Modelling two-phase flow and transport in
industrial porous media
S Majid Hassanizadeh
S.
Department of Earth Sciences
Ut ht University
Utrecht
U i
it
The Netherlands
Collaborators:
C
ll b
Simona Bottero; Utrecht University, The Netherlands
Jenny Niessner; Utrecht University
University, The Netherlands
Vahid Joekar-Niasar; Utrecht University, The Netherlands
Cas Berentsen; Delft University of Tech., The Netherlands
Rainer Helmig; University of Stuttgart, Germany
Michael A. Celia; Princeton University, USA
H l K.
Helge
K Dahle;
D hl University
U i
it off Bergen,
B
Norway
N
Outline of presentation
p
Introduction:
Darcy’s law for two-phase flow in porous media
Capillary pressure in porous media
Shortcomings of standard capillary pressure theory
Experimental evidences
New theory of capillarity
Significance of new theory
New theories of two-phase flow
Conclusions
INTRODUCTION
Theories of (multiphase) flow in porous media
have been too empirical in nature.
They have been developed through “extensions”
off simple
i l relationships
l ti hi in
i ad-hoc
d h ffashion.
hi
They are not necessarily valid for complex
systems we are considering today.
Take Darcy’s law.
DARCY TESTED HIS FORMULA ON A COLUMN OF
SAND IN A LABORATORY AND DETERMINED THE
VALUE OF CONDUCTIVITY (K):
Δh
Q = KA
L
Simple System
Complex System
METAMORPHOSIS OF DARCY’S LAW
Darcy’s law was
proposed
d ffor onedimensional steadystate flow of almost
pure incompressible
water in saturated
homogeneous
isotropic rigid sandy
soil under constant
temperature
p
It is now used in the
same form
f
f threefor
th
dimensional unsteady
flow of two or more
compressible
co
p ess b e fluids,
u ds,
with lots of dissolved
matter in anisotropic
matter,
heterogeneous
deformable porous
media under nonisothermal conditions
METAMORPHOSIS OF
DARCY’S LAW
Write the formula in terms of velocity
q = Q/A
Δh
Δh
q=K
L
METAMORPHOSIS OF
DARCY’S LAW
Δh
dh
FOR ONE-DIMENSIONAL FLOW, Change
to −
L
dx
dΔhh
dh
q = −K
dL
dx
METAMORPHOSIS OF
DARCY’S LAW
For unsteady state conditions:
h = h ( x, t )
∂dh
h
qq== −−KK
dx
∂dx
∂x
METAMORPHOSIS OF
DARCY’S LAW
FOR THREE-DIMENSIONAL FLOW
∂∂hh
qi ==−−KK
∂xi
METAMORPHOSIS OF
DARCY’S LAW
FOR ANISOTROPIC MEDIA, REPLACE K BY Kij
∂∂hh
qii = − K ijj
∂∂xxji
METAMORPHOSIS OF
DARCY’S LAW
FOR HETEROGENEOUS MEDIA
Kijj
is assumed to be a function
of position
METAMORPHOSIS OF
DARCY’S LAW
FOR FLOW IN UNSATURATED ZONE:
∂h
∂h
q i = − K ij
∂x j
METAMORPHOSIS OF
DARCY’S LAW
FOR FLOW IN UNSATURATED ZONE:
Kij
is assumed to be
a function of
water
t content
t t
θ
METAMORPHOSIS OF
DARCY’S LAW
FOR TWO-PHASE FLOW (say WATER AND OIL):
METAMORPHOSIS OF
DARCY’S LAW
FOR TWO-PHASE FLOW (say WATER AND OIL):
w
∂h
q = - K
∂x j
w
i
w
ij
METAMORPHOSIS OF
DARCY’S LAW
FOR TWO-PHASE FLOW (SAY WATER AND OIL):
w
∂h
q = - K
∂x j
w
i
w
ij
o
∂h
q = - K
∂x j
o
i
o
ij
METAMORPHOSIS OF
DARCY’S LAW
FOR TWO-PHASE FLOW (say WATER AND OIL):
K
w
ij
w
r
= k K ij
o
ij
o
r
K = k K ij
Capillary
p
y Pressure-saturation relationshipp
w
⎛
p (S )
ρ ⎞
o
w
w
h − h = o + ⎜1 − o ⎟ ( z − h )
ρ g ⎝ ρ ⎠
c
The form of Darcy’s
y Law remains unchanged:
g
α
∂h
qi = - K ij
∂x j
α
α
We are adding
W
ddi bells
b ll and
d whistles
hi tl to
t a simple
i l formula
f
l
to make it valid for much more complicated situations!
“Extended” Darcy’s
y Law
Δh
q=K
Δx
The “extended” Darcy’s Law remains unchanged:
α
∂h
qi = - K ij
∂x j
α
α
We have added bells and whistles to a simple formula
to make it valid for much more complicated situations!
Comparison of theories of Aristotle and Galileo for free fall of
objects with hypothetical measurements (after French, 1990)
Te
erminall Velocitty
GALILEO
ARISTOTLE
DATA POINTS
Time
PhD RESEARCH PROJECT
Hypothetical plot of terminal velocity
as a function of mass density for various shapes
Term
minal
Velo
ocit
Te
erminal
l Velocitty
R d shaped
Rod
h
d
Round
Pyramid
Disk
Mass Density
How many relative permeability-saturation curves
have been measured during the past few decades!?
How many capillary pressure-saturation curves have
been measured during the past few decades!?
Imbibition
scanning
curves
Drainage
scanning
curves
Relative permeability is supposed to be less than 1.
Water and oil relative permeabilities in 43 sandstone
reservoirs plotted as a function of water saturation
Problems with current theories of multiphase flow
Pressure gradient
P
di t and
d gravity
it are assumed
d tto b
be th
the only
l
driving forces. Effects of all other forces are lumped into
coefficients.
In particular, interfacial forces and interfacial energies are
not explicitly taken into account. This is unacceptable.
There is discrepancy between the theory and measurement.
Capillary pressure curves are measured under equilibrium
conditions and then used under non-equilibrium (flow)
conditions.
diti
Capillary pressure is believed to go to infinity. Nonsense!
Forces acting within interfaces control fluid
di t ib ti within
distribution
ithi the
th porous medium.
di
2σ
p =
RM
c
Mass transfer
among phases
occurs through
interfaces.
Where are interfaces in
porous media theories!?
Problems with current theories of multiphase flow
Pressure gradient
P
di t and
d gravity
it are assumed
d tto b
be th
the only
l
driving forces. Effects of all other forces are lumped into
coefficients.
In particular, interfacial forces and interfacial energies are
not explicitly taken into account. This is unacceptable.
There is discrepancy between the theory and measurement.
Capillary pressure curves are measured under equilibrium
conditions and then used under non-equilibrium (flow)
conditions.
diti
Capillary pressure is believed to go to infinity. Nonsense!
Measurement of Capillary
p
y Pressure-Saturation Curves
Capillary pressure-saturation curves for a given medium are
obtained by measurement of saturation inside and fluid
pressures outside the porous medium under quasi-equilibrium
conditions. Time to equilibrium can be 24 hours to 7 days.
Hydrophobic and hydrophilic membranes affect the results.
results
p =
c
p −p
a
Air chamber
pa
*
*
**
Porous
Medium
Scanning drying curve
*
Drying curve
*
*
*
Wetting curve
Scanning drying curve
p
Water reservoir
P
Porous
Plate
Pl t
(impermeable to air)
*
*
*
*
S
Industrial Porous Media
M
Many
industrial
i d t i l processes involve
i
l porous media
di flows.
fl
Some examples are processes in fuel cells, paper-pulp
drying, food production and safety, filtration, concrete,
ceramics, moisture absorbents, textiles, paint drying,
polymer composites,
composites and detergent tablets
There is clear difference between industrial porous media
and soils:
- fast flow
- significant mass transfer across fluid-fluid interfaces
- significant deformations
- wettability changes
- high porsosity
Dewatering of paper pulp
press nips
The paper pulp layer is transported through
the press aperture on a felt. The paper-felt
sandwich is thus compressed, so that the
water is pressed out of the paper, is
absorbed by the felt, and then flows laterally
away from the nips.
Pictures courtesy of Oleg Iliev of ITWM, Kaiserslautern
Simulation of dewateringg of paper
p p p
pulp
p
Capillary pressure curves are measured for felt
under quasi-static conditions. It takes around
24 hours to obtain them.
They are then used to model a process that
takes only seconds!
Air and oil flow in oil filters
Pictures courtesy of Oleg Iliev of ITWM, Kaiserslautern
Proton Exchange Membrane (PEM) Fuel Cells
Pc-Sw measurement of hydrophobic
y p
gas diffusion layer of a PEM fuel cell
Unsaturated flow in thin fibrous materials
Unsaturated flow in thin fibers
Unsaturated flow in pampers
Measurement of capillary
pressure curves takes about
12 hours.
Unsaturated flow in pampers
Capillary pressure-saturation experiments
(PCE and Water)
Experiments carried out at GeoDelft; Soil sample: 3 cm high and 6cm in dia.
Fluids: PCE and water;
Primary drainage, Main drainage, Main imbibition; Quasi-static and
dynamic experiments
PCE
sand
water
3 cm
Air chamber
PPT NW back
PPT W front
PCE
reservoir
Soil sample
Water
12.5
Pc-S curves based on
pressures measured
d
inside the soil sample.
Pc [(in sand]
Pc [[in fluids]]
10 0
10.0
Pcc [kPa]
7.5
Capillary pressure
DOES NOT ggo to
infinity!
5.0
2.5
Are th
A
these th
the curves
we should use in our
simulators?
i l t ?
0.0
-2.5
0.0
20.0
40.0
60.0
Water saturation
80.0
100.0
Measurement of
“Dynamic
Dynamic Pressure Difference
Difference” Curves
P −P =
n
Capillary Pressu
ure [kPa]]
10
ΔPdyn
8
6
P
4
2
0
w
D yn am ic P rim ary drain age-1
S tatic prim ary drain age
0 .2
0 .4
0 .6
0 .8
W ater S atu ration [-]
Hassanizadeh et al., 2004
1
c
Two-phase flow dynamic experiments (PCE and Water)
10.0
Pn-Pw (kPa))
8.0
6.0
4.0
2.0
S.M.I.
p
prim
drain PN~16kPa
prim drain PN~20kPa
prim drain PN~25kPa
main imb PW~0kPa
main imb PW~5kPa
main imb PW
PW~8kPa
8kPa
main imb PW~10kPa
main imb PW~0kPa last
main drain PN~16kPa
main drain PN~20kPa
main drain PN~25kPa
main drain PN~30kPa
Static Pc inside
Dyn.P.D.
Dyn.M.D.
S.P.D.
SMD
S.M.D.
Dyn.M.I.
0
-2
2.0
0
0
0.20
0.40
0.60
Saturation, S
0.80
1.00
Experiments on the uniqueness of
p − S curves
c
(drainage experiments by Topp,
Topp Klute,
Klute and Peters , 1967)
These are all drainage curves
There is no unique pc-S curve.
Dynamic Drainage Curves
Main Scanning Drainage Curves
Capilla
ary pres
ssure pc
Main Drainage Curves
Secundary Scanning Drainage Curves
Pi
Primary
Drainage
D i
Curve
C
Primary Imbibition Curve
M i Imbibition
Main
I bibiti Curve
C
Main Scanning Imbibition Curves
Secundary Scanning Imbibition Curves
Dynamic Imbibition Curves
Saturation, S
There is no unique pc-S curve.
Dynamic Drainage Curves
Main Scanning
g Drainage
g Curves
Main Drainage Curves
Secundary Scanning Drainage Curves
Ca
apillary pressurre pc
Primary Drainage Curve
Primary Imbibition Curve
Main Imbibition Curve
Main Scanning Imbibition Curves
Secundary Scanning Imbibition Curves
Dynamic Imbibition Curves
Saturation, S
A New Theoryy of
Capillarity in Porous Media
1 Difference in fluids
1.
fluids’ pressures are equal
to capillary pressure but only at
equilibrium.
2. Capillary
p
y pressure
p
is not onlyy a function
of saturation but also of interfacial area
Forces acting within interfaces control fluid
di t ib ti within
distribution
ithi the
th porous medium.
di
2σ
p =
RM
c
p −p =p
n
w
c
Microscale Dynamic Effects
w
Consider two-phase flow in a single tube:
pn − pw =
Wetting phase
2γ
⎛ μq ⎞
+ B⎜ 2 ⎟
RM
⎝r ⎠
Nonwetting phase
A
NONEQUILIBRIUM CAPILLARY EFFECT
Li
Linear
theory
h
Equilibrium capillary theory:
p − p = p (S)
n
w
c
Under non-equilibrium conditions:
p − p = p + f ( S,∂ S / ∂t)
n
w
c
Linear approx.:
p − p
n
w
∂S
= p − τ (S )
∂t
where τ is a damping coefficient.
c
Dynamic capillary pressure curves
pc=
∂S
( p − p ) − p ( S ) = −τ ∂t
a
pn − pw
w
c
This equation
Thi
ti captures
t
the
th behavior
b h i
ΔS
observed
thecurve
experiments
c
Dynamicin
drying
Δp = −τ
Drying curve
Δp
c
Wetting curve
Dynamic wetting curve
Curves measured in the lab
can be used to evaluate τ
S
Δt
Two-phase flow dynamic experiments (PCE and Water)
4.0E-02
dS/dt vs. S - all curves
3.0E-02
(-)d s/d t [s-1]
2.0E-02
1.0E-02
0.0E+00
-1.0E-02
-2.0E-02
-3.0E-02
0.00
0.20
0.40
0.60
saturation
0.80
1.00
Two-phase flow dynamic experiments (PCE and Water)
Determination of τ
Primary drainage; S=70%
2 .5
y = 6 1 .4 x + 0 .7
y = 5 4 .0 x + 1 .2
Pc(dyn)) - Pc(stat) [k
kPa]
2
1 .5
y = 5 0 .0 x + 0 .4
1
0 .5
P a ir = 1 6 k P a
P a ir = 2 0 k P a
P a ir = 2 5 k P a
0
0 .0 0 0
0 .0 0 5
0 .0 1 0
0 .0 1 5
( - ) d S /d t [s - 1 ]
0 .0 2 0
0 .0 2 5
0 .0 3 0
Utrecht University
TWO PHASE FLOW EXPERIMENTS
TWODynamic coefficient, τ
7
5
x 10
tau [Pa
a.s]
4
3
Sw [-]
[]
τ [Pa.s]
0.80
9.83 * 104
0.70
1.06* 105
0.60
1.33* 105
0 50
0.50
2 61 * 106
2.61
0.40
3.62 * 107
2
1
0
0.4
0.5
0.6
Sw [-]
0.7
0.8
Estimation of τ from literature data
(Hassanizadeh, Celia, and Dahle, Vadose Zone Journal, 2002)
Reference
Soil
Type
For Air-Water Systems:
Topp et al. (1967)
Sand
Smiles et al.
al (1971)
Sand
Stauffer (1978)
Sand
Nützmann et al.
al (1994)
Sand
Hollenbeck & Jensen(1998) Sand
For Oil-Water
Oil Water Systems:
S stems:
S dt
Sandst.
/limest.
Kalaydjian (1992b)
Type of
experiment
Value of τ
(kg/m.s)
Drainage
Drainage
Drainage
Drainage
Drainage
2*107
55*10
107
3*104
2*106
2 to 12*106
Imbibition
2*106
Significance of new theory for unsaturated flow:
Development of vertical wetting fingers in dry soil
Experiments by Rezanejad et al., 2002
Consequences of new theory for unsaturated flow:
Development of vertical wetting fingers in dry soil
David A. DiCarlo, “Experimental measurements of saturation overshoot
on infiltration” VOL. 40, 2004
Consequences of new theory for unsaturated flow:
Development of vertical wetting fingers in dry soil
David A. DiCarlo, “Experimental measurements of saturation overshoot
on infiltration” VOL. 40, 2004
Classical unsaturated flow equation
q
Flow equation:
⎞⎞
∂S
∂ ⎛ k r ( S )k ⎛ ∂ p w
n
=
+ ρg ⎟⎟
⎜
⎜
∂t
∂z ⎝ μ
⎝ ∂z
⎠⎠
Capillary pressure-saturation relationship:
p − p = p (S )
a
w
c
Richards equation:
∂S
∂ ⎛ ⎛ ∂S
⎞⎞
n
=
D⎜
+ ρg ⎟⎟
⎜
∂t
∂z ⎝ ⎝ ∂z
⎠⎠
Simulation of
vertical wetting fingers in dry soil
Stability analysis by Dautov et al. (2002) has proven that:
Richards equation is unconditionally stable.
It does not p
produce any
y fingers,
g , but a monotonicllyy
decreasing saturation profile toward the wetting front.
Modified
M
difi d Richards
Ri h d equation
i ((with
i h dynamic
d
i
effect) is conditionally unstable.
It is able to produce gravity wetting fingers.
Dynamic
y
capillary
p
y ppressure theory
y
Unsaturated flow equations:
Flow equation:
w
⎛
⎞⎞
∂S ∂ kr (S)k ⎛ ∂p
n = ⎜
+ ρg ⎟ ⎟
⎜
∂t ∂z ⎝ μ ⎝ ∂z
⎠⎠
Capillary pressure-saturation relationship:
p −p =p
a
w
c
stat
∂S
( S ) −τ
∂t
Development of vertical wetting fingers in dry soil;
Simulations based on new capillarity theory
Dautov et al. (2002)
A New Theoryy of
Capillarity in Porous Media
1 Difference in fluids
1.
fluids’ pressures are equal
to static capillary pressure but only at
equilibrium.
2. Capillary
p
y pressure
p
is not onlyy a function
of saturation but also of interfacial area
Drainage
Cap
pillary p
pressure
e head, Pc/ρg
Cap
pillary p
pressure
e head, Pc/ρg
Imbibition
Water content
Water content
CAPILLARY PRESSURE-SATURATION DATA POINTS
MEASURED IN LABORATORY
Drainage
Cap
pillary p
pressure
e head, Pc/ρg
Cap
pillary p
pressure
e head, Pc/ρg
Imbibition
Water content
Water content
CAPILLARY PRESSURE-SATURATION DATA
POINTS MEASURED IN LABORATORY
New capillarity theory
with no (need for) hysteresis
p = p ( S, a
c
c
wn
where:
p
c
is capillary pressure
S
is wetting phase saturation
awn
is the specific interfacial area between
wetting & nonwetting phases
)
pc-S-awn Relationships
55.0
9000
54.0
7000
Interrfacial
are
ea
IAVwn
(1
1/m)
Pressure
((kPa)
Cap
pillary
prressure
8000
53.0
52.0
51 0
51.0
50.0
0.4
Imbibition
Drainage
0.5
0.6
0.7
0.8
0.9
g Phase Saturation
Wetting
1
6000
5000
4000
3000
2000
1000
0.4
S t
Saturation
ti
8000
7000
Intterfacial
area
IAVwn
n (1/m)
0.5
0.6
0.7
0.8
0.9
g Phase Saturation
Wetting
S t
Saturation
ti
9000
Figures
courtesy of
Laura PyrakNolte of
Purdue
University
IAVndWN
IAVndWN
6000
5000
4000
3000
2000
Imbibition
Drainage
1000
51 51.5 52 52.5 53 53.5 54 54.5 55
Pressure (kPa)
Capillary pressure
1
pc-S-Awn Surfaces
Imbibition
Drainage
Figure courtesy of Laura Pyrak-Nolte of Purdue University
Comparison of Drainage & Imbibition Surfaces
Mean Relative Difference: -3.7%
3.7%
Standard Deviation: 10.6%
C ill
Capillary
pressure att macroscale
l
(data obtained from Vahid Joekar-Niasar et al. 2007)
New theories of two-phase flow
Extended Darcy’s law :
q = − ρ K • ( ∇G − g )
α
α
α
α
where Gα is the Gibbs free energy of a phase:
α
G =G
α
(ρ
α
, a , S ,T )
α
wn
Extended Darcy
Darcy’ss law :
q = −K • ( ∇p − ρ g −σ ∇a −σ ∇S
α
α
α
α
αa
Equation of motion for interfaces :
wn
αS
wwn = −Kwn ⎡⎣∇( awnγ wn ) +Ωwn∇Sw ⎦⎤
α
)
New theories of two-phase
p
flow
Volume balance equation for each phase
α
∂S
n
+ ∇ • qα = 0
∂t
Area balance equation for fluid-fluid interfaces
∂a
wn wn
wn
wn
w
+∇
∇• ( a w ) = r ( a , S )
∂t
wn
Summaryy of two-phase
p
flow equations
q
α
∂S
α
n
+∇•q = 0
∂t
q = −K • ( ∇p − ρ g −σ ∇a −σ ∇S
α
α
α
α
αa
wn
αS
α
)
∂a wn
wn wn
wn
wn
w
+ ∇ • (a w ) = r (a , S )
∂t
w wn = − K wn ⎡⎣∇ ( a wnγ wn ) + Ω wn∇S w ⎤⎦
w
∂
S
pn − p w = pc −τ
∂t
p c = f ( S w , a wn )
CONCLUSIONS
Darcy’s law is not really valid for complex porous
media.
Difference in fluid pressures is equal to capillary
pressure but
b only
l under
d equilibrium
ilib i
conditions.
di i
Under nonequilibrium conditions, the difference in
fluid pressures is a function of time rate of change
of saturation as well as saturation.
Fluid-fluid interfacial areas should be included in
multiphase
lti h
flow
fl theories.
th i
Hysteresis
ys e es s sshould
ou d be modelled
ode ed by introducing
oduc g
interfacial area into the two-phase flow theory