Macroscopic Description of Transport Phenomena in

Chapter 2

Macroscopic Description of Transport Phenomena in Porous Media

The objective of this chapter is to develop the mathematical models that describe transport phenomena in porous media at the macroscopic level. As will be shown, a model consists of a balance equation for each extensive quantity that is being transported, constitutive relations, describing the properties of the particular phases involved, source functions of the extensive quantities, and initial and boundary conditions, all stated at the macroscopic level. To achieve this goal, we start from a brief review of some elements of kinematics of continua. Then we develop balance equations at the microscopic level, first a general equation and then particular ones for various extensive quantities of interest. Then we develop and employ averaging rules in order to transform these microscopic balance equations into macroscopic ones. We develop expressions for the fluxes that appear in the balance equations, in terms of macroscopic state variables. Finally, a detailed discussion is presented on the nature of boundaries at the macroscopic level, and on conditions that appear on such boundaries in problems of transport of various extensive quantities. 43 J. Bear et al., Introduction to Modeling of Transport Phenomena in Porous Media © Kluwer Academic Publishers, Dordrecht, The Netherlands 1990

44

2.1

MACROSCOPIC DESCRIPTION

Elements of Kinematics of Continua

In this section, some definitions and concepts of kinematics of a single material continuum, at the microscopic level, are briefly presented. The objective of this presentation is to prepare the background for the description of transport phenomena in subsequent sections, where the considered continuum is that of a phase, or of a component that constitutes part of a (multiphase, multicomponent) material system that occupies a porous medium domain. For further information, on kinematics of continua, the reader is referred to texts on Continuum Mechanics (e.g., Aris, 1962).

2.1.1

Points and particles

Following the concepts of Continuum Mechanics, we shall use the term point to indicate a location, or place in space. The term particle will be used to denote a point in a continuum, e.g., in a mass continuum. While points are fixed in space, and independent of time, positions of particles may vary with time. In what follows we shall discuss the relationship between these two distinct, yet related, concepts-points and particles.

2.1.2

Coordinates

In Continuum Mechanics, a distinction is made between two kinds of coordinates: (a) Spatial coordinates Xi, i = 1,2,3 (or in the form of a position vector x) that define, once and for all, the location of points in space, with respect to some fixed frame of reference. The term Eulerian coordinates is often used. Henceforth, for the sake of simplicity, unless otherwise specified, Xi will denote rectangular Cartesian coordinates. (b) Material coordinates, ~i, i = 1,2,3 (or in the form of the position vector e) that are assigned, once and for all, to particles of the continuum. Usually, is sele~ted as the initial position vector of a considered particle, i.e., == xlt=o. The terms convected and Lagrangian coordinates are also used.

e e

As a particle of a continuum of an extensive quantity (say, mass) moves, the (spatial) coordinates of its position, x, vary in time, whereas its material

45

Kinematics of Continua

Figure 2.1.1: Definition sketch for particles, points and displacements. coordinates,

e, remain unchanged. We say that x is a function of both time, e, of the particle, and the motion is described by

t, and the initial position, x = x(e, t),

or

Xi = xi(6,6,6,t),

i=1,2,3.

(2.1.1 )

This description of motion is known as the Lagrangian formulation of motion. Figure 2.1.1 shows a spatial domain no occupied at t = 0 by a continuum The set of points in no specifies the initial conwith material coordinates figuration of this continuum. At some later time, t > 0, the domain occupied by the same continuum is nl . We may consider (2.1.1) as a mapping of no on nl at time t; as runs over the set of points in no, x runs over the set of points in nl . Thus, nl is the deformed configuration of the continuum initially in no. Referring to a continuous sequence of configurations as motion, (2.1.1) also describes the motion of any particle (ofthe continuum contained Le., it gives its place, x(t), as a function of time. in no) initially at Assuming that (2.1.1) can be inverted to yield the initial position (Le., material coordinates) of a particle which at time t is at position x, we have

e.

e

e,

e = e(x, t),

(2.1.2)

This description of motion is known as the Eulerian formulation of motion. It is important to emphasize that a particle here should not be identified with a small material body. Instead, it is a point that belongs to a specific

46


continuum of an extensive quantity, which at some specified (or initial) time occupies a certain finite domain. The configuration of the domain occupied by the extensive quantity may vary with time, but it will always contain the same amount of the extensive quantity. If sources andlor sinks of the extensive quantity are present, i.e., new particles are being created,or existing particles are being removed, (2.1.1) does not hold, since particles exist in the domain only instantaneously and have to be continuously redefined. We note that the concept of a particle as defined above allows particles of different continua (e.g., of mass of a phase and mass of a component) to occupy the same point, simultaneously.

2.1.3

Displacement and strain

Consider an E-continuum. A displacement, or cumulative displacement vector wE (Fig. 2.1.1) is defined as the difference between the position vector, x E , of a moving particle of the E-continuum, at a given time, and its initial position vector, E (= material coordinates of the particle)

e

(2.1.3) Two infinitesimal strain tensors may be defined: (a) If law! laffl ~ 1, for any i,j, a strain tensor, called the Lagrangian infinitesimal strain tensor, is defined by IE Cij

1 (aw!

awf)

="2 a~f + a~! .

(2.1.4)

(b) If law! laxfl ~ 1, for any i,j, a strain tensor, called the Eulerian infinitesimal strain tensor, is defined by (2.1.5) If both the displacements and the displacement gradients are small, the two infinitesimal strain tensors may be taken as equal to each other. Only this case will be considered in this book. Equation (2.1.5) will, thus be, used. It can be shown that the diagonal components, c~ (no summation on i), referred to as normal strains, represent relative stretch along the coordinate

47


£5,

axes, while each of the off diagonal components, referred to as shearing stains, represents one half the angle change between two line elements originally at a right angle to one another. The volumetric, or cubical, strain (or volumetric dilatation), which is the relative growth of volume with respect to the original one), is given by (2.1.6)

2.1.4

Processes

A process undergone by an E-continuum may be defined as a sequence of changes in the state of the continuum in the course of time. Accordingly, any process occurring in an E-continuum involves changes in both space and time of variables pertinent to the state of that continuum. Examples of state variables are velocity, strain, temperature, pressure and density. Let GE denote such a variable of an E-continuum. A process can be described in two forms:' (a) Material, or Lagrangian description, Gf = Gf(e, t), i.e., the variation of G in time as observed by following fixed particles of the continuum, identified by their material coordinates,

e.

(b) Spatial, or Eulerian description, Gf = Gf(x, t), i.e., observing the variation of GE in time at fixed places, x, in the space occupied by the continuum. Note that the symbols Gf(e, t) and Gf(x, t) represent different functions. The passage from the spatial description to the material one can be obtained from the relationship between the two respective coordinate systems. Thus, from (2.1.1) it follows that

Gf(x, t) = Gf[x(e, t), t]. 2.1.5

(2.1.7)

Material derivative

The material derivative (also called convected derivative) of a variable G with respect to a particle of a given E-continuum is the (temporal) rate of change of that variable for the considered particle. The symbols E ,where

a

48


it is verbally stated that the point of view is that of a moving E-particle, and DEG / Dt, are used to denote this derivative

(2.1.8)

e

i.e., a derivative of G with respect to time, keeping E constant. In other words, D EG /Dt gives the rate of change of G of a fixed E-continuum particle to an observer situated on that particle. Consider a mass continuum, E == m. With G = xm denoting the position vector of a particle of this continuum as it is being displaced, and denoting its material coordinates, the velocity of the particle, v m , is given by the rate of change of its position in time, i.e.,

em,

(2.1.9) We may now generalize (2.1.9) to a particle of any E-continuum. Its velocity is defined by

at Ie =const.

vE = 8xE

E

(=

xE == x).

(2.1.10)

We shall use the abbreviated symbol ()E whenever we wish to indicate that changes of ()E are observed, following particles of the E-continuum. However, whenever this fact is obvious, the superscript E will be omitted. The rate of change of the volumetric strain, defined by (2.1.6), is given by (2.1.11) i = (V.w) = V·(w) = V·V E , where, by (2.1.3),

w = xE = V E . 1

iij = 2"

Also, by (2.1.5)

(aVE aV:E) a;j + a: ' i

(2.1.12)

where the tensor i defines the rate of strain of the E-continuum. In a rigid motion, or a motion without deformation, iij == O. The material derivative of G, which is a Lagrangian concept, can also be expressed in terms of the spatial, or Eulerian, description, using the


49

relationship (2.1.7).

(jE

=

DEG{X(e E , t), t} Dt aGI

=

7ft

_ aG - at

aGI aXk(eE,t) 1° x=const. + aXk t=const. at eE=const. aG TTE

+ aXk v k

where

(2113) ..

,

t

v

aG VkE == aG kE == VE.VG; aXk k=l aXk

This abbreviated form of representing a sum of terms by a single, typical, one is known as Einstein's (double index) summation convention. Unless otherwise stated, this convention will be used throughout this book. It states that any index (called a dummy index) repeated twice and only twice in a term is held to be summed over the range of its values. Thus, for i = 1,2,3

= 2:7=1 aibi 2:J=l aijbj

=

al b1 + a2b2 + a3 b3, ail b1 + ai2 b2 + ai3 b3·

(2.1.14)

Equation (2.1.13) states that the rate of change of a quantity G, associated with a particle of an E-continuum, which at a given instant of time, t, is located at a specified point, x, in space, is represented in the spatial description as a sum of two parts: ( a) a local rate of change of G, aG / at, at the specified point x, and (b) a convective rate of change of the quantity G, VE·VG, due to the variation of G along the path of the particle, whose instantaneous velocity at the given point is V E . In the Eulerian description, a field, G(x, t), in a given domain is said to be steady, if aG / at = 0, i.e., G remains everywhere the same at all times. A field is said to be uniform if VG = 0, i.e., G is always the same at all points. Thus, a motion of a continuum is said to be steady, if its velocity field satisfies the condition a~/at = 0 at all times. A motion is said to be uniform if a~/axj = 0 at all points. The concept of a material derivative may also be applied to a material manifold of particles, i.e., a set of particles which always consists of the same

50


ones. Examples are a material line (of particles), a material surface and a material volume. A single particle is a material point. A case of material derivative of special interest is that of a material volume element, dUE, i.e., an element of volume of an E-continuum

dUE = dx{l)

X

dx(2)·dx(3).

Let dUE = (dXldx2dx3) denote the volume of a parallelepiped element in a rectangular Cartesian coordinate system (e.g., dXI == dx; dX2 == dy; dX3 == dz). Each dXi, i = 1,2,3, represents the length of a segment along the ith coordinate axis between two points: at Xi + dXi and at Xi. As the volume dUE is being displaced and deformed, containing all the time the same set of particles, the material derivative of the length dXi is related to the velocities at the end points of the segment, Xi + dXi and Xi, by

(2.1.15) where VE denotes the velocity of an E-particle, and no summation on i is invoked. Hence, according to (2.1.8), the material rate of change of dUE is given by

(dx l dx 2dx3)

(dxl)1

dX2dx3 X2,X3

+ (dx 2 )1

axIl dXI dX2dx3 ( av,E)

dXldx3 Xl ,x3

+ (dx 3 )1

dXldx2 Xl ,x2

+ (aVE) a:2 dX2 dXldx3 + (aVE) ax33 dX3 dXldx2

or (2.1.16) In rectangular Cartesian coordinates, the divergence of a vector A is defined by

Equation (2.1.16) can thus be written as (2.1.17)

51


Thus, '\l·V E represents the rate of expansion of a material volume element of the E-continuum, per unit volume. A motion in which the volume occupied by a given set of particles remains constant is called isochoric. In view of the above discussion, this means that a motion is isochoric whenever

For any element of an E-continuum, we have dE = e dUE, where e is the density of E (Le., amount of E per unit volume of the E-continuum). The material derivative of dE is given by (2.1.18) A continuum of an extensive quantity E is said to be conservative if for any volume element, dUE, of E

(dE) = O.

(2.1.19)

Hence, for the general case of a conservative E-continuum, (2.1.18) becomes (dUE) _ r7.V E __ ~ _ vE () dUE - v e - vE ' 2.1.20 where v E = l/e is the specific volume of E. From (2.1.20) it follows that the motion of a conservative E-continuum is isochoric if e=const. or e = o. For the particular case of mass of a phase (E == m), which is a conservative material quantity, dE = dm = pdU, where p denotes the mass density, we have Hence, the rate of dilatation of a conservative mass continuum, is given by

(dum) = '\l.Vm = _~ = vm , dUm p vm

(2.1.21)

where v m == 1/ P is called the specific volume (of mass). For the sake of simplicity, we shall employ the symbol v to denote v m . For p = 0, i.e., p remains unchanged for each particle as it moves, (2.1.21) reduces to (2.1.22)

52


which means that the rate of dilatation in this case vanishes. As explained above, such motion is referred to as isochoric mass motion. Note that the fact that a fluid is incompressible does not imply that its motion is necessarily isochoric. This stems from the fact that dilatation may also be produced by variations in temperature and components' concentration. Only when p = 0, the motion is always isochoric. Another case of special interest is the material derivative of a surface. Let a surface within an E-continuum be described by the equation F(x, t) = C1 = const. As the surface moves, its shape may change, yet its equation remains unchanged. Thus, F(x, t) is a conservative property of points on the surface. When the surface consists always of the same E-particles, it is called a material surface of the E-continuum. With u(x, t) denoting the velocity vector of a point x on the surface F = Ct, it follows from (2.1.13) that the material derivative of F is given by DFF

of

Dt

at

- - == -

+ u·V F = 0'

(2.1.23)

where the subscript F is introduced in the material derivative to indicate that as the points belonging to the surface, F, are displaced, they are observed by an observer moving with F. From (2.1.23) we obtain

of

of

u·VF=u - = - II av at '

of av

= IVFI,

(2.1.24)

where v is the unit vector normal to the surface (always on the same side of the latter) and u II (== u·v) is the speed of displacement of a point on the surface. Hence, U II is given by

of/at U II

= - aF/av'

VF

v = IVFI.

(2.1.25)

On the other hand, the material derivative of F, with respect to a particle of an E-continuum, instantaneously located on the surface and moving at a velocity yE, is given by DEF ==aF +yE.VF. Dt at

(2.1.26)

By subtracting (2.1.23) from (2.1.26), we obtain DEF = (yE _ u).V F = (VE _

Dt

II

U II

)aF av'

(2.1.27)

53


where (V! - u v ) is the speed at which an E-particle crosses the surface. When the considered surface is also a material surface with respect to E-particles, VvE = U v , and DEF = o. (2.1.28) Dt 2.1.6

Velocities

Let us consider a phase consisting of N components (e.g., species in solution) denoted by 'Y = 1,2, ... , N. No special symbol will be used to indicate the phase. Each component has a mass density p"Y (= mass of 'Y-component per unit volume of the phase, with L:("Y) p"Y = p) and a velocity V"Y :: vm"f at any point within the phase. This is the average velocity of the individual constituents, e.g., molecules, ions) comprising the 'Y-component within a volume element of the phase centered at the point. Several kinds of velocities (with respect to a fixed frame of reference) may be defined for the phase as a whole at any point. All of them can be written as weighted averages of the component velocities, V"Y, in the form N

N

T"Y V E = "'" L...J aE"f "'v',

(2.1.29)

L...J aE"f = 1,

"'"

"Y=1

"Y=1

where the various aE"f,s are (normalized) weights of the 'Y-components in the phase. Following are three of the more commonly used weighted velocities of a phase. (a) Mass weighted velocity, vm vm

1 N

N

"Y

N

= m LVlm"Y = L ~Vl = LW"YVl,

(2.1.30) "Y=1 "Y=1 P "Y=1 where w"Y = m"Y 1m = p"Y I p is the mass fraction of the 'Y-component in the phase, with L:("Y) w"Y = 1, p"Y = m"Y IU and p = miU. The mass weighted velocity is often referred to as barycentric velocity.

(b) Volume weighted velocity, V V

1

N

N

"Y=1

"Y=1

au

m"Y

N

= -U "'" -V L...J U"Y V"Y = "'" L...J am"Y U l = "'" L...J v"Y p"YVl =

N

E v"Ymolp"YmolV"Y,

"Y=1

"Y=1

(2.1.31)

54

MACROSCOPIC DESCRIPTION where v'Y = {)U / {)m'Y is the partial specific volume of the ,-component in the phase, U'Y = "£(-y) ({)U/{)m'Y)m'Y, v'Y mol = M'Yv'Y is the molar partial specific volume of" M'Y is the molecular weight of " and p'Y mol = p'Y / M'Y is the molar concentration of , (=number of moles per unit volume of phase). For a single component fluid phase, i.e., a fluid composed of identical constituents, N = 1, and we obtain vm = V.

(c) Molar weighted velocity, V mo1 V mo1 =

~Ol

P

N

N

L: p'Ymo1V'Y = L: X'YV'Y , "1=1

(2.1.32)

"1=1

where X'Y = p'Y mol / pmolis the mole fraction of " i.e., the ratio of the molar concentration of, to the total molar concentration of the phase, and pmol = "£(-y) p'Y mol is the total molar density of the phase.

(d) Momentum weighted velocity, V M . VMM U

or

pVmV M =

N M'Y '"'V'YL..J

"1=1

U'

N

L:p'YV'YVM"Y,

(2.1.33)

"1=1

in which M'Y /U and M/U express the momentum of the, component and the total momentum, respectively, both per unit volume of the phase, and VM"Y represents the momentum weighted velocity of the ,-component, defined by

where the ,-component is composed of N'Y individual ,-molecules per unit volume, each of mass mit and velocity Vit, with m'Y = "£0:1) mit·

55


2.1. 7

Flux and discharge

In this subsection we continue to deal only with a single phase continuum. For the sake of simplicity, no subscript will indicate this fact. The quantity (2.1.34) is defined as the flux (or total flux) of E at point x and time t, where VE is defined by (2.1.10). It is worth noting that the flux of an E-quantity can be a tensor of any rank, depending on the tensorial rank of e. Thus, if E represents linear momentum of a phase, E = M, e = pvm and jtM = pVmVM is a second rank tensor. Physically, jtE represents the quantity of E passing through a unit area of the continuum, normal to V E , per unit time, with respect to a fixed coordinate system. The elementary discharge, dQE, of E through an oriented element of area, dA, is defined by dQE =lE.dA. Through a finite surface of area A, the total discharge is given by (2.1.35) Of special interest is the particular case of a scalar E- continuum (e.g., mass, volume). In this case, the density, e, is a scalar and the flux, jtE defined in (2.1.34) is a vector. The elementary discharge, dQE, defined in (2.1.35), becomes a scalar and can be presented in the form .tE dQE J = dAlI' where dAlI is the projection of the oriented elementary area dA onto a plane normal to jtE. Let V and vm denote the volume-weighted and mass-weighted velocities at a point of a considered phase, respectively. From (2.1.36) it follows that the total flux, jtE, of E, may be decomposed into two parts: an advective flux, eV, carried by the volume weighted velocity of the phase,

56


with respect to the fixed coordinate system, and a diffusive flux, jEU, relative to the advective one. Another possible decomposition of jtE is (2.1.37) where jEm is the diffusive flux of E, relative to the mass weighted velocity of the phase. For example, when E represents the momentum of a phase, its diffusive flux, relative to the mass weighted velocity of the phase, is (2.1.38)

In a multicomponent system, diffusive fluxes of E'Y may be defined for each of the components with respect to each of the velocities V and vm of the system (2.1.39) When E'Y represents the mass, m'Y, of the ,-component of a phase (Le., e'Y == p'Y), we have, by (2.1.30) (2.1.40) whereas, by (2.1.31) E("t)jm'YU = E("t) p'Y(V'Y - V) = p(vm - V) oJ 0, where, since all,-particles have the same mass, we have replaced vm'Y by V'Y. To simplify the notation, we shall henceforth use the symbol jm to denote jmU (=diffusive mass flux), and j'Y to denote jm'YU (=diffusive mass flux of ,-component). In both cases, the diffusive flux is with respect to the volume weighted velocity.

2.1.8

Gauss' theorem

Consider a tensorial quantity, Gijk ... (of any rank), that is defined and differentiable within a regular convex spatial domain, U, bounded by a closed surface, S. The surface S consists of a finite number of parts, with a continuously turning tangent plane. We wish to calculate the integral

r oGd k ··· dU,

Ju

OXi

where the Xi'S, i = 1,2,3, are Cartesian coordinates. This integral, say for i = 1, can be evaluated by dividing the volume U into prisms of infinitesimal


57

d:Z: a

--,. --~ ----,..

Figure 2.1.2: Definition sketch for the Gauss Theorem. cross-section by means of two families of planes that are normal to the X2 and X3 axes, respectively (Fig. 2.1.2). The contribution of each of these prisms to the considered integral is made up of the contributions of the elementary volumes, dXldx2dx3, except for the edges. We note that for the elementary volume of each prism, we have 8G'kl J "'dU 8XI

Hence, the contribution of the entire prism is given by

where VI is the cosine of the angle between v and the xl-axis, the sum denoted by L is taken over all dXI 's between the edges Xl = Xi and Xl = xi*, at constant X2, X3; v is the outward normal unit vector on dS,dx2dx3 = vi*dS** = -vidS*, and dS** and dS* are the elements of area that the

58


prism cuts out of the surface, with Gjkl... and Gjkl ... being the values of Gjkl... on these elements, respectively. We note that since the two terms in the last sum have the same sign, we do not have to distinguish between the two edges by the asterisk and double asterisk superscripts. Repeating the same procedure for all prisms constituting the volume U, we obtain 8Gjkl... dU = G jkl..YI· dS (2.1.41)

1 U

Replacing

Xl

by

Xi,

8Gjkl ...

U

=j

S

i = 1,2,3, we obtain the general formula

1 Setting i

1

8 Xl

8 Xi

dU =

and summing over j

1

8Gjkl ...

U

8 Xj

1

G jkl ..YdB i.

S

(2.1.42)

= 1,2,3, we obtain

dU =

1 S

G jkl..Yj, dB

(2.1.43)

where we have made use of the double summation convention. Equation (2.1.43) is known as Gauss' Theorem. It is also called the Gauss divergence theorem. For the special case in which Gjkl. .. is a vector, G, the Gauss theorem becomes

1 U

8Gi dU = -8 Xi

1 S

GWi

dB,

or

fu V·G dU is G·v dB. =

As an example, let G denote the flux of E, i.e., G = eVE. theorem states that

fu V·eV E dU = is eVE·v dB.

(2.1.44) Gauss'

(2.1.45)

The r.h.s. of (2.1.45) represents the net efflux of E leaving the domain U through the surface S. From (2.1.45) it follows that 1 diveVE == lim U U--+O

isr eVE·v dB.

(2.1.46)

This provides a physical interpretation of V·eV E (== diveVE ) at a point, as the excess of efflux over influx of E through a closed surface surrounding a domain, per unit volume, as the latter shrinks to zero around the point.


59

If U is not a convex domain, but can be decomposed into a finite number of regular convex ones, the validity of Gauss' theorem can readily be established by writing (2.1.45) separately for each partial volume and adding the resulting equations. In particular, if U is also bounded from the inside by a surface S1, then (2.1.45) becomes

(2.1.47)

2.1.9

Reynolds' transport theorem

Consider a domain U of volume U(t) enclosed by a moving material surface Set) described by the equation F(x, t) = 0, of area Set), and let e denote the density of the extensive quantity, E. The total amount of E contained within U(t) is given by JU(t) edU. The rate of change of this quantity, in the course of time, is given by JU(t) edU, with DF( )/Dt denoting the material derivative as viewed by an observer moving with Set). Since the integral is over a varying domain, U(t), we cannot exchange the order of differentiation and integration. Figure 2.1.3a shows this volume at time t and at time t + flt. By definition

%7

DF Dt

f

edU

lU(t)

= .6.t-+O lim : { f e(t + flt) dU - f e(t) dU} I...l.t lU(t+.6.t) lU(t) =

lim :

.6.t-+O I...l.t

{f

lUi

e(t+flt)dU-

f

lUi

e(t)dU+

-L3 e(t) dU}

f

lU2

e(t+flt)dU

=lUi f {){)e dU + lim : {f e(t+flt)dU- f e(t)dU}. t .6.t-+O I...l.t lU2 lU3

(2.1.48)

With the nomenclature of Fig. 2.1.3a, we obtain,

U2 =

f

l(.6.t)

(1

(ABC)

U·1I

dB) dt,

U3

=- f

l(.6.t)

(1

(ADC)

U·1I

dS) dt.

60


(a)

(b)

Figure 2.1.3: Definition Sketches for Reynolds transport theorem. Since dU2 = (u·vdS)dt, and dU3 be rewritten for b..t -+ 0, in the form

DF [ e dU = Dt JU(t)

f

JU(t)

= -(u·vdS)dt,

equation (2.1.48) can

{Je dU + f eu.v dS. {Jt JS(t)

(2.1.49)

In words, the rate of change of the amount of E contained in a domain U(t), enclosed by a material surface Set), can be represented as the sum of two contributions: (a) The rate of change of e, within U, integrated over the (instantaneously) fixed domain U(t), and (b) the net efflux of E across the (instantaneous) surface Set). We note that (2.1.49) actually gives the material derivative of an integral. Equation (2.1.49) is known as the Reynolds transport theorem. One could obtain the same result by employing Leibnitz' rule for the differentiation of an integral, where the boundaries of the latter depend on the parameter of integration. We note that the Transport Theorem does not require that e(x, t) be differentiable in space. A particular case of (2.1.49) is when Set) is also a material surface with respect to E. Then, u·v == VE·v and DF( )/Dt == DE( )/Dt, where () represents the volume integral of e. When Set) is not necessarily a material surface with respect to E, it is convenient to rewrite (2.1.49) in the (equivalent) form

DF Dt

f

JU(t)

e dU =

f

JU(t)

{Je dU {Jt

+f

JS(t)

eVE·v dS

61

Kinematics of Continua - f

JS(t)

e(VE -

U)·V

dS.

(2.1.50)

In this form, we can account for a surface segment on which (VE - u)·v = 0, or u =1= 0, VE =1= 0, and (VE - u)·v =1= 0. We note that

DDF f

t JU(t)

e dU

= DDE

f

t JU(t)

e dU -

f

JS(t)

e(VE - u)·v dS.

(2.1.51)

Let U(t) be made up of two parts, Ul and U2, separated by a surface, S*, not necessarily material with respect to E, that moves at a velocity u (Fig. 2.1.3b), and let the densities of E in Ul and U2 be denoted by el and e2, respectively, such that e exhibits a discontinuity across S*. Let us write (2.1.49) for each of the two volumes. We obtain

DF f e dU Dt JU1(t)

=

DF f edU Dt JU2(t)

0:.U~ dU + JS*+S1 fell dS, f 0:. dU + JS*+S2 f eI 2u.,v2 dS. JU2(t) U~ f

U·Vl

JU1(t)

where eli' (i = 1,2) represents the value of e on the Ui side of S*. By adding the two equations, noting that on S* we have VI = -V2, we obtain

DF f edU Dt JU(t) = f

0:.U~ dU + JS(t) f eu·v dS + f (ell JS*(t)

= f

(){)e dU + f

JU(t) JU(t)

t

JS(t)

eu.v dS + f

JS*(t)

eI 2)u.vl dS

[eh,2 u ·v l dS,

(2.1.52)

where [eh,2 == el on U1 side - el on U2 side denotes a jump in e across S*, and v represents the outward unit normal vector on S*. Equation (2.1.52) is Reynolds transport theorem for a domain which contains a surface of discontinuity. When Set) is a material surface with respect to E, we replace DF()/Dt in (2.1.52) by DE( )/Dt, and u by V E. Finally, when Set) is material with respect to E, but S* is not, (2.1.52) becomes

-DF Dt

1

U(t)

edU =

f JU(t)

(){)e dU + f

t

eVE.v dS

JS(t)

- f [e(VE - u)h 2·V dS.

Js *

'

(2.1.53)

62


2.1.10

Green's vector theorem

Let aj = aj(x) and b = b(x) be a vector and a scalar function of position, respectively, which are continuous and at least twice differentiable, within a given closed spatial domain, U, and on its boundary S. By applying Gauss' theorem (2.1.43) to Gji == b8aj/8xi, we obtain

1

8as 8Xi

b_J Vi dB

= (2.1.54)

Likewise (2.1.55) By subtracting (2.1.55) from (2.1.54), we obtain

r (b 8aj _ ~ _) _dB _ r (b 8 aj _ 8 b -) dU 8Xi 8Xi a V~ Ju 8Xi 8 Xi 8 Xi8x a 2

Js

J

-

2

i

J

•

(2.1.56)

Equation (2.1.56) is known as Green's (vector) theorem.

2.1.11

Pathlines, transport lines and transport functions

A pathline is a curve (or line) along which a fixed particle of a continuum moves in the course of time. A pathline is thus a Lagrangian concept. i = 1.2.3, denote the material coordinates of a fixed E- particle. Let The Lagrangian description of its motion, as given by (2.1.1), is

ef,

(2.1.57) These are three equations that give the coordinates of the position-vector of the particle, as functions of time. Eliminating the time from a pair of these equations, and repeating this process for a second pair, yields two equations

Each of these equations describes a surface. Together, these two equations define the pathline of the E-particle (coinciding with the intersection of the two surfaces).

63


In the Eulerian formulation of the material derivative, the differential equation of motion of an E-continuum, is given by

or (2.1.58) The solution of these equations gives the Eulerian description of motion, (2.1.2) (2.1.59) i = 1,2,3, ~f = ~f(x, t), where the ~f's are parameters identifying a particle. By fixing the values of these parameters, and following the procedure described above, one obtains, again, the pathline of a specific particle. Obviously, (2.1.57) and (2.1.59) represent the same motion. Hence, they yield the same equation for the pathline of a particle, as long as the material coordinates of the particle are defined in the same way (Xi = ~i, at t = 0), and if the two equations are mutually invertible, i.e. i,j = 1,2,3,

where .1, referred to as Jacobian, is the determinant of a matrix in which the typical element is axil a~j. While a pathline is a curve along which a given particle moves during a sequence of times, a streamline is a curve along which a sequence of particles move at a given instant. By definition, the tangent to a streamline at each point on it is colinear with the velocity vector, VE, at that point. Accordingly, the mathematical definition of a streamline of an E-continuum at a given instant, say, t = to, is

i=1,2,3, or (2.1.60) where the dXi's are the components of an infinitesimal displacement along a streamline, and Q is a scalar. A streamline is thus an Eulerian concept. Figure 2.1.4 shows a streamline in the two-dimensional xy-plane.

64

MACROSCOPIC DESCRIPTION y

Streamline

o

Figure 2.1.4: A streamline in the xy-plane.

Once the velocity field, VE(x, to), is known, the general solution of the system (2.1.60) yields the family of streamlines, referred to as the motion pattern, of the E-continuum at the instant t = to. The general solution of (2.1.60) contains three arbitrary constants. These constants are assigned fixed values by selecting a streamline which passes through a given point. The motion pattern of a continuum may reveal the presence of special points called sources and sinks. A source is a point from which streamlines diverge radially. A sink is a point towards which they radially converge. Sources and sinks are referred to as singular points. For such points, (2.1.60) is not valid. The strength of a source (or sink) is the discharge, or, in general, the amount of an extensive quantity per unit time, passing through a surface completely enclosing it. For unsteady motion of an E-continuum (Le., 8ViE /8t :I 0), the streamlines may vary from one instant to the next, whereas for a steady motion (8Vi E /8t = 0), the streamlines remain unchanged with time. In the latter case, streamlines and pathlines coincide, and (2.1.58) and (2.1.60) become identical, since any particle, once at a point on a given streamline, will remain on the same streamline as time goes on. It is worth noting that for any scalar E-continuum (e.g., mass, energy),

65


a streamline which is a vector line of the velocity field, V E , is also a vector line of the total flux, jtE(= eVE) of that continuum, as defined by (2.1.34). This line is defined by

(2.1.61) A line defined by (2.1.61) will be called an E-transport line, or curve (abbreviated ETC), of the scalar E-continuum. Whenever the considered E is obvious, we shall refer to the curve as a transport curve. Consider a motion of a scalar E-continuum such that all E-transport curves are parallel to a given plane, say the Xlx2-plane. Furthermore, at any given time, let the motion pattern in all planes that are parallel to this plane, be identical. Such a motion is referred to as two-dimensional planar motion. Its transport curves are defined by or 11·tEdX2 - 12·tEdXl = 0.

(2.1.62)

Of particular interest is the case in which the relations or

(2.1.63)

prevail at all points in the domain of motion of the continuum. In such a case, jjE dX2 - j~E dXI is an exact differential, dW E , of a scalar function WE = WE (XI, X 2,t o ), with

(2.1.64) The general solution of (2.1.64) is

where C is an arbitrary constant and WE is a function whose value along an E-transport curve is constant. Any specific value of the constant C, yields a specific curve. Therefore, WE may be called an E-Transport Function (abbreviated ETF). We may thus conclude that if at a given instant, V'-jtE = 0 throughout a planar domain of motion of a scalar E-continuum, the pattern of motion in

66


that domain can be described by a single scalar E-transport function, WE. From Subs. 2.1.5, it follows that

.

otE _ E V·J = V·eV

ae dE - at·

1 (.)

= dUE

(2.1.65)

Hence, .WE may be defined only when the E-continuum is conservative, Le., when (dE) = 0, which corresponds to a domain that does not contain sources or sinks, and for a density field that is steady, Le., ae / at = o. In the particular case of a uniform density field, the condition V·jtE = 0 reduces to V·V E = 0 (isochoric motion) and the ETF, WE, is called streamfunction. This is the case commonly encountered in Fluid Mechanics, with e = 1 and jtE = VE (Fig. 2.1.4). The physical interpretation of WE in two-dimensional motion, follows from (2.1.64), rewritten in the form

aWE

--dX1 aX1

= h·tEdX2

aWE

+ -aX2 -dx2 -

h·tEd·X1·

(2.1.66)

Consider an oriented elementary segment ds connecting two E- transport curves, WE and WE + dw E (Fig. 2.1.5a). The discharge of E through dB, per unit distance normal to the plane of motion, Xl, X2, is given by (2.1.35)

(2.1.67) A comparison of (2.1.66) and (2.1.67) yields dQE = dw E ,

Le., the increment of the E-transport function between two transport lines is equal to"the discharge of E between the two lines, per unit width normal to the plane of motion. In general, when W~ and W~ denote the values of WE which correspond to transport lines passing through points A and B (Fig. 2.1.5b) respectively, then (2.1.68) From (2.1.68) it follows that whenever an ETF can be defined for a given pattern of motion of an E-continuum, any transport line can be chosen as a starting line for the computation of discharge between transport lines.

67


y

>---------

:1:1

o~-------_

:I:

z (a)

(b)

Figure 2.1.5: Relation between the stream function,

2.1.12

WE, and discharge, QE.

Velocity potential and complex potential

Consider a motion of an E-continuum such that or

&q,E lIiE(x, t) = ~, UXi

i = 1,2,3,

(2.1.69)

where q,E = q,E(x,t) denotes a field of a scalar quantity which is at least twice differentiable throughout the domain of motion. A motion satisfying (2.1.69) is said to be a potential motion, and q,E is referred to as the velocity potential of the E-continuum. Since

&~j (&&~~) = &~i (~!;)

for any i and j

it follows from (2.1.69) that

~ (&lIiE _ 2

&Xj

&V/) = 0 &Xi

for any i and j (~i),

or

V' X VE = O.

(2.1.70) The 1.h.s. ofthe first equation in (2.1.70) represents the mean angular rate of rotation of a material element of the E-continuum in the ij-plane, about

68


an axis normal to that plane. Thus, a potential motion is an irrotational Conversely, it can be shown that an irrotational motion is also a one. potential one. By definition a4?E

a4?E dXi uXi s

E

~ = ~-d = IV4? Icos(V4?, 15), uS

(2.1.71)

where 15 is a unit vector in the direction of a displacement, ds. From the particular case a4?E / as = 0 (Le., ds lies in the surface 4?E = constant), it follows that at every point V4?E(x, t), and hence also yE is perpendicular to a surface 4?E = const. passing through that point. Thus, the motion of an E-continuum is irrotational within a given spatial domain, during a given period of time, then at every instant, t, during this period, one can define within this domain a family of equipotential surfaces 4?E(x, t) = C (where C is an arbitrary constant) which uniquely determine the magnitude and direction of the velocity vector, yE, at each point in the domain. The above discussion can now be extended to the flux field, jtE, of a scalar E-continuum. Let this flux be such that (2.1.72) where e(x, t) is a scalar density. Then, ~E may be referred to as the potential of the E-flux field. By analogy with (2.1.70), it follows that

a( eViE) _ a( eVP) aXj

aXi

(2.1.73)

Since (2.1.74) if the motion of the continuum is irrotational (Le., V X yE port of E, as defined by jtE, is also irrotational as long as Ve

X

yE

= Ve x V4?E = 0,

= 0), the trans(2.1. 75)

69


i.e., as long as the density of E is uniform, or when the equipotential surfaces of the velocity field are also surfaces of constant density. In general, if jtE is the flux vector of a scalar contimuum and if a velocity potential, q,E, exists, (i.e. VE = - \7q,E), then q,E is called pseudopotential ofjtE( = eVE), since the surfaces q,E = canst. are also normal to jtE at every point. Of particular interest is the case of a two-dimensional planar motion of an E-continuum, which is both irrotational and isochoric, i.e. \7 X VE = 0,

and \7·V E = O.

(2.1.76)

Let X1X2 be that plane of motion. By (2.1.64) and (2.1.69), we may define at any point (Xl,X2) within the domain of motion, and at any instant, to, two functions: • a velocity potential, q,E(Xl, X2, to) = Gil!, where Gil! is an arbitrary constant, such that (2.1.77) • a streamfunction, WE(Xl,x2,to) = Gw, where Gw is an arbitrary constant, such that (2.1.78) The distinct features of such a motion are: (a) The two families of surfaces q,E( Xl, X2, to) = Gil!, and WE (Xl, X2, to) = Gw are mutually orthogonal, i.e.

oq,E oWE

-·-=0. OXi

OXi

(2.1.79)

The intersections of these two families of surfaces yield the streamlines and the equipotential lines which together constitute the so-called ftownet at time to' (b) An analytic function, wE(z), of a single complex variable, z = iX2, can be defined such that

Xl

+

(2.1.80)

70


dw E dXl dw E d( iX2) the two derivatives are equal if aipE aWE aXl - aX2 '

(2.1.81)

However, these two equalities (referred to as Cauchy-Riemann conditions), which are necessary for wE to be an analytic function, actually hold as a conclusion of comparing (2.1.77) with (2.1.78). The function wE(z) is called the complex potential of the motion of the E-continuum. The use of the complex potential facilitates the solution of problems associated with steady planar isochoric and irrotational motion of a continuum, due to the possibility of transforming the motion configuration from the physical z-plane (== xy-plane) to the plane of the complex potential, w(ip, w)-plane, and vice versa, employing the functional relations for ip(x,y) and w(x, y). (c) The functions ipE and WE are harmonic functions, i.e. (2.1.82) This follows from (2.1.76) through (2.1.78).

2.1.13

Movement of a front

A front is defined here as surface containing fluid particles that have the same value of a considered scalar property, b, bym > b, and bzm > b, we have O~/OO/ < ~. For straight tubes parallel to the axes, O~/OO/ ~ ~. The ratio O~ / 00/ is thus a measure of the tortuosity of the void space, while the term represents the effect of anisotropy on the tortuosity. We may, therefore, refer to T~ij as the tortuosity of the a-occupied portion ofUo • Note that from the definition in (2.3.49), it follows that the tortuosity, T~, depends on the spatial distribution of the oriented elementary surfaces VO/i dB, between a and all the phases in Uo • This is a basic feature of the a - phase configuration.

-;;;;V;;;O/

2.3. 7

Average of a material derivative

The material derivative of an extensive quantity E of a phase contained in Uo , is given by

(2.3.73) where we have omitted the subscript a in E and e. Making use of (2.1.17), equation (2.3.73) becomes = =

[

edUO/ + [

[

{(e)

JUoa

JUoa

JUoa

e\l·yE dUO/

+ e\l.yE} dUO/.

(2.3.74)

132

MACROSCOPIC DESCRIPTION By the definition (1.3.3) of the intrinsic phase average, (2.3.74) becomes

(Uo~ea) = or uoaia

Uoact"

+ Uoaea =

+ eV.yE a ),

Uoa(e:"

+ eV.yE a ).

(2.3.75)

We note that by (2.1.17), with dUa == dUE"" we have

(2.3.76) Hence, (2.3.75) reduces to (2.3.77) which relates the material derivative of an intrinsic phase average to the average of the material derivative. For an isochoric continuum (at the microscopic level), i.e., for V·yE == 0, we obtain e:" = ia.

2.4

Macroscopic Balance Equations

In this section, we apply the averaging rules derived in Sec. 2.3 to the microscopic balance equation of an extensive quantity E, and obtain the corresponding macroscopic balance equation. We then apply the results to particular cases of E. As in Secs. 2.2 and 2.3, the considered quantity E will be of the a-phase only. Hence, the subscript a to indicate this fact is deleted wherever there is no danger of ambiguity (Le., e == e a , Y == Va, yE == yE"" v == Va, () == ()a, p == Pa, etc.).

2.4.1

General balance equation

A possible starting point is the microscopic differential balance equation (2.2.13), written for any extensive quantity E, in the form

~;

= -V.(eY +jEU)

+ prE,

(2.4.1)

in which the total flux of E is decomposed into an advective flux, at the volume weighted velocity, y, of the a-phase, and a diffusive flux, jEU, with

133

Balance Equations

respect to V. We recall that in this equation, -V·(eV +jEU) == -V·eVE represents the net influx of E per unit volume of the phase per unit time. Hence V·eVE is an additive quantity over spatial domains. By integrating (2.4.1) over the phase present within Uo , and dividing the result by the volume Uo , we obtain

~1

Uo

Uoo.

ae dU at

= (2.4.2)

By employing (1.3.5), equation (2.4.2) takes the form

Fe -

at

-

= -V.(eV +jEU) + prE,

(2.4.3)

or, in view of (1.3.6) (2.4.4) By employing rules (2.3.9) and (2.3.25), we rewrite (2.4.4) in the form

a O..".a at e -

-

where

~(3 eu·v E

~(3 1 ( .. ) Ea(3 = U

o

or

{)oea

at

(a)

a

(3

1 (..)

dB

and

sa.{3

_---;(){(3

-V·O(eV + jEut (b)

~(V - u)·v

Ea(3

(c) (2.4.6)

~(3

-J·EU ·V "LJ a (3 (d)

+

oprEa , (e)

where: (a) Rate of increase of E (in the phase), per unit volume of porous medium.

134


(b) Net influx of E by advection and diffusion, per unit volume of porous medium. (c) Amount of E entering the phase, through the interface surface, 80/(3, of the phase within Uo , per unit volume of porous medium and per unit time, by advection with respect to the (possibly moving) 80/(3-surface. (d) Same as (c), but by diffusion through 80/(3. (e) Amount of E generated by sources of E within UOO/, per unit volume of porous medium and per unit time. By (2.3.3), the (intrinsic phase) averaged advective flux, eVO/, may be --00/ decomposed into two fluxes: a flux eV and a macroscopic advective flux eavO/. With these fluxes, (2.4.6) is rewritten in the form

V.8(eO/VO/ + eV0/ + jEUO/) r--------------&(3

{e(V - u) +jEU}.v

}:;O/(3 + 8prEO/.

(2.4.7)

Equation (2.4.7) is the general (macroscopic) differential balance equation of an extensive quantity, E, of a phase. Before looking into particular cases of interest, let us compare the macroscopic balance equation (2.4.7) with the microscopic one, (2.2.13). We note that the macroscopic equation (2.4.7) contains two additional terms, introduced as a result of the macroscopization (averaging) process: • A flux eV which is the flux of E in excess of the average advection of E by the phase. In Subs. 2.6.4 we shall refer to this flux as the dispersive flux and discuss it in more detail, and --00/

__--------~~~&(3 • A term -{e(V - u) + jEU}.v }:;O/(3 which expresses the influx of E across the 80/(3-surface, which separates the considered phase from all other phases within Uo , by advection relative to the possibly moving 80/(3-surface and by diffusion. It is of interest to note that by the averaging process, the boundary conditions on the interphase boundaries, 80/(3, became a source term in the macroscopic equation.

Balance Equations

135

Making use ofthe material derivative discussed in Subs. 2.1.5, the balance equation (2.4.7) can be rewritten in the form -0

where

n~2

kj/ = C/(aji)-lTii =

oZ} + -I P g ox/ '

n3

CI(~sl)2(aji)-lTii

(2.6.54)

(2.6.55)

is a coefficient related only to macroscopic parameters that describe the geometrical configuration of the void space. It is called the permeability, or intrinsic permeability, of the porous medium (since we have assumed here that the fluid occupies the entire void space). We note that, in general, even in the case of an isotropic homogeneous porous medium, k = const., we have V X q~ 'I o. Hence, the specific discharge, q;:n, does not possess a potential (as defined in Subs. 2.1.12), i.e., the macroscopic motion is not a potential one. Equation (2.6.54) is usually referred to as the generalized Darcy law (compare with (2.6.57) ). The determinant of aij is always positive, and therefore, the tensor aij is unconditionally invertible. Since both (aij)-l and Tii are symmetrical tensors, and assuming that they have the same principal directions (see Subs. 2.6.5), the coefficient kj/

175

Macroscopic Fluxes

is also a symmetrical tensor. A detailed discussion on second rank transport coefficients is presented in Subs. 2.6.5. Although we present the general expression for permeability in the form of (2.6.55), the actual value of kj/ components of the tensor k, for particular porous media of interest, must be determined experimentally. It may be of interest to compare the above expressions for the permeability, as given by (2.6.55), with any of the forms of Kozeny's equation (see, for example, Bear, 1972, p. 166), recalling that Cf = n/"£ 8f fl. c ' For example, one such form is k = coTn 3 / M2, where Co is a dimensionless coefficient, M == "£8f and T is a coefficient called tortuosity. Also, in (2.6.55), we note the dependence of the permeability on fl.} and on a tensorial factor that represents the geometry of the void space. More details on the properties of the permeability tensor, e.g., the effect of coordinate rotation, principal directions, directional permeability, etc., are presented in Subs. 2.6.5. Equation (2.6.54) is the common form of the motion equation for flow in saturated anisotropic porous media, when condition (2.6.53) is satisfied. Two particular cases of equation (2.6.54) are of practical interest. (a) For flow of a fluid of constant density, i.e., pf = constant, we may introduce the piezometric head -f _

cp - z

pf

+ -f p g'

(2.6.56)

that expresses the mechanical energy due to gravity and pressure of the fluid, per unit weight of fluid. Then, q~ = qr, and (2.6.54) reduces to (2.6.57) where the second rank symmetrical tensor (2.6.58) is a coefficient called hydraulic conductivity, and - Vipf is called the hydraulic gradient. We note that K depends on properties of both the fluid phase (pf / J.L / , often referred to as fluidity, equal to the reciprocal of the fluid's kinematic viscosity), and the solid matrix (through the permeability tensor, k). The motion equation (2.6.57) is usually

176


referred to as Darcy's law, as it was proposed, on the basis of experiments of water flow in a sand column, by the French engineer Henry Darcy (1856). However, the original Darcy formula is only a special case of (2.6.57), which corresponds to unidirectional flow of a homogeneous liquid through a homogeneous, fixed and nondeformable porous medium. Darcy's law, in the form of (2.6.57), states that the relative specific discharge is proportional to the hydraulic gradient. Here it was developed from first principles as an approximate macroscopic momentum balance equation. For flow in an isotropic, homogeneous, fixed and nondeformable porous medium, k = const., and for pf = const., Jlf = const., the scalar K7j5f acts as a potential. The flow as defined by (2.6.57), is referred to as potential flow. (b) For flow of a compressible fluid, in which the average density depends on pressure only, i.e., pf = pf (pf), we may use the potential, i.p*f, defined by Hubbert (1940) in the form

1

-f i.p* = z +

p!

Po

dpf gpf (pf) ,

(2.6.59)

to rewrite (2.6.54) in the form

q~ = -K.Vi.p*f,

(2.6.60)

where Po is a reference fluid pressure. The potential i.p*f is often referred to as Hubbert's potential.

CASE B. Condition (2.6.51) is valid, i.e., the inertial effects are negligible, but not (2.6.52), i.e., we do not neglect the effects of internal friction, expressed by the second term on the r.h.s. of (2.6.48). Then, we obtain kjp(T;k)-l

o2q;!j

n

OXiOXi kkj

(&Pf _

OZ)

+ Jlf oxj + pf 9 oxj + qrk = o. m

(2.6.61)

From the discussion in Sec. 3.3, it follows that (2.6.61) is a good approximation of (2.6.48), when 1 (2.6.62) ReDa 2 'T:-1'kn

"'ww f.j

=

Afi

",wn f.j

=

* ] w [ a(nw) (nw) *-1 Af.i ....-n(}2 a ·T n .kn,Tn(}n .. , J-Ln n ')

",nn

=

An, 1

(nw) "'f.j

=

] n [ a (nw) (nw) *-1 * Af.i n-w(}2 a ·Tw ·kw·Tw(}w .. ' J-Lw w I)

",!:wn

=

A~ [a(nW)a(nw>.(

f.j

f.j

f.,

f.1

J-Ln

n

[0~ + ra( } 2 a(nw),T*-l.kw w w

w

)-T:V(}w] .. , '3

]

.. ' ')

d n 1 + '£nw a(nw),T*-l.k )]

re'

n

n

r(}2

n

n

n

n

,,'

I)

187

Macroscopic Fluxes

in which

In the motion equations (2.6.95) and (2.6.96), we note two main features: • coupling between the two phases. The forces due to pressure gradient and to gravity in one fluid, cause motion in the other one (due to the momentum exchange at their common boundary).

• . Inhomogeneity in surface tension acts as an additional driving force. Surface tension, in turn, is a function of composition and temperature in the two fluids. For the sake of simplicity, let us assume that we may express the surface tension in the schematic form

where Cw and Cn represent the concentrations of components in the wetting fluid and in the nonwetting one, respectively, and T represents the temperature, assumed the same in both fluids. Then, we may ,--_ _ _A11n replace 8'Ywn/OXi

by

~_--::-_A11n

oCw ---oCw OXi

'O'Ywn

Assuming that

.....A11n

~_-::-

'O'Ywn oT + -o-T- -OX-i

188

MACROSCOPIC DESCRIPTION and similar expressions for en and T, we may approximate the expres-

_---AlIn

sion for 8'Ywn / {)Xi

,by

This expression is then introduced into the motion equations. The resulting equations indicate that, in principle, the mass transport problem for each phase is also coupled to those of component and energy transport in the two phases. Contrary to the distribution of phases within an REV, as dictated by the concept of wettability (Subs. 5.1.1), observations seem to support the notion that in multi phase flow, each fluid tends to establish its own flow paths through the void space. Accordingly, the wetting fluid tends to completely fill (and move through) the smaller pores, while the nonwetting fluid occupies the remaining, larger, pores, except for the thin film described above. If we accept this picture of phase distribution within the void space, the total surface area surrounding each fluid phase is often assumed to be such that and that, therefore, the fluid-fluid momentum transfer is much smaller than the fluid-solid one. If we also assume that

_---Mn

8'Ywn/ {)Xi

= 0,

the terms including the factor a{nw) vanish, the two motion equations developed above reduce to two uncoupled motion equations, one for each of the phases. The motion equation for the wetting phase, then, reduces to m

_

qrwi -

ew (~ vmS) _ Vwi si -

-

kwij

~ J.lw

({)PWW -w {)Z) - { .) + Pw g-{). ' XJ

XJ

(2.6.98)

where

(2.6.99)

189

Macroscopic Fluxes

Similar expressions can be written for the nonwetting phase. Because the partial area between the wetting fluid and the solid depends on the saturation, Sw, and so do ~w, a(nw) and T~£j' the permeability of each phase is a second rank tensor that varies with the saturation of that phase, viz. (2.6.100) We refer to these permeabilities as effective permeabilities to the wetting phase and to the nonwetting one, respectively. Each of the effective permeability components depends on the geometrical configuration of the void space and its characteristics (e.g., porosity) and on the saturation (that represents the fluid configuration within the void space). In general, the dependence on saturation may be different for the different tensor components. For an isotropic porous medium,

T~ = T~I,

T~ = T~I.

These expressions should be inserted in (2.6.94) through (2.6.100). In Chap. 5, that deals with multiphase flow, we shall make the assumption that the coupling between the phases is small and can be neglected. This is also the assumption that underlies modeling of multi phase flow in such disciplines as reservoir engineering and soil physics.

2.6.3

Diffusive flux

In this Subsection we consider the macroscopic flux (2.6.101) appearing in the macroscopic differential balance equation (2.4.7). We note that (2.6.102) The discussion is limited only to linear diffusive flux equations, in which the flux of E at the microscopic level is proportional to a conjugate driving force that takes the form of a gradient of a single intensive quantity ( == function of state), without coupling (Subs. 2.2.4), in a single phase. No symbol will be used to indicate this phase.

190


Rather than discuss the general case, we shall consider particular cases of diffusive fluxes of mass and of heat, from which other ones may be derived. (a) Diffusion of the mass of a i-component in a single fluid phase that occupies the entire void space, without adsorption. This phenomenon is often referred to as molecular diffusion. At the microscopic level, the flux of molecular diffusion, p, is expressed by Pick's law, (2.2.101), repeated here for convenience in the form

(2.6.103) where 1)"1 is the scalar coefficient of molecular diffusion of the i-component in the fluid phase, and p'Y is the concentration of the i-component in the fluid. Equation (2.6.103) is the simplest form of Fick's law; it expresses the flux of a single component in a single fluid that is a binary system, assuming that 1)'Y is independent of p'Y. Other forms of Fick's law for the diffusive mass flux of a i-component are jm"Ym == p'Y(V'Y _ V m ) = -p1)'YV(p'Y / p), (2.6.104) and, for a binary system (2.6.105) where p'Ymol( = p'Y / M'Y) is the molar concentration of i (= number of moles of i per unit volume of solution), pmol(= E(-y) p'Ymol) is the total molar density of the solution, and j'Y mol mol is the molar (dif fusive) flux of i (say in gr moles, per cm2 per sec), relative to the fluid moving at the molar weighted velocity. The averaging procedure outlined below for (2.6.103) can also be applied to (2.6.104) and (2.6.105). Actually, the diffusive mass flux of the i-component should be expressed in the form 'm"Ym _ 1)nr7 'Y (2.6.106) J -vJ.l, where J.l'Y = J.l'Y(p,w'Y,T) is the chemical potential of the i-component (see Subs. 2.2.4). For dilute solutions, J.l'Y = Cpw'YT, where C is a coefficient, so that (2.6.106) reduces to (2.6.104), with 1)'Y = C1)I'Y pT/ p for constant C, p, T and p. In the passage from (2.6.103) to its macroscopic counterpart, the configuration of the solid-fluid interface surface, and conditions on it, affect the transformation of the (local) gradient of concentration, appearing in

191

Macroscopic Fluxes

(2.6.103), into a gradient of the average concentration, which serves as the state variable at the macroscopic level. We shall exemplify this statement by considering three cases. Since, as explained above, our objective is to study the influence of the configuration of the solid-fluid boundary at every instant of time, it is sufficient to investigate the concentration distribution at that instant of time, assuming no 'Y-sources or sinks within the fluid. Under such assumption, a monotonous distribution of p'Y takes place within Uov , satisfying in Uov •

(2.6.107)

A more rigorous justification of this assumption of quasi-steady state within the void space, can be obtained by employing the methodology of nondimensionaiization, explained and demonstrated in Subs. 3.3.1. The starting point is the diffusion equation (= component mass balance equation) written for the fluid in the void space. In a one-dimensional domain, this equation takes the form (2.6.108) where we assume that 1)"1 is uniform within Uov • Then, following the discussion in Subs. 3.3.1, and the examples presented in Subs. 3.3.2, we introduce

where LVI) is a distance (within the void space) over which significant changes in p'Y occur. With these relations, the diffusion equation (2.6.108) can be written in the form (2.6.109)

192


For diffusion within the pore space, we may use the hydraulic radius as the characteristic length, Le. Thus, when the condition

FoZ>

= -

( ~t)(P"Y) e

~}/1)'Y

>

1

~1

(2.6.110)

prevails, where the dimensionless number FoZ> is the Fourier number associated with the fluid's diffusivity, the diffusion process can be described by the Laplace equation (2.6.107), as a quasi-steady state one. In most cases, this condition is indeed valid for the void space. Here, the Fourier number, FoZ>, gives the ratio between the time interval during which a significant change in concentration occurs and the time required for smoothing out spatial concentration differences by diffusion. The analysis can easily be extended to a three dimensional domain. When the solid-fluid surface, S1s, acts as a material surface to both the fluid as a whole and to the ,-component in it, Le., there is no mass transfer of them across it, then (2.6.111) where v is the outward normal unit vector on S1s' With V'Y = const. within U ov , equations (2.6.107) and (2.6.111) are identical to (2.3.43) and (2.3.50), respectively, that describe CASE A of Subs. 2.3.5, with G a == p'Y. Hence, by averaging (2.6.103), making use of (2.3.51), we obtain -;;:y _

J. - J

V'Y {)p'Y _

---n {)x j

V'YT* {)p'Y1 _

··----n

Jt {)Xi

(1)*'1) .. {)p'Y1

Jt--,

{)Xi

(2.6.112)

where n is the porosity and '[)*'Y = 1)'YT*, a second rank symmetric tensor, is the coefficient of molecular diffusion in a (saturated) porous medium. The definition of T* is presented in (2.3.49) and discussed in detail in Subs. 2.3.6. A detailed discussion on transport coefficients and their tensorial nature, is presented in Subs. 2.6.5. Consider a liquid a-phase that occupies only part of the void space, and a component that does not interact with the solid, nor does it cross the (microscopic) interfaces between the a-phase and the other phases present in Uo • Using f3 to denote all other phases within Uo , and Sa(3 to denote the a - f3 surface, and when condition (2.6.111) prevails on Sa(3, equation

193

Macroscopic Fluxes (2.6.112) becomes -

p. = CtJ

ap"l

ax.

-1)"1J

(2.6.113) where '[)*"1 depends on f) and on all geometrical factors which determine the value of T* defined by (2.3.49). Thus, subject to the conditions (2.6.107) in UOCt and (2.6.111) on SCtj3, equation (2.6.113) expresses the diffusive mass flux of a ,-component in an a-phase that occupies only part of the void space. In Sec. 6.4.2, we shall introduce the case of two, or three fluid phases that together occupy the void space, and a component that can diffuse in more than one of these phases and cross their (microscopic) interphase boundary.

(b) The ,-component can be adsorbed on the solid surface. For a fluid phase that completely occupies the void space, and with the assumptions introduced above with respect to the ,-distribution within Uov , equation (2.6.107) remains valid. Because the solid-fluid surface is a material surface with respect to fluid mass, the adsorbed component can reach the solid wall only by diffusion. Let us assume that this diffusive flux, normal to the solid, can be approximated by f)p"l p"lj - p"l IS -1)"I-v· 1) Is J""Iv· ·z1"I 1

aXi

Do

(2.6.114)

where Do is a microscopic length characterizing the distance between the Sjs-surface and the interior of the fluid phase within U o. From (2.6.114) it

follows that (2.6.115) By comparing (2.6.115) with (2.3.60), we conclude that the case under consideration is identical to CASE C of Subs. 2.3.5, with G Ct == p"l. Hence, using (2.3.61), we obtain

(2.6.116)

194


,--fs where M is a (vector) coefficient defined by (2.3.62), p'Y is the average value of p'Y on the surface Sfs, and ~f is the hydraulic radius (= Uof/Sfs). ,--fs . In Sec. 6.1 we shall see that p'Y is another state variable in a diffusion, or a diffusion-dispersion problem with adsorption. For a fluid that occupies only part of the void space, we replace f by a and n by Ba in (2.6.116). Also, both D~* and ~a depend on Ba.

(c) Conductive (or diffusive) heat flux. At the microscopic level, this flux is expressed by Fourier's law (2.2.100), rewritten here for convenience in the form (2.6.117) where we have used subscript f to indicate that we consider only the case in which a single fluid phase occupies the entire void space. As in the discussion on molecular diffusion presented above, we assume that >'f is constant and the distribution of Tf is quasi-steady within Uof. In the present case, following the discussion presented in (a) above, the assumption of a quasi - steady distribution is valid as long as (2.6.118) where (~t)~T) is the characteristic time for temperature changes, and FoA is the Fourier number associated with conductive heat transfer. The interpretation of Fo A is analogous so that of Fo'D presented above. It is easy to verify that this problem of heat conduction in the fluidsolid system comprising the porous medium is described by CASE B of Subs. 2.3.5, with Ga == Tf and >'01 == >'f, representing the temperature and the thermal conductivity of the fluid phase, and G(3 = Ts and >'(3 = >'s, representing the temperature and the thermal conductivity of the solid phase. We may, therefore, average (2.6.117) and employ (2.3.58) to obtain

195

Macroscopic Fluxes

For As = 0, i.e., a solid that is nonconductive, (2.6.119) reduces to (2.6.120) in which Aj = AjTj is the coefficient of heat conduction of the fluid occupying the void space of a porous medium. An expression analogous to (2.6.119), in terms of Os = 1 - n and Os T; = I -OjTj (see (2.3.56)), can be written for the macroscopic conductive heat flux in the solid phase, j!". It is interesting to compare (2.6.112) (2.6.120) and (2.6.119). Because the fluid-solid interface, Sjs, is 'impervious' to mass transfer, the relationship (2.6.112) shows that the macroscopic diffusive mass flux of the component depends on the concentration of the component within the fluid phase only. However, with respect to heat, the solid-fluid interface is a 'permeable' surface, unless the solid in nonconductive. As a result, the macroscopic conductive heat flux depends on the temperatures in both the solid and the fluid phases. Thus, (2.6.119) involves coupling between the heat transported in the two domains, Uoj and Uos, with heat being continuously exchanged between the two phases. The coefficient Tj is called tortuosity of the void space, or of the Sjs-configuration. It is the geometrical property defined by (2.3.49) and discussed in Subs. 2.3.6. For the special case of T/ ~ Ts s, i.e., when equilibrium exists between the intrinsic phase average temperatures of the fluid and of the solid, (2.6.119) reduces to (2.6.120). This reduction should have been expected, since, on the average, no heat crosses the interface between the two phases. Hence, (2.6.120) is also valid for both T/ ~ Ts s, and for As = O. For heat flux through the porous medium as a whole, jMn, we have -

-j

jlfm = nj7

-s

+ (1- n)j!"

where we have made use of (2.3.52). We note that by summing over the two phases, the surface integrals that express the heat exchange between the two phases have been eliminated. When T/ = Ts s, the total heat flux in the porous medium as a whole is given by

196

MACROSCOPIC DESCRIPTION (2.6.122)

where AH = n~j + (1 - n)~: is the thermal conductivity of the saturated porous medium as a whole. The problem of heat conduction, including the case of multiple fluid phases that occupy the void space, is further discussed in chap. 7.

2.6.4

Dispersive flux ex

--0-0'

-0

The term 8eV (== 8e V ) appearing in the general differential macroscopic balance equation (2.4.7) represents the dispersive flux (per unit area of porous medium). It is a macroscopic flux of E of an a-phase, relative to the transport of E at the average velocity, Va a, of that phase. This flux results from the variation of both the microscopic velocity and the density, ea , within the REV. We recall that (2.4.7) is an average of the balance equation (2.4.1), in which we have preferred to express the total flux of E by ea Va + jE. In order to solve mass transport problems at the macroscopic level, we ---o-Ct -a have to express a Va, in terms of average variables, such as ea a and Va. This is our objective in the present section. Subscript a will be omitted whenever it is obvious which phase is referred to. Before developing an expression for the dispersive flux in terms of averaged velocity of a phase and averaged density of an extensive quantity, let us take a second look at the concept of dispersion of a component of a fluid phase. For the sake of simplicity, we shall illustrate this concept by referring to the flow of a single fluid phase that occupies the entire void space, and to the mass of a component of that phase as the extensive quantity. Nevertheless, the discussion is equally applicable to a fluid phase in a multi phase system, and to the density of any extensive quantity. Consider the flow of a fluid phase through a porous medium. At some initial time, let a portion of the flow domain contain a certain mass of an identifiable component. This component may be referred to as a tracer. In Subs. 2.6.1, we have developed an expression for the fluid's (average) velocity. With this development in mind, let us conduct two field experiments. Figure 2.6.1a shows an (assumed) abrupt front in a two-dimensional flow domain in a porous medium at t = O. This front separates the porous medium domain occupied by a tracer labelled fluid (c = 1) from the one occupied by the same, yet nonlabelled, fluid (c = 0). If uniform flow (nor-

e

Macroscopic Fluxes

197

mal to the initial front) at an average velocity, V, takes place in the porous medium, Darcy's law provides the position ofthe (assumed) abrupt front at any subsequent time, t, through x = Vt. On the basis of Darcy's law alone, the two parts of the fluid would continue to occupy domains separated by an abrupt front. However, by measuring concentrations at a number of observation points scattered in the porous medium, we note that no such front exists. Instead, we observe a gradual transition from the domain containing fluid at c = 1, to that containing fluid at c = 0. Experience shows that as flow continues, the width of the transition zone increases. This spreading of the tracer labelled fluid, beyond the zone it.is supposed to occupy according to the description of fluid movement by Darcy's law, and the evolution of a transition zone, instead of a sharp front, cannot be explained by the averaged movement of the fluid. As a second experiment, consider the injection of a small quantity of tracer labelled fluid at point x = 0, y = 0, at some initial time t = 0, into a tracer-free fluid that is in (macroscopically) uniform flow in a two dimensional porous medium domain. Making use of the· (averaged) velocity as calculated by Darcy's law, we should expect the tracer labeled fluid to move as a volume of fixed shape, reaching point x = Vt at time t. Again, field observations (shown in Fig. 2.6.1b) reveal a completely different picture. We observe a spreading ofthe tracer, not only in the direction of the uniform (averaged) flow, but also normal to it. The area occupied by the tracer labelled fluid, which has the shape of an ellipse in the horizontal two-dimensional flow domain considered here, will continue to grow, both longitudinally, i.e., in the direction of the uniform flow, and transversally, i.e., normal to it. Curves of equal concentration have the shape of confocal ellipses. Again, this spreading cannot be explained by the averaged flow alone (especially noting that we have spreading perpendicular to the direction of the uniform averaged flow). The spreading phenomenon described above in a porous medium is called hydrodynamic dispersion (or miscible displacement). It is an unsteady, irreversible process (in the sense that the initial tracer distribution cannot be obtained by reversing the direction of the uniform flow), in which the mass of the tracer continuously mixes with the nonlabelled portion of the moving fluid. The phenomenon of dispersion may be demonstrated also by a simple laboratory experiment. Consider steady flow of water in a column of homogeneous porous material, at a constant discharge, Q. At a certain instant, t = 0, tracer-marked water (e.g., water with NaCl at a low concentration, so

198


-

-

-

Abrupt front at t = 0

···········1

1

iMil

t/j

v

-

-c=l

I....~------

L

·1

=Vt

1.0 ~----------~

Time

o

t=o

(a)

-

-

Tracer injected at t = 0

v

Contours of c = const.

(b) Figure 2.6.1: Longitudinal and transversal spreading of a tracer. (a) Longitudinal spreading of an initially sharp front. (b) Spreading of a tracer injected at a point.

199

Macroscopic Fluxes 1.0

~

I~Actual C~ith dispersion)

0.5

o

~

~

/!

I.-- Without

1

I

dispersion

3

2

4

Qt/Ucol Figure 2.6.2: Breakthrough curve in one-dimensional flow in a column of homogeneous porous material. that the effect of density variations on the flow pattern is negligible) starts to displace the original unmarked water in the column. Let the tracer concentration e = e(t) be measured at the end of the column and presented in a graphic form, called a breakthrough curve, as a relationship between the relative tracer concentration and time. In the absence of dispersion, the breakthrough curve would have taken the form of the broken line shown in Fig. 2.6.2, where Ucolumn is the pore volume in the column, and Q is the constant discharge through the column. In reality, due to hydrodynamic dispersion, it will take the form of the S-shaped curve shown in full line in Fig. 2.6.2. As stated above, we cannot explain all the above observations on the basis of the average flow velocity. We must refer to what happens at the microscopic level, viz., inside the void space. There, we observe velocity variations in both magnitude and direction across any pore cross-section. Recalling the parabolic velocity distribution in a straight capillary tube, we usually assume zero fluid velocity at the solid surface, and a maximum velocity at some internal point within the fluid. The maximum velocity itself varies according to the size of the pore. Because of the shape of the interconnected pore space, the (microscopic) streamlines fluctuate in space with respect to the mean direction of flow (Fig. 2.6.3a and b). This phenomenon, referred to as mechanical dispersion, causes the spreading of any initially close group of tracer particles. As flow continues the tracer particles will occupy an ever increasing volume of the flow domain. The two basic factors that produce mechanical dispersion are, therefore, flow and the presence of

200

MACROSCOPIC DESCRIPTION Direction of average flow ~

(a)

(b)

(c)

Figure 2.6.3: Dispersion due to mechanical spreading (a,b) and molecular diffusion (c).

a pore system through which the flow takes place. Although this spreading is in both the longitudinal direction, namely that of the average flow, and in the direction transversal to the latter, it is primarily in the former direction. Very little spreading in a direction perpendicular to the average flow is produced by velocity variations alone. Also, such velocity variations alone cannot explain the ever-growing volume fully occupied by tracer particles dispersed normal to the direction of flow. In order to explain the latter observed spreading, we must refer to an additional phenomenon that takes place in the void space, viz., molecular diffusion. Molecular diffusion, caused by the random motion of molecules in a fluid, produces an additional flux of tracer particles (at the microscopic level) from regions of higher tracer concentrations to those oflower ones. This flux is relative to the one produced by the average flow of the phase. This means, for example, that as the tracer particles spread along each microscopic streamtube, as a result of mechanical dispersion, a concentration gradient of these particles is produced, which, in turn, produces a flux of tracer by the mechanism of molecular diffusion. The latter phenomenon tends to equalize the concentration along the stream tube. Relatively, this is a minor effect. However, at the same time, a tracer concentration gradient is also produced between adjacent streamlines, causing lateral molecular diffusion across streamtubes (Fig. 2.6.3c), tending to equalize the concentration across pores. It is this phenomenon that explains the observed transversal dispersion.

Macroscopic Fluxes

201

As will be shown by (2.6.124) and (2.6.128) below, the deviations, e, develop by both the contribution of fluctuations in the advective velocity of the phase and of fluctuations in the diffusive velocity of e. It is through this reason that molecular diffusion contributes to the dispersive flux. it may thus be concluded, that even when the macroscopic effect of diffusion is relatively small, it is only the combination of microscopic velocity fluctuations and molecular diffusion that produces mechanical dispersion. Also, it is molecular diffusion which makes the phenomenon of hydrodynamic dispersion in purely laminar flow irreversible. In explaining why dispersion is an irreversible phenomenon, exhibited, for example, by the growing width of a transition zone around an initially sharp front in uniform flow, as the direction of the flow is reversed, a distinction should be made between the physical irreversibility of a process, and the irreversibility that depends on the scale and procedure selected for the description of the process. In the present case, the only possible physical cause is molecular diffusion, while the phenomenological reason is the procedure of !,\-veraging of the microscopic velocities. Hence, some theories lead to (irreversible) mechanical dispersion merely because of the averaging procedure they employ to derive a macroscopic description, even without explicitly resorting to molecular diffusion as a cause for irreversibility. We refer to the flux that causes mechanical dispersion (of a component) as dispersive flux. It is a macroscopic flux that expresses the effect of the microscopic variations of velocity in the vicinity of a considered point. We note that the decomposition of the average of the total (local) advective flux into an advective flux at the average velocity and a dispersive flux, is merely a result of the averaging process. We use the term hydrodynamic dispersion to denote the spreading (at the macroscopic level) that results from both mechanical dispersion and molecular diffusion. Actually, the separation between the two processes is rather artificial, as they are inseparable. However, molecular diffusion alone does take place also in the absence of motion (both in a porous medium and in a fluid continuum). Because molecular diffusion depends on time, its effect on the overall dispersion is more significant at low velocities. In addition to the variations in the local velocities from their average at the macroscopic level, due to the presence of pores and grains, variations may also exist in the average velocities at the megascopic level of description. These may be due, for example, to spatial variations in permeability from one portion of the flow domain to the next. Such phenomenon will manifest itself in the form of additional dispersion at the megascopic level of description

202


(see Subs. 2.6.7). Dispersion may take place both in a microscopically laminar flow regime, where a fluid moves along definite paths, and in a turbulent regime, where the turbulence may cause yet an additional mixing. In what follows, we shall focus our attention only on flow of the first type. In general, variations in tracer concentrations cause changes in the fluid's density and viscosity. These, in turn, affect the flow regime (i.e., velocity distribution) that depends on these properties. We use the term ideal tracer when the concentration of the latter does not affect the fluid's density and viscosity. At relatively low concentrations, the ideal tracer approximation is sufficient for most practical purposes. However, in certain areas, for example in the problem of sea water intrusion into a region of fresh water, the density may vary appreciably, and the ideal tracer approximation should not be used. With the above comments, it should be clear why we refer to the av--o-a erage ()eVE as the dispersive flux of E in a fluid phase. Although the above discussion uses mass of a component as an example, the conclusions are equally valid for any extensive quantity (for example heat, with heat conduction playing the role assigned above to molecular diffusion). Based on the above concepts, let us now develop an expression for the dispersive flux of an extensive quantity, E, that is being transported through a porous medium. The phase may occupy the entire void space, or only part of it. In what follows, subscript a, denoting the phase will be omitted. For example, E will stand for EO!. The microscopic velocity, V E , of any E-particle, within an a-phase, can be presented as a sum of two parts: (2.6.123)

i.e., an average velocity and a deviation relative to it. It is these deviations that affect the dispersive movement of E. We may further decompose yE by writing

(VE _ V) _ (VEO! _ yO!) + (V _ yO!)

(jE Je) + V,

(2.6.124)

i.e., a sum of deviations of two velocities: a diffusive one and an advective one.

203

Macroscopic Fluxes

The balance of an element of E, as it moves, is given by (2.2.11), rewritten here in the form

~: + V.eV E == ~: + VE·Ve + eV·V E = prE.

(2.6.125)

The corresponding averaged equation is given by

11

-8ex - 8t Uoa

eu·v dB 0

Saf)

+ VE·Ve a + eV·VE a

= ~ prE .

(2.6.126)

By subtracting (2.6.126) from (2.6.125), employing (2.3.3), and decomposing any local value into an average and a deviation, we obtain

8e 8t

~

VE·Ve a - eV·VE 0

+ (prE) 0

VE·(Ve) + VE·(Ve) 0

0

0

a

(2.6.127) Consider an ensemble of E-particles that occupy the domain Uoa within an REV. At a giyen instant, t = to, each particle is identified by its position vector, xElt=t o = x - xo(t o) = xE(t o). The state of the ensemble at t = to is described by an average density, ex, and an average velocity, VE a , which are common to all particles. This state will serve as an initial, or reference, state for describing the behavior of the ensemble at later times, t > to, and during the time increment, t!.t = t - to, over which the velocity yEa may be taken as approximately constant. During this time interval, the E-particles spread out over a larger domain, due to the velocity variations, V E , that exist within the fluid phase. The local rate of change of the density deviations

e == eIXE(t) - ea(xo(t), t), is described by (2.6.127). By integrating this equation over D.t, we obtain

e(xE(t), t = to + t!.t) t=to+.:lt = Vea.VElxE(t) dr to ft=to+.:lt ~ + lto {(prE) - eV·VE

I

0

0

+ F}lxE(t),T dr

= -(xE(t) - xE(to))·Vealto+e.:lt

ft=to+.:lt

+ lto

0

~

{(prE) - eV·VE

+ F}lxE(to),T dr, (2.6.128)

204


where E!:l.t denotes some intermediate time interval during !:l.t, and we have used the fact thatei:,cE(to),to = ea, and, hence, elxE(to),to = o. In (2.6.128), the symbol F represents all the remaining terms on the r.h.s. of (2.6.127). We note that the variation in e during the time interval, !:l.t, are caused by changes in the course of time, both in e, at a fixed point, x, and in ea(xo(t), t), at the moving centroid, xo(t). Multiplying (2.6.128) by yElxE(to),t, and averaging the resulting equation over Uooo we obtain

eyEal/1

+ V.VE a!:l.t) ~ _VE(xE -

eE)alt.vea + YE(prE)a !:l.t, (2.6.129)

where we have assumed that • all terms in the integrand on the r.h.s. of (2.6.128) can be approximated by their values at time t,

•

(2.6.130)

•

(2.6.131)

• averages of products of three deviations are negligibly small with re.spect to the other terms, and can, therefore, be neglected. Let us now focus our attention on the first term on the r.h.s. of (2.6.129). Consider a coefficient, D,E, defined by (2.6.132) or, in indicial notation

,E _ Vb E oE a D ij i Xj •

(2.6.133)

The physical meaning of D,E is illustrated in Fig. 2.6.4. According to this figure

.. {

-VEx.? = t J

Dxf 0E

a

oED 4>j oE Dt Xi

a

otXi

Xi

(xf) "2 Dt 1D

2

a

CE)2D 4>/' Dt Xi

for

i

= j,

for i f; j,

(2.6.134)

205

Macroscopic Fluxes

X' 1

T

xlP1

IFigure 2.6.4: Physical meaning of D'E. i.e., the diagonal'components of D'E represent half the average rate of growth of the square of the distance of E- particles within the REV from their instantaneous center there, measured along the respective coordinate axes. Thus, the diagonal components of D'E serve as a measure of the rate of dispersion of the E-particles relative to their average motion. On the other hand, any off-diagonal component of D'E, represents the average over the REV of a product of the angular rate of rotation of a radiusvector of an E-particle, taken along one of the coordinate axes in a given coordinate plane, multiplied by the square of its length. The macroscopic effect of such rotation is obtained by decomposing any off-diagonal c-omponent of D'E into a symmetric part and an antisymmetric one (see Fig. 2.6.4) 6

E oE OI

V, x'J

0-=:-=01

Thus, ViE xf includes both a macroscopic rate of deformation (= strain) and a macroscopic rate of rotation of right-angled configurations, of mean length a, of E-particles within an a-phase in an REV. However (see below) (2.6.136)

206


Le., DE represents a macroscopic rate of strain only. o Ci Our next task is to express the components ViE xf ,in terms of components of the average velocity vector, VCi. To this end, we note that (2.6.137) In order that the r.h.s. of (2.6.137) be a nonrandom function of time, the time interval, Llt, must be a Representative Elementary Time interval, (RET), (.~t)o. This means that (Llt)o should be sufficiently large, so as to make the average of the r.h.s. of the last equation, over (Llt)o, independent of Llt. On the other hand, (Llt)o should be sufficiently small, so that we may assign any average over Llt, say, between t - Llt /2 and t + Llt /2, to the time t. This condition is satisfied if averaged values vary linearly over Llt. If this condition is valid, we may rewrite (2.6.137) in the form

Ci ViE XjE It o

0

=

Ci ViE ltjE I/Llt)o 0

0

~

ViE ltjE 0

0

Ci

(Llt)o.

(2.6.138)

The symmetry in i and j justifies (2.6.136). The RET, (Llt)o, may be expressed as a ratio between a statistically representative length, .e~ (e.g., correlation length between velocities, yE), and a representative magnitude of the velocity, O(IYEI). By (2.6.124), and employing the linear law/of diffusive flux, e.g., (2.2.100), or (2~2.101), we have for an a-phase

and we select

.eE

Llt=(Llt)o=V;X+;~/LlCi

.eE

-V:Ci

PeE

l+;e~'

(2.6.140)

where a Peclet number is defined by (2.6.141) Note that Pe~ expresses the ratio between the rates of transport of Eparticles by advection and by diffusion.

207

Macroscopic Fluxes

;7'

a - phase

:Ill

Figure 2.6.5: Nomenclature for local and average velocities. Following Nikolaevski (1959), the local velocity, V, at a point x, belonging to the a-phase within an REV centered at xo, can be represented as a linear transformation (involving rotation and stretch) of the average velocin that REV. With and V in the directions indicated by the ity, unit vectors 1s and 1s', respectively (Fig. 2.6.5), we may express the above transformation in the form

va,

va

v:s' -- (3*~ I's' -

(3*;-;C¥V .1 S ' -- (3*~V dXi dXi , (2.6.142) ds ds' where (3* is a coefficient of proportionality. In the Xi - system, the above relation takes the form dx j VJ. -- (3*.,.....--av. s' ds'

(2.6.143) where

T. .. - (3* dXi dXj on) ds' ds' is a (microscopic) random tensor that transforms the components of the average velocity of a phase in an REV into the components of the local microscopic velocity of the phase at a point inside the REV. From (2.6.143), we obtain and

o

Vj =

Tajl VI 0

~

.

(2.6.144)

208

MACROSCOPIC DESCRIPTION Hence o

a_Q'~

0

Vi .

TOIikTOIj/ Vk

For heat transport, we replace V~ by

Pe H = 01

(2.6.145)

>"01/ POlCVOI' so that

VaOl ~OI • >"01/ POICVOI

With (2.6.145) and (2.6.140), equation (2.6.138) becomes IE Dij

-

Vi 0

Eo EOI """"Q""""""""OO Xj = {ViVj

f ETo 01

TO 01 OIik OIjl

+ (Ji.E /e)(Jj.E Ie) 0

0

-01-01 E Vk Vi Pe VOl 1 + Pe~ + Oi

(

01

}(~t)o

E .!E/oe)( .!E/oe)OI fOi Jt JJ VOl

E

Pe 1+ Pe~ Oi

•

(2.6.146) Also, by (2.1.21) and the linear law of diffusive flux, we have

V·V E

V) + V·V _V~V.(~e) ~ +V:V.(V;),

V·(V E -

=

_

(2.6.147)

where P is the mass density of the a-phase and V,,:: is the coefficient of diffusion of the total mass of the phase. From (2.6.140) and (2.6.147), we obtain

=-=w (~t)o

1 + V·V

{D~

"" 1 + ~~ +

I(.p, '["7)} Pe~ f~ v P Pe~ + 1 VOl

cE ( 1 + I ~ 6. 2 ) PeE + 1 + -s; Oi

Pefl + 1

where

I == I(p, V p) = V'[:V·

(V) : -

.01

~

(2.6.148)

(2.6.149)

Substituting (2.6.148), (2.6.146), (2.6.140) and (2.3.48) into (2.6.129), yields

209

Macroscopic Fluxes

(2.6.150) where we have deleted the terms

e ,and assumed

-et

Let us consider two cases of particular interest. CASE A. The motion of the a-phase is isochoric, the density, p, is uniform, the extensive quantity, E, is conservative and the interface, Set!], is impervious to the diffusive flux of E. Then

p = 0,

V P = 0,

(2.6.151)

prE = 0,

and (2.6.150) reduces to (2.6.152) where the coefficient

E Vk etvi'" aetiklm vet Pe~ E

aetiklm

PeE + 1 +et fg' / ~et

Wvi'" E E vet f(Pe et ,fet / ~et)

(2.6.153)

is called the coefficient of mechanical dispersion (dims. L2 T- 1 » of the extensive quantity, E, within an a-phase, under the conditions of CASE A, and (2.6.154) is a coefficient called the dispersivity (dim. L) of the extensive quantity E in the a-phase within a porous medium. CASE B. The same as CASE A, but E is nonconservative, owing to decay at a rate expressed by prE = -kEe,

where kE is a decay coefficient.

(2.6.155)

210

MACROSCOPIC DESCRIPTION In such case (2.6.156)

and (2.6.150) becomes (2.6.157) In Chap. 6, the dispersive flux presented here, will be developed for E = m'Y, i.e., to the mass of a component. Thermal dispersion will be discussed in Subs. 7.1.3.

2.6.5

Transport coefficients

The coefficients Oc", ~a.6' Tij' Qij, kij, aiklm, Oim, etc., which appear in the various macroscopic flux equations presented above, represent, at the macroscopic level, geometrical characteristics of the microscopic domain occupied by a phase (or of the void space, in the particular case of a single fluid phase that occupies the entire void space). Some of these coefficients are related to specific transport processes, e.g., advection, diffusion and mechanical dispersion, while others are encountered in a number of (or in all) processes. For a given porous medium, and for a given distribution of the phases comprising it, each of these coefficients may attain different numerical values when specified in different coordinate systems. However, since the configuration of a phase is independent of the arbitrary coordinate system selected for its description, the transformation of any such coefficient from one coordinate system to another, must obey a certain rule which en~ures that the intrinsic property described by the coefficient remains invariant. The type of rule used for transforming a coefficient upon transition from one coordinate system to another, may, thus, serve as a means for classifying the coefficients. In the case on hand, all the coefficients belong to a common category of quantities called tensors. Tensors are classified by rank, or order, which also determines the number of the tensor's components. Thus, in the physical three-dimensional space considered here, a tensor of rank n has 3n components, independent of the selected coordinate system. Accordingly, a tensor of order zero has only one component. We refer to it as a scalar. For example, the coefficients n,

211

Macroscopic Fluxes

and ~0/.6' which are fully described only by their magnitude, are scalars. Scalars are invariant under any change of the coordinate system. A tensor of order one is called a vector; it has 31 = 3 components, whereas a second rank tensor has 32 = 9 components. In this book, all tensors of rank one (Le., vectors), or higher, are represented by a bold-face letter. A typical component of a tensor is represented by a letter representing the tensor, to which subscripts, or superscripts, (often called indices) are appended. The number of indices in a component determines the tensor's rank. However, when the summation convention, introduced by (2.1.14), is invoked, the rank of a tensor is determined by the number of unrepeated (Le., unsummed, or free) indices only. Often we use a typical component to represent the tensor. Thus, Ttj, nij, kij and Dij, are components of second rank tensors, whereas the dispersivity, a, represented by a typical component, aiklm, is a fourth rank tensor, with 34 = 81 components. The following rule applies to all the coefficients mentioned above. It also serves as a mathematical definition of a Cartesian tensor in a threedimensional space. dimensional space.

(JO/

In a three-dimensional space, a Cartesian tensor, A(n), of order n, is a quantity represented in any rectangular Cartesian coordinate system, Xi, i = 1,2,3, by an ordered set of 3n numbers, Aij ...lm, called components of the tensor, which upon transition to another rectangular Cartesian coordinate system, x~, transforms (Le., the numerical values of the components transform) to a new set of 3n components, A~s'" uv, according to the rule A'

rs ... uv

~

n

indices

- {)x~ {)x~

-!l

!l

UXi UXj

'-

n

{)x~ {)x~ A ..

'J ...lm , UXl uXm ~ ~ n indices

• "!l!l v

(2.6.158)

derivatives

where

{)X~ = _ ' ) cos( lXr, lXi

~ UXi

is the cosine of the angle between the positive directions of the x~-axis in the new coordinate system and the xi-axis in the old one. Thus, tensors are identified by a linear transformation of their components upon transition from one coordinate system to another.

212


A second rank tensor A, with components if for all

Aij,

is said to be symmetric,

i:f j.

(2.6.159)

Thus, for example

*

Taij

3()~~a 3()~~a * j = TVaWa = -() VajVai = T aji , a a

(2.6.160)

defined by (2.3.49), and aij

==

~(3

Dij - VaWaj

=

~(3

Dji - VajVai

=

aji,

(2.6.161)

defined by (2.6.40), are symmetric second rank tensors, because VaWaj == The permeability of a fluid a-phase in a porous medium, is defined by (2.6.55) as * k ajf. = ()a~~ ( aji )-I T aib (2.6.162)

VajVai.

c;;-

where aij is defined by (2.6.161), and ( .. )-1 denotes an inverse of ( .. ). It is worth noting that the symmetry of T;ij, or of aij, is not sufficient for kaij to be also symmetric. We shall return to this issue later in this subsection. The definition of symmetry with respect to a pair of indices applies also to tensors of higher ranks. The mathematical form of a coefficient that appears in the description of a given transport phenomenon, may be used as a means for classifying porous media according to their effect on that phenomenon. To this end, we note that T;ij, kij, or 1)~"', act as operators which transform one vector, (e.g., a force), which may be referred to as an excitation, into another vector (e.g., a flux), which may be referred to as a response. Indeed, according to the rule of inner multiplication, of a second rank tensor, A, by a vecotr, B, we have A·B= C,

(2.6.163)

where the Cis are components of a vector, C. When B and Care colinear vectors, i.e., when the transformation imposed by the tensor A on the vector B changes only the magnitude of the latter, without changing its direction, then the direction of B is said to be a principal direction, or principal axis, of the operator A. We may then write A·B = aB,

(2.6.164)

Macroscopic Fluxes

213

where a is a scalar referred to as a principal value of A. It shows by how many times does the tensorial operator A magnify a vector which is oriented along its principal direction. Using this property of colinearity, one can find the principal values and principal directions of any second rank tensor. It can be shown that every symmetric second rank tensor always has at least one set of three mutually orthogonal principal axes, with three corresponding real principal values. If the principal values differ from each other, then there exists only one set of principal axes. In the most general case, the principal values are different from each other, and they correspond to only one set of three mutually orthogonal principal directions. It is worth noting, however, that only a symmetric second rank tensor has the particular feature of having only real principal values. Consider a tensor, A, that has three principal values, at, a2 and a3, and three corresponding principal directions indicated by the unit vectors e(l), e(2) and e(3). By using (2.6.158), we can express the tensor's components in any Cartesian coordinate system, in the form (2.6.165) If the principal axes are chosen as coordinate axes, (2.6.165) reduces to

(2.6.166) As indicated above, any coefficient, A, acts as an operator which produces a change in both magnitude and direction of a vecotr, or only a change in the latter's magnitude. If the change in magnitude is different in different directions, which, for a coefficient representing a symmetric second rank tensor, is equivalent to al t= a2 t= a3, we say that the property represented by the coefficient in anisotropic, or, alternatively, that the coefficient is anisotropic. The permeability, k, of a phase may serve as an example. In certain cases, a tensorial property of a porous material may exhibit the same response in some, but not all directions. Such behavior indicates that the microstructure of the solid matrix is such that it possesses certain features of macroscopic symmetry. The existence of such symmetries simplifies the mathematical form of the corresponding coefficient, and, hence, also the description of the process under consideration. As an example, let A be a coefficient that represents a transformation involving a change in magnitude only, which is equal in all directions. This

214


means that all directions are principal ones, with a single principal value, a = al = a2 == a3. In this case (2.6.166) reduces to the simple form (2.6.167) Thus, Aij, in (2.6.167), represents an isotropic second rank tensor, characterized by a single scalar, a. On the other hand, let A represent a (tensorial) coefficient that has one principal direction, e, with a corresponding principal value, ab while all directions in a plane normal to e are principal directions, with a common principal value, a2. Then, the components Aij can be expressed by (2.6.165), which in this case, reduces to the form (2.6.168) A tensor A that satisfies (2.6.168) is said to be an anisotropic tensor with axial symmetry. It represents an operator which produces one response along a given direction and a different, yet uniform, response in a plane normal to this direction. Indeed, the transformation (2.6.168), imposed by A on e, is given by (2.6.169)

A·e=C,

in which the vector C, with components Ci == (at + a2)ei, is colinear with e. On the other hand, the transformation imposed by A on any vector N in a plane normal to e, is expressed by A·N=C, in which C is a vector colinear with N. In a rectangular Cartesian coordinate system, with e == from (2.6.168)

IXb

we obtain (2.6.170)

Denoting (2.6.171) equation (2.6.170) can be presented in the form of a diagonal matrix

o

AT

~

o AT

].

(2.6.172)

Macroscopic Fluxes

215

The coefficient AL is the value of A along the principal direction e, namely, AL is the scalar multiplicator of any vector parallel to e. The coefficient AT is the value of A along any direction in a plane normal to e. This means that AT is the scalar multiplicator of any vector normal to e. With the above notation, (2.6.170) can be rewritten in the form (2.6.173) Equation (2.6.173) can be used to present the general form of any second rank transport coefficient which corresponds to a macroscopically anisotropic behavior of a fluid phase in a porous medium, characterized by the presence of one preferred principal direction, and a principal value in that direction, while all other principal values, in a plane normal to that direction, are equal to each other. So far we have dealt only with coefficients that represent second rank tensors. An extension to higher rank tensorial coefficients, such as dispersivity, for isotropic or anisotropic configurations of a phase in the entire void space, or only in part of it, can be obtained using a methodology developed by Robertson (1940), based on requirements of invariance. This methodology will be demonstrated below through its application to various transport coefficients of interest.

(a)

Isotropy, or full symmetry

Let Q, with components Qij, be a macroscopic tensorial property of a porous medium (Le., characterizing the void space configuration in the neighborhood of a point) or of (the configuration of ) a phase in it, employed in the description of a certain process, e.g., advection, diffusion, dispersion, compression, or shear. Let the behavior of the porous medium, in the considered process be isotropic, Le., the components Qij do not vary with direction and, hence, do not depend on any unit vector that indicates a specific direction. Then, for any pair of unit vectors, A and B, with arbitrary directions, the inner product (2.6.174) is a scalar. However, since Q is linear in A and B, it can depend only on the scalar product, AiBi. Hence (2.6.175) where Q1 is a scalar function.

216

MACROSCOPIC DESCRIPTION A comparison of (2.6.175) with (2.6.174), yields (2.6.176)

which is the general form of an isotropic second rank tensor identical to (2.6.167). This form will now be applied to some of the transport coefficients introduced in earlier subsections. (i) Tortuosity of an a-phase. In this case, Qij == T~ij. From (2.3.67), it follows that for an isotropic configuration of the a-phase, Q1 = ()~ j()a. Hence, in this case (2.6.177)

(ii) Isotropic permeability of an a - phase. The permeability of a porous medium is given by (2.6.55). When adapted to a fluid a-phase which occupies only part of the void space, it takes the form

* k ajl = ()a~; ( aji )-I T ail·

----c;-

(2.6.178)

By definition (2.6.179) which, in the case of an isotropic configuration of the phase, reduce to (2.6.180) Upon setting j

= i, and summing over j, we obtain a = ~.

Hence, (2.6.181)

By substituting (2.6.177) and (2.6.181) into (2.6.178), we obtain (2.6.182) where the scalar k = a

~ ()~ ~2 2 Ca

a

(2.6.183)

is referred to as the isotropic permeability of an a-phase in a porous medium. (In Chap. 5 we shall refer to the permeability of a phase that occupies part of the void space as effective permeability).

217

Macroscopic Fluxes

(iii) Isotropic dispersivity of an extensive quantity in a fluid phase. Let Q ikjR. represent an operator which is a fourth rank tensor. If QikjR. exhibits an isotropic behavior, then the inner product (2.6.184) must be invariant for any set of four arbitrarily oriented unit vectors, A, B, C and D. Since Q is linear in those vectors, it can depend only on the scalars that they can form, namely (A·B)(C·D),

(A·C)(B·D),

and

(A·D)(B·C),

or, in indicial notation

Hence, we may express Q as a linear combination

Q =

Q1AiBiCjDj + Q2AiBkCiDk + Q3AiBkCkDi AiBkCjDR.(Q10ikOjR. + Q20ijOkR. + Q30kjOiR.),

(2.6.185)

where Q1, Q2 and Q3 are scalar coefficients. A comparison between (2.6.184) and (2.6.185), yields (2.6.186) Equation (2.6.186) is the general form of an isotropic fourth rank tensor. If, in addition, QikjR. is symmetric in the two indices k and i, i.e., (2.6.187) then Q1 = Q3 and (2.6.186) reduces to (2.6.188) This result may now be used to obtain the form of the dispersivity tensor for a phase that has an isotropic configuration within the void space. Employing the definition of the dispersivity, aOlikR.m, in (2.6.154) and of the tortuosity, (2.6.177), we obtain (2.6.189)

218

MACROSCOPIC DESCRIPTION A comparison of (2.6.188) with (2.6.189), yields E

aO/L

_

E

= aO/iiii

_

- 2Q1

+ Q2 _-

(J~ E~

(JCX.e (Tii)

,

i = 1,2,3,

(no summation on i), E

acxT

_

E

= acxijij

(J~

o -2 cx E-

= Q2 = (Jcx.e (Tij)

(2.6.190)

,

(no summation on i or j),

i,j = 1,2,3,

i

t= j. (2.6.191)

With a~L and a~T' and applying the form (2.6.188) to (2.6.189), we finally obtain

(2.6.192) The coefficients a~L and a~T are referred to as the longitudinal and transversal dispersivities, respectively, of the isotropic dispersivity tensor, E acxiklm· (iv) The coefficient of mechanical dispersion of an extensive quantity, E, in the case of an isotropic dispersivity tensorfor isochoric fluid motion and impervious interphase surfaces. The definition of D~im' as given by (2.6.153), is rewritten here for convenience (2.6.193) Substituting (2.6.192), into (2.6.193), yields

This expression is the general form of the coefficient of mechanical dispersion of a conservative extensive quantity, E, in a fluid a-phase within a porous medium, with isotropic dispersivity.

Macroscopic Fluxes

219

Although the dispersivity is isotropic, it follows from (2.6.194) that the coefficient D~ is not isotropic. As will be shown below, this is due to the fact that the velocity vector introduces here an anisotropy by serving as an axis of symmetry.

(b)

Axial Symmetry

Let Qij denote an operator which possesses an axis of symmetry, indicated by the unit vector e. This means that the scalar Q in (2.6.174) is the same for every direction in any plane which is normal to the axis and for every reflection in that plane. This requirement is identical to the condition that (2.6.195) be invariant for any arbitrary pair of unit vectors, A and B. In order to obtain the most general form of Q, we must take into account all scalars that can be formed by the three vectors e, A and B, which are linear in A and B. Such are the products

Hence

(2.6.196) By comparing (2.6.196) with (2.6.195), we obtain (2.6.197) which is identical to (2.6.168) and can be also be expressed in the form given by (2.6.173). Let us apply these considerations to a number of second rank tensorial transport coefficients. (i) Tortuosity with axial symmetry. Let e be a unit vector representing an axis of symmetry of T:;ij. From (2.6.173), it follows that (2.6.198) where, in a rectangular Cartesian coordinate system, with e = lxl, we have

220

MACROSCOPIC DESCRIPTION From (2.3.63), we then obtain

(2.6.199) where Vcxi = cos(IR, bq) is the cosine ofthe angle between the radius-vector of a point on the Scxcx-surface of an REV, and the xi-axis. The coefficient T~L represents the tortuosity along the axis of symmetry, while T~T represents it along any direction in a plane normal to the axis of symmetry (see Subs. 2.3.6).

(ii) Permeability with axial symmetry, Qij = kcxij. By (2.6.178), the permeability tensor is an inner product of two symmetrical tensors: (aji)-l and Tt'e. Only when these two tensors have the same principal axes, will kcxij also be symmetrical and have the same principal axes. Assuming that the microscopic configuration of a phase is indeed such that this condition is satisfied, and given that Ttl and (aji)-l possess an axis of symmetry along the unit vector e, we obtain from (2.6.173) (2.6.200) In a Cartesian coordinate system with e = lXI, we have Ttl aji (ajitl

= = =

(Tl - T';)6 I i 6U + T';6il , (aL-aT)6 I j 6li+ aT6ji,

r

r

(a'Ll - a l )6l j6l i + a l 6ji'

(2.6.201)

and

(2.6.202) Hence (2.6.203)

221

Macroscopic Fluxes

(2.6.204) where

Ti (iii)

The coefficient of mechanical dispersion in the case of an isotropic dispersivity, Qim = D~im' By comparing (2.6.194) with the general form of a second rank tensor with axial symmetry, (2.6.197), we see that even when the dispersivity, aOtik£m, is isotropic, the dispersion of an extensive quantity within a fluid phase is not isotropic. Instead, it is symmetrical about the average velocity vector of the phase, with different elongations along this vector and perpendicular to it. Indeed, in a rectangular Cartesian coordinate system, with one axis, say IXl parallel to the average velocity vector, we obtain from (2.6.194)

(2.6.205) where

D~l1 =

a;;L va' f(Pe~, l~ j ~Ot)

(2.6.206)

is the principal value of D~ along the velocity vector, V Ot , and E E E -Ot ( E Ej DOt22 = DOt33 = aOtTV f PeOt,lOt ~Ot )

(2.6.207)

is the principal value of D~ in any direction perpendicular to '\F. (iv) Axially symmetrical dispersivity. Consider a fourth rank tensor, Qikj£, which possesses an axis of symmetry indicated by the unit vector e. In such case, Qikj£ = Qikj£(e), and the inner product

222


includes all scalars that can be formed by the five vectors e, A, B, C and D, which are linear in A, B, C and D. These are the products

(A·B)(C·D), (A·e)(B·e)(C·D),

(A·C)(B·D), (A·B)(C·e)(D·e),

(A·e)(B·C)(D.e), (A.C)(B·e)(D·e),

(A·D)(B·C), (A·e)(C·e)(B·D),

(A·e)(B·e)(C·e)(D·e), (A·D)(B·e)(C·e).

By a procedure similar to that employed above, we obtain

Qikj.e =

Ql bik bj.e + Q2 bij bk.e + Q3 bi.e bkj + Q4 ei ekbj.e +Qsbikeje.e + Q6eiej bk.e + Q7eibkje.e + QSeiekeje.e (2.6.208) +Q9bijeke.e + QIObi.eekej,

where the Qi'S are scalars. Equation (2.6.208) is the most general form of a fourth rank tensor with one axis of symmetry. In the case of symmetry with respect to k and f, i.e.

we obtain

Qs = QIO, and (2.6.208) reduces to the form

Qikj.e =

Ql(bikbj.e + bi.ebkj) + Q2 bijbk.e + Q4(ei ekbj.e + eie.ebkj) +Qs(bikeje.e + bi.eekej) + Q6eiejbk.e + Qs(eiekeje.e) (2.6.209) +Q9bijeke.e,

which contains seven scalar coefficients. if there exists also symmetry with respect to the indices i and j, viz.

we obtain

223

Macroscopic Fluxes and (2.6.209) reduces to Qikj£

=

QI(OikOj£ + Oi£Okj) + Q20ij Ok£ +Q4( Oikeje£ + Oif.ekej + Ojkeief. +Q60k£eiej

+ OJ£eiek)

+ Q8eiekejee + Q90ijek e£,

(2.6.210)

Finally, symmetry in the indices i, k and j, f, i.e. (2.6.211) yields Q6 = Qg,

and (2.6.210) takes on its simplest form Qikj£

=

QI(OikOjf. + Oif.Okj)

+ Q20ij Okf.

+Q4(oikeje£ + Oi£ekej + Ojkeie£ + OJ£eiek) (2.6.212) +Q6(ok£eiej + Oijekef.) + Q8eiekeje£,

which contains only five scalar coefficients. By substituting o

a

0

Qikj£ = TO/ikTO/jf. ,

(2.6.213)

and adopting a rectangular Cartesian coordinate system, with one coordinate axis taken parallel the axis of symmetry, say, lXl == e, we obtain from (2.6.209) and (2.6.213) the seven scalar coefficients -20/

(Tn)

-0

-20/ -20/ (TI2 ) = (TI3) -20/ -20/ (T22) = (T33) -20/ -20/ (T23) = (T32) -20/ -20/ (T2I) = (T3I )

-0

-0

-0

-0

-0

-0

-0

-0

ooa

TllT22

boO!

T22Tll

From

= TllT33 = T33 T ll

000'

000'

= 2QI + Q2 + 2(Q4 + Qs) + Q6 + Q8 + Qg, - QI2I2 = QI3I3 = Q2 + Q6, = Q2222 = Q3333 = 2QI + Q2, - Q2323 = Q3232 = Q2, - Q212I = Q3I3I = Q2 + Qg, - Q1l22 = Q1133 = QI + Q4, (2.6.214) - Q2211 = Q3311 = Ql + Qs. -

Qll11

-o-o-a

0

0

ex

TllT22 = T22Tll ,

224


and it follows that (2.6.215) and (2.6.212) is justified. Equations (2.6.214) make it possible to express the scalar coefficients of Qikj.e appearing in (2.6.212), in terms of the variances (,iij )2c; and the -o-o-c; covariance TuT22 , in a principal coordinate system, with e = lxI, i,j = 1,2,3. In order to arrive at an expression for the dispersivity, as defined in o a (2.6.154), we must take the inner product of Tc;ikTc;j.e and the second rank tensor, T~jm. Assuming that e also indicates a principal axis of T~jm' and employing (2.6.198) and (2.6.212), we obtain 0

=

o c; Tc;jmfc; * E = fc;E{ Q1T2*( Oik0.em + 0i.e0km ) Tc;ikTc;j.e +Q2 T ;Ok.eOim + [Ql(Tt - Tn + Q4 T Oik€.e€m +[Q2(Tt - Tn + Q6T Ok.e€i€m +Q4T;(Okm€i€.e + Om.e€i€k) + Q6T;Oim€k€.e +[(2Q4 + Qs + Q6)(Tt - Tn + QST;]€i€k€.e€m} 0

n

n

=

aNN ~ aNN' TN (Oik0.em

(

*

+ Oi.eOkm) + (aNN,TN )Ok.eOim

*)

+ CLNTL - aNN -2 aNN' TN (l:uik€.e€m +(aLNTi - aNN' TN )Ok.e€i€m + ( CLN - aNN -2 aNN') TN* (Okm€i€.e

+ 0i.e€k€m

+ Om.e€i€k

)

)

+(aNL - aNN,)TNOim€k€.e +[(aLL - 2CLN - aLN)Ti (2.6.216) where

(2.6.217)

Macroscopic Fluxes

225

and (2.6.218) and the indices correspond to a Cartesian coordinate system aligned with the principal directions, one of which, IXl = e, is the axis of symmetry. The subscript L indicates the positive direction of the axis of symmetry, while Nand N' indicate mutually orthogonal directions in a plane normal to the axis of symmetry, respectively. In the absence of an axis of symmetry, i.e., in an isotropic case, all terms containing components of e vanish, and (2.6.216) reduces to the isotropic form (2.6.192). Equation (2.6.216), which contains 7 scalar coefficients ofthe types aLLTZ, aLNTZ, aNNTN , aNN,TN , aNLTN , CLNTZ and CNLTN , represents the dispersivity tensor of an extensive quantity in an a-phase in the case of anisotropy which has an axis of symmetry indicated by a unit vector, e.

(v) The coefficient of mechanical dispersion in a phase with an axially symmetrical dispersivity. Substituting (2.6.216) into (2.6.193), yields

where at, a2, ... ,a5 are scalar coefficients defined by

al a2 a3 a4

a5

2QlT2f~ = (aNN - aNN/)TN {Q2 + Q9cos2(e, ya)}T2f~ = {aNNI + (aNL - aNN/)cos 2(e, YO

Macroscopic Description of Transport Phenomena in

Macroscopic Description of Transport Phenomena in

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