Transport in Porous Media 30: 267–299, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
267
Mechanistic Model of Steady-State Two-Phase Flow in
Porous Media Based on Ganglion Dynamics
M. S. VALAVANIDES, G. N. CONSTANTINIDES and A. C. PAYATAKES
Department of Chemical Engineering, University of Patras and Institute of Chemical Engineering
and High Temperature Chemical Processes, PO Box 1414, GR 265 00 Patras, Greece
(Received: 28 October 1996, in final form: 19 November 1997)
Abstract. Recent experimental work has shown that the pore-scale flow mechanism during steadystate two-phase flow in porous media is ganglion dynamics (GD) over a broad and practically
significant range of the system parameters. This observation suggests that our conception and theoretical treatment of fractional flow in porous media need careful reconsideration. Here is proposed
a mechanistic model of steady-state two-phase flow in those cases where the dominant flow regime
is ganglion dynamics. The approach is based on the ganglion population balance equations in combination with a microflow network simulator. The fundamental information on the cooperative flow
behavior of the two fluids at the scale of a few hundred pores is expressed through the system
factors, which are functions of the system parameters and are calculated using the simulator. These
system factors are utilized by the population balance equations to predict the macroscopic behavior
of the process. The dependence of the conventional relative permeability coefficients not only on
the wetting fluid saturation, Sw , but also on the capillary number, Ca, the viscosity ratio, κ, the wettability (θa0 , θr0 ), the coalescence factor, Co, as well as the porous medium geometry and topology
is explained and predicted on a mechanistic basis. Sample calculations have been performed for
steady-state fully developed (SSFD) and steady-state nonfully developed (SSnonFD) flow conditions. The number distributions of the moving and the stranded ganglia, the mean ganglion size, the
fraction of the nonwetting fluid in the form of mobile ganglia, the ratio of the conventional relative
permeability coefficients and the fractional flows are studied as functions of the system parameters
and are correlated with the flow phenomena at pore level and the system factors.
Key words: two-phase flow, ganglion dynamics, relative permeability, population balance equations, oil recovery, soil remediation.
Nomenclature
Latin Letters
a
Bo
bv
bwv
Ca
ij
Cf (w, v)
flow parameter vector, a = {Ca, κ, cos θa0 , cos θr0 , Co, So , x}T .
Bond number.
reduced side of the cross-sectional area of a rectangular.
parallelopiped of volume equal to that of a v-ganglion.
(used for the calculation of the ganglia collision rate kernels).
reduced side of the cross-sectional area of a conceptual
track in which v- and w-ganglia move and collide
(used for the calculation of the ganglia collision rate kernels).
capillary number, Ca = µ̃w Ũw /γ̃ow .
crowding effect factor for collisions among w- and v-ganglia
(i, j = 0, 1).
Thomson Press India, PIPS MATHKAP
tipm1265.tex; 8/04/1998; 12:03; no v.; p.1
268
Co
fw
kro
krw
l˜
Lg (v; a)
m(w, v)
n(z, t; v)1v
ñ(z̃, t˜; ṽ)1ṽ
P ij (w, v)
P̃
q̃
r
R ij (w, v; a)1w1v
So,in
S
s(v; a)
t
tR
Tm , Ts
ũg
uz (v; a)
Ũ
w, v
ṼCEVS
W (w, v : a)1w
x
z
M. S. VALAVANIDES ET AL.
coalescence factor, probability of coalescence of two colliding ganglia.
fractional flow of water, Equation (12).
conventional relative permeability to oil.
conventional relative permeability to water.
mean node-to-node distance of the pore network, characteristic length.
reduced maximum length of the v-ganglia, projected
in the direction of the macroscopic flow.
mean reduced length between two successive collisions among
w- and v-ganglia.
reduced number density of mobilized v-ganglia, Equation (3).
number of moving ṽ-ganglia per unit reservoir volume.
probability that a site of the pore network is not occupied by a ganglion
with volume other than w or v (i, j = 0).
macroscopic pressure.
flowrate.
oil/water flowrate ratio.
dimensionless rate of collisions between w- and v-ganglia.
The index i is set equal to 1 or 0 depending on whether
the w-ganglion is moving or not, whereas, the index j
pertains to the v-ganglion.
oil saturation at the inlet of the porous medium.
saturation.
probability of stranding of a newly formed v-ganglion.
reduced time.
reduced expected time between successive collisions among v- and
w-ganglia.
upper truncation classes for moving and stranded ganglia,
used for the numerical solution of the p.b.e.
instant velocity of the centroid of a ganglion.
reduced mean time-averaged velocity of the centroids of
the v-ganglia in the macroscopic flow direction.
superficial velocity.
reduced ganglion volumes, Equation (3).
average volume of a Conceptual Elemental Void Space
(mean volume of a chamber and its adjoining half throats),
characteristic volume.
probability with which a w-ganglion breaks into
a v- and a (w − v)-ganglion.
vector of all the dimensionless geometrical and topological parameters
of the porous medium affecting the flow.
reduced Cartesian coordinate, direction of the macroscopic flow.
Greek Letters
γ̃ow
θa0 , θr0
κ
λ0 (z, v; a)
λ(v; a)
λwv
µ̃
ξo
oil-water interfacial tension.
porosity.
incipient advancing and receding contact angles, respectively.
oil/water dynamic viscosity ratio.
dimensionless local stranding coefficient.
dimensionless mean stranding coefficient.
reduced mean distance between neighboring
v-ganglia (used for the calculation of the ganglia collision
2 ]−1 .
rate kernels), λwv = [n(z, t; v)1vbwv
dynamic viscosity.
fraction of oil in the form of mobile ganglia.
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
σ (z, t; v)1v
σ̃ (z̃, t˜; ṽ)1ṽ
φ 0 (z, v; a)
φ(v; a)
269
reduced number density of stranded v-ganglia, Equation (3).
number of stranded ṽ-ganglia per unit reservoir volume.
dimensionless local breakup coefficient.
dimensionless mean breakup coefficient.
Subscripts
i, j
o
w
ith or jth class of ganglion volumes.
oil.
water.
Special Symbols
hi
∼
mean value.
variables with dimensions.
1. Introduction
Two-phase flow in porous media is of great practical interest in oil and gas production
with flooding processes. During the last decade it has also become a prominent topic
in hydrologic research (Celia, Reeves and Ferrand, 1995); specifically, it is considered
to be of key value in the assessment of the risk of contamination of groundwater by
liquid organic contaminants as well as in the design of soil reconstitution flushing
processes. For simplicity, we will use the terms ‘water’ and ‘oil’ to denote the wetting
and nonwetting fluids, respectively, with the understanding that the basic results apply
to other pairs of immiscible Newtonian fluids. (For example, instead of oil one could
use NAPL, for nonaqueous phase liquid.)
Here, we focus our attention on steady-state two-phase (SS28) flow in consolidated, or unconsolidated, granular porous media (Berea type rocks, sandpacks, soils,
etc.). This is the classical starting point, as well as the usual system of reference, in
this general area. Even though SS28 flow is not burdened with many of the complications that arise from transient effects (e.g. steep concentration gradients), it remains a
very complex process that is not yet fully understood, even though it has been studied
extensively, because of its central role in theory and practice.
Until recently, virtually every work in this area was based on the longstanding dogma-like postulate that each fluid flows through its own separate network of
interconnected pathways, and that all portions of disconnected oil are completely
stranded (Richards, 1931; Craig, 1971; Dullien, 1979; Honarpour and Mahmood,
1988). According to this postulate, the numerous interfaces between the two fluids
(menisci, etc.) remain virtually immobile and, therefore, do not affect the phenomenon appreciably. The intertwined pathways of the two fluids are also assumed
implicitly or explicitly, to be nearly independent of the total flowrate within a wide
range of values. Thus, it is assumed that each fluid flows through a pore network
embedded in a matrix, composed of the solid and the other fluid (Dullien, 1979).
Based on this postulate, one would expect that the relative permeabilities to the
two fluids, for a given porous medium, would be functions of only (or mainly) the
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M. S. VALAVANIDES ET AL.
saturation of the wetting fluid Sw . Thus, for fixed saturation, one would expect the
superficial velocity of each fluid to be linearly proportional to the macroscopic pressure gradient. In reality, however, the superficial velocities are found to be strongly
non-linear functions of the macroscopic pressure gradient. Furthermore, extensive
experimental investigations (for a brief review, see Avraam and Payatakes, 1995a,
1995b) have shown that the relative permeabilities are strong functions of a large
number of parameters, in addition to Sw , specifically,
kro = kro (Sw , Ca, r, κ, cos θa0 , cos θr0 , Co, Bo; x; flow history),
(1)
krw = krw (Sw , Ca, r, κ, cos θa0 , cos θr0 , Co, Bo; x; flow history),
(2)
where Ca is the capillary number (defined as, Ca = µ̃w Ũw /γ̃ow , where µ̃w is the
viscosity of water, Ũw is the superficial velocity of water, and γ̃ow is the interfacial
tension); r = q̃o /q̃w is the oil/water flowrate ratio; q̃o and q̃w are the oil and water
flowrates across a cross-sectional area of the porous medium, respectively; κ =
µ̃o /µ̃w is the oil/water viscosity ratio; µ̃o is the viscosity of oil; θa0 and θr0 are
the advancing and receding (dynamic) contact angles; Co is the coalescence factor
(probability of coalescence of two colliding ganglia); Bo is the Bond number; x is
a parameter vector composed of all the dimensionless geometrical and topological
parameters of the porous medium affecting the flow (porosity, genus, coordination
number, normalized chamber and throat size distributions, chamber-to-throat size
correlation factors, etc.); finally, ‘flow history’ denotes the way in which the steadystate conditions have been achieved, for example, whether through initial imbibition
or drainage. (Tildes indicate dimensional variables.)
A resolution of the discrepancy between postulated and actual flow behavior was
proposed recently (Avraam and Payatakes, 1995a; Avraam et al., 1994), based on a
parametric experimental study of pore level mechanisms of SS28 flow in model pore
networks. Optical observations were used to determine the flow regimes, and pressure
drop vs flowrate measurements to obtain the corresponding relative permeabilities. It
was found that disconnected oil contributes substantially to the flow of oil. Actually,
over a wide range of the system parameters (in the domain of practical interest)
the flow of oil takes place solely through the motion of ganglia and/or droplets.
The experimentally observed rich flow behavior was roughly classified in four flow
regimes, namely, large ganglion dynamics (LGD), small ganglion dynamics (SGD),
drop traffic flow (DTF), and connected pathway flow (CPF). In the first three of these
regimes, all of the oil flow is due to the motion of disconnected bodies of oil (ganglia
or drops). Even in the fourth regime, CPF (which is achieved with relatively high Ca
values, at least one to two orders of magnitude larger than those used in waterfloods),
an appreciable portion of the oil flowrate is due to the motion of droplets and ganglia
at the fringes of the connected pathways. The values of the relative permeabilities
were found to correlate strongly with the corresponding flow regimes in a way that
makes good physical sense.
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
271
Thus, the strongly nonlinear dependence of the relative permeabilities on the
pressure gradient finds its explanation in the continual creation, motion, fission and
coalescence of ganglia and drops, or (viewed in a slightly different way) in the continual creation, motion and destruction of a dense population of menisci and other
interfaces. Furthermore, the effects of the various parameters that affect interface
motion and stability on the relative permeabilities cease to be curious and become
self-evident. These observations suggest that a novel theoretical analysis of SS28
flow in porous media must be developed, one that is based on the actual pore-scale
flow mechanisms. At this point, we should mention that W. G. Gray and S. M. Hassanizadeh, in a series of publications (Hassanizadeh and Gray, 1991, 1993a, b; Gray
and Hassanizadeh, 1991a, b), have developed a macroscale thermodynamic theory
for the analysis of multiphase flow in porous media, including interfacial phenomena, by assigning full thermodynamic properties to the various boundary surfaces
(solid-water, solid-oil, oil-water) that separate the surfaces at the microscale. Thus,
they have formulated macroscopic balance equations for mass, linear momentum and
energy, as well as an inequality for the averaged entropy. We think that there exists a
direct connection between that theory and the complex flow regime behavior that was
described above. For instance, the various terms in their energy balance equation have
a one-to-one correspondence with the experimental single-phase (connected water
and disconnected oil) and interface flow phenomena that are reported in Avraam and
Payatakes (1995a). If one could express this correspondence in quantitative terms,
then one could hope to develop a quantitative theory capable of predicting the flow
regime that would prevail in any specific case, as well as the conditions that lead
to transitions from one flow regime to another. This, however, remains a somewhat
distant goal. In the present work, we attempt to develop a theoretical analysis by
restricting our attention to the important case of ganglion dynamics (GD), which
includes LGD and SGD.
Pore-scale models (Ng and Payatakes, 1980; Payatakes et al., 1980; Dias and
Payatakes, 1986a, 1986b; Constantinides and Payatakes, 1991) are used to simulate
the complex flow phenomena at pore level, and the dynamic flow behavior of solitary ganglia. Recently, a computer-aided three-dimensional microflow simulator of
steady-state ganglion population dynamics in consolidated porous media was presented by Constantinides and Payatakes (1996). This simulator calculates ganglion
population interactions and predicts the flow regime on the mesoscopic scale and the
relative permeabilities (Figure 1). At present, the simulator can be used only for the
LGD and SGD flow regimes (see above), but an extension to the other regimes is
possible. The main disadvantage of the microflow simulator is that, it requires very
long computational time to deal with very large ganglion populations.
A new approach for the study two-phase flow processes in porous media on mesoscopic and macroscopic scales, when GD is the main flow regime, is formulated
here. The approach is based on a model for the macroscopic description of ganglion
dynamics in combination with the aforementioned microflow network simulator. The
model for the macroscopic description of GD is analytical and was introduced by
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M. S. VALAVANIDES ET AL.
Figure 1. Ranges of number of pores over which the various models are applicable and
scale-up of the models.
Payatakes et al. (1980) and Payatakes (1982). It considers ganglia as members of a
large interacting population. Ganglion motion and interactions (ganglion dynamics)
are described by two integro-differential equations of the birth-death type, called ganglion population balance equations (Payatakes, 1982). The cooperative flow behavior of the ganglia for given values of the system parameters are taken into account
through the system factors, namely, the mean time-averaged ganglion velocity, the
stranding and breakup coefficients, the mode of ganglion breakup, the probability of
stranding of a newly formed ganglion, and the mean maximum length of a ganglion
of a given volume projected on the direction of macroscopic flow. All these factors
depend on the values of the system parameters. The system factors are calculated
using the microflow network simulator for a relatively small porous medium, whereas the solution of the population balance equations can be applied to an essentially
infinite porous medium (Figure 1). The main advantages of the proposed method are
the following:
(a) It takes into account the actual flow regime (GD), that is, the mechanisms through
which the two fluids move and interact in the pore space.
(b) It utilizes quantitative pore-scale information concerning the flow behavior
of ganglia, namely, the mean time-averaged ganglion velocities, the ganglion
stranding and breakup coefficients, the mode of ganglion breakup, etc. (The
relative permeabilities of the conventional fractional flow theory are merely
phenomenological.)
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
273
(c) It explains the dependence of the relative permeabilities on the values
of all the pertinent system parameters on a mechanistic basis and in a selfconsistent way.
The new approach is applied in two limiting cases of considerable practical interest,
namely, in steady-state fully developed (SSFD) and steady-state nonfully developed (SSnonFD) flow. The effects of the main dimensionless parameters (Sw , κ,
Co) on the number distribution of the moving and stranded ganglia, the fraction of
mobile oil mass, the oil/water flowrate ratio and the relative permeabilities are investigated (the values of Ca, wettability, and the porous medium characteristics are kept
constant).
Before moving on to the next section, it should be pointed out that the theoretical method depicted in Figure 1 can be extended so as to apply to the other flow
regimes, namely, DTF and CPF. To this end, the ganglion dynamics simulator will
have to be replaced with another that can deal with droplets, as well as ganglia and
connected pathways. At this point, it seems that such a simulator could be developed
using cellular automata. This development, however, will require considerable effort
because of the several inherent limitations of current cellular automata techniques
(they restrict the choice of density and viscosity values of the two phases in relatively narrow ranges, and they still require relatively long computation times for pore
networks of reasonable size).
2. Model Formulation
2.1. population balance equations
Consider an interacting ganglion population advancing in the z-direction of a homogeneous porous medium at a given capillary number, Ca. To keep things simple, let
us characterize the state of each ganglion by its volume and by whether it is moving
or not, neglecting any effects caused by different shapes and orientation. (However,
the predominant shapes of the moving ganglia are taken into account in calculating
the collision kernels, see Appendix.) Consider now the subpopulation of ṽ-ganglia,
namely ganglia with volume in the interval (ṽ−(1ṽ)/2, ṽ+(1ṽ)/2). Only a fraction
of this subpopulation becomes mobilized in a typical flow. Let us call ñ(z̃, t˜, ṽ)1ṽ
the number of moving ṽ-ganglia per unit reservoir volume, and σ̃ (z̃, t˜; ṽ)1ṽ the
number of stranded ṽ-ganglia per unit reservoir volume. The populations of moving
and stranded ganglia interact. Moving ṽ-ganglia may become stranded, they may
fission or they may collide and coalesce with other stranded or moving ganglia.
Stranded ganglia may collide with moving ones, coalesce with them and become
mobilized. All these phenomena are described by a system of two population balance equations (Payatakes, 1982). The first equation is a population balance of the
moving ṽ-ganglia, whereas, the second one is a population balance of the stranded
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M. S. VALAVANIDES ET AL.
ṽ-ganglia. These equations can be written in dimensionless form using the following
dimensionless variables:
t = t˜
uz =
z̃
z= ,
l˜
Ũw
,
˜
l
ũz
Ũw /
,
v=
ṽ
ṼCEVS
,
˜
λ(v; a) = λ̃(ṽ; a)l,
W (w, v; a)1v = W̃ (w̃, ṽ; a)
1ṽ
ṼCEVS
w=
w̃
ṼCEVS
,
˜
φ(v; a) = φ̃(ṽ; a)l,
(3)
,
R ij (w, v; a) = R̃ ij (w̃, ṽ; a)/Ũw l˜2 ,
˜
n(z, t; v)1v = ñ(z̃, t˜; ṽ) Sl 1ṽ,
3
o,in
˜
σ (z, t; v)1v = σ (z̃, t˜; ṽ) Sl 1ṽ,
3
o,in
where l˜ and ṼCEVS are the characteristic length and volume of the porous medium
respectively, [l˜ is the mean node-to-node distance of the pore network and ṼCEVS is
the mean volume of a chamber and its adjoining half throats, as they are defined in
Payatakes et al. (1980) and Constantinides and Payatakes (1989)]; is the porosity
of the porous medium; So,in is the oil saturation at the inlet of the porous medium; t is
the dimensionless time; z is the dimensionless distance; v and w are the dimensionless volumes of v- and w-ganglia, respectively; uz (v; a) is the dimensionless mean
time-averaged velocity of the centroids of the v-ganglia in the macroscopic flow
direction; λ(v; a) and φ(v; a) are the dimensionless stranding and breakup coefficients of moving v-ganglia, respectively; W (w, v; a)1w is the probability with
which a w-ganglion breaks into a v- and a (w − v)-ganglion; R ij (w, v; a) is the
dimensionless rate of collisions between v- and w-ganglia (the indices i and j refer to
the mobility state of the v- and w-ganglion, respectively, and are set equal to 1 if the
ganglion is moving, and 0 if the ganglion is stranded), and its calculation is presented
in the Appendix. Finally, a = {Ca, κ, cos θa0 , cos θr0 , Co, So , x}T is a parameter vector consisting of the capillary number, Ca, the oil/water viscosity ratio, κ, the cosines
of the incipient advancing and receding contact angles, θa0 and θr0 , the coalescence
factor (probability of coalescence of two colliding ganglia), Co, the oil saturation,
So , and all the dimensionless geometrical and topological parameters of the porous
medium affecting the flow, denoted with the vector x.
The population balance equations in dimensionless form become
∂
∂n(z, t; v)
+ [n(z, t; v)uz (v; a)]
∂t
∂z
= − n(z, t; v)uz (v; a)[λ(v; a) + φ(v; a)] − n(z, t; v)So (z)Co ×
×
Z +∞
0
R 11 (v, w; a)n(z, t; w) dw +
Z +∞
0
R 10 (v, w; a)σ (z, t; w) dw +
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
1
+ So (z)Co
2
Z v
0
R 11 (w, v − w; a)n(z, t; w)n(z, t; v − w) dw +
+ [1 − s(v; a)]So (z)Co
+ [1 − s(v; a)]
Z +∞
v
Z v
0
R (w, v − w; a)n(z, t; w)σ (z, t; v − w) dw +
10
W (w, v; a)uz (w; a)φ(w; a)n(z, t; w) dw.
(4)
∂σ (z, t; v)
= n(z, t; v)uz (v; a)λ(v; a) −
∂t
− σ (z, t; v)So (z)Co
Z +∞
0
R01 (v, w; a)n(z, t; w) dw +
Z v
+ s(v; a)So (z)Co
+ s(v; a)
Z +∞
v
0
R10 (w, v − w; a)n(z, t; w)σ (z, t; v − w) dw+
W (w, v; a)uz (w; a)φ(w; a)n(z, t; w) dw,
(5)
where s(v; a) is the probability of stranding of a newly formed v-ganglion.
For the sake of brevity the variables uz (v; a), λ(v; a), φ(v; a), s(v; a) and
W (w, v; a) will be called system factors.
The physical meaning of the dimensionless terms appearing in (4) and (5) was
given in Payatakes (1982).
Our results (see below, Figure 15) have shown that, for SSnonFD flow the oil
saturation, So , changes along z significantly. On the other hand, the mean timeaveraged ganglion velocity, uz (v; a), depends strongly on So (Figure 2). In order to
take into account the convective net efflux of moving v-ganglia due to changes of uz
along z, the second term of the left-hand side of Equation (4) is written in its full form.
In the initially proposed population balance equations, the aforementioned effect was
neglected (uz was assumed to be independent of z) based on the assumption of sparse
ganglion populations. The present version of the model applies to dense as well as
sparse ganglion populations.
2.2. discretization and numerical solution of population
balance equations
In the present work, the population balance equations are solved for two cases,
that is, for steady-state fully developed (SSFD) and for steady-state nonfully developed (SSnonFD) flow. Under these conditions the system of the coupled integrodifferential Equations (4) and (5) is simplified significantly. Under SSFD conditions,
the number concentrations of moving and stranded ganglia, n and σ respectively, are
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M. S. VALAVANIDES ET AL.
Figure 2. Dependence of the reduced mean time-averaged ganglion velocity, uz , on the
ganglion volume, v, and the water saturation, Sw , for Ca = 10−4 , κ = 3.35, Co = 0.15,
θa0 = 45◦ and θr0 = 35◦ . The relatively long error bars corresponding to Sw = 0.6 (confidence
level 95%) show that ganglion motion depends on the initial shape and orientation of the
ganglion, the local characteristics of the porous medium and the distribution of the two phases
in the nearby area.
independent both of t and z, whereas, under SSnonFD conditions, n and σ are independent of t. However, even the simplified population balance equations cannot be
solved easily. A further simplification is obtained as follows. The continuous range
of ganglion volumes is discretized in classes (Sastry and Gaschignard, 1981). Each
class has the same width, equal to ṼCEVS . Hence, the ith class contains ganglia in the
interval (i ṼCEVS − (ṼCEVS /2), i ṼCEVS + (ṼCEVS /2)). For simplicity, the reduced
number concentrations of moving and stranded ganglia of the class i are denoted as
ni and σi respectively, the reduced mean time-averaged velocity of ganglia of class
i is denoted as uzi , and so on. Therefore, the two population balance equations are
transformed to an infinite system of equations (each of these new equations corresponds to a specific ni or σi ), whereas the integrals of the two population balance
equations are transformed to infinite sums. This system can be truncated appropriately to minimize truncation errors, assuming that ganglia larger than a critical size
will not appear. This critical size depends on the values of the system parameters and
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
is determined by trial and error. Let us call Tm and Ts the upper truncation classes of
moving and stranded ganglia respectively.
(a) Steady-State Fully Developed (SSFD) Flow
After discretization and truncation, the two population balance equations become a
system of Tm + Ts nonlinear algebraic equations, as follows:

0 = − ni uzi (λi + φi ) − ni So Co 
Tm
X
j =1
11
Rij
nj
Ts
X
j =1

10 
Rij
σj +
i−1
X
1
Rj11i−j nj ni−j +
+ So Co
2
j =1

+ (1 − si ) So Co
i−1
X
j =1
0 = ni uzi λi − σi So Co
Tm
X
j =1

+ si So Co
i−1
X
j =1
Rj10i−j nj σi−j +
Tm
X

Wj i uzj φj nj 
(6)
j =i+1
01
Rij
nj +
Rj10i−j nj σi−j +
Tm
X

Wj i uzj φj nj  .
(7)
j =i+1
The above algebraic system has Tm + Ts unknowns and Tm + Ts − 1 linearly independent equations. This can be easily proved by inductive reasoning. The additional
(Tm + Ts )th linearly independent equation comes from the total mass conservation
equation, that is
Tm
X
i=1
ini +
Ts
X
iσi = 1
(8)
i=1
and can replace one of the above equations.
The above system is solved numerically for ni and σi using the subroutine ZSPOW
of the software package IMSL. This subroutine is suitable for solving systems of
nonlinear algebraic equations. The algorithm is a variation of Newton’s method that
takes precautions to avoid large step sizes or increasing residuals (Moré et al., 1980).
The CPU time needed for the solution of the system for the values of parameters
used in this work is 5–10 min using a Hewlett Packard Apollo workstation (series
700 system).
(b) Steady-State non Fully Developed (SSnonFD) Flow
After discretization and truncation, the population balance equations for the SSnonFD
case become a system of Tm + Ts equations, from which the first Tm are first order
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M. S. VALAVANIDES ET AL.
ordinary differential equations and the rest Ts are algebraic ones, as follows:
uzi (a)
dni (z)
duzi (a)
= − ni (z)
− ni (z)uzi (a)[λi (a) + φi (a)] −
dz
dz

− ni (z)So (z)Co 
Tm
X
j =1
11
Rij
nj (z) +
Ts
X
j =1

10
Rij
σj (z) +
i=1
X
1
Rj11i−j nj (z)ni−j (z) +
+ So (z)Co
2
j =1

i−1
X
+ [1 − si (a)] So (z)Co
j =1
+
Tm
X
Rj10i−j nj (z)σi−j (z)+

Wj i (a)uzj (a)φj (a)nj (z)
(9)
j =i+1
0 = ni (z)uzi (a)λi (a) − σi (z)So (z)Co
Tm
X
j =1

+ si (a) So (z)Co
i−1
X
j =1
+
Tm
X
01
Rij
nj (z) +
Rj10i−j nj (z)σi−j (z)+

Wj i (a)uzj (a)φj (a)nj (z) .
(10)
j =i+1
This system is solved as follows. At position z, all ni populations are considered to be
known (for z = 0, the ni distribution is equal to the imposed inlet distribution). The
system of the Ts algebraic Equations (10) is solved using the ZSPOW subroutine.
Then, an explicit fourth order Runge–Kutta method with variable integration step
(Press et al., 1992) is used to solve the Tm differential Equations (9) and to calculate
the ni populations at z + 1z. During this integration, and at each internal step of the
Runge–Kutta method, the system of the σi populations (10) is also solved and the
system factors are updated accordingly, using the calculated value of the local So .
In certain cases in which the ganglion size distribution that is assumed at the inlet
(z = 0) as boundary condition differs substantially from the exact SSFD solution,
especially, when it contains too many large ganglia, a computational difficulty arises.
The combination of too many large ganglia, their large rate of breakup into small
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279
Figure 3. Dependence of the reduced stranding coefficient, λ, on the ganglion volume, v,
and the water saturation, Sw , for Ca = 10−4 , κ = 3.35, Co = 0.15, θa0 = 45◦ and θr0 = 35◦ .
daughter ganglia, Figure 4, and the large coefficient of stranding that characterizes
small ganglia, Figure 3, may lead to values of So that approach unity (oil choking).
This phenomenon was found to depend on how far from the exact SSFD solution is
the initial guess of the ni distribution. The problem was solved by assuming that the
system factors λi and si have zero values at z = 0, and reach their appropriate values
(imposed by the local oil saturation) gradually at z ≈ 2. A typical integration up to
z = 1000 (using the values Tm = 200, Ts = 50 as upper truncation limits) takes
approximately 10 h on a HP Apollo workstation (700 series).
3. Results and Discussion
3.1. system factors
Knowledge of the flow behavior of the individual ganglia as members of an interacting population is necessary to solve the population balance equations. The flow
behavior of ganglia is characterized by their velocities, the coefficients of stranding and breakup, the breakup mode, the probability of stranding of a newly formed
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Figure 4. Dependence of the reduced breakup coefficient, φ, on the ganglion volume, v, and
the water saturation, Sw , for Ca = 10−4 , κ = 3.35, Co = 0.15, θa0 = 45◦ and θr0 = 35◦ .
ganglion and the maximum length of the ganglion projected in the direction of the
macroscopic flow. These system factors depend on the values of Ca, κ, wettability
(dynamic contact angles), Co, Sw (or So ), the local geometrical and topological characteristics of the porous medium, and the size, shape and orientation of the ganglion
under consideration (keeping the porous medium fixed). The microflow network
simulator of steady-state two-phase flow in consolidated porous media developed
by Constantinides and Payatakes (1996) is used to calculate these system factors.
A detailed parametric study of the system factors has been given elsewhere (Valavanides et al., 1996). Here we restrict ourselves to certain typical results which will
help us to understand the behavior of the interacting ganglion population.
A full description of the microflow network simulator has been given elsewhere
(Constantinides and Payatakes, 1996). Here, we give just a brief description of the
simulator for the sake of completeness. The porous medium model used by the simulator is a three-dimensional network of unit cells of the constricted-tube type, suitable
for consolidated porous media (Constantinides and Payatakes, 1989). Typical experimental pore size distributions for a Berea sandstone (Dullien and Dhawan, 1975)
are used for assigning diameters to chambers and throats of the porous medium. For
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281
the purposes of the present study a pore network of 30 × 20 × 5 nodes (3000 chambers and ∼ 9000 throats) is used ( = 0.22, l˜ = 73.1 µm, ṼCEVS = 8610 µm3 ).
Under creeping flow conditions, the two-phase flow problem is solved with standard
network analysis (Dias and Payatakes, 1986a, 1986b; Constantinides and Payatakes,
1996). A typical simulation of steady-state ganglion dynamics begins by placing
ganglia of constant initial volume and random shapes at random sites in the network,
until a preselected value of oil saturation, So , is reached (the rest pore network is
filled with water). Then, a macroscopic pressure drop is applied (so that the capillary
number takes a preselected value) and the ganglia are let to move and interact. The
boundary conditions on the sides of the pore network parallel to the direction of the
macroscopic flow are assumed to be periodical. Ganglia leaving the network downstream are forced to reenter into the network upstream (for details, see Constantinides
and Payatakes, 1996). In this way, a kind of periodicity in the direction of flow and
a nearly constant So are achieved. Ganglion population motion and interactions can
be described as steady-state ganglion dynamics. Under steady-state conditions each
ganglion is identified and its migration is followed and recordered until a catastrophic event (breaking, stranding or coalescence) takes place. Each simulation lasts
for a sufficiently long time interval to take statistically reliable results for the system
factors.
3.1.1. Ganglion Velocities
One of the most fundamental system factors is the ganglion velocity. Owing to the
converging-diverging character of the flow channels and the randomness of the porous
medium, the velocity of a ganglion, ũg , varies with time and the local geometry (Dias
and Payatakes, 1986b; Hinkley et al., 1987). Hence, ganglion velocity is expressed in
terms of the time-averaged velocity with which the centroid of the ganglion migrates
downstream, ũ¯ g . For ganglia of a given volume, the time-averaged ganglion velocity depends on the initial shape and orientation of the ganglion, the local geometry
of the porous medium and the distribution of the two phases in the nearby area.
Hence, the calculated mean time-averaged ganglion velocity, ũz , is the average of
the velocities of all v-ganglion migrations that took place during the steady-state
ganglion dynamics time interval, that is ũz = hũ¯ g i. In reduced form, the mean timeaveraged ganglion velocity is expressed as uz = ũz /Ũw . The value uz is the relative
velocity of the ganglia to the interstitial velocity of water Ũw / for an oil-free pore
network. For each set of data an appropriate analytical function uz (v; a) is calculated
using weighted least-squares fitting, using as weight factors the number of ganglia
in each class. These functions are then used to solve the population balance
equations.
The dependence of uz on Sw is shown in Figure 2 (Ca = 10−4 , κ = 3.35,
0
θa = 45◦ , θr0 = 35◦ , Co = 0.15). As Sw decreases, the portion of the porous
medium through which water flows decreases and therefore the interstitial velocity
of water and the local pressure gradient increase (for Ca = constant). This, in turn,
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causes an increase of the ganglion velocities. In Figure 2, the confidence intervals
are shown for Sw = 0.6. The confidence intervals (confidence level 95%) increase as
v increases, because the ganglion population is composed mainly of relatively small
ganglia.
3.1.2. Stranding and Breakup of Ganglia
A stranding coefficient, λ0 (z, v; a), and a breakup coefficient, φ 0 (z, v; a), were
defined by Payatakes et al. (1980). These coefficients express the rates of stranding and breakup of moving v-ganglia as functions of the migration length and can be
calculated using the method described by Dias and Payatakes (1986b). In this work,
space-averaged values (independent of z) of the stranding and breakup coefficients
are used, λ(v; a) and φ(v; a), respectively. In order to calculate the space-averaged
values, λ0 and φ 0 are weighted using as weights the normalized number of ganglia at
each position z. Then, the appropriate fitting functions λ(v; a) and φ(v; a) are calculated using weighted least squares fitting, with weights the normalized number of
ganglia of each class (Valavanides et al., 1996), and are used to solve the population
balance equations.
Typical diagrams of the dependence of the stranding coefficient, λ, and the breakup
coefficient, φ, on the water saturation, Sw , are shown in Figures 3 and 4 (Ca = 10−4 ,
κ = 3.35, θa0 = 45◦ , θr0 = 35◦ , Co = 0.15; only the fitting curves are shown).
As Sw decreases, the stranding coefficient, λ, decreases, because the density ganglion population increases and the interactions between the ganglia become more
frequent. On the other hand, the breakup coefficient, φ, increases as Sw decreases
(the density of the ganglion population increases). This happens because both mechanisms for ganglion breakup, that is, dynamic breakup and pinch-off intensify as So
increases (keeping the values of the rest parameters constant). As the oil saturation
increases, the interstitial velocity of water and consequently the local pressure gradient become relatively large. These conditions favor dynamic breakup of the ganglia
(Payatakes and Dias, 1984). Moreover, as we explained above, mobilized ganglia
have to bypass many obstacles along their way, and flow not only in the direction of
flow but also in lateral directions. This kind of motion causes changes of the ganglion
shape and the formation of long oil threads which often rupture, thus intensifying
pinch-off.
Our results also show that as κ increases the stranding coefficient, λ, increases
and the breakup coefficient, φ, increases in agreement with previous works (Dias and
Payatakes, 1986b).
3.1.3. Mode of Breakup
Another important parameter in the dynamics of ganglion populations is the breakup
mode probability, W (w, v), defined as the probability that a moving w-ganglion will
break into two daughter ganglia, one of which is a v-ganglion (Dias and Payatakes,
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Figure 5. The ganglion breakup mode probability, W (w, v), used for the simulations (see
text).
1986b). To keep things simple we assume that W (w, v) remains the same for all κ
and Sw values examined. The fitting function of W (w, v) used for the solution of
ganglion population balance equations is shown in Figure 5. Relatively small ganglia
(v < 10) break mainly due to dynamic breakup (Payatakes and Dias, 1984) and the
two daughter ganglia have approximately equal volumes, whereas relatively large
ganglia break mainly due to pinch-off and the two daughter ganglia have significantly
different volumes.
3.1.4. Probability of Stranding of a Newly Formed Ganglion
A ganglion which is formed after the breakup of a mother ganglion, or after the
coalescence of two smaller ganglia, may become either mobilized or stranded. Typical
results of the probability of stranding of a newly formed ganglion, s, as a function
of v and Sw are shown in Figure 6 (Ca = 10−4 , κ = 3.35, θa0 = 45◦ , θr0 = 35◦ ,
Co = 0.15; only the fitting curves are shown). The probability s is an increasing
function of both Sw and κ (the same behavior as λ).
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Figure 6. Dependence of the probability of stranding of a newly formed ganglion, s, on the
ganglion volume, v, and the water saturation, Sw , for Ca = 10−4 , κ = 3.35, Co = 0.15,
θa0 = 45◦ and θr0 = 35◦ .
3.1.5. Maximum Length of the Ganglia Projected in the Macroscopic Flow
Direction
The reduced maximum length of the ganglia, projected in the direction of the macroscopic flow, Lg , is used in the model for the calculation of the collision rates (see
the Appendix). The results of the microflow network simulator have shown that Lg
decreases as κ increases for ganglia of the same volume, keeping the values of the
other parameters constant (Constantinides and Payatakes, 1996). For κ < 1, the ganglia show a tendency to become long and closely aligned in the direction of the
macroscopic flow. This tendency characterizes dynamic ganglion displacement (as
opposed to quasi-static), which is the predominant mode of displacement due to the
relatively high Ca value (Ca = 10−4 ) used here. Hence, for κ < 1, Lg is relatively
high. For κ > 1, the ganglia show a tendency to follow paths composed of large pores
in order to reduce viscous dissipation. This tendency increases as κ increases. Therefore, Lg decreases as κ increases. The dependence of the Lg on Sw is more complex.
On one hand, as Sw decreases, the interstitial velocity of water and the local pressure
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Figure 7. Dependence of the reduced mean maximum length of a ganglion projected in the
direction of the macroscopic flow, Lg , on the ganglion volume, v, and the water saturation,
Sw , for Ca = 10−4 , κ = 3.35, Co = 0.15, θa0 = 45◦ and θr0 = 35◦ .
gradients increase, increasing the dynamic character of the ganglion displacement.
Therefore, the tendency of the ganglia to become long and closely aligned with the
macroscopic flow (relatively high values of Lg ) increases. On the other hand, as Sw
decreases, ganglia move not only in the direction of the macroscopic flow but also
in the other directions due to the density of the population. This motion causes a
decrease of Lg , as Sw decreases. When the former mechanism is dominant, Lg is
a decreasing function of Sw . This behavior is observed for ganglia with volumes
v < 15 in Figure 7. For larger ganglia both mechanisms affect the value of Lg .
3.2. sample calculations for steady-state fully developed
flow
Here, we investigate the effects of changes in the main dimensionless flow-related
system parameters (Sw , κ, Co) on the macroscopic behavior of SSFD flow. The main
assumptions adopted are summarized below:
• The oil is totally disconnected in the form of ganglia of various sizes.
• The macroscopic flow is one-dimensional.
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• The longitudinal dispersion of ganglia is neglected.
• Gravity is neglected (but it can be readily included in the pressure gradient).
• In the integration of the population balance equations, the ganglia are considered
as points coinciding with their mass centers.
• The specific assumptions made for the calculation of the dimensionless collision
rate are summarized in the Appendix.
The assumptions employed in the development of the microflow simulator have been
described in Constantinides and Payatakes (1996).
The macroscopic quantities investigated include the number distributions of moving and stranded ganglia, the volume fraction of the mobile oil, the fractional flow,
and the relative permeabilities. The rest of the physicochemical and flow parameters
are kept constant at typical values, specifically, Ca = 10−4 , θa0 = 45◦ , and θr0 = 35◦ .
In all cases that are examined below, the porous medium is kept fixed (typical Berea
sandstone; see above).
3.2.1. Ganglion Distributions for Steady-State Fully Developed Flow
Figure 8 shows the number distributions of mobile and stranded ganglia, n and σ
respectively, under SSFD flow conditions as functions of the water saturation, Sw ,
for three characteristic oil/water viscosity ratio, κ, values (0.66, 1.45 and 3.35) and
Co = 0.15. The ganglion populations consist mainly of ganglia of relatively small
size (v < 10) in all cases. For relatively small Sw values (Sw = 0.3), the ganglion
populations are mainly composed of ganglia with v < 5. As Sw increases, the reduced
mean ganglion size hvi increases (Figure 9). As explained above, when Sw decreases, both mechanisms for ganglion breakup, that is, dynamic breakup and pinch-off,
intensify (for constant Ca = 10−4 ). Furthermore, as κ increases, the reduced mean
ganglion size hvi decreases (Figure 9). For κ > 1, ganglia tend to follow flowpaths
composed of large pores, thus reducing the rate of viscous dissipation (Constantinides and Payatakes, 1996). This tendency increases as κ increases. Consequently,
ganglia move not only in the direction of the macroscopic flow, but also in the lateral
directions and many long oil threads are formed. Hence, breakup by pinch-off is
enhanced (φ increases) for κ > 1 and the reduced mean ganglion size hvi is relatively small. The fraction of oil in the form of mobile ganglia, ξo , increases as κ
decreases. This behavior can be explained taking into account the effect of κ on the
size and the shape of the ganglia. The viscous force, which is responsible for ganglion mobilization is (roughly) proportional to the maximum length of the ganglion
projected in the direction of the macroscopic flow, Lg , and the local pressure gradient (Ng and Payatakes, 1980). As κ increases, both the reduced mean ganglion size
hvi (Figure 9) and Lg for ganglia of the same volume decrease (Constantinides and
Payatakes, 1996; see also the relevant discussion above). Consequently, the number
of stranded ganglia increases as κ increases (keeping all other parameters constant)
and ξo decreases.
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Figure 8. Steady-state fully developed mobilized, n, and stranded, σ , ganglion distributions
as functions of the water saturation, Sw , for three typical viscosity ratio values (κ = 0.66,
1.45 and 3.35) and Ca = 10−4 , Co = 0.15, θa0 = 45◦ and θr0 = 35◦ .
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Figure 9. Dependence of the reduced mean ganglion volume, hvi, on the saturation, for
steady-state fully developed flow conditions, three typical viscosity ratio values (κ = 0.66,
1.45 and 3.35) and Ca = 10−4 , Co = 0.15, θa0 = 45◦ and θr0 = 35◦ .
Figure 10 shows the dependence of the number distributions of the mobilized and
stranded ganglia, n and σ , respectively, on the water saturation, Sw , for two different
values of the coalescence factor (Co = 0.1 and 0.2; see also Figure 8 for Co = 0.15)
and κ = 1.45. For a relatively high Co value (Co = 0.2), when coalescence is
relatively prompt, the populations are composed of quite large ganglia, whereas for
a relatively small Co value (Co = 0.1), when coalescence is more difficult, the
populations are composed of substantially smaller ganglia. The microflow network
simulator data show that the system factors are roughly independent of Co. Hence,
for the simulations shown in Figure 10 we used the same system factors with those
used with Co = 0.15, κ = 1.45 (Figure 8).
The aforementioned results are in qualitative agreement with experimental results
reported by Avraam and Payatakes (1995a). (Quantitative agreement was not to be
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Figure 10. Steady-state fully developed mobilized, n, and stranded, σ , ganglion distributions
as functions of the water saturation, Sw , for two typical coalescence factor values (Co=0.1
and 0.2) and Ca = 10−4 , κ = 1.45, θa0 = 45◦ and θr0 = 35◦ .
expected, of course, given that the data under consideration were obtained using a
model pore network with 2-D topology.)
3.2.2. Relative Permeabilities for Steady-State Fully Developed Flow
The population balance equations cannot be used to calculate pressure gradients.
However, the macroscopic pressure gradients are taken from the microflow network
simulator along with the other system factors. The results of the microflow simulator
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have shown that the macroscopic pressure gradients in both fluids are almost identical (the maximum local difference is less than 5% in all cases examined). For this
reason, we can assume |∇P̃w | = |∇P̃o | for the needs of the present work. Using this
assumption, the relative permeability ratio, kro /krw , is calculated as
µ̃o Ũo
kro
=
= κr.
krw
µ̃w Ũw
(11)
Note, here, that under steady-state conditions and because both fluids are considered
incompressible, the oil/water flowrate ratio, r, is constant along the medium for given
values of the system parameters. The dependence of kro /krw on Sw and κ is shown
in Figure 11(a). These results show that kro /krw is a decreasing function of Sw and
an increasing function of κ. It is well known that kro is a decreasing function of
Sw , whereas krw is an increasing function of Sw and consequently, the ratio kro /krw
is a decreasing function of Sw . Furthermore, experimental (Avraam and Payatakes,
1995a) and theoretical works (Constantinides and Payatakes, 1996) have shown that,
as κ increases, both relative permeabilities increase, but that kro increases more than
krw , for constant Sw . Hence, the ratio kro /krw is expected to be an increasing function
of κ. The results in Figure 11(a) show these effects. To calculate kro and krw , the value
of krw is obtained from the network simulator, along with the other system factors,
and kro is set equal to κrkrw , according to Equation (11). Figure 11(b) shows the
dependence of krw and kro on Sw and κ. These results are also in qualitative (see
above) agreement with the experimental results reported by Avraam and Payatakes
(1995a).
The dependence of kro /krw on Sw and Co is shown in Figure 12(a). The ratio of
kro /krw increases substantially (12–24%) as Co increases from 0.10 to 0.20. Figure
12(b) shows the dependence of krw (calculated using the network simulator) and kro
(calculated using Equation 11) on Co and Sw . Both relative permeabilities increase
as the coalescence factor decreases (see also Constantinides and Payatakes, 1996).
3.2.3. Fractional Flows for Steady-State Fully Developed Flow
The fractional flow of water, fw , is defined by
fw =
q̃w
1
.
=
q̃w + q̃o
1+r
(12)
The dependence of fw on Sw and κ for Co = 0.15, is shown in Figure 13. As we
explained above, under SSFD conditions the flowrates of both fluids are constant at
every cross-section of the porous medium, and consequently fw is unique for a fixed
Sw value (with the other parameters constant). The fractional flow curves are usually
S-shaped, and depend strongly on the value of the viscosity ratio for Sw < 0.5,
whereas they almost coincide for Sw > 0.5.
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Figure 11. Dependence of (a) the relative permeability ratio, kro /krw , and (b) the relative
permeabilities, krw and kro , on the saturation, for steady-state fully developed flow conditions, three typical viscosity ratio values (κ = 0.66, 1.45 and 3.35) and Ca = 10−4 , Co =
0.15, θa0 = 45◦ and θr0 = 35◦ .
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Figure 12. Dependence of (a) the relative permeability ratio, kro /krw , and (b) the relative
permeabilities, krw and kro , on the saturation, for steady-state fully developed flow conditions,
for three typical coalescence factor values (Co = 0.10, 0.15 and 0.20) and Ca = 10−4 ,
κ = 1.45, θa0 = 45◦ and θr0 = 35◦ .
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Figure 13. Steady-state fully developed fractional flow curves, fw (Sw ), for three typical
viscosity ratio values (κ = 0.66, 1.45 and 3.35) and Ca = 10−4 , Co = 0.15, θa0 = 45◦ and
θr0 = 35◦ .
3.3. sample calculations for steady-state nonfully developed
flow
SSnonFD calculations are useful in a number of ways. To begin with, they can be
used to test the stability of the SSFD solutions that are obtained with the method of
the previous sections, by using the latter as inlet conditions and observing whether
they are preserved as they propagate or not. (They are.) They can also be used to study
problems related to end effects, including the following. What is the typical entrance
length required to reach the domain of fully developed flow? How does the form in
which the two fluids are fed into the porous medium affect the corresponding fully
developed flow deep in the porous medium? Is it possible to obtain different SSFD
flows by using different modes of feed, while retaining the same system parameter
values? Is it possible, under certain conditions, to have spatially chaotic behavior?
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Figure 14. The space evolution of two different inlet ganglion distributions (a uniform and
a triagonal) under steady-state nonfully developed conditions (Ca = 10−4 , κ = 3.35, Co =
0.15, θa0 = 45◦ and θr0 = 35◦ , So,in = 0.5, r = 0.3). The steady-state fully developed
ganglion distributions are the same in both cases.
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Figure 15. The space evolution of the oil saturation, So , and the fraction of oil in the form of
mobilized ganglia, ξo , for two steady-state nonfully-developed displacements with different
inlet ganglion distributions (Ca = 10−4 , κ = 3.35, Co = 0.15, θa0 = 45◦ and θr0 = 35◦ ,
So,in = 0.5, r = 0.3).
The main assumptions adopted for the SSnonFD case are the same with those for
SSFD. A typical simulation under SSnonFD conditions begins by selecting an initial
number ganglion distribution for the population of moving ganglia at the inlet, which
is then integrated with respect to position until the final steady-state fully developed
distribution is achieved. During the integration the oil/water flowrate ratio, r, remains
constant, (steady-state conditions). As a first check of the validity of the method, the
respective moving ganglion distributions calculated under SSFD flow conditions
were selected as inlet distributions. These distributions remained unchanged during
the integration in all the cases we examined, and the respective stranded ganglion
distributions were identical with those calculated under SSFD flow conditions.
Using the microflow network simulator, it has been shown that the steady-state
fully developed solution is independent (within reason) of the ganglion distribution
initially placed in the pore network (Constantinides and Payatakes, 1996). In order to
check the validity of this result with the present method, two different types of inlet
ganglion distributions were used, namely a uniform and a triangular one. Once the
bounds of such a distribution have been selected, the number of ganglia in each class
can be calculated, so that the flowrate ratio, r, be equal to the preselected value. For
a given set of values of the system parameters, we compare the developing solution
under SSnonFD conditions, with that for the corresponding SSFD flow. For κ = 3.35
the two inlet ganglion distributions (triangular and uniform) reach the corresponding
SSFD distributions for n and σ , exactly. Typical results are shown in Figure 14 for
Ca = 10−4 , κ = 3.35, Co = 0.15, θa0 = 45◦ , θr0 = 35◦ and Sw = 0.5, (r = 0.3).
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As it can be seen, the entrance length required to reach SSFD flow is approximately
˜
˜ The evolution of the respective macroscopic properties, that is, the local
500l–1000
l.
oil saturation, So , and the fraction of the mobile oil, ξo is shown in Figure 15. Both
So and ξo change rapidly near the inlet (z < 10). This is attributed to the (artificially
imposed, as explained above) gradual increase of the values of λi and si from zero
at z = 0 to the appropriate one at z = 2. Figure 15 shows that both So and ξo
show a spatially oscillating behavior during the initial steps of integration, which
disappears for large values of z. On the contrary, for κ = 0.66 the spatial oscillations
intensify as z increases, and we cannot reach a SSFD solution. For κ = 1.45 we
recover the SSFD solution for So values in the range 0.5 < So < 0.6. Although we
tried sophisticated integration procedures designed for stiff equations (semi-implicit
integration methods; Press et al., 1992), this instability problem could not be fixed.
At this point, it is not clear whether this is caused by a numerical instability, or it is
the manifestation of something inherent in the physics.
4. Conclusions
The recent experimental observations concerning the actual flow mechanisms during
steady-state two-phase flow in porous media make it necessary to reconsider the
theoretical analysis of such flows.
Here, a new mechanistic model for the analysis of two-phase flow in porous media,
when ganglion dynamics is the main flow regime, was developed. The approach is
based on the ganglion population balance equations in combination with a network
simulator. The cooperative flow behavior of the oil ganglia is expressed through the
system factors. The system factors depend on the system parameters and are calculated using the microflow simulator. The most important variables related with the flow
regime (moving and stranded ganglion number distributions, mean ganglion size,
fraction of oil in the form of moving or stranded ganglia, etc.) and the macroscopic
behavior (conventional relative permeabilities, fractional flows, etc.) can be calculated. The new approach was applied to typical cases of steady-state fully developed
and steady-state non fully developed two-phase flow in porous media. The theoretical
results show that SS28 flows are strongly affected by those parameters that affect
interface motion and stability. They also show that the relative permeabilities depend
strongly on the values of these parameters for fixed values of water (or oil) saturation,
or (viewed in a different way) they show the strong nonlinear relation between the
applied macroscopic pressure gradient and the superficial velocities of oil and water,
which is observed experimentally even under creeping flow conditions.
Appendix. Calculation of the Dimensionless Rate of Collisions, Rij (w, v; a)
For the purpose of calculating the kernels appearing in the integrals in Equations (4)
and (5) we need to take into account, among other things, the shapes of the ganglia.
To keep the analysis simple, and taking into account that ganglia show a tendency to
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MECHANISTIC MODEL OF TWO-PHASE FLOW IN POROUS MEDIA
297
become long, slender and closely aligned with the direction of the macroscopic flow
(Hinkley et al., 1987), the cruising shape of each ganglion is idealized as a rectangular
of volume equal to that of the ganglion, having its longer axis parallel to the z
direction. For a ganglion of a given volume v, the length of the parallelopiped in the
z-direction is equal to the mean maximum length of the ganglia of volume v projected
in the direction of the macroscopic flow, Lg (v). This length, Lg (v), is also a system
factor that depends on the values of the system parameters, and is calculated using
the network simulator (see above). The cross-sectional area of the parallelopiped is
square with side bv = (v/Lg )1/2 . The reduced rate of collisions between two moving
ganglia, say, a v- and a w-ganglion, R 11 (w, v; a)1w1v, is calculated as follows.
We assume (for the present purpose only) that the ganglia move along a bundle of
conceptual parallel ‘tracks’. Each track has square cross-section (space filling) with
side bwv = bw +bv . On such a track we place at random positions along the centerline
v-ganglia (in their fictitious parallelopipedal shape), in such a way that we retain
the local value of the number density of v-ganglia. The expected distance between
2 )−1 . Let us, now,
neighboring v-ganglia along the track is λwv = (n(z, t; v)1vbwv
introduce a single w-ganglion on this track. The expected time between successive
w − v collisions is tR = λwv /|uz (v) − uz (w)|. The expected number of moving
w-ganglia along a suitable length of the ‘track’, corresponding to a dimensionless
‘track’ volume equal to unity, is n(z, t; w)1w. The rate of collisions among moving
v- and w- ganglia per unit volume is given by
R 11 (w, v; a)1w1v
n(z, t; w)1w 11
Cf (w, v)
tR
2
|uz (w; a) − uz (v; a)|n(z, t; w)1w × n(z, t; v)1vCf11 (w, v),
= bwv
=
where Cf11 (w, v) is a crowding effect factor, which expresses the conditional probability that no other moving ganglion intrudes (through lateral motion from a neighboring track) in between the w- and v-ganglia that are on a collision course. This
quantity is estimated here as the probability that all the sites through which the downstream ganglion will pass during its migration and before the eventual w-v collision,
will be free of ganglia of volume other than v and w. This can be written as
Cf11 (w, v) = P 11 (w, v)m(w,v) ,
where P 11 (w, v) is the probability that a site of the pore network is not occupied
by a ganglion with volume other than w and v, and m(w, v) is the reduced length
between two successive w-v collisions. These quantities, in turn, are estimated by
P 11 (w, v) = 1 − So (1 − wn(z, t; w)1w − vn(z, t; v)1v),
m(w, v) = λwv .
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298
M. S. VALAVANIDES ET AL.
The reduced rate of collisions among moving v-ganglia and stranded w-ganglia can
be calculated in the same way, and has the form:
2
uz (v; a)n(z, t; v)σ (z, t; w)Cf01 (w, v).
R 01 (w, v; a) = bwv
It is important to note that in the case of sparse ganglion populations the crowding
effect factors tend to unity Cf11 (w, v) → 1, Cf01 (w, v) → 1, as So → 0).
Acknowledgement
This work was supported by the Institute of Chemical Engineering and High Temperature Chemical Processes.
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