P y Z Z AZ Z Ay ZE Z b

.TP y

TAP PA

PAy Pb



Deflation using Eigenvectors:

Assume that A has eigenvalues:

and we choose the corresponding eigenvectors such that

If we now define

Then:

1 2 ,n

jv .Tj j ijv v

10, ,0, , ,m nPA

1 2: ,mZ v v v



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.

i

i

1, for x

0, for x \j

j ij

z x



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.

1M PA 1M A



Results for the test problem:

Less iterates are needed

Result looks positive


Further Research


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.

Deflated Conjugate Gradient Method for modeling Groundwater Flow Near Faults

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