Numerical simulation of two-phase Darcy-Forchheimer flow during CO2 injection into deep saline aquifers
Andi Zhang Feb. 4, 2013
Darcy flow VS non-Darcy flow Darcy flow A linear relationship between volumetric flow rate (Darcy
velocity) and pressure (or potential) gradient Dominant at low flow rates v
kµ
−∇Φ =
Non-Darcy flow Any deviations from the linear relation may be defined as
non-Darcy flow Interested in the nonlinear relationship that accounts for the
extra friction or inertial effects at high pressure gradients/ high velocity
is the flow potential;
μ is the viscosity;
v is the Darcy velocity;
k is the intrinsic permeability
Φ
Non-Darcy flow equations
is the non-Darcy flow coefficient, or Forchheimer coefficient
v+ v|v| k
β
µ βρ−∇Φ =
(Forchheimer, 1901)
is absolute Darcy permeability;
is the minimum permeability ratio at high flow rate;
is the the characteristic length
v (1 )
|v|
d
mr
mrd mr
k
k
kk k
τ
µµ τ
µτ ρ
−∇Φ = −
+ + (Baree and Conway, 2004 and 2007)
Forchheimer equation
Baree and Conway equation
Darcy-Forchheimer flow Darcy-Forchheimer flow is defined as the
flow incorporating the transition between Darcy and Forchheimer flows
Transition criteria The Reynolds number (Type-I) applied mainly in the cases where the representative particle diameter is available
The Forchheimer number (Type-II) used mainly in numerical models
Re dvρµ
= d is the diameter of particles
k vf ρ βµ
= consistent definition physical meaning of the variables
Research objectives Develop a generalized Darcy-Forchheimer model
Propose a method to determine the critical
Forchheimer number for single and multiphase flows
Use the model and method to analyze the Darcy-forchheimer flow in the near well-bore area during CO2 injection into DSA
Math model for two-phase flow
θ : porosity; Sα : saturation; Qα : source and sink.
( ) ( ) ; , S v Q w ntα α
α α αθρ ρ α∂
= −∇ + =∂
; , r
dp v v v w ndx k k
α α αα α α αα
µ β ρ α− = + =
; , rkkf v w nα
α α α αα
β ρ αµ
= =1 - ( ( )) ; ,
1r dpk kv w n
f dx
αα
αα α
αµ
= =+
( ) 1( ( )) ; , 1
rS k k p Q w nt f
αα α
α α αα α
θρ ρ αµ
∂= ∇ ∇ + =
∂ +
Constitutive equations needed
1 w nS S+ =
c n wP P P= −(1/ )
c D effP P S λ−=
1
rw w
eff r rw n
S SSS S−
=− −
r
(2 3 )/k ( )weffS λ λ+=
r
2 (2 3 )/k (1 ) (1 ( ) )neff effS S λ λ+= − −
cP : capillary pressure;
DP : entry pressure;
rwS : irreducible saturation for water;
nwS : irreducible saturation for non-wetting phase;
λ : pore size distribution index.
Brooks-Corey equations :
The Forchheimer number
( ) ( ) ( ) ( )0.5 0.55.5 5.5
0.005 0.005
(1 )n n n
r w r nkk S kk Sβ
θ θ= =
−
( Geertsma, 1974)
( ) ( )0.5 5.50.005
kβ
θ=
( )( )62.923*10
kτβ
θ
−
= ( Liu et al., 1995) Is tortuosity τ
( )( )62.923*10 ; ,
r
w nkkα α
τβ αθ
−
= = ( Ahmadi et al., 2010)
( )( ) ; ,
r
Cw n
kk Sβ
α αα
τβ α
θ= =
Evans and Evans (1988) :“a small mobile liquid saturation, such as that occurring in a gas well that also produces water, may increase the non-Darcy flow coefficient by nearly an order of magnitude over that of the dry case.”
Determine the critical αf
(Chilton et al., 1931)
Type I: based on Reynolds number for single phase
The point when the linear relationship begins to deviate
Type II: based on the Forchheimer number for single phase
(Green et al., 1951)
Determine the critical αf
The point when the linear relationship begins to deviate
Intersection of two regression lines
2
2
1 : v
1: +1v
dpDarcydx f
dpnon Darcydx f
βρ
βρ
− =
− − =
2
2
1 : v
1: +1v
dpDarcydx f
dpnon Darcydx f
α
α α α α
α
α α α α
β ρ
β ρ
− =
− − =
β is not defined in the Darcy formula!!!
The friction factor is defined as
2
dpdx vβρ
−
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150
2
4
6
8
10
12
14
16
18
Fric
tion
fact
or
1/fw
Forchheimer (data point)Darcy (data point)Forchheimer (regression)Darcy (regression)
An example plot for 0.95 water saturation
2
0.8953 1.1712
0.9969
y X
R
= +
=
Data from (Sobieski and Trykozko, 2012)
The two lines intercept where 1/fw=4.825, so ( )w cf is 1/4.825=0.207
2
1.138
0.9969
y X
R
=
=
Critical Forchheimer number for H2O and CO2 at different saturation values
0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.28
0.36
0.44
0.52
0.6
Crit
ical
For
chhe
imer
num
ber f
or w
ater
((f w
) c)
Water saturation (Sw)
0.4 0.5 0.6 0.7 0.8 0.9 12
6
10
14
18
22
26
30
Crit
ical
For
chhe
imer
num
ber f
or C
O2 ((
f n) c)
WaterCO2
Numerical model
Primary variables: Pw and Sn
Fully-implicit scheme
Discretization method: CVFD Control volume finite difference
Control volume finite difference
( )
1 1 1 1 11 1
1 1 1 1 11 1
1 1 1 11 1
1 12 2
( ) ( )
( ) ( )
( ) ( )
n
n
tt t t t t t w
wi wi wi wi nw
t t t t twi wi wi wi
t t t tc cni ni ni ni
n ni i
QS a p p b p p S t
S c p p d p p
p pc S S d S SS S
θ θρ
θ
+ + + + ++ −
+ + + + ++ −
+ + + ++ −
+ −
+ − − − = − ∆
− − − −
∂ ∂− − − − ∂ ∂
tt nn
n
QS tθρ
= + ∆
2 21 12 2
2 21 12 2
1 1 ( ) 1 ( ) 1
1 1 ( ) 1 ( ) 1
w wr r
w w w wi i
n nr r
n n n ni i
kk kkt ta bx f x f
kk kkt tc dx f x f
µ µ
µ µ
+ −
+ −
∆ ∆= = ∆ + ∆ +
∆ ∆= = ∆ + ∆ +
1 1 1 1 1 11 1x = , , , ,
Tt t t t t tn ni nm w wi wmS S S p p p+ + + + + + B=AX
For 1D case, the two-phase mass conservation equations can be discretized into
Numerical algorithm
t < t_max NO
Stop
Initialization
Output
YES
Update coefficients
Solve X
|(X-X_o)/X| > tol NO
Update Variables
YES
X_o = X
Update coefficients
Solve X
t = t+dt
Output
X_o = X
mb check
For each iteration of each time step, X and other related variables in the last time step are used to update all the coefficients including a, b, c, d, dPc/dSn, fw, fn and all the elements in the right hand side B;
The right hand side term B is based on the variables in the last time step and don’t need to be updated except for the first iterative step;
Mass balance check 1
1
,1
0
1
,1 1
; , [ ]
( )
[ ]
(
) ; ,
mt t
i i ii
mt t
t i l ii l
mt
i i ii
t mj j
j i l ij i l
V S SI
t Q q
V S SC
t Q q
w n
w n
α α
α
α α
α α
α
α α
α
α
θ
θ
−
=
= ∈Γ
=
= = ∈Γ
−=
∆ +
+=
∆
=
−=
∑
∑ ∑
∑
∑ ∑ ∑
m is the number of the nodes; t is the number of time steps; Q is discharge for pumping or injecting wells; q is the flow rate through the boundaries;
Γ is the boundaries of the domain For Q and q, they are set to be positive if entering the domain while negative if leaving the domain.
Darcy-Forchheimer flow
( )( )
0 , if Darcy flow
, if Forchheimer flow c
c
f ff
f f fα α
αα α α
<= >
Validation: Buckley-Leverett problem with inertial effect
Both fluids and the porous medium are incompressible; Capillary pressure gradient is negligible; Gravity effect is negligible; Semi-analytical solution with inertial effect (Wu, 2001; Ahmadi et al., 2010)
Water flooding No flow boundary
BC type I for Pw BC type I for Sn
No flow boundary
No flow boundary
x
Comparison of saturation profiles
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Sw distribution
Distance(m)
Wat
er s
atur
atio
n
Analytical: 3000 secAnalytical: 7000 secAnalytical: 10000 secNumerical: 3000 secNumerical: 7000 secNumerical: 10000 sec
Application problem
BC type I for Pw BC type I for Sn
BC type I for Pw BC type I for Sn
BC type I for Pw BC type I for Sn
z
16 17 18
16 17
16
Row 1 1
Row 2 1
Row 15
Interface 16.5
Interface 18.5
Interface 17.5
15
15
14
I 3.5
I 1.5
I 2.5
1
16
31
x Injecting CO2
No flow boundary
Properties of soil and fluids Properties Values Comment
Soil
Soil intrinsic permeability porosity
3e-9 m2
0.37
Pore size distribution index 3.86 Brook-Corey
Water residual saturation Swr = 0.35
Non-wetting phase (NWP) residual saturation
Snr = 0.05
Fluid
Water density 994 kg/m3
NWP density 479 kg/m3
Water viscosity 7.43e-4 Pa s
NWP viscosity 3.95 e-5 Pa s
Modeling parameters Properties Values Comment
Boundary condition
Water pressure at x=0.5 m Water pressure at x=31.5 m Water pressure at z=0.5 m Water pressure at z=15.5 m
Pw = 8 M Pa, BC Type I Pw = 8 M Pa, BC Type I No flow boundary Pw = 8 M Pa, BC Type I
Left boundary Right boundary Bottom boundary Top boundary
CO2 saturation at x=0.5 m CO2 saturation at x=31.5 m CO2 saturation at z=0.5 m CO2 saturation at z=15.5 m CO2 injecting rate @ (16,1)
Sn = 0.1, BC Type I Sn = 0.1, BC Type I No flow boundary Sn = 0.1, BC Type I 1*10-3 m3/s
Per meter normal to the 2D domain
Initial condition
Water saturation NWP saturation Water pressure
Sw = 0.9 Sn = 0.1 Pw = 8 M Pa
Saturated with water initially
Space discretization Time discretization
Domain size, Length Domain size, Depth Domain size, Width Space step size
L=31 m W=15 m 1 m dx =dz=1 m
Simulation time Time step size
T= 9000 s dt=1 s
CO2 saturation and pressure profiles
Darcy-Forchheimer flow
fnx and fnz profiles with time
Darcy-Forchheimer flow
The evolution of important variables in the first row
0 2000 4000 6000 8000 100000
5
10
15
20
25
30
35fnx history
t(s)
fnx
Interface 16.5Interface 17.5Interface 18.5
0 2000 4000 6000 8000 100000.1
0.2
0.3
0.4
0.5
0.6Sn history
t(s)
Sn
Node 16Node 17Node 18
0 2000 4000 6000 8000 100008.01
8.012
8.014
8.016
8.018
8.02 x 106 Pn history
t(s)
Pn
Node 16Node 17Node 18
0 2000 4000 6000 8000 100000
0.5
1
1.5
2
2.5
3
3.5 x 10-3 vnx history
t(s)
vnx
Interface 16.5Interface 17.5Interface 18.5
Darcy-Forchheimer flow
CO2 saturation and pressure profiles
Darcy flow
Comparison of the evolution of important variables
0 2000 4000 6000 8000 100000.1
0.2
0.3
0.4
0.5
0.6Sn history
t(s)
Sn
0 2000 4000 6000 8000 100008.01
8.012
8.014
8.016
8.018
8.02 x 106 Pn history
t(s)
Pn
0 2000 4000 6000 8000 100001
1.2
1.4
1.6
1.8 x 104 Pc history
t(s)
Pc
0 2000 4000 6000 8000 100000
0.005
0.01
0.015
0.02
0.025
0.03vnx history
t(s)
vnx
D:N16D:N17F:N16F:N17
D:N16D:N17F:N16F:N17
D:N16D:N17F:N16F:N17
D:I16.5D:I17.5F:I16.5F:I17.5
CO2 saturation contour for Darcy-Forchheimer flow
0.2
0.2
0.2
0.2
0.20.4
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.2 0.2
0.4
0.40.
4
0.4
0.40.
4
0.4
0.4
Sn distribution
x(m)
z(m
)
11 12 13 14 15 16 17 18 19 20 211
1.5
2
2.5
3
3.5
4
4.5
5
Time 1000 sec (solid line), 3000 sec (dot line), 5000 sec (dash line), 7000 sec (bold solid line), 9000 sec (dash dot line)
0.2
0.2
0.2 0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.20.
2 0.2
0.40.4
0.4
0.4
Sn distribution
x(m)
z(m
)
11 12 13 14 15 16 17 18 19 20 211
1.5
2
2.5
3
3.5
4
4.5
5
CO2 saturation contour for Darcy flow
Time 1000 sec (solid line), 3000 sec (dot line), 5000 sec (dash line), 7000 sec (bold solid line), 9000 sec (dash dot line)
Overlapping them together
0.2
0.2
0.2
0.2
0.20.4
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.2 0.2
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
Sn distribution
x(m)
z(m
)
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2 0.2
0.40.4
0.4
0.4
11 12 13 14 15 16 17 18 19 20 211
1.5
2
2.5
3
3.5
4
4.5
5
Time 1000 sec (solid line), 3000 sec (dot line), 5000 sec (dash line), 7000 sec (bold solid line), 9000 sec (dash dot line); Darcy in red; Darcy-Forchheimer in black
Match Sn with Forchheimer flow regime
The white contour lines are for 0.4 saturation contour lines while the rectangles demonstrate
whether a node is of Forchheimer (grey rectangle) or Darcy (white rectangle) flow
12 14 16 18 201
2
3
4
5
x(m)
z(m
)
Sn and fnz distribution @1000 sec
12 14 16 18 201
2
3
4
5Sn and fnz distribution @3000 sec
x(m)
z(m
)
12 14 16 18 201
2
3
4
5Sn and fnz distribution @5000 sec
x(m)
z(m
)
12 14 16 18 201
2
3
4
5Sn and fnz distribution @7000 sec
x(m)
z(m
)
Comparison of CO2 pressure between Forchheimer-Darcy and Darcy flow
0 5000 100000
1000
2000
3000
4000
5000Pn history for node(16,1)
t(s)
Pn
diffe
renc
e (F
orch
heim
er-D
arcy
)
0 5000 100001
1.0001
1.0002
1.0003
1.0004
1.0005
1.0006Pn history for node(16,1)
t(s)
Pn
ratio
(For
chhe
imer
/Dar
cy)
Higher displacement efficiency In the Forchheimer regime for Darcy-Firchheimer
flow, the total CO2 saturation is 4.5916 for the nine nodes at 9100 sec.
For Darcy flow, the total CO2 saturation is 3.4072 for the same nine nodes at 9100 sec.
The displacement is 34.76% higher for Forchheimer flow than Darcy flow at 9100 sec.
Implications Important to incorporate Forchheimer effect into the
numerical simulation of multiphase flow Crucial to determine the critical Forchheimer number
and to decide the extent to which Forchheimer effect can influence the transport of CO2 in deep saline aquifers.
The higher displacement efficiency by CO2 is good news for CO2 sequestration into deep saline aquifers.
The higher injection pressure required in Forchheimer flow is bad news for CO2 sequestration.
Summary & conclusions Darcy flow is a special case of a generalized Darcy-
Forchheimer flow; Since both the Forchheimer coefficient and number
are functions of saturation, there is a critical Forchheimer number for transition for a specific saturation for each phase in multiphase flow;
The good agreement between the numerical solution and the semi-analytical solution validates the numerical tool developed in this study
Summary & conclusions The Forchheimer flow can improve the displacement
efficiency and can increase the storage capacity for the same injection rate and volume of site.
The higher injection pressure required in Forchheimer flow is bad news for CO2 sequestration because the pressure will continue to increase and might even exceed the litho-static stress and the risk for fracturing the porous media would increase.
Thank you!