GWII 9 Multiphase Flow

8/9/2019 GWII 9 Multiphase Flow

1/39

FS 2011 Groundwater II Multiphase flow 1

Mehrphasenströmungen

Multiphase Flow

Fritz Stauffer,

Institute of Environmental Engineering, ETH Zürich

Groundwater II


2/39


Capillary zone

unsaturated zone

Water

pressure

θ

z z

Capillary

zoneVol. water content

Piezometer

Groundwater table

Saturated zone

p


3/39


Two-phase flowAt the same time:

• Interconnected water phase

• Interconnected air phase

Volumetric water content θ w [L3/L3]:

Volume of water per unit volume of porous medium

θ w:≤ n n: Porosity [-]


4/39


Darcy law for the water phase• Index w

• For constant water density ρ w

( ) ww w ww

pS z g ρ

⎛ ⎞= − ∇ +⎜ ⎟⎝ ⎠

v K

vw: Specific flux of water [L/T]

pw: Water pressure [M L-1 T-2]

Kw: Hydraulic conductivity of water, Kw(S w) [L/T]S w: Saturation of water phase S w=θ w/n [-]


5/39


Darcy law for the air phase• Index a

• For constant air density ρa

( ) aa a a

a

pS z

g ρ

⎛ ⎞= − ∇ +

⎜ ⎟⎝ ⎠v K

va: spezific air flux of [L/T]

pa: Air pressure [M L-1 T-2]

Ka: Conductivity for air, Ka(S a) [L/T]

S a: Saturation of air phase S a=θ a/n [-]

Condition:

S w+ S a=1


6/39


Generalized Darcy law

• For variable density of water and air

[ ]

[ ]

( )

( )

w w

w w ww

a aa a a

a

S p

S p

ρ μ

ρ μ

= −∇ −

= −∇ −

kv g

kv g

k: Permeability [L

2

]μ : Dynamic viscosity [M T-1 L-1]


7/39


Mass balance for water and

air phase

• Without considering mass exchange between phases• Without sources and sinks

( ) ( )

( ) ( )

w w

w w

a a

a a

n S

t

n S

t

ρ ρ

ρ ρ

∂∇ ⋅ = −∂

∂∇ ⋅ = −∂

v

v

t : Time [T]


8/39


Mass balance for water and

air phase

• Darcy law inserted:

( )

( )

w ww w

w

a aa a

a

p S S z n

g t

p S S z n

g t

ρ

ρ

⎛ ⎞⎛ ⎞ ∂∇ ⋅ ∇ + =⎜ ⎟⎜ ⎟⎜ ⎟

∂⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ∂

∇ ⋅ ∇ + =⎜ ⎟⎜ ⎟⎜ ⎟ ∂⎝ ⎠⎝ ⎠

K

K

Non-linear diffential equation of second order

4 variables: S w, S a, pw, pa; 3 equations:1 relation needed: ⇒S w( pw, pa)

1w a

S S + =


9/39


If influence of air phase on water

flow is disregarded

• No friction losses in air flow considered • pa=0

( ) w ww w

w

p S S z ng t ρ

⎛ ⎞⎛ ⎞ ∂∇ ⋅ ∇ + =⎜ ⎟⎜ ⎟⎜ ⎟ ∂⎝ ⎠⎝ ⎠K

Richards equation:

( )( ) ( )w w ww w w ww w

p S pS p z n pg p t ρ

⎛ ⎞⎛ ⎞ ∂ ∂∇ ⋅ ∇ + =⎜ ⎟⎜ ⎟⎜ ⎟ ∂ ∂⎝ ⎠⎝ ⎠

K


10/39


Water retention curve• Assume: Water is wetting phase towards

solid material (controlled by wetting angle)• Interface water-air is curved

• Local radius of curvature depends on pressure difference pc at interface forhydrostatic conditions due to mechanical

equilibrium• pc: capillary pressure

= -c a w p p p

a

w


11/39


Water retention curve• Discontinuity of pressure at the interface

• Pressure on concave side is larger than on convex side

1 2

1 1 = -c a w wa p p p

R Rσ

⎛ ⎞= ⋅ +⎜ ⎟

⎝ ⎠ R1, R2: Principal radii of curvature (orthogonal sections)

σ wa: Interfacial tension (0.0729 N/m for water-air)

R1 R2

R1

Laplace equation of capillarity:


12/39


• Consider one capillary• Assume spherical interface

Capillary pressure-pore radius

Concept

2coswa

c p

r

σ α =

pc/( ρ wg)

α : Wetting angle

a

w α

r


13/39


Water retention curve

• Saturation S w is essentially a function ofcapillary pressure pc

• S w

( pc

) is to be determined experimentally

in general

• Usually it is assumed that S w( pc) is identical

for hydrostatic conditions and for steady-state and transient flow conditions


14/39


Water

retention

curve• Sand packing

• Different curves fordrainage undimbibition

• Hysteresis effect!

0

10

20

30

40

50

60

70

80

90

0 0.5 1

Sw

p c ( c m )

Drainage

Imbibition

h

c


15/39


Water retention curve: Models

• Approach of Brooks und Corey (1966):

,

,

,

,

;

1

1; 0

w w r bw e c b

w r c

w e c b

S S pS p p

S p

S p p

λ − ⎛ ⎞

= = ≥⎜ ⎟− ⎝ ⎠

= ≤ ≤

S w,e: Effective saturation [-]S w,r : Residual saturation [-]

pb: Air-entry capillary press. [M L-1

T-2

]λ : Pore distribution index [-]

pc

S wS w,r

pb

0


16/39


Water retention curve: Models

• Approach of van Genuchten (1980):

α , n und m : Parameters

Usually: m=1-1/n

,

,

,

1 ; 01

1

m

w w r

w e cn

w r c

w

S S S pS p

gα

ρ

⎛ ⎞⎜ ⎟

− ⎜ ⎟= = ≥⎜ ⎟− ⎛ ⎞⎜ ⎟+ ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

pc

S wS w,r 0


17/39


Hysteresis in Water retention curve

• Relation S w( pc) is not unique.

• Dependent on history of imbibition – and/ordrainage cycles.

• Single pore can exhibit same capillary pressure

for water filled and dry conditions. The watercontent can be different for same capillary

pressure.

• During imbibition air bubbles can be trapped

(insular air).

• For drainage pores can remain water saturated.

r

ρ


18/39


Hysteresis in Water retention curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1

S

h c

( m ) 1. Drainage

2. Drainage

Imbibition

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1

S

h c

( m )

• Sand packing

• Incl. primarywetting curves


19/39


Hydraulic conductivity

• Approach of Brooks und Corey (1966):

, ,( ) ; =3+2/w w w sat w eK S K S ε ε λ =

• Approach of van Genuchten (1980):

( )

21/ 2 1/

, , ,( ) 1 1 ; =1-1/

mm

w w w sat w e w eK S K S S m n⎡ ⎤

= − −⎢ ⎥⎣ ⎦

Hysteresis effect in K w(S w): exists, but it is relatively

small. However it may be important in K w( pc).


20/39


Hydraulic conductivity

• Sand packing

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Sw

K r

1. Drainage

2. Drainage

ImbibitionK r =K (S w)/K sat

• Relative

hydraulicconductivity


21/39


Numerical solution

• Finite difference method of FD

• Finite element method FE

• Finite volume method FV

( )( ( )w ww w ww w

p pS K p z n p

g p t ρ

⎛ ⎞ ∂∂∇ ⋅ ∇ + =⎜ ⎟ ∂ ∂

⎝ ⎠

Differential equation

for unsaturated flow:

Parameters: ( )( ) ); ; ; 0w w c w c c a w aK S p S p p p p p= − =

( )w ww

S p p

∂∂

( )( )w w cK S p


22/39


Numerical solution

• Initial condition

• Boundary conditions

• System of linear equations

• Solve linear equation system


23/39


Numerical solution

0

1

2

3

4

5

6

7

8

9

10

0.0 0.5 1.0

S

z

'

t'=1

t'=10

t'=20

t'=30

t'=40

t'=50

• Example Infiltration:

Initial condition: hydrostatic

Length of column =10 hb

hb = pb / ρ wg

Lower boundary impermeable

Infiltration rate N =0.1 K sat

Brooks-Corey-Par.: λ =2, S r =0, S max=1Result S ( z,t ) dimensionless:

'b

z

z h= ' sat

b

t K t nh= n: Porosity


24/39


Numerical solution

Infiltration into

layered sand packing:Inf.-rate = 0.082 mm/s

K sat,fine = 0.23 mm/s

K sat,coarse = 0.73 mm/s

Stauffer and Dracos, 1986

I filt ti f t ft 10 i I filt ti f t ft 30 i


25/39


Infiltration front after 10 min Infiltration front after 30 min

Stauffer and Dracos, 1986


26/39


Multiphase flow

• Existence of several non-mixing fluid phases

• Ex.: Water – air – mineral oil

• Solid in contact with two fluids:

Fluid 1

Fluid 2

fest

α σ s,1 σ s,2σ 1,2

Mechanical equilibrium, if: σ 1,2 cos α = σ s,1 - σ s,2 Young’s law

σ 1,2

: Interfacial tension between fluids 1 und 2

σ s,1: Interfacial tension between solid and fluid 1


27/39


Multiphase flow

• No equilibrium, if: σ 1,2 cos α


28/39


Wetting hierarchy

• If several fluid phases are present

• One phase is wetting, one phase is non-wetting, the

remaining phases are ambivalent, wetting-non-wetting

Ex.: For mineral solid (e.g., quartz sand) water is wetting, air

is non-wetting, oil is wetting if water is absent and oil is non-

wetting if air is absent.

a) b) c)

Mineral

surface

Organic

surface


29/39


Multiphase flow

• A fluid phase gets immobile, if thesaturation is smaller than the residual

saturation

• The residual saturation of mineral oilmay be very small, if water and air are

present in the pore.

• Flux equations and mass balance equations are similar to

equations for two phase flow.


30/39


Multiphase flow

• Relation between capillary pressure and fluid content is

analogue to two-phase flow.1,2

1,2,

2

cosc s p ar

σ

=

S oS w

pc= pa- po

0

pc= po - pw

0

o

w

a

o p

bwo pboa

S w

pc= pa - pw

0

a

w p

bwa


31/39


Static distribution of light fluid

• ρ a


32/39


Rough estimate of mineral oil

migration

R0

hcwa

hs

Groundwater

Capillary fringe

H 1

S o1 z

Oil spill

Phase 1: Cylindrical oil spill, essentially vertical migration

( )

( )

1

1

1 ,max

,max

( )

/ 2 ln 1

/ 2

o

o o

s w coa

s w coa

n S t zK S

z z h h

h h

= ⋅

⎡ ⎤⎛ ⎞⋅ − + +⎢ ⎥⎜ ⎟

⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦


33/39



migrationPhase 2: Mobile oil plug leaves behind practically immobile trace

R0hcwa

Groundwater

Capillar fringe

H 1S o1 z

S ro1

mobile

immobile


34/39



migration

hcwa

Groundwater

Capillary fringe

H 1S ro1

S o1 H

Phase 3: Mobile oil plug reaches capillary fringe

( )

112

1 1 0 1 1

o ro

o ro o ro

V S H H

n S S R S S π

= −

− −


35/39



migration

Phase 4: Mobile oil plug sinks into groundwater (swim condition)

hcwa

H 1 S ro1

S o1

S o3 zmax

H D

max3

1 1

o d coa o cwa w

ow o o

o ro

H h h z

S

S S

ρ ρ ρ

ρ ρ ρ

− +=− +

−


36/39



migrationPhase 5: Mainly radial migration within capillary fringe.

Sinking oil goes up and leaves behind immobile trace

S ro1

S

o1S ro3

R0 Rmax

d hcoaS o2S o2

S o2 w

hcoa

zmax

2 2 1 1 max 3

max 02

1 o d coa o ro

w coa ro

HS h S z S

R R h S

⎛ ⎞− −

= ⋅ +⎜ ⎟⎝ ⎠



37/39


g

migrationPhase 6: Migration within capillary fringe in the flow direction of

groundwater until all mineral oil is immobile (slow process)

S ro1

So1

S ro3 zmax

R0 Rmax

S ro2

L( y)

y Rmax

whcoa

( ) 2 22 2 max2

2( )

für < y

o ro

ro

0 max

S S R y L y

S

R R

⋅ − ⋅ −=

≤

Additional:

Influence of water

table fluctuations


38/39


Static distribution of heavy

fluid

• ρ a


39/39


Infiltration of dense fluid

Chlorinated hydrocarbon

Migration in groundwater is highly influenced by heterogeneities

GWII 9 Multiphase Flow

Documents

Transcript of GWII 9 Multiphase Flow