Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:
Advances in Water Resources 24 (2001) 385±407
www.elsevier.com/locate/advwatres
True-to-mechanism model of steady-state two-phase ¯ow in porous
media, using decomposition into prototype ¯ows
M.S. Valavanides 1, A.C. Payatakes *
Department of Chemical Engineering, Institute of Chemical Engineering and High Temperature Chemical Processes, University of Patras,
GR 265 00 Patras, Greece
Received 29 August 1999; received in revised form 30 March 2000; accepted 31 August 2000
Abstract
A true-to-mechanism model is proposed, which considers steady-state two-phase ¯ow in porous media (SS2uPM) as a composition of two prototype ¯ows, namely ganglion dynamics (GD) and connected-oil pathway ¯ow (CPF). Coupling of the prototype
¯ows is eected with the simple rule that the macroscopic pressure gradient is the same in both. For a given set of values of the ¯ow
system parameters, a domain of admissible ¯ow combinations is obtained. The solution is determined by assuming that each point in
this domain has equal probability of being `visited'. This leads to unique values for the ¯ow arrangement variables (FAV), the rate of
mechanical energy dissipation, and the relative permeabilities. The new model accounts for the non-linearity of the ¯ow as well as
for the eects of all the system parameters (notably those aecting interfaces), and its predictions are in very good agreement with
existing data. Ó 2001 Published by Elsevier Science Ltd.
Keywords: Porous media; Two-phase ¯ow; Relative permeabilities
1. Introduction
Two-phase ¯ow in porous media (2uFPM) occupies a
central position in enhanced oil recovery, the behavior
of liquid organic pollutants near the source in contaminated soils, etc. Conventional fractional ¯ow theory
assumes that the relative permeabilities are only functions of the water saturation S w (for a given porous
medium). This approach has been based on Richards's
postulate that each ¯uid ¯ows through its own separate
network of connected pathways and that all portions of
the disconnected non-wetting phase (say oil) are stranded
[10,17].
However, experimental investigations [1,2,4] have
shown that the disconnected oil contributes signi®cantly
(even exclusively) to the ¯ow and that relative permeabilities are strong functions of a large number of
other parameters
*
Corresponding author.
E-mail addresses: marval@iceht.forth.gr (M.S. Valavanides),
acp@iceht.forth.gr (A.C. Payatakes).
1
Now with IRC Help-Forward, Xenofontos 5, GR 105 57 Athens,
Greece.
kri kri S w ;Ca;r;j;cos h0A ;cos h0R ;Co;Bo;xpm ;flow history;
i o;w:
1
Here, Ca is the capillary number (de®ned as
cow , where l~w is the viscosity of water, U~ w
Ca l~w U~ w =~
the super®cial velocity of water, and c~ow is the interfacial
qw the oil/water ¯owrate ratio,
tension), r q~o =~
lw the oil/water viscosity ratio, h0A and h0R the
j l~o =~
limiting values of the advancing and receding contact
angles at nil velocity, Co the coalescence factor, Bo the
bond number, xpm a parameter vector composed of all
the dimensionless geometrical and topological parameters of the porous medium aecting the ¯ow (porosity,
genus, coordination number, normalized chamber and
throat size distributions, chamber-to-throat size correlation factors, etc.) and ®nally, `¯ow history' denotes the
way in which the steady-state conditions have been
achieved, for example, whether through initial imbibition or drainage. Similar observations hold for the
capillary pressure. In the following, a tilde is used to
characterize any dimensional physical quantity.
Note, in particular, that kro and krw increase strongly as
Ca increases (at ®xed saturation), which means that the
¯owrate vs pressure gradient relation is strongly nonlinear. The key question, then, is `What are the sources of non-linearity and of the other complex eects
0309-1708/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 6 3 - 4
386
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Notation
General notes:
A tilde () on top of a character
denotes a dimensional variable. Absence
of a tilde denotes that the variable is
dimensionless.
Superscipt A superscript attached to a variable
denotes characterization on the
macroscopic scale.
Subscript A subscript attached to a variable
denotes characterization on the microscopic scale.
0
0
Variables with prime (S w ; b0 ) are the
physically admissible values of the
corresponding plain variables.
Latin characters
a
crowding eect factor
A~
macroscopic cross-sectional area, normal
to the macroscopic ¯ow direction
reduced resistance of a jik-class cell in
A1U
jik
one-phase ¯ow
reduced bulk phase resistance of a
Abjik
jik-class cell for dierent ¯ow con®gurations
mean of Abjik over all possible jik combiAbm
nations, Eq. (30)
reduced interphasial resistance of any
Bbn
conducting cell of an n ganglion for
dierent ¯ow con®gurations
Bo
bond number
Ca
capillary number
pressure ¯uctuation factor, Eq. (27)
Cfl
CiX;G
fractional distributions of blocked cells
(side-gate unit cells) in any i-class
ganglion
fractional distributions of core cells in
CiC;G
any i-class ganglion
fractional distributions of end-gate cells
CiE;G
in any i-class ganglion
Co
coalescence factor
~T
uniform maximum pore-network depth
D
~ Cj
class-j chamber diameter
D
Gm
ganglion mobilization number, Eq. (22)
g~
gravity
reduced conductance of the eective
gef
porous medium
reduced eective conductance of the
g~efDOF
porous medium DOF region
1U
1U
; g~jik
conductance of a jik-class cell in onegjik
phase ¯ow (reduced and dimensional)
b
gjik;n
reduced conductance of a b-type
(b `E,G' , `C,G' or `X,G') jik-class cell
of a n-ganglion
g0
fj
fjek
Imax
J~jGC ~z; h
J~iGT ~z; h
G
Jjik
J~dr;i
J~min;j
k~
k~ef
kro ; krw
`~
L~G
n
mG
n
nG
i
n~G
i
n~G
Nd
NiG
P y
p~i ; p~j
pnG
b
Pjik;n
q~1U
q~o
q~w
q~o;CPF
q~o;DOF
q~w;DOF
q~1U
q~1U
uc
reduced uniform conductance of all cells
in the equivalent network
occurrence probabilities of class j
occurrence probabilities of any jik-class
cell
maximum ganglion size class
mean surface curvature of the o/w
interphase, as a function of its position in
a j-class chamber
mean surface curvature of the o/w
interphase, as a function of its position in
an i-class throat
mean (along a jik cell) reduced o/w mean
interphase curvature
drainage curvature of the receding o/w
interphase in an i-class throat
minimum curvature value of the o/w
interphase advancing in a j-class chamber
absolute permeability
eective porous medium absolute
permeability
relative permeabilities in oil and water
lattice network constant (node-to-node
distance)
n-class ganglion's length, projected in the
macroscopic ¯ow direction
assigned mobilization probability to any
n-class oil ganglion
reduced number density of the i-class
ganglia
number density of the i-class ganglia
characteristic distribution
network lattice dimensionality
total number of unit cells occupied by
any i-class ganglion
normal distribution of y
actual pressures in water next to
menisci
occurrence probability of any n-ganglion
in the ganglion cells domain
occurrence probabilities of the various
ganglion cells in the ganglion cells
domain
equivalent macroscopic ¯owrate of
one-phase in the DOF region
macroscopic ¯owrate of oil
macroscopic ¯owrate of water
mean oil ¯owrate in CPF
mean oil ¯owrate in DOF
mean water ¯owrate in DOF
¯owrate of the equivalent ¯uid in the
DOF region
¯owrate of an incompressible ¯uid
through a cell
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
q~G
uc;i
r
Sw
0
Sw
Sio;G
S o;G
S o;DOF
u1 ; u0ij
u~G
i
U~ o
U~ w
U~ o;G
U~ o;CPF
U~ o;DOF
U~ w;DOF
V~
V~ uc
V~i G
VmmG
VmG
W_
W_ o;CPF
W_
o;DOF
W~Tj
x
x0ij
xmic
xpm
y
~z
ÿo~
p=o~z
total ¯owrate through any i-class
ganglion conducting cells
oil/water ¯owrate ratio
water saturation
physically admissible value of water
saturation, see also Eq. (51)
speci®c oil saturation in any i-class
ganglion
average oil saturation in the ganglion
cells domain
disconnected-oil saturation in the DOF
region
bipolar coordinates
mean velocity of an i-class ganglion's
mass center
super®cial velocity of oil
super®cial velocity of water
oil super®cial velocity in the domain of
ganglion cells, Eq. (32)
oil mean super®cial velocity in CPF
oil mean super®cial velocity in DOF
water mean super®cial velocity in DOF
macroscopic control volume
average (over all classes) volume of the
unit cell
average actual volume of an i-class
ganglion
expected mean ganglion size of the
mobilized ganglia
expected mean ganglion size of the
overall ganglion population
reduced total rate of mechanical energy
dissipation
reduced rate of mechanical energy
dissipation in CPF
reduced rate of mechanical energy
dissipation in DOF
class-j throat width
reduced common pressure gradient
Cartesian coordinates
reduced microscopic pressure gradient
parameter vector composed of all the dimensionless geometrical and topological
parameters of the porous medium aecting the ¯ow (porosity, genus, coordination number, normalized chamber and
throat size distributions, chamber-tothroat size correlation factors, etc.)
reduced variable
macroscopic ¯ow direction
common pressure gradient
Greek symbols
b
the porous medium volume fraction
occupied by the connected oil
b0
bI;J
c~ow
1U
D~
pjik
b
D~
pjik;n
f
go;CPF
j
eA ; eR
h0A ; h0R
k~o ; k~w
j
l~o ; l~w
nij
q~o ; q~w
r
ra
U
vG
x
XPA
387
physically admissible value of b
factor de®ned by Eq. (25)
oil/water interfacial tension
pressure drop along any jik-class unit cell
during one-phase ¯ow through it
pressure drop along any jik-class cell of
an n-ganglion for dierent ¯ow
con®gurations
free parameter which regulates the
steepness of the ganglion size distribution
expected mean macroscopic connectedoil ¯ow fraction
contact angle
advancing and receding contact angles
limiting values of the advancing and receding contact angles (at nil velocity)
mobility of oil and water, respectively
oil/water viscosity ratio
dynamic viscosity of oil and water,
respectively
aspect ratio between an i-class throat and
a j-class chamber
mass density of oil and water, respectively
pore-network coordination number
variance of the distribution of the
logarithm of the microscopic pressure
gradient
expected mean macroscopic ¯ow quantity
tortuosity of the spine of any ganglion
fraction of all the ganglion cells over all
the DOF region cells
integration domain corresponding to all
physically admissible solutions
Abbreviations
CPF
connected-oil pathway ¯ow
C,G
core unit cell of a ganglion
DeProF
decomposition in prototype ¯ows
DTF
drop trac ¯ow
DOF
disconnected-oil ¯ow
E,G
end-gate unit cell of a ganglion
EMT
eective medium theory
FAV
¯ow arrangement variables
GCD
ganglion cells domain
GD
ganglion dynamics
LGD
large ganglion dynamics
PA
physically admissible
SFA
smooth ®eld approximation
SGD
small ganglion dynamics
SSFD
steady-state fully developed
WSCD
water saturated cell domain
X,G
non-conducting unit cell of a ganglion
2uPM
two-phase ¯ow in porous media
388
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
indicated by Eq. (1), and how can these phenomena be
modeled?'
2. Experimental observations on steady-state two-phase
¯ow in porous media (SS2uFPM)
Visual experimental studies [1,2] have shown that the
disconnected oil contributes substantially to the overall
¯ow of oil. The observed ¯ow behavior has been roughly
classi®ed into four ¯ow regimes, namely large ganglion
dynamics (LGD), small ganglion dynamics (SGD), drop
trac ¯ow (DTF), and connected-oil pathway ¯ow
(CPF). In the ®rst three of these regimes, all of the oil
¯ow is due to the motion of disconnected bodies of oil
(ganglia or drops). Even in the fourth regime, CPF
(which is achieved with relatively large Ca values), an
appreciable portion of the oil ¯owrate is due to the
motion of droplets and ganglia in between the connected-oil pathways. In many cases, the overall ¯ow is a
mixture of two or more of the aforementioned ¯ow
patterns. For reasons of economy, in the present work,
we often combine LGD and SGD into an overall ¯ow
regime, namely, ganglion dynamics (GD).
The complex behavior indicated by Eq. (1) becomes
comprehensible once the active participation of ganglia
and droplets in the ¯ow is acknowledged and taken into
account explicitly.
3. Decomposition in prototype ¯ows (DeProF) for steadystate two-phase ¯ow in porous media (SS2uFPM): model
formulation
A theoretical model is developed here which accounts
for the pore-scale mechanisms and the network wide
cooperative eects, and which is suciently simple and
fast for practical purposes. This model is based on the
concept of decomposition in prototype ¯ows (DeProF)
and is an improved version of an earlier work presented
in [15].
Brie¯y, we assume that the macroscopic ¯ow can be
decomposed into two prototype ¯ows, namely CPF and
disconnected-oil ¯ow (DOF). The latter comprises
LGD, SGD and DTF (although in the present work we
omit DTF; the contribution of DTF will be included in a
future work). Each prototype ¯ow has the essential
characteristics of the corresponding experimentally observed ¯ow patterns, in suitably idealized form, and so
the pore-scale mechanisms are incorporated in the prototype ¯ows. The cooperative eects among ganglia are
also incorporated in DOF making suitable use of eective medium theory (EMT). A key feature of the proposed model is that the two prototype ¯ows are
entwined by imposing the condition that the pressure
gradient is the same in both. This is a condition that is
supported both by theory and experiment [4]. We ®nd
that there is a domain of admissible combinations of the
prototype ¯ows that corresponds to any given set of
Fig. 1. (a) Actual ¯ow and its theoretical decomposition into prototype ¯ows: CPF and DOF; (b) conceptual replacement of the actual DOF region
with an eective porous medium layer, of absolute permeability k~ef , through which ¯ows `eective water', with a total ¯owrate equal to the sum of the
¯owrates of oil and water through DOF, Eq. (9). The eective permeability of this layer is de®ned by Eq. (10).
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
values of the system parameters. The solution is determined by assuming equal probability for each admissible combination and averaging. The solution depends
on the values of all the system parameters (see RHS of
Eq. (1)). A more detailed description follows.
Consider the general ¯ow in Fig. 1(a). Connected
pathways of oil (when they exist) occur in roughly
`parallel' pathways. In between these pathways, a fraction of the population of oil ganglia are mobilized by the
water-induced pressure gradient and migrate downstream. The total ¯ow is a mixture of two ¯ow patterns,
speci®cally CPF and DOF. The key dierence between
these ¯ow patterns is that in DOF, numerous menisci
are in motion and many `catastrophic' (irreversible) ¯ow
phenomena are dominant [2,4]. This aects the relative
magnitude of the rate of mechanical energy dissipation
caused by capillary eects compared to that caused by
bulk viscous stresses. Each ¯ow pattern prevails over
mesoscopic regions of the porous medium (ranging
from, say, 104 ±109 pores), whereas the macroscopic ¯ow
is homogeneous. (Macroscopic inhomogeneities are not
considered here.)
Consider, now, a macroscopic control volume, V~ ,
having a cross-sectional area A~ normal to the macroscopic ¯ow direction, ~z, Fig. 1. The physical system of
the porous medium, oil and water is characterized by the
values of the physicochemical parameters, namely
c~ow ; l~o ; l~w ; q~o ; q~w ; h0A ; h0R ; xpm . Oil and water are
continuously fed into the control volume with constant
¯owrates q~o and q~w , so that the operational (dimensionless) parameters Ca and r have constant values. In
the present analysis, disconnected oil is considered to be
composed of ganglia only. (The omission of droplets is
an acceptable ®rst approximation as it will become evident below.) In the CPF region, the oil retains its
connectivity and ¯ows with virtually one-phase ¯ow.
The porous medium volume fraction occupied by the
connected oil is denoted as b. The DOF regime is de®ned as the region composed of the rest of the unit cells,
and the DOF volume fraction equals 1 ÿ b. S w and b
are called ¯ow arrangement variables (FAV) because
they give a coarse indication of the prevailing ¯ow
pattern. One of the objectives of the present model
(DeProF) is to determine the values of S w and b that
conform with the externally imposed conditions. The
analysis is carried out on two length scales: a macroscopic scale (1012 pores, or more) and a microscopic
scale.
3.1. Macroscopic scale, and connection with the porenetwork quantities
The overall oil and water ¯owrates are partitioned
into the contributions of the connected and disconnected-oil regions: q~o;CPF ; q~o;DOF are the mean oil ¯owrates in CPF and DOF, respectively, whereas q~w;DOF is
389
the water ¯owrate in DOF, Fig. 1(a). Oil and water
mean super®cial velocities in CPF and DOF are de®ned
~
~
as U~ o;CPF q~o;CPF = bA,
U~ o;DOF q~o;DOF = 1 ÿ bA,
w;DOF
w;DOF
~
~
q~
= 1 ÿ bA, respectively. It is exand U
pedient to use reduced mean super®cial velocities de®ned by U m;n U~ m;n =U~ m , where m o, w and n CPF,
DOF. The disconnected-oil saturation in DOF is denoted by S o;DOF . The absolute permeability of the po~
rous medium is denoted by k.
As already mentioned, an essential feature of the
proposed method is that the two prototype ¯ows are
entwined by having common pressure gradient,
ÿo~
p=o~z. Strictly speaking, this holds true for steadystate fully developed two-phase ¯ow in porous media
(SSFD2uFPM) [2,4] (and only approximately so in regions with weak saturation gradients).
Using this condition, we obtain
!
~
o~
p
q~i
i K
i o; w
2
kr
ÿ
l~i
o~z
A~
as well as the following relation between the mobility
ratio, k~o =k~w , and the ¯owrate ratio, r:
k~o q~o
lo
k~ro =~
r:
krw =~
lw k~w q~w
3
We de®ne the reduced pressure gradient, x, as
!
k~
o~
p
ÿ
:
x
o~z
c~ow Ca
4
Assuming that in the CPF region oil ¯ows with virtually
one-phase ¯ow (i.e., neglecting the possible lubrication
eect of wetting ®lms), we can use Darcy's law to obtain
!
~
o~
p
k
:
5
ÿ
U~ o;CPF
o~z
l~o
Assuming that there is no water present in the CPF
region, a total water mass balance yields
q~w;DOF q~w ) 1 ÿ bU~ w;DOF U~ w
6
and a total oil mass balance yields
q~o;CPF q~o;DOF q~o ) bU~ o;CPF 1 ÿ bU~ o;DOF U~ o :
7
An overall oil balance gives
b 1 ÿ bS o;DOF 1 ÿ S w :
8
In the ®nal analysis, the macroscopic behavior of the
¯ow is the manifestation of pore-scale ¯ow phenomena
and pore-network cooperative eects.
Scaling-up of the ¯ow in the CPF region is relatively
simple. Oil ¯ow in the pores is virtually one-phase ¯ow
and, under creeping ¯ow conditions, the hydraulic
conductivity of each unit cell, Fig. 2, is a function of
local geometrical parameters only. The permeability of
390
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Fig. 2. A typical jik-class unit cell of the examined model porous medium. All dimensions are normalized with the lattice constant `~ 1221 lm. The
thick inclined arrow represents the orientation of the lattice skeleton relative to the macroscopic pressure gradient. In each chamber a bipolar
cylindrical system is used. The bipolar system is aligned so that its cylindrical axis coincides with the longitudinal axis of the chamber and its poles
reside at the intersection of the annulus of the cylinder with the longitudinal axes of the adjacent throats. u1 ; uoij and uoik are the limit surfaces of the
unit cell expressed in bipolar coordinates.
the CPF region can be obtained from the hydraulic
conductances of the unit cells through application of
EMT. This is the absolute permeability of the porous
medium, and it is strictly a function of pore geometry
and pore-network topology (or connectivity).
Scaling-up of the ¯ow in the DOF region is more
complex, because the two dierent ¯uids are simultaneously arranged within the pores separated by a
multitude of advancing, receding and stranded menisci.
Here, the hydraulic conductivity of any individual unit
cell is not only a function of the appropriate local geometrical parameters, but also of the ¯uid arrangement
within it, Fig. 3. Furthermore, even under creeping ¯ow
conditions, the hydraulic conductance in a unit cell will
be a function of the velocity of the menisci, because of
the dependence of the contact angles (advancing and
Fig. 3. A microscopic scale representation of a DOF region. An oil ganglion of size class 5 is shown. The vector at the bottom of the ®gure represents
~ is shown expanded. The thick
the oil-ganglion mean velocity. For simpler representation, all cells are shown identical and the lattice constant, `,
dashed line separates the ganglion cells domain (GCD) and the water saturated cells domain (WSCD).
391
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
receding) on the local displacement velocity. In order to
deal with this problem, we proceed as follows. First, we
replace the DOF region with an eective porous medium, through which ¯ows a one-phase ¯uid with
¯owrate, q~1U , that is equal to the sum of the ¯owrates of
water, q~w;DOF , and oil, q~o;DOF , through the DOF region,
Fig. 1(b). In order to ®x ideas, we assume that this
conceptual ¯uid has viscosity equal to that of water, and
we will call it `water'. (This choice does not cause loss of
generality, because the viscosity that is assigned to the
conceptual ¯uid is taken into account properly.) Thus,
we have
q~1U q~o;DOF q~w;DOF :
9
ef
~
Second, we de®ne the eective permeability, k , of
this conceptual medium by imposing Darcy's law for
one-phase ¯ow
!
~ef
o~
p
k
ÿ1
~ ÿ b
:
10
ÿ
q~1U A 1
o~z
l~w
Eqs. (9) and (10) along with (5)±(7) give
l~
k~ef g~efDOF w
`~
"
#
b
c~ow
ÿ k~ 1 ÿ bÿ1 ;
Ca 1 r
j
ÿo~
p=o~z
11
where g~efDOF is, by de®nition, the eective hydraulic
conductance. Eq. (11) shows that k~ef is a function of the
absolute permeability (of the actual porous medium) as
well as of the pressure gradient, the capillary number,
the ¯owrate ratio, the viscosity ratio, the interfacial
tension and the FAV, b.
Now, a basic tenet of the present theory is that the
eective hydraulic conductance, g~efDOF (and therefore,
k~ef ), can be calculated from the hydraulic conductances
of the unit cells composing the (actual) pore-network in
the DOF region by pressing into service EMT. To this
end, one needs to have the topology of the skeleton of
the pore network, and the size distribution of the hydraulic conductances of the unit cells. The latter, as
already mentioned, are functions not only of the unit
cell geometry, but also of the ¯ow arrangement in each
unit cell and the probability of occurance of each ¯ow
arrangement. These results are developed in the following section. Before proceeding, however, we must
note that EMT in its conventional form is used for
networks, each branch (here unit cell) of which has a
random but constant conductance. In the present work,
the conductance of each unit cell can only be considered
at any given instant as momentarily constant. Furthermore, the ¯ow arrangement in each unit cell and the
probability of occurrence of each ¯ow arrangement depend, as we shall see, on the pressure gradient itself.
However, we expect that EMT is still valid in a narrow
domain around the solution, because one can assume
that in such a narrow domain all hydraulic conductances are constant and equal to the values that correspond to the pressure gradient given by the solution.
(The results bear out this expectation.)
3.2. Microscopic scale
3.2.1. Model porous medium. Geometric and topologic
characteristics
The microscopic scale analysis involves the speci®c
geometric and topologic characteristics of the porous
medium under consideration. In order to be able to
validate the theoretical model by comparing its predictions with available experimental results, we carried out
the analysis adopting the topological and geometric
characteristics of the glass pore-network model that was
used in the experiments of Avraam and Payatakes [2]. It
should be noted that this choice does not constitute a
constraint on the validity of the approach. A similar
analysis can be made with 3-D pore networks, even
networks with coordination number r that varies from
node to node (within reason of course). The pore geometry can also be dierent, but the basic approach
would still be the same. As shown in [1], two-phase ¯ow
in 3-D networks diers from than in 2-D ones quantitatively, but not qualitatively.
The model pore network of [2] is of the chamber-andthroat type. The network skeleton is a square lattice
with node-to-node distance `~ 1221 lm, and arranged
so that the macroscopic ¯ow direction is parallel to one
family of diagonals. For this particular network, the
lattice dimensionality is Nd 2, and the coordination
number is r 4. (For a cubic network, we would have
Nd 3 and r 6.) The maximum pore depth is nearly
uniform and was experimentally determined to be
~ T 115 lm. The chamber size (diameter) distribution
D
and the throat size (width) distribution have discrete
(composed of ®ve classes each) nearly normal size dis~ Cj i 610 lm,
tributions with mean values hD
~
hWTj i 169 lm and standard deviations equal to 1/4 of
the corresponding mean values. The occurrence probabilities fj of each class along with the respective actual
characteristic dimensions of the chambers (diameters),
~ Cj , and the throats (widths), W~Tj , are given in Table 1.
D
It should be noted that the chamber and throat diameter values in Table 1 are somewhat dierent from
those reported in [2]. This is because [2] reported
Table 1
Occurrence probabilities fj of individual classes j of chambers and
throats and the associated actual characteristic dimensions, rep~ Cj , and the throat width, W~Tj
resented by the chamber diameter, D
j
1
2
3
4
5
fj %
~ Cj (lm)
D
W~Tj (lm)
16
330
111
21
470
139
26
610
169
21
750
195
16
890
222
392
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
nominal dimensions, i.e., the dimensions on the negative
mask that was used in the etching of the glass model.
The dimensions given here are the measured actual dimensions of the model pore network that was used in
the experiments of [2].
The chambers are modeled as normal circular cylinders with their longitudinal axes oriented perpendicular
to the network plane and located at the network nodes.
The throats are modeled as normal elliptic cylinders
with their longitudinal axes placed on the network
bonds. The basic unit (building block) of the pore network is the unit cell that comprises of a throat connecting the corresponding sectors (quarters) of the
chambers adjacent to it. A typical unit cell is depicted in
Fig. 2. It is denoted as a jik-class cell, because it is
composed of a j-class chamber and a k-class chamber
connected with an i-class throat. The thick dashed lines
represent the unit cell limits, and the dashed±dotted lines
represent the network lattice. The solid matrix of the cell
is shaded. V~ uc is the average (over all classes) volume of
~ the
the unit cell. All lengths are normalized with `,
network lattice constant. The basic geometrical characteristics of the cell, i.e., the reduced chamber diameters,
DCj , DCk , and the reduced throat width, WTi , are also
de®ned in Fig. 2. All other dimensions shown in this
®gure are explicit functions of DCj , DCk and WTi . In the
present work, we assume that there is no correlation
between the classes of throats, chambers and cells. Thus,
the occurrence probability of any jik-class unit cell, fjik ,
is given by
fjik fj fi fk :
12
1U
3.2.2. Reduced conductances for one-phase ¯ow, gjik
When the problem of one-phase ¯ow through this
network is considered, one can calculate the conductances of each unit cell solely from its geometrical
characteristics. These conductances represent the transfer function of each cell for one-phase creeping ¯ow. All
cells that belong to the same jik-class have the same one1U
1U
. The reduced conductances, gjik
,
phase conductance, g~jik
are given by
1U
1U
g~jik
gjik
1
l~
q~1U
l~
q~1U
l~
1Uuc 1U
1U uc
1U
3
3
3
1U
3
~
~
~
~
A
D~
p
~
q
Ajik l~q~uc =` `
`
jik
jik
uc `
1
12
u0ij u0ik ÿ 2u1
DT
ÿ1
64 WTi2 D2T
1 ÿ x0ij ÿ x0ik ;
13
p WTi3 D3T
where `~ is the network lattice constant, l~ the dynamic
viscosity of the ¯uid, q~1U
uc the ¯owrate through the unit
1U 1U
1U ~3
~
~
l
q
~
quc A1U
cell, D~
pjik
jik
uc =` the corresponding pressure
drop along the unit cell, and A1U
jik is the reduced resistance of the jik-class cell. Details concerning the
1U 1U
~
quc are given in
derivation of the transfer function D~
pjik
the Appendix A. The ®rst term in the brackets on the
RHS of Eq. (13) is the contribution of the ¯ow in the
two chambers, and the second term is the contribution
of the ¯ow in the throat.
1U
It is clear that gjik
is a function of only the geometrical characteristics of the jik-class cell. This is a typical
characteristic of creeping one-phase ¯ow.
3.2.3. Eective medium theory
The overall conductance of a network, such as the
one previously described, can be calculated numerically
by applying eective medium theory (EMT). The ®rst
step is to replace the original network with an equivalent
network that has the same skeleton as the prototype, but
is composed of uniform unit cells. In order to achieve
the same overall conductance, the uniform cells of the
equivalent network must have reduced conductances
equal to g0 , which is determined by solving the EMT
equation [13]
X
j;i;k
fjik
1U
g0 ÿ gjik
0;
1U
r=2 ÿ 1g0 gjik
14
where r is the pore-network coordination number. The
equivalence of these networks is expressed on the
macroscopic scale: their macroscopic transfer functions
relating super®cial velocities and pressure gradients
(Darcy's law) are identical. When Eq. (14) is solved in
~ of the original
terms of g0 , the absolute permeability, k,
network is analytically recovered through the relation
k~ gef `~2 ;
15
where gef is the reduced eective conductance of the
equivalent network, and is related to g0 by (Burganos
and Sotirchos [5])
r
g0 :
16
gef
2Nd
Note that for the network adopted here, and for all
regular orthonormal networks, gef g0 .
Using Eqs. (13)±(16) and the data in Table 1, we
obtain the theoretical value k~ 8:89 lm2 , which is in
excellent agreement with the experimental value
8:90 lm2 reported in [2].
When the problem of one-phase creeping Newtonian
¯ow in a cell is considered, the resistance to the ¯ow is
caused by viscous stresses only, and so the transfer
function of the cell is linear. In the case of DOF, where
two-phase ¯ow in a cell is common, things are more
complicated and non-linearity appears even in the case
of very slow Newtonian ¯ows. This case is treated
below.
3.2.4. Reduced conductances for the disconnected-oil ¯ow
(DOF) domain
A microscopic scale representation of a typical DOF
region is shown in Fig. 3. An oil ganglion having a
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
typical `cruising' con®guration [16,18] is shown at the
center. All the cells that accommodate parts of this (or
any other) oil ganglion are called ganglion cells and are
outlined with a thick dashed line.
For the purposes of the present work, we will use the
term ganglion in an extended way (rather than saying
network-ganglion) to denote the entire part of the pore
network that is composed of the ganglion cells of any
given oil ganglion (see, for example, the region outlined
with the thick dashed line in Fig. 3). The DOF region is
partitioned into two sub-regions: the sub-region that
comprises all ganglion cells and the sub-region that
comprises unit cells containing only water. These will be
called ganglion cell domain (GCD) and water saturated
cell domain (WSCD), respectively. The fraction of all the
ganglion cells over all the DOF region cells is denoted by
x, and will be called the GCD fraction. Thus, the
fraction of all the water saturated cells in the DOF region equals 1 ÿ x. (As it will be shown below, the
variable x will be eliminated in the system of DeProF
equations presented in this work. We introduce it here
for the sake of clarity in further developments. In a future extension of the present work in which drop trac
¯ow (DTF) will be included in the DOF domain, x will
be shown to be the appropriate third ¯ow-arrangement
variable.)
The ganglion sizes have a distribution that depends
on the ¯ow conditions (for a given porous medium and
¯uids system). Determination of this distribution will
be achieved along with the rest of the solution. Each
oil ganglion is said to belong to a certain class i with
i 1; Imax based on the number of chambers it occupies. At any instant, the total number of unit cells
occupied by any i-class ganglion (having reduced volume i) is NiG i 2Nd ÿ 1 1. The average actual
volume of an i-class ganglion is given by V~i G NiG V~ uc .
Imax is the upper truncation class of the ganglion size
distribution, and so no ganglia with volume greater
than NIGmax V~ uc appear. It is important to note that Imax is
an implicit function of x, and is a derived parameter. In
a typical situation, only a fraction of the ganglion
population will be mobilized. Moving ganglia have a
tendency to become aligned with the macroscopic ¯ow
direction, acquiring the so-called `cruising shape'
[16,18]. In the present work, this is taken into account
by assuming that all ganglia have a zigzag spine that is
aligned along the macroscopic pressure gradient. An
important parameter associated with the `zigzag' motion of the ganglion is the tortuosity of its spine, vG .
This is also the ratio of the length of its actual migration path over the eective migration length in the
macroscopic ¯ow direction. This parameter depends on
the porous medium geometrical and topologic characteristics and will be considered to be equal for all
ganglion sizepclasses.
For the case under consideration,
we set vG 2.
393
Unit cells in which ganglion menisci exist are called
gate unit cells in accordance with [14,16]. In the present
work, we assume that all ganglia are `wormlike' and
aligned (as much as their zigzag shape allows) with the
macroscopic ¯ow direction. Among all gate unit cells of
a mobilized ganglion, there are only two gate unit cells
in which rheons take place (one cell for the `xeron' and
one cell for the `hygron' [16]). Such cells will be denoted
with the index `E,G' (`end-gate,ganglion'). We assume
that the only potential end-gate unit cells are those located at the foremost part (for the hygron) and at the
aft-most part of the ganglion (for the xeron); for example unit cells 1 or 7 for the xeron and 6 or 12 for the
hygron in Fig. 3. All cells of any mobilized ganglion that
are completely saturated with oil, e.g., cells 8, . . ., 11 in
Fig. 3 will be called `core cells', and will be denoted with
the index `C,G' (`core,ganglion'). Any i-class mobilized
ganglion has i ÿ 1 `C,G' cells. The rest of the ganglion
cells (which are neither end-gate cells nor core cells)
contain oil and water separated by virtually immobile
menisci and are called side-gate unit cells.
Ganglion migration induces a corresponding ¯ow,
q~G
uc;i , of oil in the core cells and an equal total ¯owrate of
water and oil in the end-gate cells. This ¯ow must be
consistent with the mean velocity of the mass center of
the ganglion, u~G
i , i.e.,
~2 G
~G
q~G
uc;i u
i ` v :
17
The rest of the cells of a mobilized ganglion (side-gate
unit cells) are considered as non-conducting, and they
are denoted with the index `X,G' (short for `non-conducting,ganglion'). Any i-class mobilized oil ganglion
occupies 2i Nd ÿ 1 `X,G' cells, and, of course, all of the
NiG cells of any i-class stranded oil ganglion are `X,G'
cells. Therefore, the fractional distributions of the various ¯ow con®gurations of any i-class ganglion cells
(mobilized as well as stranded ganglia taken into account) are explicitly given by
G
CiX;G i 2Nd ÿ 1 1 ÿ mG
i i 1 =Ni ;
G
CiC;G mG
i i ÿ 1=Ni ;
18
G
CiE;G 2mG
i =Ni ;
where mG
i is the mobilization probability assigned to the
i-class oil ganglia, and will be discussed below. It is
easily veri®ed that CiX;G CiC;G CiE;G 1 given the
aforementioned expression for NiG .
The number density of the i-class ganglia,
n~G
;
i 1; Imax , is de®ned as the number of i-class gani
glia per unit volume of all the ganglion cells, or, better,
per GCD unit volume. Expressing the volume of ganglion cells in the DOF region in terms of the ganglion
size distribution, we obtain
394
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
x 1 ÿ bV~
Imax
X
i1
However, it will have a more substantial role to play in
the extended model formulation in which DTF will be
taken into account.
G ~ uc
n~G
x 1 ÿ bV~
i Ni V
)
Imax
X
i1
G ~ uc
1:
n~G
i Ni V
19
If all of the disconnected oil were redistributed into
class-1 ganglia (singlets), then the number density of
~ uc
singlets would be n~G
1= 2Nd V . We de®ne the reduced number density of the i-class ganglia,
G
~ uc
~G
nG
nG
fnG
i ; i 1; Imax g, as ni n
i =~
2~
i Nd V . Clearly,
G
ni is equal to the ratio of the total number of i-class
ganglion cells over the total number of all ganglion cells
in the DOF region. All fnG
i i 1; Imax g are unknown
variables, which will be determined in the context of the
DeProF model.
By de®nition, the GCD fraction, x, in terms of the
ganglion size distribution, is given by
PImax G G X
Imax
n~i V~i
1 ÿ bV~ i1
~G
20
n~G
x
i Vi :
1 ÿ bV~
i1
Sio;G ,
corresponds to the i-class
A speci®c oil saturation,
of ganglia. This saturation is easily determined by assuming that all core cells (`C,G' cells) are completely
saturated with oil whereas all other ganglion cells are
saturated half with oil and half with water. By making
an oil volume balance over all cells of this ganglion class,
one obtains Sio;G iNd =NiG . The average oil saturation
in the GCD, S o;G , in terms of the reduced ganglion size
distribution is given by
PImax G G o;G
n~i V~i Si
x 1 ÿ bV~ i1
) S o;G
S o;G
x 1 ÿ bV~
Imax
1X
S o;DOF
:
inG
i
2 i1
x
21
As in the case of x; S o;G is also a variable of temporary
usefulness if one neglects oil droplets (as we do here).
3.2.5. Probability of oil ganglion mobilization, mG
n
In a typical situation, only a fraction of the oil ganglion population is mobilized. We assign a probability of
mobilization, mG
n , to ganglion class-n. This probability is
estimated by applying, in a statistical sense, the generalized oil ganglion mobilization criterion ([14,16]) appropriately extended to account for the crowding eects
that emerge in the presence of a dense population of oil
ganglia.
Consider a typical ganglion that is placed within a
pore network through which water ¯ows from left to
right driven by a macroscopic pressure gradient, ÿr~
p,
Fig. 4. In order to obtain a conservative estimate of the
pressure gradient that is required to cause mobilization
of the ganglion, we consider the case in which the endmenisci oer maximum capillary resistance. Thus, the
downstream meniscus is in a throat (denoted with i),
while the upstream meniscus is in a chamber (denoted
with j).
The curvature of the downstream meniscus, J~dr;i , is a
function of the size of the throat (not to be confused
with that of the i-class of throats) and the receding
contact angle, hR . For simplicity, we use the limiting
value of hR (as the receding velocity go to zero), h0R , irrespective of the actual velocity of the meniscus. The
minimum possible curvature of the upstream meniscus,
J~min;j , is a function of the size of the local chamber (not
to be confused with that of the j-class of chambers) and
the advancing contact angle, hA . Again, for simplicity,
we use the limiting value of hA , namely h0A , irrespective
of the actual velocity of the meniscus. (The dependence
of hR and hA on the corresponding meniscus velocity can
be introduced readily; this would produce a small bene®t
Fig. 4. A ganglion at a `hard-to-move' con®guration in the time interval between two rheons. (Emphasis on the end-gate unit cells.)
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
at the expense of substantially more cumbersome calculations.)
Following [14,16], the appropriate ganglion mobilization number is de®ned by
Gm max
i;j
h
~
pj ÿ p~i
c~ow J~dr;i h0R ÿ J~min; j h0A
i;
22
where p~i and p~j are the actual pressures in the water next
to the menisci with indices i and j, respectively. In (22),
index i scans all the throats in the gate unit cells, whereas
index j scans all the chambers. According to [14,16], the
criterion for ganglion mobilization is
If Gm > 1 ) mobilization;
If Gm < 1 ) stranding:
23
Calculation of the actual values of p~i and p~j requires
detailed pore-network simulations as in [6,7,9]. Whenever `exact' values of p~i and p~j are available, they should
be used. Otherwise, a rough estimate of the pressure
dierence ~
pj ÿ p~i can be obtained by applying smooth
®eld approximation (SFA).
For a solitary ganglion, Eq. (22) and use of SFA give
Ca
k
~
with k k=`~2 , and
Gm bI;J
DL~ij cos hji
i:
bI;J max h
i; j
`2 J~dr;i h0R ÿ J~min; j h0A
24
25
Here, DL~ij cos hji is the length of the line connecting
points i and j projected along the macroscopic pressure
gradient, Fig. 4; I and J are the values of i and j, respectively, for which the RHS of (25) becomes maximum. (In the case of `wormlike' ganglia that are aligned
with the macroscopic pressure gradient, positions I and
J usually correspond to the `nose' and `tail' of the ganglion, respectively. This is adopted in the present work
as the standard case.)
If one introduces typical parameter values in (24) and
(25) and applies the mobilization criterion (23), then it is
easy to conclude that a typical ganglion remains
stranded unless the value of Ca becomes of the order of
(10ÿ4 ) to (10ÿ3 ) or larger depending on ganglion size.
However, ganglion mobilization becomes possible at
much smaller Ca values as in the case of dense populations of ganglia. As discussed in [14,16], in the case of
populations of ganglia, Eq. (24) must be modi®ed as
follows:
Gm bI;J
Ca
population of ganglia;
kkrw
26
where krw is the relative permeability to water. Here, krw
accounts for the `crowding eect' and it can take values
much smaller than unity when the population of ganglia
becomes very dense. Consequently, in the case of
395
ganglion populations, ganglia can move even at Ca
10ÿ5 . On the pore-scale, the explanation is that the presence of a dense population of ganglia reduces the number
of pathways that are available for water ¯ow to relatively
few. Thus, the interstitial velocity of water in these
pathways is large even for small Ca values. This, in turn,
requires large pressure gradients in the water even for
small Ca values, and this causes mobilization of ganglia.
A further improvement in the calculation of Gm is
proposed here. Pore network calculations [7,9] have
shown that the local pressure of water at any node, even
under steady-state macroscopic two-phase ¯ow, ¯uctuates rapidly and by orders of magnitude as a result of
the cooperative eects of the motion of a multitude of
menisci throughout the network. These ¯uctuations
have an assymetric eect on ganglion mobilization.
Negative ¯uctuations in the value of ~
pj ÿ p~i do not
aect any ganglion that would not be mobilized by the
time-averaged value. On the contrary, a positive ¯uctuation may mobilize a ganglion that would not be
mobilized by the time-averaged value. On the other
hand, a mobilized ganglion is relatively unaected by
pressure ¯uctuations when its downstream meniscus is
not in a throat, which is a considerable part of its lifetime. Thus, pressure ¯uctuations facilitate and promote
ganglion mobilization. This can be taken into account
by introducing a pressure ¯uctuation factor Cfl in
Eq. (26) to obtain
Gm bI;J
CaCfl
kkrw
27
in which Cfl can take values substantially larger than
unity.
In order to estimate Cfl in the present work, we assume that the local (dimensionless) pressure gradient on
the pore-scale, xmic , ¯uctuates around the macroscopic
pressure gradient, x, with a probability distribution P y,
where y ln xmic ÿ ln x=ra . We assume that the distribution F y is normal with mean 0 and variance 1.
The value of ra is determined by setting xmic ax;
y 3, where `a' is a `gain' parameter. (The eect of the
gain parameter will be examined using sensitivity analysis.) Clearly, the distribution of xmic is lognormal.
Lognormal distributions are characteristic of populations of entities that interact through birth±death phenomena. Eqs. (25) and (26) [or (27)], and the criterion
(23) form the basis for the estimation of the mobilization
probability for the various size classes of ganglia.
In order to estimate the n-class ganglion mobilization
probability, mG
n , we examine, for all ij combinations,
whether the estimated pressure dierence along the
ganglion is larger than the critical pressure dierence
required to induce mobilization. Using Eqs. (23), (25)
and (27), we examine whether the following inequality
holds:
396
L~G
n
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Z
1
ÿ1
o~
h
i
p
P ydy P c~ow J~dr;i h0R ÿ J~min; j h0A
o~z
8j; i 1; . . . ; 5:
28
The LHS of the above inequality is the smooth-®eld
approximation of the local pressure dierence along the
ganglion. jo~
p=o~zj is the macroscopic pressure gradient
~
and L~G
n n` is the ganglion length projected on the
macroscopic ¯ow direction. It should be noted that the
pressure gradient is calculated along with the rest of
the variables in the context of the present model, and it
inherently accounts for the crowding eect. P y is the
aforementioned normal distribution with y in the range
ÿ1 6 y 6 1. The RHS of inequality (28) expresses
the pressure dierence required to cause mobilization of
the ganglion. Based on the above, the oil ganglion mobilization probability of each ganglion class is a function
0
0
G
of the form mG
n mn x; a; hA ; hR ; xpm .
In Fig. 5, the mobilization probability for a class-3 oil
ganglion, mG
3 , is plotted against the reduced macroscopic
pressure gradient x for a typical set of system parameters used in the experiments in [2], and for dierent
values of the gain parameter a. It is clear that the
crowding eect has a strong impact on the range of the
pressure gradient over which ganglion mobilization is
prompt, namely the mobilization zone. On the left end
of this zone, oil ganglia as members of a dense population become mobilized at much lower pressure gradients compared with solitary ones. On the right end of
the zone, a higher pressure gradient is required for total
mobilization of the ganglion population. It is important
to note that the pressure ¯uctuation eect is relatively
insensitive to the gain parameter a, and so use of this
parameter introduces little arbitrariness. In all our simulations below, we set a 10.
3.2.6. Oil ganglion mean velocity, uG
n
From a pore-scale linear momentum balance (see
Appendix A), the mean reduced velocity for class-n
ganglia, uG
n x, is obtained
"
#
u~G
`~2 G
1
1
G
G
n x
n 1Bn
G x Ln ÿ
un x
v
Ca
U~ w
k~
ÿ
ÿ1
n ÿ 1AC;G
;
29
2AG;G
m
m
where
Abm hAbjik ijik
5
X
j;i;k1
C;G
BnE;G :
BG
n Bn
Abjik fj fi fk ;
30
31
Abjik and Bbn represent the reduced eects of the bulk
phases and the interphases on the total pressure drop in
b
[see, also Eq. (35)], when a
the corresponding cell, D~
pjik;n
G
total ¯owrate q~uc;n is maintained through the ganglion
cell; index `jik' speci®es the geometry of each cell,
whereas `b' speci®es the two-phase ¯ow pattern in it.
Index `n' speci®es the ganglion size class. Explicit expressions for Abjik and Bbn are given in the Appendix A
[see Eqs. (A.5) and (A.6)]. The mean of Abjik over all
possible jik combinations, i.e., Abm , is used to account for
Fig. 5. The mobilization probability of a class-3 oil ganglion, mG
3 , as a function of the reduced macroscopic pressure gradient, x. The results shown
are for a class-3 ganglion, this being either a solitary ganglion (a 1) or a member of a dense population and for various values of the crowding eect
coecient, a. For smaller oil ganglia, the curves are translated to the left, for larger ganglia, the curves are translated to the right.
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
the uncorrelated path of a typical ganglion as it migrates
downstream.
The disconnected-oil super®cial velocity, U~ o;DOF , may
www.elsevier.com/locate/advwatres
True-to-mechanism model of steady-state two-phase ¯ow in porous
media, using decomposition into prototype ¯ows
M.S. Valavanides 1, A.C. Payatakes *
Department of Chemical Engineering, Institute of Chemical Engineering and High Temperature Chemical Processes, University of Patras,
GR 265 00 Patras, Greece
Received 29 August 1999; received in revised form 30 March 2000; accepted 31 August 2000
Abstract
A true-to-mechanism model is proposed, which considers steady-state two-phase ¯ow in porous media (SS2uPM) as a composition of two prototype ¯ows, namely ganglion dynamics (GD) and connected-oil pathway ¯ow (CPF). Coupling of the prototype
¯ows is eected with the simple rule that the macroscopic pressure gradient is the same in both. For a given set of values of the ¯ow
system parameters, a domain of admissible ¯ow combinations is obtained. The solution is determined by assuming that each point in
this domain has equal probability of being `visited'. This leads to unique values for the ¯ow arrangement variables (FAV), the rate of
mechanical energy dissipation, and the relative permeabilities. The new model accounts for the non-linearity of the ¯ow as well as
for the eects of all the system parameters (notably those aecting interfaces), and its predictions are in very good agreement with
existing data. Ó 2001 Published by Elsevier Science Ltd.
Keywords: Porous media; Two-phase ¯ow; Relative permeabilities
1. Introduction
Two-phase ¯ow in porous media (2uFPM) occupies a
central position in enhanced oil recovery, the behavior
of liquid organic pollutants near the source in contaminated soils, etc. Conventional fractional ¯ow theory
assumes that the relative permeabilities are only functions of the water saturation S w (for a given porous
medium). This approach has been based on Richards's
postulate that each ¯uid ¯ows through its own separate
network of connected pathways and that all portions of
the disconnected non-wetting phase (say oil) are stranded
[10,17].
However, experimental investigations [1,2,4] have
shown that the disconnected oil contributes signi®cantly
(even exclusively) to the ¯ow and that relative permeabilities are strong functions of a large number of
other parameters
*
Corresponding author.
E-mail addresses: marval@iceht.forth.gr (M.S. Valavanides),
acp@iceht.forth.gr (A.C. Payatakes).
1
Now with IRC Help-Forward, Xenofontos 5, GR 105 57 Athens,
Greece.
kri kri S w ;Ca;r;j;cos h0A ;cos h0R ;Co;Bo;xpm ;flow history;
i o;w:
1
Here, Ca is the capillary number (de®ned as
cow , where l~w is the viscosity of water, U~ w
Ca l~w U~ w =~
the super®cial velocity of water, and c~ow is the interfacial
qw the oil/water ¯owrate ratio,
tension), r q~o =~
lw the oil/water viscosity ratio, h0A and h0R the
j l~o =~
limiting values of the advancing and receding contact
angles at nil velocity, Co the coalescence factor, Bo the
bond number, xpm a parameter vector composed of all
the dimensionless geometrical and topological parameters of the porous medium aecting the ¯ow (porosity,
genus, coordination number, normalized chamber and
throat size distributions, chamber-to-throat size correlation factors, etc.) and ®nally, `¯ow history' denotes the
way in which the steady-state conditions have been
achieved, for example, whether through initial imbibition or drainage. Similar observations hold for the
capillary pressure. In the following, a tilde is used to
characterize any dimensional physical quantity.
Note, in particular, that kro and krw increase strongly as
Ca increases (at ®xed saturation), which means that the
¯owrate vs pressure gradient relation is strongly nonlinear. The key question, then, is `What are the sources of non-linearity and of the other complex eects
0309-1708/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 6 3 - 4
386
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Notation
General notes:
A tilde () on top of a character
denotes a dimensional variable. Absence
of a tilde denotes that the variable is
dimensionless.
Superscipt A superscript attached to a variable
denotes characterization on the
macroscopic scale.
Subscript A subscript attached to a variable
denotes characterization on the microscopic scale.
0
0
Variables with prime (S w ; b0 ) are the
physically admissible values of the
corresponding plain variables.
Latin characters
a
crowding eect factor
A~
macroscopic cross-sectional area, normal
to the macroscopic ¯ow direction
reduced resistance of a jik-class cell in
A1U
jik
one-phase ¯ow
reduced bulk phase resistance of a
Abjik
jik-class cell for dierent ¯ow con®gurations
mean of Abjik over all possible jik combiAbm
nations, Eq. (30)
reduced interphasial resistance of any
Bbn
conducting cell of an n ganglion for
dierent ¯ow con®gurations
Bo
bond number
Ca
capillary number
pressure ¯uctuation factor, Eq. (27)
Cfl
CiX;G
fractional distributions of blocked cells
(side-gate unit cells) in any i-class
ganglion
fractional distributions of core cells in
CiC;G
any i-class ganglion
fractional distributions of end-gate cells
CiE;G
in any i-class ganglion
Co
coalescence factor
~T
uniform maximum pore-network depth
D
~ Cj
class-j chamber diameter
D
Gm
ganglion mobilization number, Eq. (22)
g~
gravity
reduced conductance of the eective
gef
porous medium
reduced eective conductance of the
g~efDOF
porous medium DOF region
1U
1U
; g~jik
conductance of a jik-class cell in onegjik
phase ¯ow (reduced and dimensional)
b
gjik;n
reduced conductance of a b-type
(b `E,G' , `C,G' or `X,G') jik-class cell
of a n-ganglion
g0
fj
fjek
Imax
J~jGC ~z; h
J~iGT ~z; h
G
Jjik
J~dr;i
J~min;j
k~
k~ef
kro ; krw
`~
L~G
n
mG
n
nG
i
n~G
i
n~G
Nd
NiG
P y
p~i ; p~j
pnG
b
Pjik;n
q~1U
q~o
q~w
q~o;CPF
q~o;DOF
q~w;DOF
q~1U
q~1U
uc
reduced uniform conductance of all cells
in the equivalent network
occurrence probabilities of class j
occurrence probabilities of any jik-class
cell
maximum ganglion size class
mean surface curvature of the o/w
interphase, as a function of its position in
a j-class chamber
mean surface curvature of the o/w
interphase, as a function of its position in
an i-class throat
mean (along a jik cell) reduced o/w mean
interphase curvature
drainage curvature of the receding o/w
interphase in an i-class throat
minimum curvature value of the o/w
interphase advancing in a j-class chamber
absolute permeability
eective porous medium absolute
permeability
relative permeabilities in oil and water
lattice network constant (node-to-node
distance)
n-class ganglion's length, projected in the
macroscopic ¯ow direction
assigned mobilization probability to any
n-class oil ganglion
reduced number density of the i-class
ganglia
number density of the i-class ganglia
characteristic distribution
network lattice dimensionality
total number of unit cells occupied by
any i-class ganglion
normal distribution of y
actual pressures in water next to
menisci
occurrence probability of any n-ganglion
in the ganglion cells domain
occurrence probabilities of the various
ganglion cells in the ganglion cells
domain
equivalent macroscopic ¯owrate of
one-phase in the DOF region
macroscopic ¯owrate of oil
macroscopic ¯owrate of water
mean oil ¯owrate in CPF
mean oil ¯owrate in DOF
mean water ¯owrate in DOF
¯owrate of the equivalent ¯uid in the
DOF region
¯owrate of an incompressible ¯uid
through a cell
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
q~G
uc;i
r
Sw
0
Sw
Sio;G
S o;G
S o;DOF
u1 ; u0ij
u~G
i
U~ o
U~ w
U~ o;G
U~ o;CPF
U~ o;DOF
U~ w;DOF
V~
V~ uc
V~i G
VmmG
VmG
W_
W_ o;CPF
W_
o;DOF
W~Tj
x
x0ij
xmic
xpm
y
~z
ÿo~
p=o~z
total ¯owrate through any i-class
ganglion conducting cells
oil/water ¯owrate ratio
water saturation
physically admissible value of water
saturation, see also Eq. (51)
speci®c oil saturation in any i-class
ganglion
average oil saturation in the ganglion
cells domain
disconnected-oil saturation in the DOF
region
bipolar coordinates
mean velocity of an i-class ganglion's
mass center
super®cial velocity of oil
super®cial velocity of water
oil super®cial velocity in the domain of
ganglion cells, Eq. (32)
oil mean super®cial velocity in CPF
oil mean super®cial velocity in DOF
water mean super®cial velocity in DOF
macroscopic control volume
average (over all classes) volume of the
unit cell
average actual volume of an i-class
ganglion
expected mean ganglion size of the
mobilized ganglia
expected mean ganglion size of the
overall ganglion population
reduced total rate of mechanical energy
dissipation
reduced rate of mechanical energy
dissipation in CPF
reduced rate of mechanical energy
dissipation in DOF
class-j throat width
reduced common pressure gradient
Cartesian coordinates
reduced microscopic pressure gradient
parameter vector composed of all the dimensionless geometrical and topological
parameters of the porous medium aecting the ¯ow (porosity, genus, coordination number, normalized chamber and
throat size distributions, chamber-tothroat size correlation factors, etc.)
reduced variable
macroscopic ¯ow direction
common pressure gradient
Greek symbols
b
the porous medium volume fraction
occupied by the connected oil
b0
bI;J
c~ow
1U
D~
pjik
b
D~
pjik;n
f
go;CPF
j
eA ; eR
h0A ; h0R
k~o ; k~w
j
l~o ; l~w
nij
q~o ; q~w
r
ra
U
vG
x
XPA
387
physically admissible value of b
factor de®ned by Eq. (25)
oil/water interfacial tension
pressure drop along any jik-class unit cell
during one-phase ¯ow through it
pressure drop along any jik-class cell of
an n-ganglion for dierent ¯ow
con®gurations
free parameter which regulates the
steepness of the ganglion size distribution
expected mean macroscopic connectedoil ¯ow fraction
contact angle
advancing and receding contact angles
limiting values of the advancing and receding contact angles (at nil velocity)
mobility of oil and water, respectively
oil/water viscosity ratio
dynamic viscosity of oil and water,
respectively
aspect ratio between an i-class throat and
a j-class chamber
mass density of oil and water, respectively
pore-network coordination number
variance of the distribution of the
logarithm of the microscopic pressure
gradient
expected mean macroscopic ¯ow quantity
tortuosity of the spine of any ganglion
fraction of all the ganglion cells over all
the DOF region cells
integration domain corresponding to all
physically admissible solutions
Abbreviations
CPF
connected-oil pathway ¯ow
C,G
core unit cell of a ganglion
DeProF
decomposition in prototype ¯ows
DTF
drop trac ¯ow
DOF
disconnected-oil ¯ow
E,G
end-gate unit cell of a ganglion
EMT
eective medium theory
FAV
¯ow arrangement variables
GCD
ganglion cells domain
GD
ganglion dynamics
LGD
large ganglion dynamics
PA
physically admissible
SFA
smooth ®eld approximation
SGD
small ganglion dynamics
SSFD
steady-state fully developed
WSCD
water saturated cell domain
X,G
non-conducting unit cell of a ganglion
2uPM
two-phase ¯ow in porous media
388
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
indicated by Eq. (1), and how can these phenomena be
modeled?'
2. Experimental observations on steady-state two-phase
¯ow in porous media (SS2uFPM)
Visual experimental studies [1,2] have shown that the
disconnected oil contributes substantially to the overall
¯ow of oil. The observed ¯ow behavior has been roughly
classi®ed into four ¯ow regimes, namely large ganglion
dynamics (LGD), small ganglion dynamics (SGD), drop
trac ¯ow (DTF), and connected-oil pathway ¯ow
(CPF). In the ®rst three of these regimes, all of the oil
¯ow is due to the motion of disconnected bodies of oil
(ganglia or drops). Even in the fourth regime, CPF
(which is achieved with relatively large Ca values), an
appreciable portion of the oil ¯owrate is due to the
motion of droplets and ganglia in between the connected-oil pathways. In many cases, the overall ¯ow is a
mixture of two or more of the aforementioned ¯ow
patterns. For reasons of economy, in the present work,
we often combine LGD and SGD into an overall ¯ow
regime, namely, ganglion dynamics (GD).
The complex behavior indicated by Eq. (1) becomes
comprehensible once the active participation of ganglia
and droplets in the ¯ow is acknowledged and taken into
account explicitly.
3. Decomposition in prototype ¯ows (DeProF) for steadystate two-phase ¯ow in porous media (SS2uFPM): model
formulation
A theoretical model is developed here which accounts
for the pore-scale mechanisms and the network wide
cooperative eects, and which is suciently simple and
fast for practical purposes. This model is based on the
concept of decomposition in prototype ¯ows (DeProF)
and is an improved version of an earlier work presented
in [15].
Brie¯y, we assume that the macroscopic ¯ow can be
decomposed into two prototype ¯ows, namely CPF and
disconnected-oil ¯ow (DOF). The latter comprises
LGD, SGD and DTF (although in the present work we
omit DTF; the contribution of DTF will be included in a
future work). Each prototype ¯ow has the essential
characteristics of the corresponding experimentally observed ¯ow patterns, in suitably idealized form, and so
the pore-scale mechanisms are incorporated in the prototype ¯ows. The cooperative eects among ganglia are
also incorporated in DOF making suitable use of eective medium theory (EMT). A key feature of the proposed model is that the two prototype ¯ows are
entwined by imposing the condition that the pressure
gradient is the same in both. This is a condition that is
supported both by theory and experiment [4]. We ®nd
that there is a domain of admissible combinations of the
prototype ¯ows that corresponds to any given set of
Fig. 1. (a) Actual ¯ow and its theoretical decomposition into prototype ¯ows: CPF and DOF; (b) conceptual replacement of the actual DOF region
with an eective porous medium layer, of absolute permeability k~ef , through which ¯ows `eective water', with a total ¯owrate equal to the sum of the
¯owrates of oil and water through DOF, Eq. (9). The eective permeability of this layer is de®ned by Eq. (10).
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
values of the system parameters. The solution is determined by assuming equal probability for each admissible combination and averaging. The solution depends
on the values of all the system parameters (see RHS of
Eq. (1)). A more detailed description follows.
Consider the general ¯ow in Fig. 1(a). Connected
pathways of oil (when they exist) occur in roughly
`parallel' pathways. In between these pathways, a fraction of the population of oil ganglia are mobilized by the
water-induced pressure gradient and migrate downstream. The total ¯ow is a mixture of two ¯ow patterns,
speci®cally CPF and DOF. The key dierence between
these ¯ow patterns is that in DOF, numerous menisci
are in motion and many `catastrophic' (irreversible) ¯ow
phenomena are dominant [2,4]. This aects the relative
magnitude of the rate of mechanical energy dissipation
caused by capillary eects compared to that caused by
bulk viscous stresses. Each ¯ow pattern prevails over
mesoscopic regions of the porous medium (ranging
from, say, 104 ±109 pores), whereas the macroscopic ¯ow
is homogeneous. (Macroscopic inhomogeneities are not
considered here.)
Consider, now, a macroscopic control volume, V~ ,
having a cross-sectional area A~ normal to the macroscopic ¯ow direction, ~z, Fig. 1. The physical system of
the porous medium, oil and water is characterized by the
values of the physicochemical parameters, namely
c~ow ; l~o ; l~w ; q~o ; q~w ; h0A ; h0R ; xpm . Oil and water are
continuously fed into the control volume with constant
¯owrates q~o and q~w , so that the operational (dimensionless) parameters Ca and r have constant values. In
the present analysis, disconnected oil is considered to be
composed of ganglia only. (The omission of droplets is
an acceptable ®rst approximation as it will become evident below.) In the CPF region, the oil retains its
connectivity and ¯ows with virtually one-phase ¯ow.
The porous medium volume fraction occupied by the
connected oil is denoted as b. The DOF regime is de®ned as the region composed of the rest of the unit cells,
and the DOF volume fraction equals 1 ÿ b. S w and b
are called ¯ow arrangement variables (FAV) because
they give a coarse indication of the prevailing ¯ow
pattern. One of the objectives of the present model
(DeProF) is to determine the values of S w and b that
conform with the externally imposed conditions. The
analysis is carried out on two length scales: a macroscopic scale (1012 pores, or more) and a microscopic
scale.
3.1. Macroscopic scale, and connection with the porenetwork quantities
The overall oil and water ¯owrates are partitioned
into the contributions of the connected and disconnected-oil regions: q~o;CPF ; q~o;DOF are the mean oil ¯owrates in CPF and DOF, respectively, whereas q~w;DOF is
389
the water ¯owrate in DOF, Fig. 1(a). Oil and water
mean super®cial velocities in CPF and DOF are de®ned
~
~
as U~ o;CPF q~o;CPF = bA,
U~ o;DOF q~o;DOF = 1 ÿ bA,
w;DOF
w;DOF
~
~
q~
= 1 ÿ bA, respectively. It is exand U
pedient to use reduced mean super®cial velocities de®ned by U m;n U~ m;n =U~ m , where m o, w and n CPF,
DOF. The disconnected-oil saturation in DOF is denoted by S o;DOF . The absolute permeability of the po~
rous medium is denoted by k.
As already mentioned, an essential feature of the
proposed method is that the two prototype ¯ows are
entwined by having common pressure gradient,
ÿo~
p=o~z. Strictly speaking, this holds true for steadystate fully developed two-phase ¯ow in porous media
(SSFD2uFPM) [2,4] (and only approximately so in regions with weak saturation gradients).
Using this condition, we obtain
!
~
o~
p
q~i
i K
i o; w
2
kr
ÿ
l~i
o~z
A~
as well as the following relation between the mobility
ratio, k~o =k~w , and the ¯owrate ratio, r:
k~o q~o
lo
k~ro =~
r:
krw =~
lw k~w q~w
3
We de®ne the reduced pressure gradient, x, as
!
k~
o~
p
ÿ
:
x
o~z
c~ow Ca
4
Assuming that in the CPF region oil ¯ows with virtually
one-phase ¯ow (i.e., neglecting the possible lubrication
eect of wetting ®lms), we can use Darcy's law to obtain
!
~
o~
p
k
:
5
ÿ
U~ o;CPF
o~z
l~o
Assuming that there is no water present in the CPF
region, a total water mass balance yields
q~w;DOF q~w ) 1 ÿ bU~ w;DOF U~ w
6
and a total oil mass balance yields
q~o;CPF q~o;DOF q~o ) bU~ o;CPF 1 ÿ bU~ o;DOF U~ o :
7
An overall oil balance gives
b 1 ÿ bS o;DOF 1 ÿ S w :
8
In the ®nal analysis, the macroscopic behavior of the
¯ow is the manifestation of pore-scale ¯ow phenomena
and pore-network cooperative eects.
Scaling-up of the ¯ow in the CPF region is relatively
simple. Oil ¯ow in the pores is virtually one-phase ¯ow
and, under creeping ¯ow conditions, the hydraulic
conductivity of each unit cell, Fig. 2, is a function of
local geometrical parameters only. The permeability of
390
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Fig. 2. A typical jik-class unit cell of the examined model porous medium. All dimensions are normalized with the lattice constant `~ 1221 lm. The
thick inclined arrow represents the orientation of the lattice skeleton relative to the macroscopic pressure gradient. In each chamber a bipolar
cylindrical system is used. The bipolar system is aligned so that its cylindrical axis coincides with the longitudinal axis of the chamber and its poles
reside at the intersection of the annulus of the cylinder with the longitudinal axes of the adjacent throats. u1 ; uoij and uoik are the limit surfaces of the
unit cell expressed in bipolar coordinates.
the CPF region can be obtained from the hydraulic
conductances of the unit cells through application of
EMT. This is the absolute permeability of the porous
medium, and it is strictly a function of pore geometry
and pore-network topology (or connectivity).
Scaling-up of the ¯ow in the DOF region is more
complex, because the two dierent ¯uids are simultaneously arranged within the pores separated by a
multitude of advancing, receding and stranded menisci.
Here, the hydraulic conductivity of any individual unit
cell is not only a function of the appropriate local geometrical parameters, but also of the ¯uid arrangement
within it, Fig. 3. Furthermore, even under creeping ¯ow
conditions, the hydraulic conductance in a unit cell will
be a function of the velocity of the menisci, because of
the dependence of the contact angles (advancing and
Fig. 3. A microscopic scale representation of a DOF region. An oil ganglion of size class 5 is shown. The vector at the bottom of the ®gure represents
~ is shown expanded. The thick
the oil-ganglion mean velocity. For simpler representation, all cells are shown identical and the lattice constant, `,
dashed line separates the ganglion cells domain (GCD) and the water saturated cells domain (WSCD).
391
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
receding) on the local displacement velocity. In order to
deal with this problem, we proceed as follows. First, we
replace the DOF region with an eective porous medium, through which ¯ows a one-phase ¯uid with
¯owrate, q~1U , that is equal to the sum of the ¯owrates of
water, q~w;DOF , and oil, q~o;DOF , through the DOF region,
Fig. 1(b). In order to ®x ideas, we assume that this
conceptual ¯uid has viscosity equal to that of water, and
we will call it `water'. (This choice does not cause loss of
generality, because the viscosity that is assigned to the
conceptual ¯uid is taken into account properly.) Thus,
we have
q~1U q~o;DOF q~w;DOF :
9
ef
~
Second, we de®ne the eective permeability, k , of
this conceptual medium by imposing Darcy's law for
one-phase ¯ow
!
~ef
o~
p
k
ÿ1
~ ÿ b
:
10
ÿ
q~1U A 1
o~z
l~w
Eqs. (9) and (10) along with (5)±(7) give
l~
k~ef g~efDOF w
`~
"
#
b
c~ow
ÿ k~ 1 ÿ bÿ1 ;
Ca 1 r
j
ÿo~
p=o~z
11
where g~efDOF is, by de®nition, the eective hydraulic
conductance. Eq. (11) shows that k~ef is a function of the
absolute permeability (of the actual porous medium) as
well as of the pressure gradient, the capillary number,
the ¯owrate ratio, the viscosity ratio, the interfacial
tension and the FAV, b.
Now, a basic tenet of the present theory is that the
eective hydraulic conductance, g~efDOF (and therefore,
k~ef ), can be calculated from the hydraulic conductances
of the unit cells composing the (actual) pore-network in
the DOF region by pressing into service EMT. To this
end, one needs to have the topology of the skeleton of
the pore network, and the size distribution of the hydraulic conductances of the unit cells. The latter, as
already mentioned, are functions not only of the unit
cell geometry, but also of the ¯ow arrangement in each
unit cell and the probability of occurance of each ¯ow
arrangement. These results are developed in the following section. Before proceeding, however, we must
note that EMT in its conventional form is used for
networks, each branch (here unit cell) of which has a
random but constant conductance. In the present work,
the conductance of each unit cell can only be considered
at any given instant as momentarily constant. Furthermore, the ¯ow arrangement in each unit cell and the
probability of occurrence of each ¯ow arrangement depend, as we shall see, on the pressure gradient itself.
However, we expect that EMT is still valid in a narrow
domain around the solution, because one can assume
that in such a narrow domain all hydraulic conductances are constant and equal to the values that correspond to the pressure gradient given by the solution.
(The results bear out this expectation.)
3.2. Microscopic scale
3.2.1. Model porous medium. Geometric and topologic
characteristics
The microscopic scale analysis involves the speci®c
geometric and topologic characteristics of the porous
medium under consideration. In order to be able to
validate the theoretical model by comparing its predictions with available experimental results, we carried out
the analysis adopting the topological and geometric
characteristics of the glass pore-network model that was
used in the experiments of Avraam and Payatakes [2]. It
should be noted that this choice does not constitute a
constraint on the validity of the approach. A similar
analysis can be made with 3-D pore networks, even
networks with coordination number r that varies from
node to node (within reason of course). The pore geometry can also be dierent, but the basic approach
would still be the same. As shown in [1], two-phase ¯ow
in 3-D networks diers from than in 2-D ones quantitatively, but not qualitatively.
The model pore network of [2] is of the chamber-andthroat type. The network skeleton is a square lattice
with node-to-node distance `~ 1221 lm, and arranged
so that the macroscopic ¯ow direction is parallel to one
family of diagonals. For this particular network, the
lattice dimensionality is Nd 2, and the coordination
number is r 4. (For a cubic network, we would have
Nd 3 and r 6.) The maximum pore depth is nearly
uniform and was experimentally determined to be
~ T 115 lm. The chamber size (diameter) distribution
D
and the throat size (width) distribution have discrete
(composed of ®ve classes each) nearly normal size dis~ Cj i 610 lm,
tributions with mean values hD
~
hWTj i 169 lm and standard deviations equal to 1/4 of
the corresponding mean values. The occurrence probabilities fj of each class along with the respective actual
characteristic dimensions of the chambers (diameters),
~ Cj , and the throats (widths), W~Tj , are given in Table 1.
D
It should be noted that the chamber and throat diameter values in Table 1 are somewhat dierent from
those reported in [2]. This is because [2] reported
Table 1
Occurrence probabilities fj of individual classes j of chambers and
throats and the associated actual characteristic dimensions, rep~ Cj , and the throat width, W~Tj
resented by the chamber diameter, D
j
1
2
3
4
5
fj %
~ Cj (lm)
D
W~Tj (lm)
16
330
111
21
470
139
26
610
169
21
750
195
16
890
222
392
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
nominal dimensions, i.e., the dimensions on the negative
mask that was used in the etching of the glass model.
The dimensions given here are the measured actual dimensions of the model pore network that was used in
the experiments of [2].
The chambers are modeled as normal circular cylinders with their longitudinal axes oriented perpendicular
to the network plane and located at the network nodes.
The throats are modeled as normal elliptic cylinders
with their longitudinal axes placed on the network
bonds. The basic unit (building block) of the pore network is the unit cell that comprises of a throat connecting the corresponding sectors (quarters) of the
chambers adjacent to it. A typical unit cell is depicted in
Fig. 2. It is denoted as a jik-class cell, because it is
composed of a j-class chamber and a k-class chamber
connected with an i-class throat. The thick dashed lines
represent the unit cell limits, and the dashed±dotted lines
represent the network lattice. The solid matrix of the cell
is shaded. V~ uc is the average (over all classes) volume of
~ the
the unit cell. All lengths are normalized with `,
network lattice constant. The basic geometrical characteristics of the cell, i.e., the reduced chamber diameters,
DCj , DCk , and the reduced throat width, WTi , are also
de®ned in Fig. 2. All other dimensions shown in this
®gure are explicit functions of DCj , DCk and WTi . In the
present work, we assume that there is no correlation
between the classes of throats, chambers and cells. Thus,
the occurrence probability of any jik-class unit cell, fjik ,
is given by
fjik fj fi fk :
12
1U
3.2.2. Reduced conductances for one-phase ¯ow, gjik
When the problem of one-phase ¯ow through this
network is considered, one can calculate the conductances of each unit cell solely from its geometrical
characteristics. These conductances represent the transfer function of each cell for one-phase creeping ¯ow. All
cells that belong to the same jik-class have the same one1U
1U
. The reduced conductances, gjik
,
phase conductance, g~jik
are given by
1U
1U
g~jik
gjik
1
l~
q~1U
l~
q~1U
l~
1Uuc 1U
1U uc
1U
3
3
3
1U
3
~
~
~
~
A
D~
p
~
q
Ajik l~q~uc =` `
`
jik
jik
uc `
1
12
u0ij u0ik ÿ 2u1
DT
ÿ1
64 WTi2 D2T
1 ÿ x0ij ÿ x0ik ;
13
p WTi3 D3T
where `~ is the network lattice constant, l~ the dynamic
viscosity of the ¯uid, q~1U
uc the ¯owrate through the unit
1U 1U
1U ~3
~
~
l
q
~
quc A1U
cell, D~
pjik
jik
uc =` the corresponding pressure
drop along the unit cell, and A1U
jik is the reduced resistance of the jik-class cell. Details concerning the
1U 1U
~
quc are given in
derivation of the transfer function D~
pjik
the Appendix A. The ®rst term in the brackets on the
RHS of Eq. (13) is the contribution of the ¯ow in the
two chambers, and the second term is the contribution
of the ¯ow in the throat.
1U
It is clear that gjik
is a function of only the geometrical characteristics of the jik-class cell. This is a typical
characteristic of creeping one-phase ¯ow.
3.2.3. Eective medium theory
The overall conductance of a network, such as the
one previously described, can be calculated numerically
by applying eective medium theory (EMT). The ®rst
step is to replace the original network with an equivalent
network that has the same skeleton as the prototype, but
is composed of uniform unit cells. In order to achieve
the same overall conductance, the uniform cells of the
equivalent network must have reduced conductances
equal to g0 , which is determined by solving the EMT
equation [13]
X
j;i;k
fjik
1U
g0 ÿ gjik
0;
1U
r=2 ÿ 1g0 gjik
14
where r is the pore-network coordination number. The
equivalence of these networks is expressed on the
macroscopic scale: their macroscopic transfer functions
relating super®cial velocities and pressure gradients
(Darcy's law) are identical. When Eq. (14) is solved in
~ of the original
terms of g0 , the absolute permeability, k,
network is analytically recovered through the relation
k~ gef `~2 ;
15
where gef is the reduced eective conductance of the
equivalent network, and is related to g0 by (Burganos
and Sotirchos [5])
r
g0 :
16
gef
2Nd
Note that for the network adopted here, and for all
regular orthonormal networks, gef g0 .
Using Eqs. (13)±(16) and the data in Table 1, we
obtain the theoretical value k~ 8:89 lm2 , which is in
excellent agreement with the experimental value
8:90 lm2 reported in [2].
When the problem of one-phase creeping Newtonian
¯ow in a cell is considered, the resistance to the ¯ow is
caused by viscous stresses only, and so the transfer
function of the cell is linear. In the case of DOF, where
two-phase ¯ow in a cell is common, things are more
complicated and non-linearity appears even in the case
of very slow Newtonian ¯ows. This case is treated
below.
3.2.4. Reduced conductances for the disconnected-oil ¯ow
(DOF) domain
A microscopic scale representation of a typical DOF
region is shown in Fig. 3. An oil ganglion having a
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
typical `cruising' con®guration [16,18] is shown at the
center. All the cells that accommodate parts of this (or
any other) oil ganglion are called ganglion cells and are
outlined with a thick dashed line.
For the purposes of the present work, we will use the
term ganglion in an extended way (rather than saying
network-ganglion) to denote the entire part of the pore
network that is composed of the ganglion cells of any
given oil ganglion (see, for example, the region outlined
with the thick dashed line in Fig. 3). The DOF region is
partitioned into two sub-regions: the sub-region that
comprises all ganglion cells and the sub-region that
comprises unit cells containing only water. These will be
called ganglion cell domain (GCD) and water saturated
cell domain (WSCD), respectively. The fraction of all the
ganglion cells over all the DOF region cells is denoted by
x, and will be called the GCD fraction. Thus, the
fraction of all the water saturated cells in the DOF region equals 1 ÿ x. (As it will be shown below, the
variable x will be eliminated in the system of DeProF
equations presented in this work. We introduce it here
for the sake of clarity in further developments. In a future extension of the present work in which drop trac
¯ow (DTF) will be included in the DOF domain, x will
be shown to be the appropriate third ¯ow-arrangement
variable.)
The ganglion sizes have a distribution that depends
on the ¯ow conditions (for a given porous medium and
¯uids system). Determination of this distribution will
be achieved along with the rest of the solution. Each
oil ganglion is said to belong to a certain class i with
i 1; Imax based on the number of chambers it occupies. At any instant, the total number of unit cells
occupied by any i-class ganglion (having reduced volume i) is NiG i 2Nd ÿ 1 1. The average actual
volume of an i-class ganglion is given by V~i G NiG V~ uc .
Imax is the upper truncation class of the ganglion size
distribution, and so no ganglia with volume greater
than NIGmax V~ uc appear. It is important to note that Imax is
an implicit function of x, and is a derived parameter. In
a typical situation, only a fraction of the ganglion
population will be mobilized. Moving ganglia have a
tendency to become aligned with the macroscopic ¯ow
direction, acquiring the so-called `cruising shape'
[16,18]. In the present work, this is taken into account
by assuming that all ganglia have a zigzag spine that is
aligned along the macroscopic pressure gradient. An
important parameter associated with the `zigzag' motion of the ganglion is the tortuosity of its spine, vG .
This is also the ratio of the length of its actual migration path over the eective migration length in the
macroscopic ¯ow direction. This parameter depends on
the porous medium geometrical and topologic characteristics and will be considered to be equal for all
ganglion sizepclasses.
For the case under consideration,
we set vG 2.
393
Unit cells in which ganglion menisci exist are called
gate unit cells in accordance with [14,16]. In the present
work, we assume that all ganglia are `wormlike' and
aligned (as much as their zigzag shape allows) with the
macroscopic ¯ow direction. Among all gate unit cells of
a mobilized ganglion, there are only two gate unit cells
in which rheons take place (one cell for the `xeron' and
one cell for the `hygron' [16]). Such cells will be denoted
with the index `E,G' (`end-gate,ganglion'). We assume
that the only potential end-gate unit cells are those located at the foremost part (for the hygron) and at the
aft-most part of the ganglion (for the xeron); for example unit cells 1 or 7 for the xeron and 6 or 12 for the
hygron in Fig. 3. All cells of any mobilized ganglion that
are completely saturated with oil, e.g., cells 8, . . ., 11 in
Fig. 3 will be called `core cells', and will be denoted with
the index `C,G' (`core,ganglion'). Any i-class mobilized
ganglion has i ÿ 1 `C,G' cells. The rest of the ganglion
cells (which are neither end-gate cells nor core cells)
contain oil and water separated by virtually immobile
menisci and are called side-gate unit cells.
Ganglion migration induces a corresponding ¯ow,
q~G
uc;i , of oil in the core cells and an equal total ¯owrate of
water and oil in the end-gate cells. This ¯ow must be
consistent with the mean velocity of the mass center of
the ganglion, u~G
i , i.e.,
~2 G
~G
q~G
uc;i u
i ` v :
17
The rest of the cells of a mobilized ganglion (side-gate
unit cells) are considered as non-conducting, and they
are denoted with the index `X,G' (short for `non-conducting,ganglion'). Any i-class mobilized oil ganglion
occupies 2i Nd ÿ 1 `X,G' cells, and, of course, all of the
NiG cells of any i-class stranded oil ganglion are `X,G'
cells. Therefore, the fractional distributions of the various ¯ow con®gurations of any i-class ganglion cells
(mobilized as well as stranded ganglia taken into account) are explicitly given by
G
CiX;G i 2Nd ÿ 1 1 ÿ mG
i i 1 =Ni ;
G
CiC;G mG
i i ÿ 1=Ni ;
18
G
CiE;G 2mG
i =Ni ;
where mG
i is the mobilization probability assigned to the
i-class oil ganglia, and will be discussed below. It is
easily veri®ed that CiX;G CiC;G CiE;G 1 given the
aforementioned expression for NiG .
The number density of the i-class ganglia,
n~G
;
i 1; Imax , is de®ned as the number of i-class gani
glia per unit volume of all the ganglion cells, or, better,
per GCD unit volume. Expressing the volume of ganglion cells in the DOF region in terms of the ganglion
size distribution, we obtain
394
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
x 1 ÿ bV~
Imax
X
i1
However, it will have a more substantial role to play in
the extended model formulation in which DTF will be
taken into account.
G ~ uc
n~G
x 1 ÿ bV~
i Ni V
)
Imax
X
i1
G ~ uc
1:
n~G
i Ni V
19
If all of the disconnected oil were redistributed into
class-1 ganglia (singlets), then the number density of
~ uc
singlets would be n~G
1= 2Nd V . We de®ne the reduced number density of the i-class ganglia,
G
~ uc
~G
nG
nG
fnG
i ; i 1; Imax g, as ni n
i =~
2~
i Nd V . Clearly,
G
ni is equal to the ratio of the total number of i-class
ganglion cells over the total number of all ganglion cells
in the DOF region. All fnG
i i 1; Imax g are unknown
variables, which will be determined in the context of the
DeProF model.
By de®nition, the GCD fraction, x, in terms of the
ganglion size distribution, is given by
PImax G G X
Imax
n~i V~i
1 ÿ bV~ i1
~G
20
n~G
x
i Vi :
1 ÿ bV~
i1
Sio;G ,
corresponds to the i-class
A speci®c oil saturation,
of ganglia. This saturation is easily determined by assuming that all core cells (`C,G' cells) are completely
saturated with oil whereas all other ganglion cells are
saturated half with oil and half with water. By making
an oil volume balance over all cells of this ganglion class,
one obtains Sio;G iNd =NiG . The average oil saturation
in the GCD, S o;G , in terms of the reduced ganglion size
distribution is given by
PImax G G o;G
n~i V~i Si
x 1 ÿ bV~ i1
) S o;G
S o;G
x 1 ÿ bV~
Imax
1X
S o;DOF
:
inG
i
2 i1
x
21
As in the case of x; S o;G is also a variable of temporary
usefulness if one neglects oil droplets (as we do here).
3.2.5. Probability of oil ganglion mobilization, mG
n
In a typical situation, only a fraction of the oil ganglion population is mobilized. We assign a probability of
mobilization, mG
n , to ganglion class-n. This probability is
estimated by applying, in a statistical sense, the generalized oil ganglion mobilization criterion ([14,16]) appropriately extended to account for the crowding eects
that emerge in the presence of a dense population of oil
ganglia.
Consider a typical ganglion that is placed within a
pore network through which water ¯ows from left to
right driven by a macroscopic pressure gradient, ÿr~
p,
Fig. 4. In order to obtain a conservative estimate of the
pressure gradient that is required to cause mobilization
of the ganglion, we consider the case in which the endmenisci oer maximum capillary resistance. Thus, the
downstream meniscus is in a throat (denoted with i),
while the upstream meniscus is in a chamber (denoted
with j).
The curvature of the downstream meniscus, J~dr;i , is a
function of the size of the throat (not to be confused
with that of the i-class of throats) and the receding
contact angle, hR . For simplicity, we use the limiting
value of hR (as the receding velocity go to zero), h0R , irrespective of the actual velocity of the meniscus. The
minimum possible curvature of the upstream meniscus,
J~min;j , is a function of the size of the local chamber (not
to be confused with that of the j-class of chambers) and
the advancing contact angle, hA . Again, for simplicity,
we use the limiting value of hA , namely h0A , irrespective
of the actual velocity of the meniscus. (The dependence
of hR and hA on the corresponding meniscus velocity can
be introduced readily; this would produce a small bene®t
Fig. 4. A ganglion at a `hard-to-move' con®guration in the time interval between two rheons. (Emphasis on the end-gate unit cells.)
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
at the expense of substantially more cumbersome calculations.)
Following [14,16], the appropriate ganglion mobilization number is de®ned by
Gm max
i;j
h
~
pj ÿ p~i
c~ow J~dr;i h0R ÿ J~min; j h0A
i;
22
where p~i and p~j are the actual pressures in the water next
to the menisci with indices i and j, respectively. In (22),
index i scans all the throats in the gate unit cells, whereas
index j scans all the chambers. According to [14,16], the
criterion for ganglion mobilization is
If Gm > 1 ) mobilization;
If Gm < 1 ) stranding:
23
Calculation of the actual values of p~i and p~j requires
detailed pore-network simulations as in [6,7,9]. Whenever `exact' values of p~i and p~j are available, they should
be used. Otherwise, a rough estimate of the pressure
dierence ~
pj ÿ p~i can be obtained by applying smooth
®eld approximation (SFA).
For a solitary ganglion, Eq. (22) and use of SFA give
Ca
k
~
with k k=`~2 , and
Gm bI;J
DL~ij cos hji
i:
bI;J max h
i; j
`2 J~dr;i h0R ÿ J~min; j h0A
24
25
Here, DL~ij cos hji is the length of the line connecting
points i and j projected along the macroscopic pressure
gradient, Fig. 4; I and J are the values of i and j, respectively, for which the RHS of (25) becomes maximum. (In the case of `wormlike' ganglia that are aligned
with the macroscopic pressure gradient, positions I and
J usually correspond to the `nose' and `tail' of the ganglion, respectively. This is adopted in the present work
as the standard case.)
If one introduces typical parameter values in (24) and
(25) and applies the mobilization criterion (23), then it is
easy to conclude that a typical ganglion remains
stranded unless the value of Ca becomes of the order of
(10ÿ4 ) to (10ÿ3 ) or larger depending on ganglion size.
However, ganglion mobilization becomes possible at
much smaller Ca values as in the case of dense populations of ganglia. As discussed in [14,16], in the case of
populations of ganglia, Eq. (24) must be modi®ed as
follows:
Gm bI;J
Ca
population of ganglia;
kkrw
26
where krw is the relative permeability to water. Here, krw
accounts for the `crowding eect' and it can take values
much smaller than unity when the population of ganglia
becomes very dense. Consequently, in the case of
395
ganglion populations, ganglia can move even at Ca
10ÿ5 . On the pore-scale, the explanation is that the presence of a dense population of ganglia reduces the number
of pathways that are available for water ¯ow to relatively
few. Thus, the interstitial velocity of water in these
pathways is large even for small Ca values. This, in turn,
requires large pressure gradients in the water even for
small Ca values, and this causes mobilization of ganglia.
A further improvement in the calculation of Gm is
proposed here. Pore network calculations [7,9] have
shown that the local pressure of water at any node, even
under steady-state macroscopic two-phase ¯ow, ¯uctuates rapidly and by orders of magnitude as a result of
the cooperative eects of the motion of a multitude of
menisci throughout the network. These ¯uctuations
have an assymetric eect on ganglion mobilization.
Negative ¯uctuations in the value of ~
pj ÿ p~i do not
aect any ganglion that would not be mobilized by the
time-averaged value. On the contrary, a positive ¯uctuation may mobilize a ganglion that would not be
mobilized by the time-averaged value. On the other
hand, a mobilized ganglion is relatively unaected by
pressure ¯uctuations when its downstream meniscus is
not in a throat, which is a considerable part of its lifetime. Thus, pressure ¯uctuations facilitate and promote
ganglion mobilization. This can be taken into account
by introducing a pressure ¯uctuation factor Cfl in
Eq. (26) to obtain
Gm bI;J
CaCfl
kkrw
27
in which Cfl can take values substantially larger than
unity.
In order to estimate Cfl in the present work, we assume that the local (dimensionless) pressure gradient on
the pore-scale, xmic , ¯uctuates around the macroscopic
pressure gradient, x, with a probability distribution P y,
where y ln xmic ÿ ln x=ra . We assume that the distribution F y is normal with mean 0 and variance 1.
The value of ra is determined by setting xmic ax;
y 3, where `a' is a `gain' parameter. (The eect of the
gain parameter will be examined using sensitivity analysis.) Clearly, the distribution of xmic is lognormal.
Lognormal distributions are characteristic of populations of entities that interact through birth±death phenomena. Eqs. (25) and (26) [or (27)], and the criterion
(23) form the basis for the estimation of the mobilization
probability for the various size classes of ganglia.
In order to estimate the n-class ganglion mobilization
probability, mG
n , we examine, for all ij combinations,
whether the estimated pressure dierence along the
ganglion is larger than the critical pressure dierence
required to induce mobilization. Using Eqs. (23), (25)
and (27), we examine whether the following inequality
holds:
396
L~G
n
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
Z
1
ÿ1
o~
h
i
p
P ydy P c~ow J~dr;i h0R ÿ J~min; j h0A
o~z
8j; i 1; . . . ; 5:
28
The LHS of the above inequality is the smooth-®eld
approximation of the local pressure dierence along the
ganglion. jo~
p=o~zj is the macroscopic pressure gradient
~
and L~G
n n` is the ganglion length projected on the
macroscopic ¯ow direction. It should be noted that the
pressure gradient is calculated along with the rest of
the variables in the context of the present model, and it
inherently accounts for the crowding eect. P y is the
aforementioned normal distribution with y in the range
ÿ1 6 y 6 1. The RHS of inequality (28) expresses
the pressure dierence required to cause mobilization of
the ganglion. Based on the above, the oil ganglion mobilization probability of each ganglion class is a function
0
0
G
of the form mG
n mn x; a; hA ; hR ; xpm .
In Fig. 5, the mobilization probability for a class-3 oil
ganglion, mG
3 , is plotted against the reduced macroscopic
pressure gradient x for a typical set of system parameters used in the experiments in [2], and for dierent
values of the gain parameter a. It is clear that the
crowding eect has a strong impact on the range of the
pressure gradient over which ganglion mobilization is
prompt, namely the mobilization zone. On the left end
of this zone, oil ganglia as members of a dense population become mobilized at much lower pressure gradients compared with solitary ones. On the right end of
the zone, a higher pressure gradient is required for total
mobilization of the ganglion population. It is important
to note that the pressure ¯uctuation eect is relatively
insensitive to the gain parameter a, and so use of this
parameter introduces little arbitrariness. In all our simulations below, we set a 10.
3.2.6. Oil ganglion mean velocity, uG
n
From a pore-scale linear momentum balance (see
Appendix A), the mean reduced velocity for class-n
ganglia, uG
n x, is obtained
"
#
u~G
`~2 G
1
1
G
G
n x
n 1Bn
G x Ln ÿ
un x
v
Ca
U~ w
k~
ÿ
ÿ1
n ÿ 1AC;G
;
29
2AG;G
m
m
where
Abm hAbjik ijik
5
X
j;i;k1
C;G
BnE;G :
BG
n Bn
Abjik fj fi fk ;
30
31
Abjik and Bbn represent the reduced eects of the bulk
phases and the interphases on the total pressure drop in
b
[see, also Eq. (35)], when a
the corresponding cell, D~
pjik;n
G
total ¯owrate q~uc;n is maintained through the ganglion
cell; index `jik' speci®es the geometry of each cell,
whereas `b' speci®es the two-phase ¯ow pattern in it.
Index `n' speci®es the ganglion size class. Explicit expressions for Abjik and Bbn are given in the Appendix A
[see Eqs. (A.5) and (A.6)]. The mean of Abjik over all
possible jik combinations, i.e., Abm , is used to account for
Fig. 5. The mobilization probability of a class-3 oil ganglion, mG
3 , as a function of the reduced macroscopic pressure gradient, x. The results shown
are for a class-3 ganglion, this being either a solitary ganglion (a 1) or a member of a dense population and for various values of the crowding eect
coecient, a. For smaller oil ganglia, the curves are translated to the left, for larger ganglia, the curves are translated to the right.
M.S. Valavanides, A.C. Payatakes / Advances in Water Resources 24 (2001) 385±407
the uncorrelated path of a typical ganglion as it migrates
downstream.
The disconnected-oil super®cial velocity, U~ o;DOF , may