Shuyu Sun
Earth 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:

t
    u   q m
• Volumetric/phase behaviors – from
thermodynamic modeling:

   (T , P )   ( P )
• Constitutive equation – Darcy’s law:

u
K

p
Incompressible Single Phase Flow
• Continuity equation


u

q
(
x
,
t
)



(
0
,
T
]
• Darcy’s law
K
u



p

(
x
,
t
)



(
0
,
T
]
• Boundary conditions:
p  pB
( x , t )   D  (0,T ]
u  n  uB
( x , t )   N  (0,T ]
DG scheme applied to flow equation
• Bilinear form
K
K
K
a
(
p
,)



p




{

p

n
}
[

]

s
{



n
}
[
p
]





 
K
K
[


p

n


s



n
p

p
]
[

]



1






h

s
 1
E
E

T
h
e
e
e

E
h
f
o
r
m
e
e

E
h
e
e
e
e


D
e
f
o
r
m
e
e


D
e
e

E
he
e
form
• Linear functional




0

 0

  0
 0


K
l
(
)

(
q
,)

s

n
p

u


form
e
B
B


e
e


D
• Scheme: seek
SIPG
OBB-DG, NIPG
IIPG
OBB-DG
NIPG
SIPG,IIPG
e
e


N
p

D
(
T
)
h
k h such
that
a
(
p
,
v
)

l
(
v
)
v

D
(
T
)
h
k
h
Transport in Porous Media
• Transport equation


c
*





u
c

D
(
u
)

c

qc

r
(
c
) (
x
,
t
)



(
0
,
T
]

t
• Boundary conditions
uc  D  c   n  c B u  n
t  (0,T ], x  in (t )
D  c   n  0
• Initial condition

c
(
x
,
0
)

c
(
x
)
0
t  (0,T ], x  out ( t)
x





• Dispersion/diffusion tensor




D
(
u
)

D
I

u
E
(
u
)

I

E
(
u
)
ml
t
DG scheme applied to transport equation
• Bilinear form






c
u

n
[

]

c
u

n


cq


[
c
][

]





 

h


B
(
c
,;
u
)

D
(
u
)

c

c
u



{
D
(
u
)

c

n
}[
]

s
{
D
(
u
)


n
}[
c
]



e
form
e



E
E

T
h
e
e

E
h
e
e

E
h

*
e
e

E
h
e
e
e

E
h
,
out
e

e
e
e

E
e
h
• Linear functional
L
(
;
u
,
c
)

c
q

c
u

n

r
(
M
(
c
))




• Scheme: seek c
(

,
t
)

D
(
T
) s.t.
h
r h
I.C. and
 


w
B
e

e
e

E
h
,
in


c
(
h,
)

B
(
c
,

;
M
(
u
))

L
(

;
M
(
u
))
h
h
h

t



D
(
T
)
t

(
0
,
T
]
r h
Example: importance of local conservation
Example: Comparison of DG and FVM
Upwind-FVM on 40 elements
Linear DG on 20 elements
Advection of an injected species from the left boundary under
constant Darcy velocity. Plots show concentration profile at 0.5 PVI.
Example: Comparison of DG and FVM
FVM
Linear DG
Advection of an injected species from the left. Plots
show concentration profiles at 3 years (0.6 PVI).
Example: flow/transport in fractured media
Locally refined mesh:
FEM and FVM are better than FD
for adaptive meshes and complex geometry
Example: flow/transport in fractured media
L2(L2) Error Estimators
Adaptive DG example
A posteriori error estimate
in the energy norm for all primal DGs



1
/
2
2



C

c

D
(
u
)

C

c

K

E
2
2

2
 
L
(
L
)
L
(
L
)
E

Ε
h 


DG
1
/
2


DG
 h R
 h
I L(L(E
R
B
1 L(L())
))
2
E
2
E
2
2
2
2
2
2


E



1
2
1
1
2
h
  R
2 2
2 2

R
B
1L
B
0L
(L())
(L())
2
2
h

E





E



1
2
1
2
h
 h
R
/
t L2(L2())
 2

R
B
0L

B
0
(L())
2
2

E




E



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.
Anisotropic mesh adaptation
Adaptive DG example (cont.)
L2(L2) Error Estimators on 3D
Adaptive DG example in 3D
T=1.5
T=2.0
T=0.1
T=0.5
T=1.0
Two-Phase Flow Governing Equations
• Mass Conservation
 

(p
S
)
p





u

0
,
p
p

t
p

n
,
w
• Darcy’s Law

k
rp


u


K

P

g

D
,
p

n
,
w
p
p p

p
• Capillary Pressure
P

P

P

P
(
S
)
c n w cw
• Saturation Summation Constraint
S

S

1
w
n
DG-MFEM IMPES Algorithm – Pressure Equ
• If incompressible (otherwise treating it with a source term):


S
p



u

0
,
p

t

u
0
t
p

n
,
w

• Total Velocity:


k
rp




,





u

u

u


K



K


w
n
t a
c t
w
n c
p  t

p





P

g

D
,p

n
,
w
ppp

• Pressure Equation:
i

1
i

1




K





u



K


t
w a
n c
i
i
i
• MFEM Scheme:
– Apply MFEM
– Two unknown variables: Velocity Ua and Water potential
DG-MFEM IMPES Algorithm – Saturation Equ
• Solve for the wetting (water) phase equation:

S
w


u

0
w

t

• Relate water phase velocity with total velocity:

u
fw
u
 wu
w
a
a

t
• Saturation Equation (if using Forward Euler):




S


fu
S
i

1
i w

t
i i
w
a
i
i w

t
• DG Scheme:
– Apply DG (integrating by parts and using upwind on element
interfaces) to the convection term.
Reservoir Description (cont.)
• Relative permeabilities (assuming zero residual
saturations):
m
m
k rw  S we , k rn  1  S we  , S we  S w , m  2
• Capillary pressure
p c ( S we )   B c log S we , S we  S w , B c  5 and 50 bars
K=100md
K=1md
Comparison: if ignore capillary pressure …
With nonzero
capPres
With zero capPres
Saturation at 10 years: Iter-DG-MFE
Saturation at 3 years
Notice that Sw is continuous within each rock,
but Sw is discontinuous across the two rocks
Iter-DG-MFE Simulation
Saturation at 5 years
Notice that Sw is continuous within each rock,
but Sw is discontinuous across the two rocks
Iter-DG-MFE Simulation
Saturation at 10 years
Notice that Sw is continuous within each rock,
but Sw is discontinuous across the two rocks
Iter-DG-MFE Simulation
Compositional Three-Phase Flow
• Mass Conservation (without molecular diffusion)
Ui 
 c x
i ,
u
  w ,o , g
• Darcy’s Law
u  
k r

K  P    g ,
28
  w, o, g
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
0
0
• k 0  0 . 3 ; k rg0  0 . 3 ; k row
; k rog
 0 .3

0
.
3
rw
29
Example (cont.)
MFE-dG 0.1 PVI.
MFE-dG 0.2 PVI.
30
MFE-dG 0.5 PVI.
Example 3 (cont.)
nC10 at 10% PVI CO2
nC10 at 200% PVI CO2
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
32
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
Gibbs Ensemble Monte Carlo simulation
E  E
I
I
N ,V
E
I
I
N ,V
E
I
II
I
I
E  E
II
II
N ,V
II
II
II
N ,V
II
E  E
E  E
I
II
I
II
N ,V V N ,V V
I
II
I
E  E
I
N  1,V
I
II
I
E  E
I
N 1,V
II
II
II
II
Three Monte Carlo movements in simulation
Particle displacements
Volume Change
Particle Transfer
Microstructure from the ab initio calculation
The microstructure of the molecular models form the ab initio calculation
Bond length(Å)
OH(H2O)
OH(H2O) 2
CH(C2H6)
CH(H2O----C2H6)
(H2O)

H
O
H

H
O
H
(H2O) 2
Angle(degree)
0.9619
0.9698
1.0938
1.0940

H
C
H
T-shaped pair of water
molecules
Hydrogen length(Å)
1.9321
105.06
105.28
107.5
The nearest neighbor
interaction between
the Water and Ethane
Water-ethane high pressure equilibria at T=523 K
Experimental data are from Chemie-Ing. Techn. (1967), 39, 816
EoS: Statistical-Associating-Fluid-Theory (SAFT)