Coupling of MFE or Mimetic Finite Differences with Discontinuous Galerkin for Poroelasticity

Center forSubsurfaceModeling

Coupling of MFE or Mimetic Finite Differences with Discontinuous Galerkin for Poroelasticity

Mary F. Wheeler

Ruijie Liu

Phillip Phillips

Center for Subsurface Modeling

The University of Texas at Austin


Vertical Subsidence due to 100 million barrels of fluid (and sand) extracted from the Goose Creek oil field near Galveston, Texas (Pratt and Johnson, 1926, p.582). Water-covered areas are shown in black.

Vertical Subsidence


Cross-section of a long bone (Fritton, Wang, Weinbuam, and Cowin, 2001 Bioengineering Conference, ASME 2001).

Remark: very low permeability

Bone Poroelasticity


0( ( ) ( , ) ( , )) ( , )

where

1s f

f

c x p x t u x t z q x tt

f g

Governing Equations:

Constitutive Laws:

( ) 2 ( )

( )

ij ij kk ij ij

f

f

u u p

z k p g

kk

Poroelasticity Theory


Poroelasticity Theory

ˆ( )

ˆ

ˆ( )

ˆ

D

T

p

f

u t u on

n t on

p t p on

Z n q on

Boundary conditions:

Initial Conditions:

)()(

)()(

)()(

00

00

00

xptp

xt

xutu

t

t

t


Notations, Spaces and Norms for Nonconforming Spaces

1

1 1

1 2

2

2 2

,1

2 2

,1

: { , , , }

: { , , , , , }

: ( ) { ( ) : , 0}

( ) ( ( )) 2,3

:

( ) ( )

( ) ( )

h

h h h

j

h N

P P M

s sh E

s s dh h

Nh mhm Em jj

Nh mhm Em jj

Subdivisions of E E E

Edges or faces e e e e e

Norms H v L v H j

H d

Broken norms

w w w H

Averag

H

w w w H

1 2

1 2

:

1 1{ } ( ) ( )

2 2

[ ] ( ) ( )

a aa a

a aa a

e eE E

e eE E

e and jump

w w w

w w w


MFE/Mimetic Galerkin Formulation for Poroelasticity

Find such that:

where

( , , ) h h hu p z V W S

,0

0

1

( , ) ( , ) ( , ) ( , ), ( )

( ( ), ) ( , ) ( , ),

ˆ, ( , )p

u t r h

h

f h

A u v J u v v p f v v D

c p u w z w q w w Wt

k z s p s ps g s s S

( )fz k p g

(1)

(2)

(3)

}))((:{)( hd

rEhr EEPvvD Discontinuous space of piecewise polynomials:


Bilinear Form:

where

Bilinear Form for Elasticity

0

0

1

1

,0

( , ) ( ) ( ) { ( ) }[ ]

{ ( ) }[ ] ( ( ) )

( ) ( , )

h

a

h

a aa

aa

Pa

u ijkl kl kl ijkl kl j iE ea

h

Pa a

DG ijkl kl j i ijkl kl j ie ea e

aDG ijkl kl j ie

e

A u v D u v D u p vE

D v u D u p v

D v u J u v

2 2

,0

1

DG DG

DG DG a

( , ) [ ] [ ]

1 : SIPG; 0 : IIPG

1 : NIPG; 1 and =0 : OBB

h

a aa D

Pa a

e ea ea a

r rJ u v u v u v

e e

Ref.: Riviere and Wheeler; Hansbo and Larson; Liu, Phillips and Wheeler, …


Auxiliary and Projection Error

2: projection

A Ih h

A I

A I

U R u u U u R u

P Pp p P p Pp

Z z z Z z z

Pp p L


References for Approximation Assumptions

1 3

1

1

1

1

1

1, ,

Let 1,2, or 3;

There exists { ( ( )) , ( )}

such that .

(i) ( ( ) ) 0, ( )

( ), ,

(ii) [ ( )] 0, ( ( ))

(iii) , where : suitable

h r

h h h kE

h

dh h h kE

sh EE s E

r

R L H D

v H

q R v v q P E

v H e

q R v q P e

v R v Ch v E

2

0, 0,

* element or macroelement

1(iv) [ ] ( )

1 : triangular or tetrahedral, (Crouzeix and Raviart)

2 : triangular (Fortin and Soulie); rectangular (Girault)

3 : (Crouzeix and Falk

ih

h hee

v R v C v R ve

k

k

k

)

* : (Girault and Scott)


Error Estimates

Main Results:

22 1/ 21/ 20 0 0

, ,0 00

2 21/ 2

0 0 00

2 22

0 0

22

0

22*

0

( )

( , ) ( , )

( ) ( )

(0)

(0)

{

}

u

uu

u u

u u

TA A A

A

TA A A At t

T TI It

TI I I

AA

TI It

A A

TA A

A A

U c P T k Z

J U U J U U

const k Z P

P T U U T

U U

U const U


Main Results (Continued):

Applying Gronwall's inequality:

Displacement Flow and pressure (s : optimal exponent for flow)

21/ 21/ 2

0 0 0 0( ) ( ) ( )

1

( ) 1

u

T

h h hA

r s

u u T c p p k z z

Ch for r

and

C h h for r

Error Estimates


• If then the coupled model with mixed or mimetic finite elements for pressure and conforming Galerkin converges with optimal rates in energy and in L2 for flow pressure and velocity. Estimates depend on C*. Flow is locally conservative.

• If discontinuous Galerkin is used and the approximation assumption holds, then couple mixed/mimetic or DG for flow and DG for displacements converge independent of C0(x). Flow is locally conservative.

Summary

** 00 ,C C x C x


P0=1 psi

Fixed displacement

boundary

Red Line:

No flow boundary

E: 1.0e+4 psi

K=1.0e-6, 0.1

Kw =1.0e+12

P0=1

time

Pressure Output Flag

X

Y

Numerical Example


DG: Solid is solved by discontinuous Galerkin finite element

Mixed finite element for flow: piece-wise constant for pressure

DG for Solid Flow

Numerical Example


-1

-0.5

0

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

P

Pressure Distribution at 0.01 seconds

U----Continuous linearP----Piece-wise constant

High Permeability

CG for Solid and MFE for Flow


-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

P


U----Continuous linearP----Piece-wise constant

Low Permeability



DG: Low Permeability at earlier stage

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X/H

P/P

0


U----discontinuous Galerkin (linear) (NIPG)P----Piece-wise constant



CG: Low Permeability DG: Low Permeability

Pressure Contour


Mandel Problem

2F

2F

2a

2b

X

Y


Analytical Pressure Solution of Mandel Problem

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

4.0E+05

-100 -80 -60 -40 -20 0 20 40 60 80 100

Marching in timeMarching in time

Time =1.0e+3 1.0e+5 8.0e+5 2.0e+6 3.0e+6 5.0e+6 1.0e+7


CG: Linear-Linear; Low Permeability

Incompressible Case Compressible Case

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

8.E+05

9.E+05

1.E+06

-100 -80 -60 -40 -20 0 20 40 60 80 100

Time step = 1000Time = 1000 200000 500000 1000000 7000000

E = 1.0e+7Poisson ratio=0.2α = 1.01/M = 0 k/η = 1.0e-12

Linear-Linear Element

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

8.E+05

9.E+05

1.E+06

-100 -80 -60 -40 -20 0 20 40 60 80 100

Time step = 1000Time = 1000 200000 500000 1000000 7000000

E = 1.0e+7Poisson ratio=0.2Undrained Poissonratio=0.4B=0.8α = 0.89M = 1.75e+7k/η = 1.0e-12

Linear-Linear Element

Numerical Results (CG for Solid and Flow)


Numerical Results (DG for Both Solid and Flow)

Linear Elements

Red line: CG

Green Line: DG


50 mm

Z

pressure: 0.006 Mpa

5 mm

10 mm

10 mm

Y

Width in X direction is 1 mm; E = 55 Mpa; Poisson's ratio =0.3 or 0.499; Uniform pressure = 0.006 Mpa

Numerical Example—Bracket


(a) Poisson Ratio = 0.3

Continuous Galerkin

(b) Poisson Ratio = 0.499


(a) OBB

Discontinuous Galerkin: 0.499

(b) NIPG

(c) SIPG

(d) IIPG


Pure Bending Beam—CG and DG Simulations

High Strength Materials

Low Strength Material with Ideal Plasticity (Von-Mises )


Pure Bending Beam—Meshing

Area where DG is applied


CG Simulation on Plastic Zone Development


DG Simulation on Plastic Zone Development


Breast Reconstruction Model

1. Elasticity Model

2. Large Deformation

3. Updated Geometry

4. Materials are close to incompressible

Poisson ratio = 0.499

Gravity loading only

Domain in red is in tension

Continuous Galerkin Finite Element Solution



Discontinuous Galerkin Finite Element Solution

Poisson ratio = 0.499

Gravity loading only

Domain in red is in tension



Geometry updating for continuous gravity loading


• Coupling of DG and CG in Geomechanics/Multiphase simulator

• Extensions to include plasticity (Valhall Oil Reservoir)

• Error estimators for adaptivity

Current and Future Work

Coupling of MFE or Mimetic Finite Differences with Discontinuous Galerkin for Poroelasticity

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