Numerical simulation of two -phase Darcy-Forchheimer flow

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




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


fnx and fnz profiles with time


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


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: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


x(m)

z(m

)

12 14 16 18 201

2

3

4


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!

Documents

Numerical simulation of two -phase Darcy-Forchheimer flow