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8/9/2019 GWII 9 Multiphase Flow
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FS 2011 Groundwater II Multiphase flow 1
Mehrphasenströmungen
Multiphase Flow
Fritz Stauffer,
Institute of Environmental Engineering, ETH Zürich
Groundwater II
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FS 2011 Groundwater II Multiphase flow 2
Capillary zone
unsaturated zone
Water
pressure
θ
z z
Capillary
zoneVol. water content
Piezometer
Groundwater table
Saturated zone
p
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FS 2011 Groundwater II Multiphase flow 3
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 [-]
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FS 2011 Groundwater II Multiphase flow 4
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 [-]
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FS 2011 Groundwater II Multiphase flow 5
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
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FS 2011 Groundwater II Multiphase flow 6
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]
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FS 2011 Groundwater II Multiphase flow 7
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]
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FS 2011 Groundwater II Multiphase flow 8
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 + =
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FS 2011 Groundwater II Multiphase flow 9
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
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FS 2011 Groundwater II Multiphase flow 10
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
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FS 2011 Groundwater II Multiphase flow 11
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:
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FS 2011 Groundwater II Multiphase flow 12
• 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
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FS 2011 Groundwater II Multiphase flow 13
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
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FS 2011 Groundwater II Multiphase flow 14
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
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FS 2011 Groundwater II Multiphase flow 15
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
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FS 2011 Groundwater II Multiphase flow 16
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
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FS 2011 Groundwater II Multiphase flow 17
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
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FS 2011 Groundwater II Multiphase flow 18
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
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FS 2011 Groundwater II Multiphase flow 19
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).
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FS 2011 Groundwater II Multiphase flow 20
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
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FS 2011 Groundwater II Multiphase flow 21
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
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FS 2011 Groundwater II Multiphase flow 22
Numerical solution
• Initial condition
• Boundary conditions
• System of linear equations
• Solve linear equation system
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FS 2011 Groundwater II Multiphase flow 23
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
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FS 2011 Groundwater II Multiphase flow 24
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
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FS 2011 Groundwater II Multiphase flow 25
Infiltration front after 10 min Infiltration front after 30 min
Stauffer and Dracos, 1986
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FS 2011 Groundwater II Multiphase flow 26
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
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FS 2011 Groundwater II Multiphase flow 27
Multiphase flow
• No equilibrium, if: σ 1,2 cos α
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FS 2011 Groundwater II Multiphase flow 28
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
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FS 2011 Groundwater II Multiphase flow 29
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.
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FS 2011 Groundwater II Multiphase flow 30
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
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FS 2011 Groundwater II Multiphase flow 31
Static distribution of light fluid
• ρ a
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FS 2011 Groundwater II Multiphase flow 32
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
= ⋅
⎡ ⎤⎛ ⎞⋅ − + +⎢ ⎥⎜ ⎟
⎜ ⎟+⎢ ⎥⎝ ⎠⎣ ⎦
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FS 2011 Groundwater II Multiphase flow 33
Rough estimate of mineral oil
migrationPhase 2: Mobile oil plug leaves behind practically immobile trace
R0hcwa
Groundwater
Capillar fringe
H 1S o1 z
S ro1
mobile
immobile
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FS 2011 Groundwater II Multiphase flow 34
Rough estimate of mineral oil
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
= −
− −
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FS 2011 Groundwater II Multiphase flow 35
Rough estimate of mineral oil
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
− +=− +
−
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FS 2011 Groundwater II Multiphase flow 36
Rough estimate of mineral oil
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
⎛ ⎞− −
= ⋅ +⎜ ⎟⎝ ⎠
Rough estimate of mineral oil
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FS 2011 Groundwater II Multiphase flow 37
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
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FS 2011 Groundwater II Multiphase flow 38
Static distribution of heavy
fluid
• ρ a
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FS 2011 Groundwater II Multiphase flow 39
Infiltration of dense fluid
Chlorinated hydrocarbon
Migration in groundwater is highly influenced by heterogeneities