numerical simulation of two phase porous media flow models
TRANSCRIPT
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Numerical simulation of two phase porous media flow models with
application to oil recovery
Roland MassonIFP New energies
ENSG course 201118/04 - 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
gPKSk
V
owcwo
ooro
www
wwrw
ρµ
ρµ
)()(
)(
,
,
( ) ( )( ) ( )
=+∂
∂
=+∂
∂
0
0
oooo
wwww
VdivtS
VdivtS
ρφρρφρ
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
pp
S
==1
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Water injection in a 1D reservoir
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Five Spots simulation in 2D
1000 m
1000 m
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
PressureWater front
1000 m
100 m
50 m
Aquifer
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Coning: stratified reservoir
Permeability
Water frontPressure
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SINGLE PHASE DARCY FLOWSINGLE PHASE DARCY FLOW
( ) ( ) qVdivt
=+∂
∂ ρφρ
)( gPK
V
ρµ
−∇−=
φKρµ
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Incompressible Darcy single phase flow
• Diffusion equation
Ω∂=∇−
Ω∂=
Ω=∇−
N
DD
ongnpK
onpp
onfpK
div
.
)(
µρ
µρ
!
"!#
!
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Compressible Darcy single phase flow
• Parabolic equation(linearized)
Ω=
×Ω∂=∇−
×Ω∂=
×Ω=∇−+∂
= onpp
TongnpK
Tonpp
TonpK
divpdpd
t
N
DD
t
00
0
000
),0(.
),0(
),0(0)()1
(
µρ
µρρρ
ρ
$%
!
"!#
00 pp t ==
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NOTATIONS
objectlgeometrica & "
σ
!'(!)(" !*κ
21xx
!)(" !*(*
" !*
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Finite Volume Discretization• Finite volume mesh
– Cells– Cell centers– Faces
• Degrees of freedom:
• Discrete conservation law
=∇−=∆−= κκκσ σ
κκκ
fdxdsnudxu'
'.
'κκ κκσ ′=
κx 'κx
κu
'κκn
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Two Point Flux Approximation (TPFA)
• TPFA
• Flux Conservativity
• Flux Consistency
),(. ''' κκκκσ
κκ uuFdsnu ≈∇−
0),(),( '''' =+ κκκκκκκκ uuFuuF
( ) +∇−=−=σ
κκκκκκ
κκκκ σσ
)(.),( '''
'' hOdsnuuuxx
uuF
''' κκκκ ⊥xx'κxκx
'κκn
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Two Point Flux Approximation• Boundary faces
σσκ ⊥xx '
( ) +∇−=−=σ
σσκσκ
σκσ σσ
)(.),( hOdsnuuuxx
uuF
σx
κxσn
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Two Point Flux Approximation• Finite Volume Scheme
''
'
κκκκ
κκxx
T =
( ) ( ) κκσ
σκσκκκκσ
κκκκ
κσσ
fguxx
uuxx
bord
=−+− Σ∩∂∈Σ∩∂∈= int'
''
Ω∂=Ω=∆−
surgu
surfu
σκκσ
σxx
T =
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Exemples of admissible meshes
" " 2/π≤
+
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Corner Point Geometries and TPFA
Assumption that the directions of the CPG are aligned with the principal directions of the permeability field
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Corner Point GeometriesStratigraphic grids with erosions
Examples of degenerate cells(erosions)
• Hexahedra
• Topologicaly Cartesian
• Dead cells
• Erosions
• Local Grid Refinement (LGR)
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CPG faults
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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
LL
L uTF = '' κκκκκ
'κ
LL
L
L TTT κκκκκκ ''' ,0 −==
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2D example
Ω∂=Ω=∆−
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)
'κκ σσx
σxσκ
κxκ
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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)|
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Computation of the Jacobian sparsematrix and the right hand side JU = B
( ) ( ) κκσ
σκκκκσ
κκκκ κσ
fguTuuTbound
=−+− Σ∩∂∈Σ∩∂∈= int'
''
( )( )
−−
=
=
κκκκσ
κκκκσ
κκ
uuTline
uuTline
''
''
:':
.
( )σκσκ guTline −:
.
'κuκu
σσ
κuκκκ fline :
.
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Computation of the Jacobian sparsematrix and the right hand side: JU = B
( ) ( ) κκσ
σκκκκσ
κκκκ κσ
fguTuuTbound
=−+− Σ∩∂∈Σ∩∂∈= 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))
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TPFAIsotropic Heterogeneous media
• FV scheme
)()('
)('
''''
'' κκκκκσσκ
κσκσκ
κκκκκκκ
uuTuuxx
Kuuxx
KF −=−=−=
Ω∂=Ω=∇−
surgu
surfuKdiv )(
'κxκxσx
κK'κK
κu
'κuσu
''1
'
'
' κκκκ κ
σκ
κ
σκ
κκ K
xx
K
xx
T+=
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TPFAIsotropic heterogeneous permeability
κu
'κuσu
''1
'
'
' κκκκ κ
σκ
κ
σκ
κκ K
xx
K
xx
T+=
''
'
'
'
''
''
κκκκ
κκ
κ
σκ
κ
σκ
κκκκ
κκκκxx
Kxx
Kxx
Kxx
xxT =
+=
'κxκx σxκK'κK
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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
rrpp
rrpK
).(
0
)/ln(2
)( ww
w rrK
qprp
π=−
wp
wq
wrr =
wn
rq
nrpKrq wr π2
).()( =∇−=
)(rp
wrr /1 100
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Numerical well index
• Cartesian mesh∆x,∆y >> rw
( ) ( ) 0int'
'' =+−+− =Σ∩∂∈Σ∩∂∈= κκκσ
σκσκκκσ
κκκκwbord w
wqppTppT
Well w
Well cell
)/ln(2 0 w
ww rr
Kq
ppw πκ =−
2/1220 )(14.0 yxr ∆+∆≈
y∆x∆
Pressure Numerical computation with specified well flow rate and pressure boundary condition given by the analytical solution
with
wκ
wκ
analytical solution
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Well flow rate with specified pressure
( ) ( ) 0)(,
,'
''int
=−+− =Π∈Σ∩∂∈= κκ
κκκκσ
κκκκii
iwi ppWIppT
/ 0
)()/ln(
2
0w
ww pp
rrK
qw
−= κπ
)/ln(2
0 wrrK
WIπ= Well index
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Computation of the Jacobian matrix and right hand side JU = B with wells
( ) ( ) 0)(,
,'
''int
=−+− =Π∈Σ∩∂∈= κκ
κκκκσ
κκκκii
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 schemeto an analytical well solution
≥+
≤≤=−
112
11
11
)/ln(2
)/ln(2
)/ln(2)(
rrifrrK
qrr
Kq
rrrifrrK
q
prpw
ww
www
w
ππ
π
rq
nrprKrq wr π2
).()()( =∇−=
)(rp
wrr /1 1000
)/ln(2
)( ww
w rrK
qprp
π=−
rq
nrpKrq wr π2
).()( =∇−=
)(rp
wrr /1 1000
wrr1
10/)( 12 KKrK ==
1)( KrK =
K
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Proof of Peaceman well index: uniformcartesian mesh, well at the center of the cell
)(wprp=−pqr=npKrq)(∇−=
wrxy >>∆=∆
<=>−=
w
w
rru
rrppu
0
ruK ∀=∆− 0
κκp
=
∇−=wrr
ww dsnpKq .
wp
1
2
$
0.' '
' =+∇− =
wqdsnpKκ κκσ
κκ
κ
κ)/ln(
2)( w
ww rr
Kq
prpπ
=−
wp
wq
wrr =
wn
rq
nrpKrq wr π2
).()( =∇−=
κ 'κwp
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Proof of Peaceman well index formula
===
−+∇−=∇−'
''
''
' .2
..κκσ
κκκκσ
κκκκσ
κκ πdsnn
rq
dsnuKdsnpK rw
κpκ κnnκ
4)(. '
'''
wquu
xxdsnpK +−≈∇−
=κκ
κκκκσκκ
σ
4))/ln(
2(0. '
'''
ww
ww
qrx
Kq
ppxx
dsnpK +
∆−−−≈∇−= π
σκ
κκκκσκκ
( )'''
'. κκκκκκσ
κκσ
ppxx
dsnpK −≈∇−=
" ( )ww
w rxK
qpp /)2/exp(ln
2∆−+= π
πκ
'κwp
'κκn
rnκ
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Vertical well with hydrostatic pressure drop
( ))1()()1()( 2/1 −−−−= − iZiZgipip iww ρ
220
0
14.0),()/ln(
))((2)( yxriH
rr
imKiWI
w
∆+∆==π
!*(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(1 5
6 !*
-
1
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Analysis of TPFA discretization
– Discrete norms: on each cell
– Discrete Poincaré Inequality
κuu h =2/1
22
= Κ∈
κκ
κ uulh
2/1
2
)()(
2'
' ')(
int
10
+−=
Σ∈Σ∈=σκ
σ σσκκκ
κκσ κκ
σσu
xxuu
xxu
boundhThh
10
2 )(hhlh uDu Ω≤
'κxκx
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Analysis of TPFA discretization
• A priori estimate:
210
)()( lhThh fDu
hΩ≤
( ) ( ) =
−+−
Σ∩∂∈= κκκ
κ κσκ
σκκκσκκ
κκκ κ
σσufu
xxuu
xxu
bound
0'
''
( )2/1
22/1
222
''
'
≤+− Σ∈=
κκ
κκσ
κσκκκσ
κκκκ
κκσσ
ufuxx
uuxx
bound
, (%(
Ω∂=Ω=∆−
suru
surfu
0
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Analysis of TPFA discretization• Error estimate κκκ uxue −= )(
0')('
'''
'
=
+−
κκκκκ
κκ
κκκκ
Reexx
dsnuxx
xuxuR '
''
'' .
'1)()(
κκκκκκ
κκκκ κκ ∇−−−=
)(, ''' hORRR =−= κκκκκκ
( ) κκκσ
κκκκ
κσ
fuuxx
=−= '
''
κκκσ
κκκκ
κ fdsnu =∇− = '
''
.
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Analysis of TPFA discretization• Error estimate κκκ uxue −= )(
0')('
'''
'
=
+−
κκκκκ
κκ
κκκκ
Reexx
)(, ''' hORRR =−= κκκκκκ
ChehThh ≤
)(10
( ) '''
'2
)(''1
0κκ
σκκ
κκκ
κκ κκκκ ReeRee
hThh −−=−=
hxxeCehh ThhThh
≤ ')(
2
)('1
010
κκσ
κκ
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49
TPFA discretization
• Discrete linear system:
– Coercivity:
– Symmetry:
– Monotonicity: ( Ah=M-Matrice)
hhh FUA =
Thh AA =
01 ≥−hA
2
)(min 10
),(hThhhhh uKUUA ≥
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50
M- Matrice monotonicity
01 ≥−A
>∃
≥≤> ≠
jji
jji
ijiii
Athatsuchi
A
AA
0
00,0
,
,
,,
0≥=+≠
iij
jijiii SUAUA
0min0
<= iii UUif
0
0
0000)( i
ijjijii
jji SUUAUA +−=
≠
" " "" #
0>
jijA
( )
( ) κκσ
σσκσσκ
κκσκκ
κκ
κσ
σ
fguxx
uuxx
bord
=−
+−
Σ∩∂∈
=
)()(
''
'
7 8 " %
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51
Finite volume schemes
• Parabolic Equations: time discretization– Implicit Euler integration in time– Stability analysis
![Page 52: Numerical simulation of two phase porous media flow models](https://reader033.vdocuments.us/reader033/viewer/2022051217/62789aa7bb672440004cd139/html5/thumbnails/52.jpg)
52
Parabolic model
Ω=×Ω∂=∇−
×Ω=∇−+∂
= onuu
TonnuK
TonfuKdivu
t
t
00
),0(0.),0()(
![Page 53: Numerical simulation of two phase porous media flow models](https://reader033.vdocuments.us/reader033/viewer/2022051217/62789aa7bb672440004cd139/html5/thumbnails/53.jpg)
53
Finite volume space and time discretizations
( )[ ] 0)(1
=−∇−∂ +n
n
t
t
t dxdtftuKdivuκ
0).()()()(1
''
1 =
∇−++−
+
=
+ dtdsntuKtfdxtudxtun
n
t
t
nn
κκσ σκκ
κ κ
)()( 1
1
+∆≈+
nt
t
ttYdttYn
n
/ $ "
)(tY
tttt nn ∆=−= +10 ,0
+ ∆
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54
Finite volume space and time discretizations
( ) κκκσ
κκκκκκ κκ fuuT
tuu nn
nn
=−+∆
−=
+++
'
1'
1'
1
![Page 55: Numerical simulation of two phase porous media flow models](https://reader033.vdocuments.us/reader033/viewer/2022051217/62789aa7bb672440004cd139/html5/thumbnails/55.jpg)
55
Stability analysis: discrete energyestimate
( ) ( )
=−+
∆−
=
+++
+κ
κκσκκκκ
κκ
κκ κκ fuuT
tuu
u nnnn
n
'
1'
1'
11
22
10
222
1
2121221
2
2
l
nhlh
h
nhl
nh
nhl
nhl
nh
uft
utuuuu
+
+++
∆
≤∆+−+−
222 )()(2 bababaa −+−=−
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56
Stability: discrete energy estimate
2212212222 lhl
nh
nhl
nhl
nh ftuuuu ∆≤−+− ++ γ
2202222 lh
N
lhl
Nh ftuu γ+≤
, L2
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57
Stability analysis: discrete maximum principle(f=0, zero flux BC)
nnn uuTt
Tt
u κκκκσ
κκκκσ
κκκ κκ+∆=
∆+ +
==
+ 1'
''
''
1 1
κκ allforMum n ≤≤
Then κκ allforMum n ≤≤ +1
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58
Stability analysis: discrete maximum principle(f=0, zero flux BC)
Muuif nn >= ++ 11 sup0 κ
κκProof:
lead to a contradiction
( ) ( )MuuuTt
Mu nnnn −+−∆=− ++
=
+ 00
0
00
11'
''
0
1κκκ
κκσκκκ κ
![Page 59: Numerical simulation of two phase porous media flow models](https://reader033.vdocuments.us/reader033/viewer/2022051217/62789aa7bb672440004cd139/html5/thumbnails/59.jpg)
59
Exercize: well test withcompressible Darcy single phase flow
• Parabolic equation(linearized)
Ω=
×Ω∂=∇−
×Ω∂=
×Ω=∇−+∂
= onpp
TongnpK
Tonpp
TonpK
divpdpd
t
N
DD
t
00
0
000
),0(.
),0(
),0(0)()1
(
µρ
µρρρ
ρ
$%
!
"!#
00 pp t ==