single phase darcy 2011

7/28/2019 Single Phase Darcy 2011

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Numerical simulation of two phaseporous media flow models with

application to oil recovery

Roland Masson

IFP New energies

ENSG course 2011

18/04 - 19/04 -20/04 -21/04


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Outline: 18-19/04

Discretization of single phase flows

Two Point Flux Finite Volume Approximation

of Darcy Fluxes Homogeneous case

Heterogeneous case

Exercise: single phase incompressible Darcy

flow in 1D (using Scilab)


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Outline: 19-20/04 Discretization of two phase immiscible

incompressible Darcy flows

Hyperbolic scalar conservation laws IMPES discretization of water oil two phase flow

Exercise: Impes discretization of water oil twophase flow in 1D (using Scilab)


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Outline: 20-21/04 Discretization of wells

Exercise: Five spots water oil simulation

Description of the Research Project


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Examination: 15/06 By binoms

Written report on the Project

Oral examination

Presentation of the report Run tests of the prototype code

Questions on numerical methods used in the

simulation


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Finite Volume Discretization of single

phase Darcy flows

Darcy law and conservation equation

Two Point Flux Discretization (TPFA) of diffusion

fluxes on admissible meshes

Exercice: single phase incompressible Darcy

flow in 1D


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Oil recovery by water injection

( )

( )

+=

=

gSPPKSk

V

gPKSkV

owcw

o

oor

o

ww

w

wwr

w

)()(

)(

,

,

( ) ( )( ) ( )

=+

=+

0

0

oooo

wwww

Vdivt

S

Vdiv

t

S

1=+ ow SS Capillary pressure PcRelative permeabilities kr,w and kr,o


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1D test caseInjection of water in a reservoir

prodpp=inj

w

ppS=

=1


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Water injection in a 1D reservoir


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Five Spots simulation in 2D

1000 m

1000m

Pressure

Water front


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Heterogeneities

Water front Pressure

Permeability


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Heterogeneities


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Coning: aquifer and vertical well

Pressure

Water front

1000 m100m

50m

Aquifer


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Coning: stratified reservoir

Permeability

Water front

Pressure


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SINGLE PHASE DARCY FLOWSINGLE PHASE DARCY FLOW

( ) ( ) qVdivt

=+

)( gP

K

V

=

K


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Incompressible Darcy single

phase flow Diffusion equation

=

=

=

N

DD

ongnp

K

onpp

onfpKdiv

.

)(

!

"!#

!


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Compressible Darcy single phase flow

Parabolic equation

(linearized)

=

=

=

=+

=onpp

TongnpK

Tonpp

TonpK

divpdp

d

t

N

DD

t

00

0

00

0

),0(.

),0(

),0(0)()1

(

$%

!

"!#

00 pp t ==


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NOTATIONSobjectlgeometrica

& "

!'(!)(" !*

21xx

!)(" !*(*

" !*


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Finite Volume Discretization Finite volume mesh

Cells Cell centers

Faces

Degrees of freedom:

Discrete conservation law

===

fdxdsnudxu

'

'.

' =

x 'x

u

'n


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Two Point Flux Approximation (TPFA)

TPFA

Flux Conservativity

Flux Consistency

),(. '''

uuFdsnu

0),(),( '''' =+ uuFuuF

( ) +==

)(.),( '''

'' hOdsnuuu

xx

uuF

''' xx'xx

'n


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Two Point Flux Approximation Boundary faces

xx '

( ) +==

)(.),( hOdsnuuuxx

uuF

x

xn


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Two Point Flux Approximation

Finite Volume Scheme

'

'

'

xx

T =

( ) ( )

fguxx

uuxx

bord

=+ = int'

'

'

=

=

surgu

surfu

xx

T =


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Exemples of admissible meshes

"" 2/

+


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Corner Point Geometries and

TPFA

Assumption that the directions of the CPGare aligned with the principal directions ofthe permeability field


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Corner Point Geometries

Stratigraphic grids with erosions

Examples of degenerate cells(erosions)

Hexahedra

Topologicaly Cartesian

Dead cells

Erosions

Local Grid Refinement (LGR)


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Cell Centered FV: MultiPoint Flux

Approximation (MPFA)

Example of the "O" scheme

Exact on piecewise linear functions

Account for discontinuous diffusion tensors

Account for anisotropic diffusion tensors

L

L

L

uTF ='' '

LL

L

L

TTT ''' ,0==


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2D example

=

=

surgu

surfu

( )yxeu += sin

, "

-


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Comparison of MPFA "O" scheme and TPFA

order 2

+ $

Non convergent


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Cell-Face data structure List of cells: m=1,...,N

Volume(m) Cell center X(m)

List of interior faces: i=1,...,Nint cellint(i,1) = m1, cellint(i,2)=m2

surfaceint(i)

Xint(i)

List of boundary faces: i=1,...,Nbound cellbound(i)

surfacebound(i)

Xbound(i)

'

x

x

x


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Computation of interior and

boundary face transmissibilities

Interior faces: i=1,...,Nint

m1 = cellint(i,1)

m2 = cellint(i,2)

Tint(i) = surfaceint(i)/|X(m2)-X(m1)|

Boundary faces: i=1,...,Nbound

m = cellbound(i)

Tbound(i) = surfacebound(i)/|X(m)-Xbound(i)|

Computation of the Jacobian sparse


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matrix and the right hand side JU = B

( ) ( )

fguTuuT

bound

=+ =

int'

''

( )

( )

=

=

uuTline

uuTline

''

''

:'

:

.

( ) guTline :

.

'uu

u fline :

.



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matrix and the right hand side: JU = B

( ) ( )

fguTuuT

bound

=+ = int'

''

Cell loop: m=1,...,N B(m) = Volume(m)*f(X(m))

Interior face loop: i=1,...,Nint m1 = cellint(i,1), m2 = cellint(i,2)

J(m1,m1) = J(m1,m1) +Tint(i)

J(m2,m2) = J(m2,m2) +Tint(i)

J(m1,m2) = J(m1,m2) -Tint(i)

J(m2,m1) = J(m2,m1) -Tint(i)

Boundary face loop: i=1,...,Nbound m = cellbound(i)

J(m,m) = J(m,m) +Tbound(i)

B(m) = B(m) + Tbound(i)*g(Xbound(i))

TPFA


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TPFA

Isotropic Heterogeneous media FV scheme

)()(')(' ''''

''

uuTuuxx

Kuuxx

KF ===

=

=

surgu

surfuKdiv )(

'xx

xK'K

u

'u

u

''

1

'

'

'

K

xx

K

xx

T+=


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TPFA

Isotropic heterogeneous permeability

u

'

u

u

''

1

'

'

'

K

xx

K

xx

T+=

'

'

'

'

'

'

'

''

xxK

xx

K

xx

K

xx

xxT =

+

=

'xxxK

'K


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Well discretization Radial stationary analytical solution for vertical wells in

homogeneous porous media

Numerical Peaceman well index for well discretization withimposed pressure

Proof of Peaceman formula for uniform cartesian meshes

Pressure drop for vertical single phase wells


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Stationary radial analytical solution

in homegeneous media

=

==

>=

= wrr

ww

ww

w

qdsnpK

rrpprrpK

).(

0

)/ln(2

)( ww

w rrK

qprp

=

wp

w

q

wrr=

wn

rqnrpKrq wr2

).()( ==

)(rp

wrr/1 100


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Numerical well index Cartesian mesh

x,y >> rw

( ) ( ) 0int'

'' =++ ==

wbord w

wqppTppT

Well w

Well cell

)/ln(2

0 ww

w rrK

qpp

w =

2/1220 )(14.0 yxr +

yx

Pressure Numerical computation with specified well flow rate and pressure

boundary condition given by the analytical solution

with

w

w

analytical solution


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Well flow rate with specified pressure

( ) ( ) 0)(,

,

'

''

int

=+ ==

ii

iwi ppWIppT

/ 0

)(

)/ln(

2

0

w

w

w pp

rr

Kq

w=

)/ln(

2

0 wrr

KWI

= Well index

C i f h J bi i d


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Computation of the Jacobian matrix and

right hand side JU = B with wells( ) ( ) 0

)(,

,

'

''

int

=+ ==

ii

iwi ppWIppT

Loop on interior faces: i=1,...,Nint

m1 = cellint(i,1), m2 = cellint(i,2)

J(m1,m1) = J(m1,m1) +Tint(i)

J(m2,m2) = J(m2,m2) +Tint(i)

J(m1,m2) = J(m1,m2) -Tint(i)

J(m2,m1) = J(m2,m1) -Tint(i)

Loop on wells: i=1,...,Nwell m = cellwell(i)

J(m,m) = J(m,m) + WI(i)

B(m) = B(m) + WI(i)*pw(i)


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Exercice: convergence of the scheme

to an analytical well solution

+

=

11

2

1

1

1

1

)/ln(2)/ln(2

)/ln(2

)(

rrifrrK

qrrK

q

rrrifrrK

q

prpw

ww

www

w

r

qnrprKrq wr

2).()()( ==

)(rp

wrr/1 1000

)/ln(2

)( ww

w rrK

qprp

=

rqnrpKrq wr2

).()( ==

)(rp

wrr /

1 1000

wr

r1

10/)( 12 KKrK ==

1)( KrK =

K

Proof of Peaceman well index: uniform


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Proof of Peaceman well index: uniform

cartesian mesh, well at the center of the cell

)(wprp=pqr=npK rq)(=

wrxy >>=

=

w

w

rru

rrppu

0

ruK = 0

p

=

=

wrr

ww dsnpKq .

wp

1

2

$

0.' '

' =+ =

wqdsnpK

)/ln(2

)( ww

w rrK

qprp

=

wp

wq

wrr=

wn

r

qnrpKrq wr

2).()( ==

'wp


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Proof of Peaceman well index formula

=== += ''

'

'

'

' .2..

dsnnr

q

dsnuKdsnpK rw

p nn

4)(. '

''

'wquu

xxdsnpK +

=

4))/ln(

2(0. '

''

'w

ww

w

qrx

K

qpp

xxdsnpK +

=

( )'''

'.

pp

xxdsnpK

=

" ( )ww

w rx

K

qpp /)2/exp(ln

2

+=

'wp

'n

rn


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Vertical well with hydrostatic pressure drop

( ))1()()1()( 2/1 = iZiZgipip iww

22

00 14.0),()/ln(

))((2

)( yxriHrr

imK

iWI w+==

!*(3334

List of well perforations from bottom to top:i=1,...,Np

m(i) = cell of perforation i

WI(i) = Well index of perforation i pw(i) = pressure of perforation i

BHPw pp =)1(15

6 !*

-

1

Analysis of TPFA discretization


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Discrete norms: on each cell

Discrete Poincar Inequality

uu h =2/1

2

2

=

uulh

2/1

2

)(

)(

2

'

' ')(

int

10

+=

=

u

xxuu

xxu

bound

hThh

10

2 )( hhlh uDu

'xx

Anal sis of TPFA discreti ation


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A priori estimate:

210

)()( l

hThhfDu

h

( ) ( ) =

+

=

ufuxx

uuxx

ubound

0'

'

'

( )

2/1

2

2/1

2

22

''

'

+ =

ufuxxuuxxbound

,(%(

=

=

suru

surfu

0


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Error estimate uxue = )(

0')('

'

''

'

=

+

Reexx

dsnuxx

xuxuR '

''

'' .

'

1)()(

=

)(, ''' hORRR ==

( )

fuuxx == ''

'

fdsnu = = '

'

'

.


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Error estimate uxue = )(

0')(''

''

'

=

+

Reexx

)(, ''' hORRR ==

Che hThh

)(10

( ) '''

'

2

)(''1

0

ReeReehTh

h ==

hxxeCehh Th

hThh

')(

2)(

'10

10


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TPFA discretization Discrete linear system:

Coercivity:

Symmetry:

Monotonicity: ( Ah=M-Matrice)

hhh FUA =

T

hh AA =

01 hA

2

)(min 10),(

hThhhhh uKUUA

M- Matrice monotonicity


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M Matrice monotonicity

01 A

>

>

j

ji

j

ji

ijiii

Athatsuchi

A

AA

0

0

0,0

,

,

,,

0=+

i

ij

jijiii SUAUA

0min0

j ij

A

( )

( )

fguxx

uuxx

bord

=

+

=

)(

)(

'

'

'

78" %


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Finite volume schemes

Parabolic Equations: time discretizationImplicit Euler integration in time

Stability analysis


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Parabolic model

=

=

=+

=onuu

TonnuK

TonfuKdivu

t

t

00

),0(0.

),0()(

Finite volume space and time discretizations


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Finite volume space and time discretizations

( )[ ] 0)(1

= +n

n

t

t

t dxdtftuKdivu

0).()()()(

1

'

'

1=

++

+

=

+dtdsntuKtfdxtudxtu

n

n

t

t

nn

)()( 11

+

+

n

t

t

ttYdttY

n

n

/ $"

)(tY

ttttnn

==

+10

,0

+

Finite volume space and time


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Finite volume space and time

discretizations

( )

fuuT

t

uu nnnn

=+

=

++

+

'

1

'

1

'

1

Stability analysis: discrete energy


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Stability analysis: discrete energy

estimate

( ) ( )

=+

=

++

+

+

fuuTtuuu nn

nn

n

'

1'

1'

1

1

22

10

222

1

2121221

2

2

l

n

hl

h

h

n

hl

n

h

n

hl

n

hl

n

h

uft

utuuuu

+

+++

++

222)()(2 bababaa +=

Stability: discrete energy estimate


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Stability: discrete energy estimate

221221

2222 lh

l

n

h

n

hl

n

hl

n

h ftuuuu +++

2202

222 lh

N

lh

l

N

h ftuu +

, L2

Stability analysis: discrete maximum principle


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Stability analysis: discrete maximum principle

(f=0, zero flux BC)

nnn uuTt

Tt

u

+

=

+ +

==

+ 1''

'

'

'

1 1

allforMumn

Then allforMum n +

1

Stability analysis: discrete maximum principle(f fl BC)


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(f=0, zero flux BC)

Muuif

nn>=

++ 11

sup0 Proof:

lead to a contradiction

( ) ( )MuuuTt

Mu

nnnn+

=

++

=

+

00

0

00

11

''

'0

1

Exercize: well test with


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e c e e test t

compressible Darcy single phase flow Parabolic equation

(linearized)

=

=

=

=+

=onpp

TongnpK

Tonpp

TonpK

divpdp

d

t

N

DD

t

00

0

00

0

),0(.

),0(

),0(0)()1

(

$%

!

"!#

00 pp t ==

single phase darcy 2011

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