Frontiers and Future of Multiphase Fluid Flow Modeling in Oil Reservoirs
description
Transcript of Frontiers and Future of Multiphase Fluid Flow Modeling in Oil Reservoirs
Shuyu SunEarth Science and Engineering program, Division of PSE, KAUST
Applied Mathematics & Computational Science program, MCSE, KAUST
Acknowledge: Mary F. Wheeler, The University of Texas at Austin
Abbas Firoozabadi, Yale & RERI; Joachim Moortgat, RERI
Mohamed ElAmin, Chuanxiu Xu and Jisheng Kou, KAUST
Presented at 2010 CSIM meeting, KAUST Bldg. 2 (West), Level 5, Rm 5220, 11:10-11:35, May 1, 2010.
Energy and Environment Problems
Single-Phase Flow in Porous Media
• Continuity equation – from mass conservation:
• Volumetric/phase behaviors – from thermodynamic modeling:
• Constitutive equation – Darcy’s law:
t
u qm
(T,P) (P)
u K
p
Incompressible Single Phase Flow
• Continuity equation
• Darcy’s law
• Boundary conditions:
],0(),( Ttxq u
],0(),( Ttxp K
u
p pB (x, t) D (0,T]
un uB (x, t) N (0,T]
DG scheme applied to flow equation
• Bilinear form
• Linear functional
• Scheme: seek such that
form
form
( , ) { }[ ] { }[ ]
[ ][ ]
h h h
D D h
e eE e eE T e E e E
ee ee e e
e e e E e
K K Ka p p p s p
K Kp s p p
h
n n
n n
ND e
e Be
e Be upK
sql
nform),()(
)( hkh TDp
)()(),( hkh TDvvlvpa
IIPG SIPG,0
NIPG0
DG-OBB0
IIPG0
NIPG DG,-OBB1
SIPG1
form
s
Transport in Porous Media
• Transport equation
• Boundary conditions
• Initial condition
• Dispersion/diffusion tensor
],0(),()()( * Ttxcrqccct
c
uDu
uc Dc n cBun t (0,T], x in (t)
Dc n 0 t (0,T], x out (t)
xxcxc )()0,( 0
)()()( uEIuEuIuD tlmD
DG scheme applied to transport equation
• Bilinear form
• Linear functional
• Scheme: seek s.t. I.C. and
houthh
hhh
Eee
e
e
Eee e
Eee e
Eee e
Eee e
TEE
ch
cqcc
csccccB
]][[][
]}[)({]}[)({)();,(
,
*
form
nunu
nuDnuDuuDu
))((),;(,
cMrcqccLinhEe
e eBw nuu
],0()(
))(;())(;,(),(
TtTD
MLMcBt
c
hr
hhhh
uu
)(),( hrh TDtc
Example: importance of local conservation
Example: Comparison of DG and FVM
Advection of an injected species from the left boundary under constant Darcy velocity. Plots show concentration profile at 0.5 PVI.
Upwind-FVM on 40 elements Linear DG on 20 elements
Example: Comparison of DG and FVM
Advection of an injected species from the left. Plots show concentration profiles at 3 years (0.6 PVI).
FVM Linear DG
Example: flow/transport in fractured media
Locally refined mesh:
FEM and FVM are better than FDfor adaptive meshes and complex geometry
Example: flow/transport in fractured media
Adaptive DG example
L2(
L2)
Err
or E
stim
ator
s
A posteriori error estimate in the energy norm for all primal DGs
2/1
2
)(
2/1
)(222
)(
hEELL
DG
LL
DG KcCcCΕ
uD
ELLB
ELLB
ELLB
ELLB
ELLBELLIEE
tRhRh
Rh
Rh
RhRh
2
))((0
2
))((0
2
))((0
2
))((1
2
))((1
2
))((
22
222
2222
2222
/2
1
2
1
1
2
1
2
1
Proof Sketch: Relation of DG and CG spaces through jump terms
S. Sun and M. F. Wheeler, Journal of Scientific Computing, 22(1), 501-530, 2005.
Adaptive DG example (cont.)
Ani
sotr
opic
mes
h ad
apta
tion
Adaptive DG example in 3DL
2(L
2) E
rror
Est
imat
ors
on 3
D
T=1.5
T=2.0
T=0.1 T=0.5 T=1.0
Two-Phase Flow Governing Equations
• Mass Conservation
• Darcy’s Law
• Capillary Pressure
• Saturation Summation Constraint
)( wcwnc SPPPP
wnpDgPk
ppp
rpp ,,
Ku
1 nw SS
wnpt
Spp
pp ,,0)(
u
DG-MFEM IMPES Algorithm – Pressure Equ
• If incompressible (otherwise treating it with a source term):
• Total Velocity:
• Pressure Equation:
• MFEM Scheme: – Apply MFEM – Two unknown variables: Velocity Ua and Water potential
wnpt
Sp
p ,,0
u
0 tu
cnwtcat KKuuu
nwt
p
rpp
k
,
ic
in
ia
iw
it KuK 11
wnpDgP ppp ,,
DG-MFEM IMPES Algorithm – Saturation Equ
• Solve for the wetting (water) phase equation:
• Relate water phase velocity with total velocity:
• Saturation Equation (if using Forward Euler):
• DG Scheme: – Apply DG (integrating by parts and using upwind on element
interfaces) to the convection term.
0
w
w
t
Su
at
waww f uuu
iwi
ia
iw
iwi
St
fSt
u1
• Relative permeabilities (assuming zero residual saturations):
• Capillary pressure
Reservoir Description (cont.)
2,,1, mSSSkSk wwem
wernmwerw
bars50 and5,,log)( cwwewecwec BSSSBSp
K=100md
K=1md
Comparison: if ignore capillary pressure …
Saturation at 10 years: Iter-DG-MFE
With nonzero capPres
With zero capPres
Saturation at 3 years
Iter-DG-MFE Simulation
Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks
Saturation at 5 years
Iter-DG-MFE Simulation
Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks
Saturation at 10 years
Iter-DG-MFE Simulation
Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks
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Compositional Three-Phase Flow
• Mass Conservation (without molecular diffusion)
• Darcy’s Law
gowPkr ,,,
gKu
gow
ii xc,,
,
uU
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Example of CO2 injection
• Initial Conditions: C10+H2O(Sw=Swc=0.1), 100 bar,160 F.
• Inject water (0.1 PV/year) to 2 PV, then inject CO2 to 8 PV. Poutlet= 100 bar
• Relative permeabilities:
– Quadratic forms except nw=3.
– Residual/critical saturations:
• Sor = 0.40; Swc = 0.10; Sgc = 0.02
• Sgmax = 0.8; Somin = 0.2
• ; ; ;3.00
rwk 3.00rgk 3.00
rowk 3.00rogk
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Example (cont.)
MFE-dG 0.1 PVI. MFE-dG 0.2 PVI. MFE-dG 0.5 PVI.
Example 3 (cont.)
nC10 at 10% PVI CO2 nC10 at 200% PVI CO2
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Remarks for Multiphase Flow
• Framework has been established for advancing dG-MFE scheme for three-phase compositional modeling. In our formulation we adopt the total volume flux approach for the MFE.
• dG has small numerical diffusion
• CO2 injection – Swelling effect and vaporization– Reduction of viscosity in oil phase – Recovery by CO2 injection > Recovery by
C1 > Recovery by N2
EOS Modeling of Phase Behaviors
• PVT modeling: EOS– Peng-Robinson EOS – Cubic-plus-association EOS
• Thermodynamic theory• Stability calculation
– Tangent Phase Distance (TPD) analysis– Gibbs Free Energy Surface analysis
• Flash calculation– Bisection method (Rachford-Rice equation) – Successive Substitution – Newton’s method
,
I
I I
E
N V
,
II
II II
E
N V
,
I I
I I
E E
N V
,
II II
II II
E E
N V
,
I I
I I
E E
N V V
,
II II
II II
E E
N V V
1,
I I
I I
E E
N V
1,
II II
II II
E E
N V
Gibbs Ensemble Monte Carlo simulation
Particle displacements
Volume Change
Particle Transfer
Three Monte Carlo movements in simulation
The microstructure of the molecular models form the ab initio calculation
T-shaped pair of water molecules
The nearest neighbor interaction between the Water and Ethane
Microstructure from the ab initio calculation
Bond length(Å) Angle(degree) Hydrogen length(Å)OH(H2O) 0.9619OH(H2O) 2 0.9698 1.9321CH(C2H6) 1.0938CH(H2O----C2H6) 1.0940 (H2O) 105.06 (H2O) 2 105.28
107.5
HOH
HCHHOH
Water-ethane high pressure equilibria at T=523 K
EoS: Statistical-Associating-Fluid-Theory (SAFT)
Experimental data are from Chemie-Ing. Techn. (1967), 39, 816