Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

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 com-position of two prototype ¯ows, namely ganglion dynamics (GD) and connected-oil pathway ¯ow (CPF). Coupling of the prototype ¯ows is e€ected 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 e€ects of all the system parameters (notably those a€ecting 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 contam-inated soils, etc. Conventional fractional ¯ow theory assumes that the relative permeabilities are only func-tions of the water saturation Sw _{(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 thatallportions of thedisconnectednon-wetting phase (say oil) arestranded

[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 per-meabilities are strong functions of a large number of other parameters

kriˆk i

r…Sw;Ca;r;j;cosh0A;cosh 0

R;Co;Bo;xpm;flow history†;

iˆo;w: …1†

Here, Ca is the capillary number (de®ned as Caˆl~wU~

w_=~

cow, where l~w is the viscosity of water,U~ w

the super®cial velocity of water, and~cowis the interfacial

tension), rˆq~o_=q~w _{the oil/water ¯owrate ratio,}

jˆl~o=l~w the oil/water viscosity ratio, h 0

A and h

0

R the

limiting values of the advancing and receding contact angles at nil velocity, Cothe coalescence factor, Bothe bond number, xpm a parameter vector composed of all

the dimensionless geometrical and topological parame-ters of the porous medium a€ecting the ¯ow (porosity, genus, coordination number, normalized chamber and throat size distributions, chamber-to-throat size corre-lation factors, etc.) and ®nally, `¯ow history' denotes the way in which the steady-state conditions have been achieved, for example, whether through initial imbib-ition 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, thatko

r andk

w

r increase stronglyas

Ca increases (at ®xed saturation), which means that the ¯owrate vs pressure gradient relation is strongly non-linear. The key question, then, is `What are the sourc-es of non-linearity and of the other complex e€ects

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*

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.

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 micro-scopic scale.

0 _{Variables with prime (S}w0_;b0_{) are the} physically admissible values of the corresponding plain variables.

Latin characters

a crowding e€ect factor

~

A macroscopic cross-sectional area, normal to the macroscopic ¯ow direction A1U

jik reduced resistance of a jik-class cell in

one-phase ¯ow Ab

jik reduced bulk phase resistance of a

jik-class cell for di€erent ¯ow con®gura-tions

Ab

m mean of A

b

jik over all possiblejik

combi-nations, Eq. (30) Bb

n reduced interphasial resistance of any

conducting cell of an nganglion for di€erent ¯ow con®gurations

Bo bond number Ca capillary number

Cfl pressure ¯uctuation factor, Eq. (27)

CXi;G fractional distributions of blocked cells

(side-gate unit cells) in any i-class ganglion

CCi;G fractional distributions of core cells in

any i-class ganglion

CEi;G fractional distributions of end-gate cells

in any i-class ganglion Co coalescence factor

~

DT uniform maximum pore-network depth ~

DCj class-jchamber diameter

Gm ganglion mobilization number, Eq. (22)

~

g gravity

gef reduced conductance of the e€ective

porous medium

~

gDOFef reduced e€ective conductance of the

porous medium DOF region g1U

jik;~g 1U

jik conductance of a jik-class cell in

one-phase ¯ow (reduced and dimensional) gb

jik;n reduced conductance of a b-type

(bˆ`E,G' , `C,G' or `X,G') jik-class cell of an-ganglion

g0 reduced uniform conductance of all cells

in the equivalent network

fj occurrence probabilities of classj

fjek occurrence probabilities of any jik-class

cell

Imax maximum ganglion size class ~

JjGC…~z;h† mean surface curvature of the o/w

interphase, as a function of its position in a j-class chamber

~

JGT

i …~z;h† mean surface curvature of the o/w

interphase, as a function of its position in ani-class throat

J_jikG mean (along ajik cell) reduced o/w mean interphase curvature

~

Jdr;i drainage curvature of the receding o/w

interphase in ani-class throat

~

Jmin;j minimum curvature value of the o/w

interphase advancing in aj-class chamber

~

k absolute permeability

~

kef e€ective porous medium absolute permeability

kro;k w

r relative permeabilities in oil and water ~

` lattice network constant (node-to-node distance)

~

LG_n n-class ganglion's length, projected in the macroscopic ¯ow direction

mG

n assigned mobilization probability to any

n-class oil ganglion

nGi reduced number density of the i-class

ganglia

~

nG

i number density of thei-class ganglia

~

nG

characteristic distribution Nd network lattice dimensionality

NG

i total number of unit cells occupied by

anyi-class ganglion

n occurrence probability of anyn-ganglion

in the ganglion cells domain Pb

jik;n occurrence probabilities of the various

ganglion cells in the ganglion cells domain

~

q1U _{equivalent macroscopic ¯owrate of}

one-phase in the DOF region

~

qo _{macroscopic ¯owrate of oil} ~

qw _{macroscopic ¯owrate of water} ~

q1U _{¯owrate of the equivalent ¯uid in the}

DOF region

~

q1U

uc ¯owrate of an incompressible ¯uid

~

qG

uc;i total ¯owrate through any i-class

ganglion conducting cells r oil/water ¯owrate ratio Sw _{water saturation}

Sw0

physically admissible value of water saturation, see also Eq. (51)

Sio;G speci®c oil saturation in any i-class

ganglion

So;G _{average oil saturation in the ganglion}

cells domain

So;DOF _{disconnected-oil saturation in the DOF}

region

u1;u0ij bipolar coordinates

~

uG

i mean velocity of an i-class ganglion's

mass center

~

Uo super®cial velocity of oil

~

Uw super®cial velocity of water

~

Uo;G oil super®cial velocity in the domain of ganglion cells, Eq. (32)

~

Uo;CPF oil mean super®cial velocity in CPF

~

Uo;DOF oil mean super®cial velocity in DOF

~

Uw;DOF water mean super®cial velocity in DOF

~

V macroscopic control volume

~

Vuc _{average (over all classes) volume of the}

unit cell

~

VG

i average actual volume of ani-class

ganglion VmG

m expected mean ganglion size of the

mobilized ganglia VG

m expected mean ganglion size of the

overall ganglion population

_

W reduced total rate of mechanical energy dissipation

_

Wo;CPF _{reduced rate of mechanical energy}

dissipation in CPF

_

Wo;DOF _{reduced rate of mechanical energy}

dissipation in DOF

~

WTj class-jthroat width

x reduced common pressure gradient x0ij Cartesian coordinates

xmic reduced microscopic pressure gradient

xpm parameter vector composed of all the

di-mensionless geometrical and topological parameters of the porous medium a€ect-ing the ¯ow (porosity, genus, coordina-tion number, normalized chamber and throat size distributions, chamber-to-throat size correlation factors, etc.) y reduced variable

~z macroscopic ¯ow direction

…ÿop~=o_{~z† common pressure gradient}

Greek symbols

b the porous medium volume fraction occupied by the connected oil

b0 physically admissible value ofb bI;J factor de®ned by Eq. (25)

~

cow oil/water interfacial tension D~p1U

jik pressure drop along anyjik-class unit cell

during one-phase ¯ow through it

D~pb

jik;n pressure drop along any jik-class cell of

ann-ganglion for di€erent ¯ow con®gurations

f free parameter which regulates the steepness of the ganglion size distribution go;CPF _{expected mean macroscopic}

connected-oil ¯ow fraction j contact angle

eA;eR advancing and receding contact angles

h0A;h 0

R limiting values of the advancing and

re-ceding contact angles (at nil velocity)

~

ko;k~w mobility of oil and water, respectively j oil/water viscosity ratio

~

l_o;l~_w dynamic viscosity of oil and water, respectively

nij aspect ratio between ani-class throat and

a j-class chamber

~

qo;q~w mass density of oil and water, respectively

r pore-network coordination number ra variance of the distribution of the

logarithm of the microscopic pressure gradient

U expected mean macroscopic ¯ow quantity vG _{tortuosity of the spine of any ganglion}

x fraction of all the ganglion cells over all the DOF region cells

XPA 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 trac ¯ow

DOF disconnected-oil ¯ow

E,G end-gate unit cell of a ganglion EMT e€ective 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

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 trac ¯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 con-nected-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 steady-state 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 e€ects, and which is suciently 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 ob-served ¯ow patterns, in suitably idealized form, and so the pore-scale mechanisms are incorporated in the pro-totype ¯ows. The cooperative e€ects among ganglia are also incorporated in DOF making suitable use of e€ec-tive medium theory (EMT). A key feature of the pro-posed 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 e€ective porous medium layer, of absolute permeability~_kef_{, through which ¯ows `e€ective water', with a total ¯owrate equal to the sum of the}

values of the system parameters. The solution is deter-mined by assuming equal probability for each admiss-ible 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 frac-tion of the populafrac-tion of oil ganglia are mobilized by the water-induced pressure gradient and migrate down-stream. The total ¯ow is a mixture of two ¯ow patterns, speci®cally CPF and DOF. The key di€erence between these ¯ow patterns is that in DOF, numerous menisci are in motion and many `catastrophic' (irreversible) ¯ow phenomena are dominant [2,4]. This a€ects the relative magnitude of the rate of mechanical energy dissipation caused by capillary e€ects 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 macro-scopic ¯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

~

continuously fed into the control volume with constant ¯owrates q~o _{and ~q}w_{, so that the operational}

(dimen-sionless) parameters Ca andr 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 ev-ident 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 equals1_ÿb_†.Sw andb 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 Sw _{and b that}

conform with the externally imposed conditions. The analysis is carried out on two length scales: a macro-scopic scale (1012 _{pores, or more) and a microscopic}

scale.

3.1. Macroscopic scale, and connection with the pore-network quantities

The overall oil and water ¯owrates are partitioned into the contributions of the connected and discon-nected-oil regions:q~o;CPF_{; ~q}o;DOF _{are the mean oil}

¯ow-rates in CPF and DOF, respectively, whereasq~w;DOF _is

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;CPFˆ~qo;CPF_=…bA~_{†, U}~o;DOF

ˆ~qo;DOF_=‰…1ÿb†A~_Š,

and _U~w;DOF

ˆ~qw;DOF_=‰…1ÿb†A~_{Š, respectively. It is}

ex-pedient to use reduced mean super®cial velocities de-®ned byUm;n_ˆU~m;n_=~

Um, wheremˆo, w andnˆCPF, DOF. The disconnected-oil saturation in DOF is de-noted by So;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,

…ÿop~=o~z_†. Strictly speaking, this holds true for steady-state fully developed two-phase ¯ow in porous media (SSFD2uFPM) [2,4] (and only approximately so in re-gions withweaksaturation gradients).

Using this condition, we obtain

~

as well as the following relation between the mobility ratio,k~o=k~w, and the ¯owrate ratio, r:

We de®ne the reduced pressure gradient,x, as

xˆ_~ ~k

Assuming that in the CPF region oil ¯ows with virtually one-phase ¯ow (i.e., neglecting the possible lubrication e€ect of wetting ®lms), we can use Darcy's law to obtain

~

Assuming that there is no water present in the CPF region, a total water mass balance yields

~

qw;DOF_ˆq~w₎1_ÿb_†U~w;DOF_ˆU~w 6_† and a total oil mass balance yields

~

qo;CPF‡q~o;DOFˆq~o)bU~o;CPF‡_ÿ1 b†U~o;DOFˆU~o: …7_† An overall oil balance gives

b‡ …1_ÿb†So;DOFˆ1_ÿSw: …8_†

In the ®nal analysis, the macroscopic behavior of the ¯ow is the manifestation of pore-scale ¯ow phenomena and pore-network cooperative e€ects.

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 di€erent ¯uids are simul-taneously 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 geo-metrical 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 the oil-ganglion mean velocity. For simpler representation, all cells are shown identical and the lattice constant,~`, is shown expanded. The thick dashed line separates the ganglion cells domain (GCD) and the water saturated cells domain (WSCD).

Fig. 2. A typicaljik-class unit cell of the examined model porous medium. All dimensions are normalized with the lattice constant`~ˆ1221lm. 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;uoijanduoikare the limit surfaces of the

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 e€ective porous me-dium, 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

~

q1U

ˆq~o;DOF

‡q~w;DOF_:

…9_† Second, we de®ne the e€ective permeability, k~ef, of this conceptual medium by imposing Darcy's law for one-phase ¯ow

~

q1U‰_A~_1ÿb†Šÿ1 ˆ~k

ef

~

lw ÿo_op_~~

z

!

: …10†

Eqs. (9) and (10) along with (5)±(7) give

~

kef g~DOF_ef l~_~w `

ˆ Ca…1

"

‡r† c~ow …ÿop~=o~z_†ÿ

~

kb j

#

…1ÿb†ÿ1; …11†

where g~DOF

ef is, by de®nition, the e€ective hydraulic

conductance. Eq. (11) shows thatk~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 e€ective hydraulic conductance, g~DOF

ef (and therefore, ~

kef), 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 hy-draulic 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 fol-lowing 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. Further-more, the ¯ow arrangement in each unit cell and the probability of occurrence of each ¯ow arrangement de-pend, 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 conduc-tances are constant and equal to the values that corre-spond 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 predic-tions 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 numberr that varies from node to node (within reason of course). The pore ge-ometry can also be di€erent, but the basic approach would still be the same. As shown in [1], two-phase ¯ow in 3-D networks di€ers from than in 2-D ones quanti-tatively, but not qualitatively.

The model pore network of [2] is of the chamber-and-throat type. The network skeleton is a square lattice with node-to-node distance `~ˆ1221lm, and arranged so that the macroscopic ¯ow direction is parallel to one family of diagonals. For this particular network, the lattice dimensionality is Ndˆ2, and the coordination

number is r_ˆ4. (For a cubic network, we would have Nd ˆ3 and rˆ6.) The maximum pore depth is nearly

uniform and was experimentally determined to be

~

DT ˆ115 lm. The chamber size (diameter) distribution

and the throat size (width) distribution have discrete (composed of ®ve classes each) nearly normal size dis-tributions with mean values _hD~Cji ˆ610lm, h_W~_{Tji ˆ169lm and standard deviations equal to 1/4 of}

the corresponding mean values. The occurrence prob-abilitiesfjof each class along with the respective actual

characteristic dimensions of the chambers (diameters),

~

DCj, and the throats (widths),W~Tj, are given in Table 1.

It should be noted that the chamber and throat di-ameter values in Table 1 are somewhat di€erent from those reported in [2]. This is because [2] reported

Table 1

Occurrence probabilities fj of individual classesj of chambers and

throats and the associated actual characteristic dimensions, rep-resented by the chamber diameter,_D~_Cj, and the throat width,_W~_Tj

j 1 2 3 4 5

fj…%† 16 21 26 21 16 ~

DCj(lm) 330 470 610 750 890 ~

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 di-mensions of the model pore network that was used in the experiments of [2].

The chambers are modeled as normal circular cylin-ders 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 net-work is the unit cell that comprises of a throat con-necting 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 ani-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 unit cell. All lengths are normalized with `~, the network lattice constant. The basic geometrical charac-teristics 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 andWTi. In the

present work, we assume that there is no correlation between the classes of throats, chambers and cells. Thus, the occurrence probability of anyjik-class unit cell,fjik,

is given by

fjikˆfjfifk: …12†

3.2.2. Reduced conductances for one-phase ¯ow,g1U jik

When the problem of one-phase ¯ow through this network is considered, one can calculate the conduc-tances of each unit cell solely from its geometrical characteristics. These conductances represent the trans-fer function of each cell for one-phase creeping ¯ow. All cells that belong to the samejik-class have the same one-phase conductance,g~1U

jik. The reduced conductances,g 1U

where `~is the network lattice constant, l~ the dynamic viscosity of the ¯uid,q~1U

uc the ¯owrate through the unit

cell,D~p1U

3_{the corresponding pressure}

drop along the unit cell, and A1U

jik is the reduced

re-sistance of the jik-class cell. Details concerning the derivation of the transfer functionD~p1U

jik…~q 1U

uc†are given in

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.

It is clear that g1U

jik is a function of only the

geomet-rical characteristics of the jik-class cell. This is a typical characteristic of creeping one-phase ¯ow.

3.2.3. E€ective medium theory

The overall conductance of a network, such as the one previously described, can be calculated numerically by applying e€ective 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]

wherer 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 terms ofg0, the absolute permeability,k, of the original~

network is analytically recovered through the relation

~

kgef`~2; …15†

where gef is the reduced e€ective conductance of the

equivalent network, and is related to g0 by (Burganos

and Sotirchos [5])

gef ˆ

r 2Nd

g0: …16_†

Note that for the network adopted here, and for all regular orthonormal networks,gef ˆg0.

Using Eqs. (13)±(16) and the data in Table 1, we obtain the theoretical value ~kˆ8:89 lm2_{, which is in}

excellent agreement with the experimental value 8:90lm2_{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

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 calledganglion cellsand 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 givenoil 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 re-gion equals …1ÿx†. (As it will be shown below, the variable xwill 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 fu-ture extension of the present work in which drop trac ¯ow (DTF) will be included in the DOF domain,xwill 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 iˆ1; Imax based on the number of chambers it

occu-pies. At any instant, the total number of unit cells occupied by any i-class ganglion (having reduced vol-ume i) is NG

i ˆ ‰i…2Ndÿ1† ‡1Š. The average actual

volume of ani-class ganglion is given byV~_iGˆNG i V~

uc_.

Imax is the upper truncation class of the ganglion size

distribution, and so no ganglia with volume greater thanNG

Imax

~

Vucappear. It is important to note thatImaxis

animplicitfunction ofx, and is aderivedparameter. 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' mo-tion of the ganglion is the tortuosity of its spine, vG_.

This is also the ratio of the length of its actual mi-gration path over the e€ective mimi-gration length in the macroscopic ¯ow direction. This parameter depends on the porous medium geometrical and topologic charac-teristics and will be considered to be equal for all ganglion size classes. For the case under consideration, we set vG

ˆp2.

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 lo-cated at the foremost part (for the hygron) and at the aft-most part of the ganglion (for the xeron); for ex-ample 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…iÿ1†`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,

~

qG

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, ~uGi , i.e.,

~

qG_uc;iˆu~G_i `~2

vG: …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-con-ducting,ganglion'). Any i-class mobilized oil ganglion occupies 2i…Ndÿ1†`X,G' cells, and, of course, all of the

NG

i cells of any i-class stranded oil ganglion are `X,G'

cells. Therefore, the fractional distributions of the vari-ous ¯ow con®gurations of any i-class ganglion cells (mobilized as well as stranded ganglia taken into ac-count) are explicitly given by

CiX;Gˆ i…2Nd

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‡C

C;G

i ‡C

E;G

i ˆ1 given the

aforementioned expression for NG i .

The number density of the i-class ganglia,

~

nG

i ; iˆ1; Imax, is de®ned as the number of i-class

x1_ÿb_†V~X

Imax

iˆ1 ~

nGi N G i V~

uc

ˆx1_ÿb_†V~

)X

Imax

iˆ1 ~

nG_i N_iGV~ucˆ1: …19_†

If all of the disconnected oil were redistributed into class-1 ganglia (singlets), then the number density of singlets would be n~G ˆ1=…2NdV~uc†. We de®ne the

re-duced number density of the i-class ganglia,

fnGi ; iˆ1; Imaxg, as nGi ˆ~n G i =n~

G

ˆ2~n

G

i NdV~uc. Clearly,

nG

i 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 …iˆ1;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

x_ˆ…1ÿb†V~

PImax

iˆ1 ~n G i V~

G i …1_ÿb_†V~ ˆ

XImax

iˆ1 ~

nGi V~ G

i : …20†

A speci®c oil saturation,Sio;G, corresponds to thei-class

of ganglia. This saturation is easily determined by as-suming 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 Soi;GˆiNd=NiG. The average oil saturation

in the GCD,So;G_{, in terms of the reduced ganglion size}

distribution is given by

So;G

ˆx…1ÿb†V~ PImax

iˆ1 ~n G i V~

G i S

o;G i

x…1ÿb†_V~ )S

o;G

ˆ1₂X Imax

iˆ1

inGi ˆ

So;DOF

x : …21†

As in the case ofx; So;G is also a variable of temporary usefulness if one neglects oil droplets (as we do here).

However, it will have a more substantial role to play in the extended model formulation in which DTF will be taken into account.

3.2.5. Probability of oil ganglion mobilization,mG n

In a typical situation, only a fraction of the oil gan-glion 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 gener-alized oil ganglion mobilization criterion ([14,16]) ap-propriately extended to account for the crowding e€ects 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,_ÿrp~†, 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 end-menisci o€er maximum capillary resistance. Thus, the downstream meniscus is in a throat (denoted with i), while the upstream meniscus is in a chamber (denoted withj).

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 ofhR (as the receding velocity go to zero),h0R,

ir-respective of the actual velocity of the meniscus. The minimum possible curvature of the upstream meniscus,

~

Jmin;j, is a function of the size of the local chamber (not

to be confused with that of thej-class of chambers) and the advancing contact angle, hA. Again, for simplicity,

we use the limiting value ofhA, namely h0A, irrespective

of the actual velocity of the meniscus. (The dependence ofhRandhAon the corresponding meniscus velocity can

be introduced readily; this would produce a small bene®t

at the expense of substantially more cumbersome cal-culations.)

Following [14,16], the appropriate ganglion mobil-ization number is de®ned by

Gmmax

i;j

…p~jÿp~i†

~

cow J~dr;i…hR† ÿ0 J~min;j…h0A†

h i; …22†

wherep~iandp~jare the actual pressures in the water next

to the menisci with indicesiandj, respectively. In (22), indexiscans all the throats in the gate unit cells, whereas indexjscans 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]. When-ever `exact' values ofp~iandp~jare available, they should

be used. Otherwise, a rough estimate of the pressure di€erencep~jÿp~i†can be obtained by applying smooth

®eld approximation (SFA).

For asolitaryganglion, Eq. (22) and use of SFA give

Gmb_I;JCa

k …24†

withkˆk~=`~2_{, and}

b_I;J max

i;j

DL~ijcoshji `2 _J~_dr;i…h0

R† ÿJ~min;j…h 0 A†

h i: …25†

Here, DL~ijcoshji is the length of the line connecting

pointsiandjprojected along the macroscopic pressure gradient, Fig. 4; I and J are the values of i and j, re-spectively, for which the RHS of (25) becomes maxi-mum. (In the case of `wormlike' ganglia that are aligned with the macroscopic pressure gradient, positionsIand

Jusually correspond to the `nose' and `tail' of the gan-glion, 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 ofCabecomes of the order of (10ÿ4_{) to (10}ÿ3_{) or larger depending on ganglion size.}

However, ganglion mobilization becomes possible at

much smaller Ca values as in the case ofdense popula-tions of ganglia. As discussed in [14,16], in the case of populations of ganglia, Eq. (24) must be modi®ed as follows:

Gmb_I;J Ca kkw r

…population of ganglia_†; …26_†

wherekw

r is the relative permeability to water. Here,k w r

accounts for the `crowding e€ect' and it can take values much smaller than unity when the population of ganglia becomes very dense. Consequently, in the case of

ganglion populations, ganglia can move even at Ca

10ÿ5_{. On the pore-scale, the explanation is that the}

pres-ence 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 smallCavalues. This, in turn, requires large pressure gradients in the water even for smallCavalues, 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, ¯uctu-ates rapidly and by orders of magnitude as a result of the cooperative e€ects of the motion of a multitude of menisci throughout the network. These ¯uctuations have an assymetric e€ect on ganglion mobilization. Negative ¯uctuations in the value of …p~jÿp~i† do not

a€ect any ganglion that would not be mobilized by the time-averaged value. On the contrary, a positive ¯uc-tuation may mobilize a ganglion that would not be mobilized by the time-averaged value. On the other hand, a mobilized ganglion is relatively una€ected by pressure ¯uctuations when its downstream meniscus is not in a throat, which is a considerable part of its life-time. 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

kkw r

…27_†

in which Cfl can take values substantially larger than

unity.

In order to estimate Cfl in the present work, we

as-sume that the local (dimensionless) pressure gradient on the pore-scale, xmic, ¯uctuates around the macroscopic

pressure gradient,x, with a probability distributionP…y†, where yˆ …lnxmicÿlnx†=ra. We assume that the

dis-tribution F…y† is normal with mean 0 and variance 1. The value of ra is determined by setting xmicˆax;

y ˆ3, where `a' is a `gain' parameter. (The e€ect of the gain parameter will be examined using sensitivity anal-ysis.) Clearly, the distribution of xmic is lognormal.

Lognormal distributions are characteristic of popula-tions of entities that interact through birth±death phe-nomena. 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 then-class ganglion mobilization probability, mG

n, we examine, for all ij combinations,

~

LG n

Z ‡1

ÿ1

op~ o_~z

Py_†dyP~cow J~dr;i…h0R†

h

ÿJ~min;j…h0A†

i

8j; iˆ1;. . .;5: …28_†

The LHS of the above inequality is the smooth-®eld approximation of the local pressure di€erence along the ganglion. jop~=o~zj is the macroscopic pressure gradient and L~G_n ˆn`~ 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 e€ect. P…y†is the aforementioned normal distribution withyin the range

ÿ16y6_{‡ 1}. The RHS of inequality (28) expresses the pressure di€erence required to cause mobilization of the ganglion. Based on the above, the oil ganglion mo-bilization probability of each ganglion class is a function of the formmG

n ˆm

G

n…x; a; h 0

A; h

0 R; 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 par-ameters used in the experiments in [2], and for di€erent values of the gain parameter a. It is clear that the crowding e€ect 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 popu-lation become mobilized at much lower pressure gradi-ents 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 e€ect is relatively insensitive to the gain parameter a, and so use of this parameter introduces little arbitrariness. In all our sim-ulations below, we setaˆ10.

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

uGn…x† ˆ

~

uG n…x†

~

Uw ˆ 1 vG x

~ `2

~

kL

G n

"

ÿ 1

Ca…n‡1†B

G n

#

2AGm;G ÿ

‡ …nÿ1†ACm;G ÿ1

; …29†

where

Ab

mˆ hA

b jikijikˆ

X 5

j;i;kˆ1

Ab

jikfjfifk; …30†

BG_n ˆBC_n;GˆBE_n;G: …31†

Ab

jik and B b

n represent the reduced e€ects of the bulk

phases and the interphases on the total pressure drop in the corresponding cell,D~pb

jik;n[see, also Eq. (35)], when a

total ¯owrate ~qG

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 ex-pressions for Ab

jik and Bbn are given in the Appendix A

[see Eqs. (A.5) and (A.6)]. The mean of Ab

jik over all

possiblejikcombinations, i.e.,Ab

m, 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

the uncorrelated path of a typical ganglion as it migrates downstream.

The disconnected-oil super®cial velocity,U~o;DOF, may be expressed in terms of the oil ganglion velocities. Be-fore proceeding, an intermediate step should be made. The super®cial velocity in the domain of ganglion cells,

~

An oil mass balance in the DOF domain gives

x1_ÿb_†U~o;Gˆ …1_ÿb_†U~o;DOF)xU~o;GˆU~o;DOF: …33_† Combining Eqs. (32) and (33) we obtain

~

3.2.7. Unit cell conductances in the DOF region

At any instant, the pressure drop along any cell in the DOF region depends not only on the local geometrical con®guration (jik-class), but also on its ¯ow con®gura-tion, i.e., on whether this cell belongs to the ganglion cells domain (`C,G', `E,G', or `X,G' cell), or to the water saturated cells domain (one-phase ¯ow in a water satu-rated cell). The conductance of cells containing a single phase is given by Eq. (13). We turn now to unit cells containing menisci. Along any ganglion cell, the total pressure drop has two contributions. One contribution is caused by the viscous forces that resist the motion of each bulk phase. The other contribution is caused by the capillary action of the menisci. In the Appendix A, the transfer functionD~pb

jik;n…~q G

uc;n†is derived for all ganglion

cells. The reduced conductances of the cells in the GCD domain (Fig. 3) are given by the general expression

gb

n…x†is then-ganglion reduced mean velocity

and depends on the pressure gradientx, Eq. (29). Thus, the hydraulic conductances of the ganglion cells are functions of the pressure gradient. This is one of the main causes of the non-linearity of the ¯ow. (Another source of non-linearity is, as we shall see below, that the ¯ow arrangement is a sensitive function of the pressure gradient.)

3.2.8. Application of e€ective medium theory (EMT) in the disconnected-oil ¯ow (DOF) region

The EMT equation, appropriately modi®ed to ac-count for the di€erent size classesj;i;k†, ganglion sizes

…n†and ¯ow con®gurations…b†of the unit cell conduc-tances in the DOF region, takes the form

X

The ®rst term in the brackets corresponds to the water saturated cells in the DOF region, whereas the second to the ganglion cells.Pb

jik;n are the occurrence probabilities

of the various ganglion cells in the ganglion cells domain (GCD), and are given by

P_jikb_;nˆC_nbpG_nfjik; …37†

wherepG

n is the occurrence probability of anyn-ganglion

in the GCD region given by

pG_n ˆnG_n X

g0is the reduced conductance of the uniform cells of the

network that is macroscopically equivalent to the actual network and ¯ow arrangement in analogy with Eq. (14). A brief description of Eq. (36) is necessary at this point. Strictly speaking, EMT has been developed for networks with conductances that are not functions of the corre-sponding ¯uxes and, therefore of the gradient of the potential that drives the ¯ow. In Eq. (36), the conduc-tances gb

jik;n are functions of the pressure gradient x as

shown by Eq. (35). One might say that EMT is pressed into service to deal with a non-linear ¯ow system. This, however, is permissible in the neighbourhood of the solution, because one can think ofgb

jik;n asconstantover

a suciently narrow neighborhood around the solution. The e€ective permeability of this equivalent network,

~

kef, is related tog0through a relation similar to Eqs. (15)

and (16)

so that Eqs. (10) and (11) hold. Inserting Eq. (39) in (36), we obtain a strongly non-linear equation in x

4. The DeProF equations

The basic equations of the previous analysis comprise a system of equations in reduced form

Uo;CPFˆ 1 The physical meaning of each one of Eqs. (41)±(47) is as follows. Eq. (41): Darcy's law in the CPF region (the transfer function of the particular region); Eq. (42): overall water mass balance; Eq. (43): overall oil mass balance; Eq. (44): overall oil (or water) mass arrange-ment condition; Eq. (45): oil ganglion volume normal-ization condition; Eq. (46): pore-scale vs macroscopic oil ganglion mass balance in DOF; Eq. (47): EMT equation for the DOF region. Eqs. (42)±(44) are mass balance and ¯ow arrangement relations on a macroscopic scale, whereas Eqs. (45) and (46) express the consistency be-tween the microscopic and macroscopic scale in the DOF description. Eq. (47) implicitly represents the

transfer functionfor the DOF region, whereas in Eq. (41) the e€ect ofxis explicit and linear, in Eq. (47) the e€ect of x is implicit and strongly non-linear. Actually, the non-linearity of SSFD2uFPM is incorporated in Eq. (47). The above system of seven equations relates

…Imax‡5†dependent variables, namelyx,Uo;CPF,Uo;DOF,

Uw;DOF_,So;DOF_andfnG

i; iˆ1; Imaxg, along with the two

FAVs, namelySw _andb.

Closure of the above system can be accomplished by imposing certain relations between thenG

i s in the form

of a pre-selectedtypeof ganglion size distribution. Two arguments lead to the appropriate closure equations.

The distribution type should be in accordance with the available experimental [1,2] and theoretical [7,18] ob-servations. In addition, an order-of-magnitude-examin-ation of Eqs. (45) and (46) shows that these are consistent only with classes of sharply decreasing ganglion size distributions. Based on these arguments, we adopt the following distribution:

nG_i ˆ

nG 1;n

G

2 ˆparameters to be determined;

nG

1; nG2 are variables to be determined. In solving

the problem,Imaxtakes the value for which all oil ganglia

with iPImax have probability of mobilization equal to

unity based on the ganglion mobilization criterion, Eq. (23), (27) and (25). The rationale behind this is that large ganglia having Pr(mobil)_ˆ1 ®ssion into smaller ones very rapidly [16], therefore they are very rare. Here, fis a parameter that regulates the steepness of the size distribution foriP3. During the solution of the DeProF equations, fis allowed to take all permissible values in the interval 0<f<1. (As permissible value offis de-®ned one that leads to a physically admissible 2u¯ow.)

The closure scheme has a serious impact on the pre-dictions of the algorithm. The scheme proposed here gives predictions that are substantially better than those that were obtained using a simpler closure scheme in Payatakes and Valavanides [15]).

5. Algorithm for the solution of the DeProF equations

5.1. Physically admissible solutions

Let us consider a system of SSFD2uFPM for which the following information is given: (a) porous medium structure (xpm); (b) physicochemical parametersfc~ow;l~o;

Rg; (c) operational parameters Ca and r.

We know from experience that a certain steady-state fully developed ¯ow will be established. What we need to predict is the ¯ow arrangement and the pressure gradient that correspond to this steady-state.

To solve the problem, we proceed with the following overall method. First, we partition the domain {Sw_{; b}}

of all possible ¯ow arrangement parameter values using suciently ®ne steps to obtain a 2-D grid. Then, we select any pair of values for the ¯ow arrangement parameters {Sw_{;b} from this grid, and seek a solution to}

the system of the DeProF equations (see below). Now, if a solution to the DeProF equations that is compatible with the selected {Sw_{; b} values, exists, this is allowed as}

a physically admissible solution and is denoted as {Sw0

; b0}. Otherwise, the set of {Sw_{;b} values is rejected}

corresponds to the physically admissible solutions,xPA.

A prime is used to denote physically admissible values of any quantity. [The symbol without a prime is reserved for the expected value of the quantity, Eq. (51).]

We claim that on a mesoscopic scale (say 104_±109

pores), the actual ¯ow at a given region of the porous medium `wanders' over the domain of physically ad-missible solutions `visiting' any one with equal prob-ability, or frequency (ergodicity). The di€erence between any two physically admissible solutions consists in the di€erent ganglion size distributions prevailing in this region at the time (namely in the values ofnG

1; n

G 2, and

z). However, on the macroscopic scale, the ¯ow is steady and uniform (fully developed). Experimental observa-tions [1±4] are in accord with this claim.

Now, we turn to the method that is used to solve the DeProF equations for any given set of {Sw_{, b} values.}

The solution is obtained iteratively. We start by making an initial guess for the value ofxand using this tentative value forx, the system of linear Eqs. (41)±(46) is solved analytically to obtain tentative values for the other six

dependent variables, namely Uo;CPF_{, U}o;DOF_{, U}w;DOF_,

So;DOF_,nG

1 andn

G

2.Imaxis determined as discussed in the

above in connection with Eq. (48). These values are used to calculate the values of x, gb

jik;n and P b

jik;n from Eqs.

(21), (35) and (37), respectively. Then, the non-linear Eq. (47) is solved numerically and a corrected value of xis obtained. A few iterations usually e€ect convergence. Thus, the seven dependent variables are determined, the ¯ow is fully characterized, and the transfer function

x0ˆx0…Ca; r; j; h0A; h 0

R; xpmjS

w0

; b0† …49_†

for the examined steady-state two-phase ¯ow in the porous medium is determined.

The reduced rate of mechanical energy dissipation in SS2uFPM, de®ned asW_ ˆW~_k~l~w=…~cowCa†

2

, is calculat-ed as the weightcalculat-ed sum of the rcalculat-educcalculat-ed rates of me-chanical energy dissipation in each prototype ¯ow

_

W ˆbrUo;CPF‡1

‡ …1ÿb†rUo;DOF

x: …50†

A unique value ofW_0ˆW_0…Sw0

; b0_†is obtained for every physically admissible couple_fSw0_{; b}0_g.

Fig. 6. Reduced rate of mechanical energy dissipation,W_, for various values of the system parameters. For each set of system parameters values the two-phase ¯ow visits a continuum of physically admissible ¯ow arrangements [which correspond to the indicated domain on the (Sw_{;b) plane]. Each}

such ¯ow arrangement has its own value ofW_0_{. A unique set of values forS}w_{andbare obtained by averaging over all possible values, Eq. (51). These}

Several examples of domains of physically admissible solutions along with the corresponding values ofW_ are shown in Figs. 6(a) and (b).

5.2. Expected macroscopic ¯ow arrangement

By assuming that each physically admissible solution is visited with the same probability, or frequency (as-sumption of ergodicity), and averaging over their domain, XPA, a unique solution for the macroscopic ¯ow is

ob-tained. For any physical quantity, U0_{, the corresponding} expected mean macroscopic ¯ow quantity,U, is de®ned as

Uˆ

Z

XPA

Z

U0…Sw0; b†dSw0

db0

Z

XPA

Z

dSw0

db0

:

…51_† The rationale behind relation (51) is as follows. For systems in stationary but non-equilibrium states, the macroscopically established arrangement corresponds to the minimum entropy production that is compatible with the externally imposed constraint, so long as the phenomenological coecients of the system can be supposed to be constants (or almost constants) ([8]); in

addition, these stationary states are stable with respect to internal changes. Nevertheless, the aforementioned theorem does not apply to the system we consider here, because the ¯ow system is far-from-equilibrium, and the conditions under which the Onsager reciprocal relations hold are not met [2±4]. The DOF patterns (GD, DTF) are dissipative structures typical of such systems. The ¯ow arrangement that is obtained by averaging over the entire domain of physically admissible solutions, ac-cording to Eq. (51), is proposed as the expected macroscopic solution, and the transfer equation be-comes

xˆx…Ca; r; j; h0A; h 0

R; xpm†: …52†

The relative permeabilities are obtained readily as a byproduct of the calculation. Below we give a few examples of the use of the proposed theory.

6. Sample calculations

The results of several calculations that correspond to typical experiments in [2] are presented here. The geo-metrical and topological characteristics of the pore

network that were used in the theoretical calculations are those of the model pore network of [2] so that direct quantitative comparisons can be made (see also Section 3.2).

Three systems of ¯uids identical to those used in the experiments of Avraam and Payatakes (1995) [2] were examined. Their physicochemical properties are listed in Table 2.

In all systems, the static advancing and receding contact angles were assumed equal to h0_Aˆ45° _and

h0_{R ˆ}39°_{, respectively. The operational parameters take}

values in the regions 10ÿ7_<Ca<510ÿ6_{and 0:1<r<}

12:0. The e€ects of the viscosity ratio, j, the capillary number, Ca, and the ¯owrate ratio, r, are examined separately.

In Figs. 6(a) and (b), the cloud of the small gray solid spheres represents the reduced total mechanical energy dissipation, W_0, for every physically admissible combi-nation of the ¯ow arrangement variables {Sw0

; b0}, and for various values ofj(j_ˆ0:66; 1:45; 3:35),Caandr. Each diagram corresponds to a SS2u¯ow experiment in [2]. The domain of the physically admissible ¯ow ar-rangements is also shown as the support of the spheres on the (Sw_{; b) plane. In each diagram, the large solid}

sphere indicates the expected reduced mean macroscopic total mechanical energy dissipation, W_, and it corre-sponds to the expected mean macroscopic ¯ow ar-rangement indicated by the large circle on the (Sw_{; b)}

plane (the projection of the large sphere). We observe that for low intensity SS2u¯ows the ¯ow arrangement is mainly or completely DOF (here GD). As the ¯ow in-tensi®es by increasingCa orr, the domain of physically admissible solutions progressively expands and `moves'

Table 2

Physicochemical properties of the three systems of ¯uids used in the simulations using the DeProF theory. They correspond to the systems of ¯uids reported in [2]

System I System II System III

~

lo(Pa s) 3:1510ÿ3 1:3610ÿ3 1:3610ÿ3

~

lw(Pa s) 0:9410ÿ3 0:9410ÿ3 2:0710ÿ3

j 3.35 1.45 0.66

~

cow(N/m) 2510ÿ3 2510ÿ3 2210ÿ3

he(°) 42 40 40

Fig. 7. Mean macroscopic ¯ow arrangement variables (FAV)Sw_{andbfor di€erent values of the operational parametersCaandr. The strong}

non-linearity betweenSw_{andris clearly represented by the trace of the FAV in the (S}w_{; r) plane. Spheres indicate the simulations corresponding to the}

towards CPF-dominated con®gurations. We also ob-serve that small j values favor DOF, whereas large jvalues increase the contribution of CPF.

In Fig. 7, the expected mean macroscopic ¯ow ar-rangement {Sw_{; b} is shown for several values ofj,Ca}

and r. The spheres represent the expected ¯ow ar-rangements that correspond to the conditions of all the experiments in [2]. The gray curve is the trace of the expected ¯ow arrangement line in the (Sw_{; b) plane, and}

shows the strongly non-linear relation betweenSw_andr.

For low Ca values, two-phase ¯ow takes place only through DOF (b0), whereas for moderate to highCa values, the ¯ow consists of a mixture of CPF and DOF depending onr. These results are in full agreement with the experimental observations reported in [2].

The capability of the DeProF method for quantita-tive predictions is shown in Figs. 8 and 9 through di-rect comparison of the theoretical values of the mechanical power dissipation, W_, and the relative

permeabilitiesko

r; k

w

r, with the experimental ones,W_ exp_,

ko

r; k

w

r, for several di€erent values of the system and

operational parameters. In each diagram, the experi-mental results are shown with `solid' symbols (triangles and circles), while the theoretical results are shown with thick lines. In Fig. 8, the contribution of each prototype ¯ow, W_CPF_{; W}_DOF_{, to the total mechanical}

power dissipation is also presented. In Figs. 8 and 9, it is clear that the predictions of the DeProF theory are in very good to excellent agreement with the experi-mental values. The predictive capability of the DeProF method may be improved somewhat with the inclusion of the DTF regime in the calculations. (This is part of current work.) Similarly good results were obtained for

allthe experimental systems reported in [2], but are not presented here because of space limitations. (Available on request.) In order to gain more insight into SS2uFPM, we introduce, here, three quantities. The ®rst is the connected-oil fraction,go;CPF_{, and is de®ned by}

go;CPF ˆq~o;CPF=q~o: …53_†

The other two quantities are the expected mean ganglion sizes of the mobilized and overall ganglion populations, VmG

m andVmG, de®ned, respectively, as

VmmGˆ XImax

iˆ1

imGi p G i ;

V_mGˆX Imax

iˆ1

ipG_i :

…54_†

The theoretical dependence ofgo;CPF _{onr, for variousj}

andCa values, is given in Fig. 10. We see that for low intensity 2u ¯ows, the main (or only) ¯ow regime is DOF (go;CPF_{1). As the ¯ow intensi®es (largeCaorr,}

and also larger j), the contribution of CPF increases, but still a signi®cant oil ¯ow fraction occurs through

DOF. Only in extreme cases (very large Ca and/or r) doesgo;CPF _{approach unity.}

In Fig. 11, DeProF predictions of the expected mean ganglion sizes,VmG

m andVmG, are plotted as a function of

rfor variousjandCavalues. We observe that LGD is predicted for small values of Ca and r as observed in [2,4]. As Ca and/or r increase, SGD sets in again as observed in [2,4]. Finally, indication of severe oil gan-glion fragmentation during relatively intense two-phase ¯ows in porous media indicates that an extension of this DeProF model to account for drop trac ¯ow (DTF) would be useful.

7. Conclusions

A true to mechanism theoretical model of steady-state two-phase ¯ow in porous media has been

Fig. 9. Comparison between the DeProF predictions and the actual experiments in [2] as a function of the operational parametersCaandr, for the relative permeabilities of oil and waterko

r; k w

r. Solid symbols represent experimental values while open symbols represent the corresponding DeProF

developed based on the novel concept of DeProF. The actual pore-scale ¯ow mechanisms and the associated network-wide cooperative e€ects are incorporated into

two prototype ¯ows (CPF, and DOF). The two proto-type ¯ows are interlocked through the condition of common pressure gradient (which is strictly valid for steady-state fully developed two-phase ¯ow in porous media, and approximately so for steady-state two-phase ¯ow in porous media). The e€ect of the porous medium geometry and topology, the physicochemical properties of the two phases and the pressure gradient are consis-tently incorporated in the modeling of all relevant as-pects of disconnected-oil ¯ow dynamics, including ganglion size distribution, ganglion mobilization prob-ability and ganglion velocity. An appropriate ¯ow analysis on a macroscopic scale, complemented by mi-cro±macroscopic consistency relations, and use of EMT in appropriately extended form lead to a system of equations (DeProF equations). The DeProF equations can be solved readily, in part analytically and in part numerically. The solution is relatively fast (3±5 min with a good workstation).

Unlike conventional fractional ¯ow theory, the present model of steady-state two-phase ¯ow in porous media predicts thestrong non-linearityof the ¯owas well as the e€ects of all the system parameters. It is also suciently simple for practical use.

Oil ¯ow during steady-state fully developed two-phase ¯ow in porous media takes place (to a large ex-tent, and in many cases completely) in disconnected form, through the motion of ganglia and/or droplets. The motion of disconnected oil runs completelycounter

to the conventional assumption, but it has been proven experimentally and has a simple theoretical explanation (generalized ganglion mobilization criterion along with the crowding e€ect).

Fig. 10. The expected connected-oil ¯owrate,go;CPF_{as a function ofrfor di€erent ¯uid systems (j}

ˆ0.66, 1.45, 3.35) and variousCavalues. Symbols represent the DeProF theory predictions corresponding to the experiments in [2].

Fig. 11. DeProF theory predictions of the mean size of the entire ganglion population,VG

m, and the mean size of the mobilized ganglion

population, VmG

m , as a function of r for di€erent ¯uid systems