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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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Computation of the Jacobian sparse
matrix and the right hand side JU = B
( ) ( )
fguTuuT
bound
=+ =
int'
''
( )
( )
=
=
uuTline
uuTline
''
''
:'
:
.
( ) guTline :
.
'uu
u fline :
.
Computation of the Jacobian sparse
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Computation of the Jacobian sparse
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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Analysis of TPFA discretization
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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Analysis of TPFA discretization
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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Analysis of TPFA discretization
Error estimate uxue = )(
0')('
'
''
'
=
+
Reexx
dsnuxx
xuxuR '
''
'' .
'
1)()(
=
)(, ''' hORRR ==
( )
fuuxx == ''
'
fdsnu = = '
'
'
.
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Analysis of TPFA discretization
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 ==