introduction to modeling of transport phenomena in porous media || macroscopic description of...

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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 exten- sive 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 kine- matics 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 or- der 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 con- ditions 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

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Page 1: Introduction to Modeling of Transport Phenomena in Porous Media || Macroscopic Description of Transport Phenomena in Porous Media

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

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44 MACROSCOPIC DESCRIPTION

2.1 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, e is sele~ted as the initial position vector of a considered particle, i.e., e == xlt=o. The terms convected and Lagrangian coordi­nates are also used.

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

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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, t, and the initial position, e, of the particle, and the motion is described by

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 with material coordinates e. The set of points in no specifies the initial con­figuration 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 e 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 in no) initially at e, Le., it gives its place, x(t), as a function of time.

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(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

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46 MACROSCOPIC DESCRIPTION

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, eE (= material coordinates of the particle)

(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 1 (aw! awf) Cij ="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

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Kinematics of Continua 47

axes, while each of the off diagonal components, £5, 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]. (2.1.7)

2.1.5 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 aE ,where

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48 MACROSCOPIC DESCRIPTION

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)

i.e., a derivative of G with respect to time, keeping eE 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 em, 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.,

(2.1.9)

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

vE = 8xEI (= xE == x). at eE =const. (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

i = (V.w) = V·(w) = V·VE ,

where, by (2.1.3), w = xE = V E . Also, by (2.1.5)

1 (aVE aV:E) iij = 2" a;j + a:i '

(2.1.11)

(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

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relationship (2.1.7).

DEG{X(eE , t), t} Dt

(jE =

aGI aGI aXk(eE,t) 1°

= 7ft x=const. + aXk t=const. at eE=const.

_ aG aG TTE (2113) - at + aXk v k , ..

where

aG VkE == t aG vkE == 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 + a3b3, ail b1 + ai2b2 + ai3b3·

(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

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50 MACROSCOPIC DESCRIPTION

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 dx2dx3)

(dxl)1 dX2dx3 + (dx2 )1 dXldx3 + (dx3 )1 dXldx2 X2,X3 Xl ,x3 Xl ,x2

( av,E) (aVE) (aVE) axIl dXI dX2dx3 + a:2 dX2 dXldx3 + 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)

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Kinematics of Continua 51

Thus, '\l·VE 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) be-comes

(dUE) _ r7.VE __ ~ _ vE () dUE - v - e - vE ' 2.1.20

where vE = 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 conserva­

tive 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 vm == 1/ P is called the specific volume (of mass). For the sake of simplicity, we shall employ the symbol v to denote vm .

For p = 0, i.e., p remains unchanged for each particle as it moves, (2.1.21) reduces to

(2.1.22)

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52 MACROSCOPIC DESCRIPTION

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 -- == - + u·V F = 0

Dt at ' (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 = IVFI, av

(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 _ U )aF Dt II II av'

(2.1.27)

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Kinematics of Continua 53

where (V! - uv ) 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. Dt

(2.1.28)

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 E "'" E"f "' T"Y "'" E"f V = L...J a v', L...J a = 1, (2.1.29)

"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

1 N N "Y N vm = 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

1 N N au m"Y N V = - "'" U"Y V"Y = "'" --Vl = "'" v"Y p"YVl U L...J L...J am"Y U L...J

"Y=1 "Y=1 "Y=1 N

= E v"Ymolp"YmolV"Y, (2.1.31) "Y=1

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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'Ymol = M'Yv'Y is the molar partial specific volume of" M'Y is the molecular weight of " and p'Ymol = 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

N N

V mo1 = ~Ol L: p 'Ymo1V'Y = L: X'YV'Y , P "1=1 "1=1

(2.1.32)

where X'Y = p'Ymol / 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'Ymol is the total molar density of the phase.

(d) Momentum weighted velocity, V M .

or

VMM U

N M'Y '"'V'Y-L..J U' "1=1

N pVmVM = L:p'YV'YVM"Y,

"1=1

(2.1.33)

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·

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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,

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56 MACROSCOPIC DESCRIPTION

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 oGdk ··· 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

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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

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58 MACROSCOPIC DESCRIPTION

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

1 8Gjkl... dU 1 G dS 8 = jkl..YI·

U Xl S (2.1.41)

Replacing Xl by Xi, i = 1,2,3, we obtain the general formula

1 8Gjkl ... dU 1 G dB 8 = jkl .. Yi.

U Xi S (2.1.42)

Setting i = j and summing over j = 1,2,3, we obtain

1 8Gjkl ... dU 1 G dB 8 = jkl..Yj,

U Xj S (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 8Gi 1 -8 dU = GWi dB, U Xi S

or fu V·G dU = is G·v dB. (2.1.44)

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

fu V·eVE dU = is eVE·v dB. (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

diveVE == lim U1 r eVE·v dB. U--+O is (2.1.46)

This provides a physical interpretation of V·eVE(== 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.

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Kinematics of Continua 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 %7 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

DF f edU Dt lU(t)

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

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

-L3 e(t) dU}

= f {){)e dU + lim : {f e(t+flt)dU- f e(t)dU}. (2.1.48) lUi t .6.t-+O I...l.t lU2 lU3

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

U2 = f (1 U·1I dB) dt, l(.6.t) (ABC)

U3 = - f (1 U·1I dS) dt. l(.6.t) (ADC)

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60 MACROSCOPIC DESCRIPTION

(a) (b)

Figure 2.1.3: Definition Sketches for Reynolds transport theorem.

Since dU2 = (u·vdS)dt, and dU3 = -(u·vdS)dt, equation (2.1.48) can be rewritten for b..t -+ 0, in the form

DF [ e dU = f {Je dU + f eu.v dS. Dt JU(t) JU(t) {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 f e dU = f {Je dU + f eVE·v dS Dt JU(t) JU(t) {Jt JS(t)

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Kinematics of Continua 61

- f e(VE - U)·V dS. JS(t)

(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 e dU = DDE f e dU - f e(VE - u)·v dS. (2.1.51) t JU(t) t JU(t) JS(t)

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 = f 0:. dU + fell U·Vl dS, Dt JU1(t) JU1(t) U~ JS*+S1

DF f edU f 0:. dU + f eI2u.,v2 dS. Dt JU2(t) JU2(t) U~ JS*+S2

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:. dU + f eu·v dS + f (ell - eI2)u.vl dS JU(t) U~ JS(t) JS*(t)

= f (){)e dU + f eu.v dS + f [eh,2u ·v l dS, (2.1.52) JU(t) t JS(t) JS*(t)

where

[eh,2 == elon U1 side - elon 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

- edU = DF 1 Dt U(t)

f (){)e dU + f eVE.v dS JU(t) t JS(t)

- f [e(VE - u)h 2·V dS. Js * '

(2.1.53)

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62 MACROSCOPIC DESCRIPTION

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

Likewise

1 8a-b_J Vi dB =

s 8Xi

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

r (b 8aj _ ~ _) _ dB _ r (b 82aj _ 82b -) dU Js 8Xi 8Xi aJ V~ - Ju 8Xi8Xi 8Xi8x i aJ •

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

(2.1.54)

(2.1.55)

(2.1.56)

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.

Let ef, i = 1.2.3, denote the material coordinates of a fixed E- particle. The Lagrangian description of its motion, as given by (2.1.1), is

(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).

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Kinematics of Continua 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)

~f = ~f(x, t), i = 1,2,3, (2.1.59)

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.

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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 (8ViE /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),

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Kinematics of Continua 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

·tEd ·tEd 0 or 11 X2 - 12 Xl = . (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, dWE , of a scalar function WE = WE (XI, X2,to ), 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

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66 MACROSCOPIC DESCRIPTION

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

. otE _ E 1 (.) ae V·J = V·eV = dUE dE - at· (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·VE = 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 aWE --dX1 + --dx2 aX1 aX2

·tEd ·tEd = h X2 - h ·X1· (2.1.66)

Consider an oriented elementary segment ds connecting two E- transport curves, WE and WE + dwE (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)

A comparison of (2.1.66) and (2.1.67) yields

dQE = dwE,

(2.1.67)

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.

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Kinematics of Continua 67

y

>--------- :1:1 o~-------_ :I: z

(a) (b)

Figure 2.1.5: Relation between the stream function, WE, and discharge, QE.

2.1.12 Velocity potential and complex potential

Consider a motion of an E-continuum such that

&q,E or 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 _ &V/) = 0 2 &Xj &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

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68 MACROSCOPIC DESCRIPTION

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

By definition

a4?E a4?E dXi E ~ = ~-d = IV4? Icos(V4?, 15),

uS uXi s (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( e ViE) _ a( e VP) aXj aXi

(2.1.73)

Since

(2.1.74)

if the motion of the continuum is irrotational (Le., V X yE = 0), the trans­port of E, as defined by jtE, is also irrotational as long as

Ve X yE = Ve x V4?E = 0, (2.1. 75)

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Kinematics of Continua 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·VE = 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 = Xl + iX2, can be defined such that

(2.1.80)

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70 MACROSCOPIC DESCRIPTION

dwE

dXl dwE

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, <p. Such a front is also called an isotimic surface. For example, <p may represent an intensive quantity such as density, p. However, it may also represent pressure, so that a surface at every point of which p = constant, is an isotimic surface.

At time t = 0, the configuration of a front is described by the equation

<p = <p(f.) = <Po = const. (2.1.83)

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Kinematics of Continua

This equation represents the locus of all particles with <P = <Po. Another form of this equation is

71

(2.1.84)

In the absence of sources or sinks, the motion of any particle belonging to a considered front is given by (2.1.1)

i=1,2,3. (2.1.85)

Equations (2.1.84) and (2.1.85) represent four equations in seven vari­ables, Xi, ~i and t. By eliminating the three ~i'S, we obtain one equation in four variables

9 = g(x, y, z, t; <Po) = o. (2.1.86)

This equation describes the movement of the <po-front in the course of time.

Before leaving this section, it would be appropriate to draw the reader's attention to the fact that, although this chapter is dealing with the kine­matics of continua at the microscopic level, most of the concepts, definitions and relations included in it apply as well to the macroscopic level considered in the remaining sections of this chapter and in the rest of the book. For a given porous medium domain, once a continuum of some extensive quantity of a phase has been defined, the concepts presented above can be applied to it. For example, the definitions of particles, types of coordinates, x E and eE ,

velocities, V E , material derivative, D( )/Dt, flux and discharge, E-transport function, WE, streamlines and pathlines, the potential, iPE, and complex po­tential, are all continuum concepts, applicable also to the continuum level of describing transport in porous media. Some of these concepts are presented here, although they are not explicitly used in later chapters, for the sake of completeness and as tools to be used whenever necessary.

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72 MACROSCOPIC DESCRIPTION

Figure 2.2.1: A control domain U bounded by a surface S.

2.2 Microscopic Balance and Constitutive Equations

8

Because in this section we continue to deal only with a single phase contin­uum, no special symbol will be used to denote the considered phase.

2.2.1 Derivation of balance equations

The description of a process undergone by an extensive quantity, E, in a single phase continuum, starts with the formulation of a balance equation for that quantity in the neighborhood of an arbitrary point, P, within the phase. The point is defined by its position vector, x. This equation can be derived in several ways. One way is to use the Eulerian approach in which we focus our attention on a fixed finite domain, U, referred to as control domain, or control volume, of arbitrary shape, bounded by a closed surface, S. Figure 2.2.1 shows such a control domain, with U and S denoting its volume and the area of the surface bounding it, respectively.

The instantaneous balance of an extensive quantity, E within U can be

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Microscopic Description

expressed, verbally, by

{ ::~::ulation of } E within U

(a)

{Net influx of } E intoU through S

(b)

73

Net rate of } production of . E within U

(c)

Let us express these balance components in a mathematical form.

(a) The rate of increase in the amount of E within U, is expressed by

(2.2.1)

where the exchange of integration and differentiation is permitted in view of the fact that the boundary of the domain U is fixed.

(b) The net influx (= total influx minus total efflux) of E into U, through S, is given by

dE e = dU'

where e is the density of E, i.e., the amount of E per unit volume of phase, and v is the outward normal unit vector on the elemental area dS.

(c) The net rate of production of E from sources within U, is expressed by

LprE dU,

where p is the mass density of the phase and rE denotes the rate of internal production of E, per unit mass of the phase.

Altogether, the balance of E within U is expressed by

L ~: dU = - is eyE·v dS + LprE dUo (2.2.2)

Assuming that eyE is differentiable within U, we apply Gauss' theorem, (2.1.45), to the first term on the r.h.s. of (2.2.2), and rewrite this equation in the form

L (~: + V'·e V E - prE) dU = O. (2.2.3)

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74

.tEI J:r

z

y

.tEI Jz :r,y,z+ ~az

I r , I

-----II--~ I , :r -! D..:c,y,z

/ /

/

/ /

/

I }-

MACROSCOPIC DESCRIPTION

.tEI J:r x+! a:r,y,z

-+--C1x --..._---,

.tEI Jy :r,y- ~ay,z

~------~x

Figure 2.2.2: Control box for deriving an expression for the divergence of a flux

JE(= eVE) in a rectangular Cartesian coordinate system.

By shrinking the volume U to zero around an arbitrary point, P(x), we obtain

oe + "V.eVE _ prE = 0 ot '

(2.2.4)

where all terms refer to the point. Equation (2.2.4) is the microscopic dif­ferential balance equation of any extensive quantity, E, in a continuum.

A simple illustration of the physical meaning of the divergence of a flux, "V.eVE, appearing in (2.2.4), can be obtained by considering the particular case of a parallelepiped control box of volume U (= ~x ~y ~z), in a Cartesian coordinate system (Fig. 2.2.2). We return to the use of the symbol jtE to denote the total flux of E.

For this particular control domain, the limit as U --* 0 of the r.h.s. of (2.1.46), expressing the excess of the rate of efflux of E into U over influx

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Microscopic Description

from it, per unit volume, takes the form

lim U1 f jtE.v dB U-+O is

1· 1 { ( ·tE I ·tE I ) A A = 1m J.D.x - J .D.x uYuz ~x,~y,~z-+O ~x~y~z x x- 2 ,y,z x x+ 2 ,y,z

+ (jytEI _.D.y _jytEI +.D.y )~x~z x,y 2 ,z x,Y 2 ,z

+ (j;Elx,y,z_~z - j;Elx,y,z+~z ) ~x~y}

= lim 2 ' , 2 ' , {j;Elx_.D.X y z - j!Elx+.D.X Y z

~x,~y,~z-+O ~x

75

·tEI ·tEI ·tEI ·tEI } Jy xy-~z -Jy xy+.D. y z Jz xyz_.D.z -Jz xy,z+.D.z + '2' '2' + "2 '2

~Y ~z

= _ (OJ!E + oj~E + OJ!E). . ax oy OZ

Hence, by (2.1.46)

a ·tE a ·tE a ·tE d· .tE(_ 0 .tE) _ Jx + Jy + Jz lVJ = V'J - -- -- --,

ax oY oz (2.2.5)

where it is assumed that jtE is differentiable at all points of the considered domain.

In any orthogonal curvilinear coordinate system Yi, let an element of length, dfi, along a coordinate line, be given by dfi = hidYi, where hi is a scale factor for the Yi coordinate. In such a coordinate system

div jtE = h ~ h {a a (jiEh2h3)+ 00 (j~Ehlh3)+ 00 (j~Ehlh2)}' (2.2.6) 1 2 3 Yl Y2 Y3

This expression can be used in (2.2.4) if we wish to write this equation in the Yi coordinate system.

Another way of developing (2.2.4) is to start from the Lagrangian ap­proach in which we follow an extensive quantity, E, within a domain U E

enclosed by a material surface, SE, as it travels. Both UE and SE may vary in the course of time. At any time, t, the amount of E within UE(t) is given by

E = 1 edU. UE(t)

(2.2.7)

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76 MACROSCOPIC DESCRIPTION

Since no amount of E can cross the boundary, SE, the quantity E can grow within UE only as a result of production of E within UE(t). Hence

DE r e dU = r prE dUo Dt JUE(t) JUE(t)

(2.2.8)

Application of Reynolds transport theorem, (2.1.49), with u·v = VE.v, to the 1.h.s. of (2.2.8), yields

DE r e dU = r ae dU + r eVE.v dB. Dt JUE(t) JUE(t) at JSE(t)

(2.2.9)

If the flux, jtE(= eVE), is also differentiable within UE(t), we may apply Gauss' theorem, (2.1.45), in order to transform the area integral in (2.2.9) into a volume integral. We obtain

DE r e dU = r ae dU + r V.e V E dU, Dt JUE(t) JUE(t) at JUE(t)

which, upon substitution into (2.2.8), becomes

l (~e + V.eVE _ prE) dU = O. JUE(t) vt

(2.2.10)

Since the domain UE(t) is arbitrary, the integrand itself must vanish every­where. Hence, the differential balance of E is expressed by

(2.2.11)

which is identical to (2.2.4), obtained by the Eulerian approach. Following (2.1.36), the total flux of E can be expressed as the sum oftwo

fluxes (2.2.12)

Le., an advective flux, eV, and a diffusive fluxjEU(= e(VE - V)). Then, the balance equation (2.2.11) can be rewritten as

ae = -V.(eV +jEU) + prE. at

Another form of (2.2.13) is

DEe 'EU E Dt = -eV·V - V·J + pr .

(2.2.13)

(2.2.14)

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Microscopic Description 77

While (2.2.13) is an Eulerian form of the balance equation, (2.2.14) is written as a Lagrangian equation. Its I.h.s. represents the rate of change in e as a particle moves, while its r.h.s., which is an Eulerian expression, represents the sources that cause this change. They include the effects of diffusion, dilatation and production.

Equation (2.2.14) explains why e of a particle cannot be maintained unchanged as it moves, unless no sources exist in its neighborhood.

Equation (2.2.13) is another general form of the differential balance equa­tion of an extensive quantity in a phase continuum at the microscopic level. It states that in the close vicinity of any considered point in a phase contin­uum, the excess of influx over efflux of E, per unit volume of the phase, by both advection by the phase and diffusion, plus the rate of production of E per unit volume, is equal to the rate of increase in the amount of E per unit volume. It is important to note the physical interpretation of the divergence of a flux, as well as of the other terms in (2.2.13). This observation may aid the reader in deriving a balance equation for any specific extensive quantity. We also note tha:t the divergence of a flux in (2.2.13), written in the symbolic (vector) form, is invariant under changes in the coordinate system. We may use this fact to rewrite the balance equation (2.2.13) in particular coordinate systems. For example

• In a rectangular Cartesian coordinate system, Xi, (employing Einstein's summation convention)

O 0 'tE ~ - _ --.!.L rE ot - OXi + p ,

or, with Xl = X,X2 = y,X3 = Z

oe oj~E oj~E oj;E E ot = ----a;- -Ty - Tz+pr .

• In cylindrical coordinates (r, 0, z)

oe _ 1 0 (.tE) 1 0 ( .tE) 0 ( .tE) rE - - --- rJ - -- J() - - J +p . ot r or r r 00 OZ z

• In spherical coordinates (r, (}, 't/J )

oe 1 0 ( 2 .tE) 1 0 (.tE . ot r2 or r Jr - r sin 000 J() sm 0)

1 oj~E E -. -0 Onl. + pr . rsm 'f'

(2.2.15)

(2.2.16)

(2.2.17)

(2.2.18)

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78 MACROSCOPIC DESCRIPTION

• In any curvilinear orthogonal coordinate system Yi, i = 1,2,3

8e 8t

where the hi'S are the scale factors.

2.2.2 Particular cases of balance equations

(2.2.19)

Let us apply (2.2.11), or (2.2.13), to several extensive quantities of interest. Since we continue to deal only with a single phase continuum, we shall continue to omit the subscript denoting the phase.

(a) E represents the mass, m, of a phase. Then e = p, rE = rm = 0 (by the principle of mass conservation), jEU = jm and VE = vm = mass weighted velocity of the phase, as defined by (2.1.30). For this case, the diffusive mass flux, jm, is expressed in the form

(2.2.20)

Hence, (2.2.11) reduces to

8p = -V'.pVm

8t ' (2.2.21)

or, in indicial notation 8p 8p~m 8t - ---ax;-. (2.2.22)

Equation (2.2.21) may also be written in the form

or (2.2.23)

which is identical to (2.1.21) as could be expected. For p = const., or for Dmp/Dt = 0, equation (2.2.23) reduces to V'·vm =

0, describing isochoric motion, i.e., motion at constant volume.

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Microscopic Description 79

An alternative form of the mass balance equation (2.2.21), in terms of the volume weighted velocity, v, is

(2.2.24)

which can also be obtained directly from (2.2.13). In each case, the decision on whether to use a mass-weighted velocity, or a

volume weighted one depends on the kind of velocity that is being measured, or calculated by a motion equation.

Equations (2.2.21) and (2.2.24) are two forms of the (microscopic) differ­ential mass balance equation of a phase. In general, the density, p, appearing in the equation of mass balance may be a function of pressure, temperature and concentration of components of the considered phase.

(b) E represents linear momentum of a phase. Then e = pVm ,

rE == rM = F = the resultant body force per unit mass of the phase, and by (2.1.38), jEm = jMm = -q = pvm(vM - Vm), where 0' is the stress tensor that represents the diffusive flux of momentum carried by the molecules of the phase, with respect to the mass weighted velocity. For this case, (2.2.13) becomes

8p;;m = _V'·(pvmvm _ 0') + pF, (2.2.25)

where vmvm is the dyadic product of the two vectors. In indicial notation, (2.2.25) takes on the form

where:

(a) (b) (c) (d)

(a) rate of accumulation of momentum, (b) rate of momentum gained by advection, (c) rate of momentum gained by diffusive

momentum transfer, (d) rate of supply of momentum

by the body force,

(2.2.26)

and all terms are per unit volume of the phase. Equation (2.2.26) is the (microscopic) differential balance equation of linear momentum of a phase.

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80 MACROSCOPIC DESCRIPTION

In view of the mass balance equation (2.2.21), equation (2.2.26) can be rewritten in the form

aVkm m aVkm a p-- = -pl/i -- + pFk + -erik,

at aXi aXi (2.2.27)

or

(2.2.28)

also known as the equation of motion. If gravity is the only body force, then F == g = -g'lz, with g denoting gravitational acceleration and 'lz denoting a unit vector directed upwards.

(c) E represents energy of a phase. Consider a domain U bounded by a surface S containing a single phase. The total energy of the phase consists of its thermodynamic internal energy (due to thermal agitation and short range intermolecular forces), and its kinetic energy. The potential energy of the body as a whole does not appear explicitly in the balance equation, as we choose to include it in the work term of the body forces (see below). Accordingly, the energy density is given by

(2.2.29)

where I is the specific internal energy (Le., internal energy per unit mass) and ym is the (mass weighted) velocity.

Energy is supplied to the phase contained in U through its surface, S, by advection through S, expressed by

-is {pI + p(vm)2}ym.v dB

and by the (diffusive) flux of heat, jH, through S, expressed by - Is jH.v dB. The minus sign results from the definition of v as the outward normal unit vector on S.

Transforming the sum of the last two integrals into a volume one by employing the divergence theorem, (2.1.45), we obtain

The rate of production of energy within U, is expressed by Iu prH dU, where rH is the rate of heat produced within U, per unit mass, e.g., by chemical reactions.

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Microscopic Description 81

Finally, energy is added to the domain U by the work of the forces act­ing on the phase contained in U. These include (a) Is vm ·pF dU, where F represents body force per unit mass, expressing the rate of supply of kinetic energy by the body force acting on the phase contained in U, and (b) - Is V m .( -o}1I dS, expressing the work done by the surface force acting on the surface S of U. Employing the divergence theorem, this term can be replaced by Iu V'·(cr·Vm) dUo

By combining all the above terms, dividing by U and passing to the limit as U ~ 0, we obtain the differential energy balance equation in the form

:t {p(I + Hvm)2)}

(a)

= -V'.{p(I + Hvm)2)vm +jH} + Vm.pF + V'.(cr.Vm) + prH,

where

(b)

(a) rate of energy accumulation, (b) rate of net energy influx by advection

and heat conduction,

(c)

(c) rate of energy supplied to the phase by the work of mechanical forces and heat sources,

and all terms are per unit volume. In indicial notation, (2.2.30) is written in the form

(2.2.30)

! {p(I + Hvm)2)} = - {)~i {p(I + Hvm)2rVr} - {)~/f

+ {pl-'rFi+prH}+ {)~.(VrCTij). (2.2.31) J

Equation (2.2.31) can be written in the form

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82 MACROSCOPIC DESCRIPTION

where the last two terms on the l.h.s. vanish by virtue of (2.2.22) and (2.2.27), respecti vely.

Hence, (2.2.31) reduces to

DmI m·H H PDt = 0' : VV - V'J + pf ,

or, in indicial notation

(a)

where

avm (7"_'_

tJ ax' J

(b) (c) (d)

(a) = material rate of growth of internal energy,

(2.2.32)

(2.2.33)

(b) rate of increase of internal energy by the work done in producing strain (= (7ik€ik),

(c) net influx of internal energy by heat conduction, (d) rate of increase of internal energy from internal sources,

and all terms are per unit volume of a moving phase. In a fluid continuum, 0' = T - pI, where T is the deviator stress, p is the

pressure (positive for compression) and I is the unit tensor. The physical interpretation of the first term on the r.h.s. of (2.2.32) is obtained from

(2.2.34) (a) (b)

where e is defined by (2.1.12). In indicial notation, (2.2.34) takes the form

av:m avm J • • t

(7ij-ll- = (7ijeij = Tijeij - P-ll-' UXi UXi

(2.2.35)

(2.2.36)

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Microscopic Description

In (2.2.34)

( a) irreversi ble rate of internal energy (=heat) gain due to shear,

(b) reversible rate of internal energy gain by compression.

Both terms are per unit volume of the phase. By combining (2.2.32) with (2.2.34), we obtain

For a multi component system, we should add

83

(2.2.37)

where F"Y is the body force acting on the ,-component, provided F"Y is dif­ferent for different components.

In order to rewrite the internal energy balance equation (2.2.37) in terms of the absolute temperature, T, and the heat capacity, we note that according to a basic postulate of thermodynamics, the specific internal energy of a phase, I, is a single valued function of a set of specific values (per unit of mass) of extensive state variables, e.g., I = 1(s,v,w"Y), where s is the specific entropy, v( = 1/ p) is the specific volume, and w"Y( = p"Y / p) is the specific mass of a ,-component, , = 1,2, ... , all of a phase. Hence

d1 = Tds - pdv + LJ.t"Ydw"Y, ("Y)

(2.2.38)

where 81/8slv,w"l = T, 81/8vls,w"l = -p, and 81/8w"Yls,v,w6,6:f:."Y = J.t"Y is the chemical potential of the ,-component.

From (2.2.38) it follows that

T = T(s, v,w"Y) or s = s(T, v,w"Y) ,=1,2, ...

Hence, neglecting changes in component concentration, we have

8s 1 8s1 ds = 8T dT + 8v dv. V,W"l T,w"l

(2.2.39)

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84 MACROSCOPIC DESCRIPTION

Since (see any text on reversible thermodynamics)

as I ap I av T,w'Y = aT v,w'Y'

C - aI I - T as I v - - , aT v,w'Y aT v,w'Y

(2.2.40)

where Cv is the specific heat of the considered phase per unit mass at con­stant volume. By inserting (2.2.39) and (2.2.40) into (2.2.38), we obtain

dI = (T ~; I - p) dv + CvdT + "'£ p"l dw"I. (2.2.41) v,w'Y ("I)

Hence, the l.h.s. of (2.2.37) becomes

DmI _ ( ap I ) Dmv DmT '"' "IDmw"l PDt - P T aT - p Dt + pCv]5t + P L.,.P ])t'

v~ ~

(2.2.42)

From (2.2.23), or (2.1.21), we also have

~ Dmv == pDm1/P = _~ DmP = V'.vm. v Dt Dt P Dt

(2.2.43)

By substituting (2.2.43) in (2.2.42), and combining the resulting equation with (2.2.37), we obtain

DmT ·H ap I '"' Dmw"l H pCv-- = T : V'Vm - V"J - T- V'·Vm - P L.,.P"l-- + pr . Dt aT v,w'Y ("I) Dt

(2.2.44) This is the (microscopic) differential equation of heat balance, written in

terms of the temperature of the considered phase. Another useful form of this equation is

DmT ·H ap I '"' Dmw"l H pCp-- = T: V'Vm - V"J + T- V'·Vm - p L.,.P"l-- + pr , Dt aT p,w'Y ("I) Dt

(2.2.45) where Cp is the specific heat at constant pressure.

It is possible to express the heat balance equation in terms of the specific enthalpy, h, (see any text on thermodynamics), of the considered phase, defined by

h = I + pv. (2.2.46)

Then, (2.2.37) becomes

Dmh .H DmP H p-- = T: V'vm - V"J + -- + pr .

Dt Dt (2.2.47)

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Microscopic Description 85

By adding h(ap/at + V·pym) = 0 to the l.h.s. of the last equation, we obtain

(2.2.48)

It is often more convenient to use specific enthalpy as the dependent variable, especially when dealing with liquids and gases which undergo phase change.

(d) E represents the entropy of a phase. The entropy balance takes the form

as ym '("7 '("7 oS rS p at = -p . v S - V'J + p , (2.2.49)

where S represents entropy, s is the specific entropy (per unit mass of the phase), jS is the diffusive entropy fiux, rS is the rate of production of entropy per unit mass, and we have taken the mass balance equation of the phase into account.

Another form of the entropy balance is obtained by comparing the ex­pressions for pDm I/D t and d I in (2.2.37) and (2.2.38). Employing (2.2.63), we obtain

pDms = ~(T: Vym _ V-jH + prH + Lf.t""Yv.jm"tm - p Lf.t""Yrm"t) , Dt T (""Y) (""Y)

(2.2.50) where rm"t is the rate of production of the ,-component, per unit mass of the phase.

The terms on the r.h.s. of (2.2.50), can be regrouped in two parts: V-js and prs. Then, by comparing (2.2.49) with (2.2.50), we find that

oS 1 (oH ~ ""Yom"tm) J = T J - L...Jf.t J , (""Y)

(2.2.51)

(2.2.52)

By the second law of thermodynamics, rS 2': o.

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86 MACROSCOPIC DESCRIPTION

(e) E represents the mass ofa I-component, (e.g., a solute) of an a-phase. Then, omitting the symbol a that denotes the phase, e = p'Y is the concentration of the I-component (= mass of I-component per unit volume of the phase), rm'Y is the rate of production of the considered component (e.g., by chemical reaction, or radioactive decay, i.e., negative production) per unit mass of the phase, and jm'Ym represents the diffusive flux of the mass ofthe I-component (see (2.1.40)), with respect to phase mass particles. This flux is defined by

(2.2.53)

where ym'Y = y'Y. Then, the mass balance equation for the considered component takes the form

fJp'Y = _V.(p'Yym +jm'Ym) + prm'Y. fJt

(2.2.54)

Note that, unlike (2.2.13), the balance equation is written here in terms of ym.

Another form of the mass balance equation for a component is obtained by using the diffusive flux P (with respect to the volume weighted velocity of the phase), viz.

P == jm'YU = p'Y(ym'Y _ V). (2.2.55)

We obtain from (2.2.13)

fJp'Y 'Y fJt = -V·(p'YY +p) + prm . (2.2.56)

By summing (2.2.54) for all I-components, we obtain

fJL~7P'Y = -V.{ (L:p'Y)ym + L:jm'Ym} + p L:rm'Y, (-y) (-y) (-y )

(2.2.57)

h " 'Y . " 'Yvm'Y - ym were L.,.,('Y) p = p, l.e., L.,.,(-y) p = p . As indicated by (2.1.40), L(-y)jm'Ym = 0, i.e., the sum of diffusive fluxes

for all components, with respect to the mass weighted velocity, is identi­cally zero. Also, since the mass of the system as a whole is conserved, the combined mass production must vanish, i.e., L(-y) rm'Y = O. Hence, (2.2.57) becomes the (total) mass conservation equation

fJp = _V.pym fJt '

(2.2.58)

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Microscopic Description 87

which is identical to (2.2.21). On the other hand, if we sum (2.2.56) over all,-components, we obtain

()Lb~) p'Y = -V.{ (I: p'Y) V + I:F} + p I:rm'Y, ("I) ("I) ("I)

(2.2.59)

where j'Y is defined by (2.2.55), or

~ = -V·(PV + I:F) + P I:rm'Y. ("I) ("I)

(2.2.60)

In this case L("I)P = L("I) p'YV'Y -pV = p(vm_ V) i- O. This conclusion is drawn from the definition of the mass and volume weighted velocities in (2.1.30) and (2.1.31).

It is of interest to note that (2.2.54) may also be written in the forms

Dmp'Y _ 'Y"n Vm "n -m'Ym + rm'Y ~ - -p v· - V·J p, (2.2.61)

or, with DmPIDt = 0

Dm(P'Y I p) V -m'Ym + rm'Y p Dt = -.J p. (2.2.62)

For an isochoric motion of a phase, v·vm == 0, and (2.2.61) reduces to

Dmp'Y _ "n -m'Ym + rm'Y ----v·J p. Dt

(2.2.63)

The last three equations are presented in a mixed Eulerian-Lagrangian form.

If rm'Y denotes the rate of production of mass of the ,-component, per unit mass of the phase, in r chemical reactions, then

r

prm'Y = I: VZJk, (2.2.64) k=l

where vZ IM'Y, with M'Y denoting the molar mass of the ,-component, is pro­portional to the stoichiometric coefficient appearing with the ,-component in the equation that describes the kth chemical reaction. This coefficient is positive when mass of the component is produced, and negative when it is lost. The symbols Jk and vZ Jk denote the rate of the kth reaction and the corresponding rate of production of the mass of the ,-component, respectively, per unit volume of the phase, per unit time.

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88 MACROSCOPIC DESCRIPTION

2.2.3 Constitutive equations

Since in this subsection we deal only with a single phase, we have omitted the symbol denoting the phase.

In Subs. 2.2.2, we have developed the following system of microscopic balance equations, applicable to any phase continuum.

Equation of mass balance (2.2.22)

Op OpV,r ot -~'

(2.2.65)

which is a single equation in the four unknown variables: mass density, p, and three velocity components, Vr.

Equation of linear momentum balance (2.2.27)

oV,r _ _ V!7t oVr F: OO"ji p ot - P J ox' + p t + ox''

J J

i,j=1,2,3, (2.2.66)

which, given the components, Fk, of the body force, represents three equa­tions in the six additional variables, O"ij (since O"ij = O"ji).

Equation of internal energy balance (2.2.33)

01 um 01 o"l-T ojr rH p- = -pY; - + O"i'-- - -- + p ,

ot t OXi J OXj OXi i,k = 1,2,3, (2.2.67)

which, for a known energy source, rH, adds one more equation in four additional variables: 1 and the three components of the conductive heat fl. ·H ux, Ji .

Equation of mass balance of a component of a phase (2.2.54)

(2.2.68)

which, for each ')'-component and for a known source rm"Y of mass of the ')'-component, adds one more equation in four additional variables: p'Y and the three components of the diffusive mass flux, fr"f m •

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Microscopic Description 89

Equation of entropy balance (2.2.50)

(2.2.69)

which is one more equation with N +8 additional variables: p,"I (N variables), Tij (6 variables), T and s.

Altogether, for an N-component phase, since 2:~=1 p"l = p, we have N + 5(== (N - 1) + 6) independent balance equations in the 17 + 5N (== 22 + 5(N - 1)) variables that are functions of time and position:

( (- 1/ )) vm .. I ·H "I 'm"lm "I .. T d p orv - p, i ,(Ttl' ,Ji ,p ,Ji ,p, ,Ttl' an s.

The equations given above are nothing but general balance statements which are valid for any phase. The 12 + 4N additional relationships that are needed in order to complete the description of any specific transport phenomenon, or process, under consideration, in the form of a closed set of equations, are relations that characterize the behavior of the particular material comprising the phase under consideration. These relations are re­ferred to as constitutive equations. As we shall see in the examples below, constitutive equations take the forms of (static) equations of state, as well as stress-strain relationships and flux equations, which may be regarded as dynamic equations of state.

Accordingly, let us add six additional equations

(6 equations),

which decompose the components of the stress into a deviatoric part Tij,

and a pressure, p (taken positive for compression). We now have 11 + N equations, but the total number of variables has been increased to 18 + 5N, as we have added the pressure, p, as a variable.

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90 MACROSCOPIC DESCRIPTION

The 7 + 4N relationships still required to obtain a closed set of equations, are

(a) Equations of state

p s

or J

p"Y

p(p, p"Y, T) seT, v,w'Y)

J(T, v,w'Y) p'Y (p, w'Y, T)

(one equation), (one equation),

(one equation), (N - 1 equations),

(2.2.70)

where p = l/v, p"Y is the chemical potential of the ,-component, and w"Y = p"Y / p. The symbols p"Y and w"Y stand for all the p"Y's and w"Y's of all the ,-components present in the system.

(b) Stress-strain relationships

( 6 equations). (2.2.71)

(c) Flux equations

Heat conduction: jf! = jf! (T) (3 equations).

Diffusive mass flux: ji'Ym ji'Ym(p'Y) (3N - 3 equations).

The entire discussion so far, as well as the examples of constitutive equa­tions presented below, are related to a single phase continuum, viz., at the microscopic level. However, the same ideas are also applicable to the macroscopic level. Just as we obtain the macroscopic balance equations by averaging microscopic ones, the macroscopic constitutive equations for the multiphase system called porous medium are obtained by averaging the ap­propriate microscopic ones. In Sec. 2.6, we shall employ this procedure to develop macroscopic flux equations for the various fluxes that appear in the macroscopic balance equations considered in Sec. 2.4.

In spite of what was said above about the derivation of macroscopic con­stitutive equations by averaging the corresponding microscopic ones, very often it is assumed that the general structure of a macroscopic equation is the same as that of the microscopic one, without going through the averag­ing procedure. It then remains only to determine experimentally the values

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Microscopic Description 91

of the coefficients that appear in the assumed macroscopic constitutive rela­tion. We shall see in Secs. 2.3 and 2.4 that by this approach, avoiding the averaging procedure, we lose the interactions across interphase boundaries.

Following are several examples of constitutive relations for fluid and solid materials of interest (at the microscopic level). We continue to omit the symbol that indicates the phase. Further details can be found in texts on Continuum Mechanics.

(a) Equations of state for fluids

The general equation of state for a fluid phase is

p = p(p,p'Y,T), or p = p(p'\ v, T), (2.2.72)

where p"Y stands for the p"Y's of all components present in the fluid. Equation (2.2.72) states that the density, p, is a function of pressure, p, concentration of various components, p"Y, say dissolved solids, and absolute temperature, T. For a homogeneous, single component fluid phase, the equation of state reduces to p = p(p, T). Under isothermal conditions, this general expression for a compressible fluid phase, is further reduced to p = p(p). Sometimes the specific volume v( = 1/ p), is used in the above relationships instead of the density.

From (2.2.72), it follows that

dp = 8PI dp + 2: 8p I dp"Y + 8p I dT 8p T,p"l ("'I) 8p"Y p,T 8T p"l,p

P ((3p dp + 2: (3p"l dp"Y - (3T dT), (2.2.73) ("'I)

where f3 = ~ 8p I __ ~ 8v I

p - p 8p T,p"l - v 8p T,p"l (2.2.74)

is the coefficient of compressibility of the fluid, at constant temperature and concentration ,while

1 8p I 1 8v I (3p"l == P 8p"Y p,T = -1) 8p"Y p,T

(2.2.75)

introduces the effect of a change in p, or v, as a result of a change in con­centration of a ,-component at constant temperature and pressure, and

1 8p I 1 8v I (3T == - P 8T p,p"l = :;; 8T p,p"l

(2.2.76)

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92 MACROSCOPIC DESCRIPTION

is called the coefficient of thermal expansion at constant pressure and con­centration. We note that, in general, f3p, f3p"l, and f3T vary with p, pry and T.

If, in certain ranges of p, pry and T, the coefficients f3p , f3p"l and f3T are constants (or approximated as such) for a given fluid, the equation of state (2.2.72) takes on the specific form

p = poexp{f3p(p - Po) + Lf3p"l(P'Y - pJ) - f3T(T - To)}, (-y)

where p = Po for p = Po, pry = pJ and T = To.

(2.2.77)

Special cases of (2.2.77) arise for uniform concentrations, pry = pJ = const., and isothermal conditions, T = To = const.

Some fluids, in certain ranges of p, pry and T, obey empirical relationships, for example

p = Po{l + f3;(p - Po) + Lf3~"I(P'Y - pJ) - f3T(T - To)}, (-y)

where

(2.2.78)

f3' = J:. ap I ' p Po ap p"l,T

, 1 ap I f3 p"l = - a 'Y ' po P p,T

f3' - 1 ap I T - - Po aT p,p"l'

are assumed constant. Equation (2.2.78) may be considered as an approxi­mate linear form of (2.2.77).

A particular fluid is the ideal gas for which the equation of state, takes the form

pM p = p(p,T) = RT' (2.2.79)

called the ideal gas law, where R is the universal gas constant and M is the molecular weight of the gas.

The petroleum industry has adopted the concept of a compressibility factor, Z, for describing the behavior of mixtures of gases at moderate to high pressures. The compressibility factor is simply an empirical correction factor to the perfect gas law (2.2.79). With this correction, (2.2.79) is replaced by

pM p = p(p,T) = ZRT. (2.2.80)

In general, Z = Z(p, T). This is a rather complex relationship, usually presented in the form of tables and graphs. For an isothermal process, Z = Z(p). For an ideal gas, Z = 1.

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Microscopic Description 93

For an incompressible fluid, ap/ap = o. Muskat (1937) suggests the following general equation of state for homo­

geneous fluids of practical interest

p = Po (~) m exp{;Jp(p - Po)}, (2.2.81)

where m = O,and ;JP = 0 for an incompressible liquid. For a compressible liquid, m = 0, ;JP ::j:. O. For gases under isothermal conditions, m = 1, ;JP = 0 and for gases under adiabatic conditions, ;JP = 0, and m = Cv /Cp, i.e., the ratio between the coefficients of specific heat at constant volume and at constant pressure. Another example is (3.3.24).

(b) Stress-strain relations for fluids

Real fluids are both compressible and viscous. To facilitate the discussion on their stress-strain relationships, it is convenient to express the stress, O'ij, in the form

(2.2.82)

where the deviator Tij is the viscous stress tensor, p = -l( O'ii - Tii) == -~ i:(i) ( O'ii - Tid is the pressure and Dij is the Kroenecker delta. In (2.2.82), the pressure p is considered positive for compression, while O'ij and Tij are considered positive for tension.

In (2.2.82), one may interpret 0' (and hence T and p) as force per unit area. However, in developing the microscopic balance equation for linear momentum ( Sec. 2.2.2(b », the stress tensor, 0', was introduced as a diffusive flux of momentum, carried by the molecules of the phase.

The viscous stress tensor, Tij, represents the diffusive flux (Subs. 2.1.7) of linear momentum across a material surface element, due to velocity vari­ations across the element. As such, it acts as a frictional force between adjacent layers of the fluid, per unit area. A typical component of the veloc­ity gradient, relevant to Tij, is aVi/axj. Therefore, it is generally assumed that for fluids, Tij = Tij(ckl), where Ekl represents the rate of strain defined by (2.1.12), with

. _ ~ (avr avr) Ckl - 2 a + a .

Xl Xk (2.2.83)

When the relation Tij = Tij (E kl) is nonlinear, the fluid is called a Stoke­sian fluid (or non-Newtonian fluid).

For sufficiently small values of aV;,/ ax j, we assume that

C' . Tij = ijkC ckl, (2.2.84)

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94 MACROSCOPIC DESCRIPTION

where Cfjkf denotes components of the fluid's viscosity coefficient. A fluid that obeys the linear relationship (2.2.84) is called a Newtonian fluid.

When the viscous stress generated in a fluid element is independent of the orientation ofthe latter, Le., when the molecular structure of the fluid is statistically isotropic (which is the case of all gases and simple liquids, unlike suspensions and solutions that contain very large chain - like molecules), the coefficient Cfjkf is an isotropic tensor that can be written in the form

(2.2.85)

where a' and a" are scalar coefficients (see Subs. 2.6.5). Hence, by inserting (2.2.85) into (2.2.84), we obtain for an isotropic, single component, com­pressible Newtonian fluid

2 · \" . .c Tij = J1,Eij + /\ E kk Vij , Tij = Tji, (2.2.86)

and (2.2.87)

where J1, = a' is the fluid's dynamic viscosity (also called shear coefficient of viscosity) and A" = a" is another viscosity coefficient of the fluid. Water is a typical example of a Newtonian fluid that obeys (2.2.86). The coefficient (A" + ~ J1,) is called the coefficient of bulk (or dilatational) viscosity.

For an incompressible, isotropic, single component Newtonian fluid, un­der isothermal conditions, or under conditions of isochoric mass flow (Le., \7·vm = 0; see Subs. 2.1.5), ikk == 0, and (2.2.86) reduces to

( 8Vr 8V:m ) Tij = 2J1,iij = J1, 8; j + O;i . (2.2.88)

Then, the pressure p in (2.2.87) reduces to p = -~O'ii and is called the mean normal stress. Equation (2.2.88) is often called the generalized Newton law, describing the molecular flux of linear momentum.

For a fluid at rest, we always have Tij = 0, and O'ij = -POij, P = -~O'ii. In this case, the mean negative normal stress, p, is called the hydrostatic pressure.

A perfect fluid is a nonviscous (or in viscid) fluid that cannot sustain a viscous stress, i.e.,

Tij = 0 and (2.2.89)

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Microscopic Description

Shear stress, T

I

Perfectly I Plastic elastic solid / ~

/ /

I

I

" /

95 Ideal plastic

(Non-viscous) ideal fluid

----------------------~ Rate of strain (E)

Figure 2.2.3: Schematic stress-rate of strain relationships for a number of materials.

Figure 2.2.3 shows the stress-rate of strain relationships for a number of materials.

With stress interpreted as the diffusive flux 0; momentum, equations (2.2.86), (2.2.88) are constitutive equations that take the form of flux equa­tions. In this case, the flux is of linear momentum. We may therefore refer to them as dynamic equations of state. On the other hand, the relationship p = p(p, p'Y, T) may be regarded as a static equation of state.

( c) Stress-strain relationships for solids

While for fluids, the deviatoric stress-tensor is regarded as a force which arises in a moving fluid and depends on the rate of strain, a solid, once deformed, is subject to stress even if it is at rest. Thus, for a linearly elastic solid under isothermal conditions, the stress-strain relation is given by the generalized Hooke's law

(2.2.90)

where Cijk£ is the elasticity tensor and ck£ denotes components of the (Eule-

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96 MACROSCOPIC DESCRIPTION

rian infinitesimal) strain tensor which is related to the displacement vector, w, by (2.1.5) rewritten here in the form

Ckf- = ~ (OWk + OWf-). 2 OXf- OXk

(2.2.91)

The solid's velocity, Vs is related to the displacement, w, by

Vs = D~~ :=w. (2.2.92)

The dilatation, c, is given by

OWi e = eii = -,

OXi (2.2.93)

recalling that the summation convention is implied. For a linearly elastic isotropic solid, (2.2.90) reduces to

,(OWi OWj) >.."OWk 8 (J'ij = J.l s ~ + -;:;-- + s ~ ij, UXj uXi UXk (2.2.94)

or 3>"" + 2 ' = (J'ii s J.l s ,

ekk where J.l~ and >..~ are called Lame constants of the elastic solid. In the par­ticular case of a uniaxial state of stress in the xl-direction (i.e. (J'n =J 0, (J'22

= (J'33 = 0), it is customary to define two constants of an isotropic elastic solid

E = (J'n/en

where E is called Young's modulus of elasticity, and v is called Poisson's ratio. In this case

>.." v = s 2(>"~ + J.l~) (2.2.95)

Other examples of stress-strain relations of solids are given in 2.2.4. For a linearly thermoelastic solid, i.e., when changes in temperature are

taken into account, the stress-strain relationship (2.2.90) is extended to the form

(2.2.96)

where 11kl!(T - To) is the strain contributed by the temperature field and To is a reference temperature.

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Microscopic Description 97

For an isotropic solid (2.2.97)

where aT is the coefficient of linear thermal expansion. By analogy with (2.2.94) the stress-strain relationship then becomes

O"ij = 2Jt~{€ij - aT(T - To)Oij} + A~{€ - 3aT(T - To)}Oij

2J.L~€ij + A~€Oij - (3A~ + 2J.L~)aT(T - To)Oij. (2.2.98)

Equation (2.2.98) is the constitutive equation of a linearly thermoelastic isotropic solid. It may be inverted to yield the strain as a function of the stress

(2.2.99)

(d) Conductive heat flux

The conductive (or the diffusive) flux of heat, jH, in a phase is expressed by Fourier's Law

(2.2.100)

where A, a second rank tensor, is the thermal conductivity of the phase; for an isotropic phase, this coefficient reduces to a scalar, A.

( e) Molecular diffusion

The diffusive flux (= molecular diffusion), JY, of the mass of a i- component in a fluid phase, defined by (2.1.40), is expressed by Fick's law

(2.2.101)

where 1)7 is called the coefficient of molecular diffusion of the ,-component in the fluid phase and p'Y is the concentration of the i-component in that phase. Actually, the diffusive flux should be related to the gradient of the chemical potential, Jt'Y.

Equation (2.2.101) is valid for a binary (two-component) fluid phase. In a multi component fluid phase, the flux of the i-component is expressed by

j'Y = - L V f3T\1 pf3 ,

(13)

(2.2.102)

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98

Stress, (T

(a)

Perfectly elastic

Strain, £

MACROSCOPIC DESCRIPTION

Stress, (T

Rigid, perfectly plastic, with strain hardening

Strain, £

(b)

Stress, (T Rigid, perfectly plastic

Stress, (T

Strain, £ Strain, £ Strain, £

(c) (d)

Figure 2.2.4: Stress-strain relations for various solids.

where Dfh is a coefficient of diffusion that expresses the contribution to the flux of the i-component by a unit gradient of the ,6-component. We note that (2.2.101) and (2.2.102) define the diffusive flux relative to the volume weighted velocity of the phase (see (2.2.55)). In both equations, the symbol a denoting the phase has been omitted.

In a binary system, the diffusive mass flux, (2.2.53), relative to the mass weighted velocity, is given by

jml'm p'Y(yml' _ ym) = p'Y(yml' _ Y) _ p'Y p(ym _ Y) P

-pDT'\lw'Y, (2.2.103)

where w'Y = p'Y / p is the mass fraction of the i-component in the phase and D'Y is the same for both components. We use the symbols D~, pcx and w~ (=pU Pcx), when we wish to indicate that we consider an a-phase.

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Microscopic Description 99

It is worth noting that while from the mathematical standpoint, a con­stitutive equation is a postulate, or a definition, the first guide in selecting a particular constitutive equation for a specific material is experimental data. However, in general, experimental evidence alone is insufficient to determine the constitutive equations of a given material. As an aid in formulating the constitutive equations, Truesdell and Toupin (1960) proposed seven rules, or criteria (often referred to as invariance principles) that these equations must obey . The reader is referred to the above mentioned work of Truesdell and Toupin (1960) for a detailed discussion of these principles and of constitutive relations in general.

2.2.4 Coupled transport phenomena

The generalized Newton's law, (2.2.88), that describes the molecular fiux of linear momentum, Fourier's law, (2.2.100), that describes the conductive heat fiux, and Fick's law (2.2.101) that describes the diffusive mass fiux (= molecular diffusion), are particular cases of the general linear law

J'?J- = _ ~ L',:!J)~n ~ L...J ~J ax"

j=l J

i,j = 1,2,3, (2.2.104)

where jF'(= j:pn) denotes the ith component of the fiux of an extensive quantity En of a phase, ~n is a state variable associated with En and Lft is a coefficient of proportionality which in an isotropic phase reduces to a scalar. Since only one phase is being considered, no special symbol will be used in this paragraph to indicate this fact.

The three linear diffusive fiux laws mentioned above, also called phe­nomenologicallaws, state that a nonuniform distribution of a state variable, ~n (e.g., temperature), produces a fiux of only the corresponding exten­sive quantity (e.g., heat). However, experimental evidence suggests that, in principle, gradients of state variables, ~r, r i- n, corresponding to other extensive quantities, may also contribute to the fiux of En. Phenomena of this kind are referred to as coupled phenomena (or cross-effects). Common examples of such phenomena are the Soret (or thermodiffusion) effect, in

which mass fiux of a solute in a liquid phase is produced by a temperature gradient, in addition to the fiux produced by the gradient of the solute's concentration according to Fick's law, and the Dufour effect, in which heat fiux is caused by a concentration gradient, in addition to the heat fiux caused by temperature gradient. Thermodynamic, thermoelectric, thermomagnetic

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100 MACROSCOPIC DESCRIPTION

and galvanomagnetic effects, are other examples of coupled phenomena. The description of coupled phenomena requires a generalization of the linear law (2.2.104).

In order to derive such a law, we recall that the theories of mechanics and thermodynamics of material systems, are based on the postulate that a finite number, say N, of variables is sufficient to completely define the motion and physico-chemical properties of an element of a considered system, say, a phase. Let these variables, referred to as parameters of state, be denoted by 'lj;n n = 1,2, ... , N. In a system that constitutes a continuum, the various 'lj;'s are specific values of the extensive quantities pertinent to the system, i.e., 'lj;n = dEn / dm( == en / p). Examples of 'lj;n 's are the specific mass, w'Y (= p'Y / p), of a {'-component of a phase, the specific volume, v( = 1/ p), of a phase, the specific momentum of a phase, vm (= mass weighted velocity = p vm / p), and the specific entropy of a phase, s. These 'lj;'s constitute tensorial fields that are functions of the spatial coordinates and of time.

The behavior of a system (here, a phase) is characterized by its consti­tutive equations. The most fundamental one is the caloric equation of state that relates the specific internal energy, J, of a system to the complete set

of the N parameters of state, 'lj;n, through a single valued function which is independent of time, position, motion, or stress. We may express this relation in the general form

J = f('lj;t, 'lj;2, ... ,'lj;N,e),

where the ~i'S are the material coordinates (Subs. 2.1.2) of an element of the system.

A differential increment in the (thermodynamic) state of an element is expressed by

dJ oj 1 oj 2 oj N

o'lj;l d'lj; + o'lj;2 d'lj; + ... + o'lj;N d'lj;

N oj N L: 0 n d'lj;n == L: cPn d'lj;n, n=l 'lj; n=l

cPn = oj n( 1 2 N)I o'lj;n = cP 'lj;, 'lj; , ... , 'lj; e' n = 1,2, ... ,N, (2.2.105)

where cPn is the increment of internal energy per unit increment in the value of the parameter of state 'lj;n. The various cPn's are thus functions of the state of the system as expressed by the 'lj;T's. Accordingly, we refer to the N equations for the cPn's appearing in (2.2.105) as equations of state.

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Microscopic Description 101

For example, by comparing (2.2.105) with (2.2.38), we obtain in the latter

where w'Y = p'Y / p.

ds, T,

-dv, p,

In general, for a given n, q,n need not depend on all the N parameters of state, but only on a subset of interdependent, or coupled parameters. In­cluded in that subset are only those parameters that are needed in order to describe the particular behavior of the specific material and conditions under consideration.

Referring to the transport and/or transformation of an extensive quan­tity, En as representing a degree of freedom of the system, each q,n is a single-valued function of all the parameters of state that belong to the group of coupled degrees of freedom of the system. Let e( < N) denote the number of coupled degrees of freedom of a given system. Then, the set of equations of state that include the relevant parameters, is given by

q = 1,2, ... ,e. (2.2.106)

For example:

T T(s, v,w'Y), p pes, v,w'Y),

j.l'Y j.l'Y(s, v,w'Y),

where, when necessary, w"l stands for the w of all the ,-components involved, and j.l'Y denotes the chemical potential of the ,-component.

It is worth noting that a successive elimination of the parameters of state from the set of equations (2.2.106), reduces the latter to a single equation such as

For example

F(v,p,T,j.l'Y) = 0, or p = p(p,T,j.l'Y). (2.2.107)

This may explain the relationship between the flux of a given extensive quantity and the associated functions of state.

Assuming that the linear relationship of the type (2.2.104) is valid, and continuing to omit the subscript that denotes a phase, we find that the flux

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102 MACROSCOPIC DESCRIPTION

of an extensive quantity, Eq, belonging to the coupled subset, is given by

with no summation on q. Defining (Veynik, 1961)

{)if! T {)7/JT X":=-----,

J {)7/JT {)x j

(2.2.108)

r = 1,2, ... ,f, (2.2.109)

with no summation on r, as the thermodynamic force which is conjugate to the gradient of 7/JT, and

qT _ qq {)if!q / {)7/JT Lij - Lij {)if!T / {)7/JT ' r = 1,2, ... ,f, (2.2.110)

with no summation on r, we may rewrite (2.2.108) in the form of the '-equations

i = 1,2,3; q= 1,2, ... ,i. (2.2.111)

We recall that '- depends on the considered q. From (2.2.111) it follows that the flux of an extensive quantity,Eq, is

a single-valued function of all the (coupled) thermodynamic forces associ­ated with Eq. Equations (2.2.111), for the various Eq,s, are also called the phenomenological equations, or the kinetic equations of state of a system possessing'- coupled degrees of freedom. They express linear relationships between fluxes and thermodynamic forces. In general, however, the relation­ships between fluxes and thermodynamic forces may be nonlinear.

While (2.2.109) serves as a definition for the thermodynamic force, XT, equation (2.2.110) determines the nature of the phenomenological coeffi­cients, LqT.

In 1851, Stokes postulated that in (2.2.108), the coefficients LrJ are sym­metric with respect to the coordinates i and j, i.e.

L~~ = L~~ tJ Jt . (2.2.112)

This postulate states that the transformation of a unit force along one axis into a conjugate flux along another axis remains unaltered when those axes are interchanged.

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Microscopic Description 103

Of special interest are the cross coefficients for q =1= r, which give the flux of Eq caused by the force, XT, associated with the gradient of eT (== the density of E T ). Employing the principle of microscopic reversibility of processes, and methods of statistical mechanics, Onsager (1931) showed that for the linear equations (2.2.111), and provided a proper choice is made for the fluxes, jq, and forces, XT, the phenomenological coefficients are also sym­metric in rand q, i.e.

q =1= r. (2.2.113)

These relationships are known as Onsager's, or Onsager-Casimir's, re­ciprocal relations (or Onsager's law). They express a relationship between any pair of cross-phenomena (e.g., thermal diffusion and Dufour effect) aris­ing from simultaneously occurring irreversible processes (e.g., heat conduc­tion and molecular diffusion).

Together, the relationships (2.2.112) and (2.2.113) take the form

(2.2.114)

According to Onsager, the reciprocal relations (2.2.112), hold under two conditions:

(a) The relationship between each individual flux and its conjugate ther­modynamic force is linear.

Indeed, experimental evidence suggests that for a wide range of irre­versible processes and experimental conditions, fluxes are linear func­tions of thermodynamic forces (e.g., Fourier's law, Fick's law). Al­though some irreversible processes (e.g., chemical reactions) may be nonlinear, they still obey the linear relationships, as an approximation, in the neighborhood of equilibrium.

(b) The fluxes, jq, and their conjugate forces, xq, should be selected such that

R. 3

Tr S = LLj{X'!, (2.2.115) q=l i=l

where Trs is the rate of production of thermal energy per unit mass of the system.

Equation (2.2.115) implies that each jq and its conjugate force, xq, must be of the same tensorial rank (not necessarily a vector, as indicated by the single subscript).

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104 MACROSCOPIC DESCRIPTION

At this point it is worth recalling the concept of thermodynamic equilib­rium.

Any interaction between a thermodynamic system and its environment, is accompanied by the transfer of an extensive quantity between them. The same is true for interactions between parts of a system. In the absence of such interactions, a system is said to be in a state of equilibrium. Thus, equilibrium requires that any function of state, q>, relevant to a considered extensive quantity, be uniform throughout the system.

By the second law of thermodynamics

where rS is defined in (2.2.52). Also, by (2.2.111) and (2.2.115)

" L~r:Xr:X~ > 0 L...J tJ J t - •

(q,r,i,j)

Hence, a sufficient condition for the validity of (2.2.117) is

L~~ > 0 tJ '

for all i,j

1 (Lrq Lqr) 4" ij + ij •

For an isotropic medium, LIj = Lqr Oij, and

Lqq > 0 - ,

(2.2.116)

(2.2.117)

(2.2.118)

(2.2.119)

(2.2.120)

As to the identification of the phenomena to be coupled, one must con­sider the Curie-Prigogine symmetry principle (Curie, 1894), that states that in isotropic systems, extensive quantities whose tensorial ranks differ by an odd integer cannot interact. In other words, (2.2.111) can only include ther­modynamic forces, X q , which are tensors of the same rank, or when the difference in rank is even. From Curie's principle it follows that one must consider separately scalar, vector and tensor extensive quantities, and write (2.2.111) separately for each tensorial rank.

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Microscopic Description 105

Let us consider several examples.

(a) A thermo-mechanical system

Let us consider the fluxes of volume, U, and entropy, S, in a fluid phase possessing two coupled degrees of freedom: one mechanical and the other thermal. The corresponding equations of state are

p = p(s,v), } T = T(s,v),

(2.2.121)

where v = 1/ P is the specific volume, s is the specific entropy and T is the absolute temperature. For this case, following (2.2.38), equation (2.2.105) yields

q,2 T, 'lj;2 = s,

with jI denoting volumetric flux, and j2 denoting entropy flux. From (2.2.109) through (2.2.111), we obtain

LH ~)

Lll (op/ov)l s - Lll ij (op/ov)is - ij'

L~7 ~)

Lll (op/os)lv ij (oT/os)iv'

X~ _oPI ~ __ op I X7 aTlas aT I = - as vOXj = - OXj v' ) ov s ax j - ax j s' )

and the flux

J'! = LHX~ + LPX~ = -LH op I _ LP aT I ~ ~)) ~)) ~) ax ' ') ox· .

) s ) v (2.2.122)

Thus, in (2.2.122) the total volume flux is produced by both a pressure gradient and a temperature gradient.

In a similar way, the flux of entropy is given by

,2 21 1 22 2 21 op I 22 aT I J·=L .. X·+L .. X·=-L .. - -L .. -, ')) ~)) ') OX· ') OX· ' ) s ) v

(2.2.123)

where

L~! = L~~(oT/ov)ls t) I) (op/ov)ls ' (2.2.124)

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106 MACROSCOPIC DESCRIPTION

(b) A thermo-diffusive system

Consider the fluxes of mass and entropy in a binary system containing a solvent and a solute (,), The equations of state of this system are

p"Y = p"Y(s,w"Y), T = T(s,w"Y),

(2.2.125)

where p"Y represents the chemical potential of the solute, and w"Y(= p"Y / p) is its specific mass (=mass of solute per unit mass of the system).

For this case we have

... 2 T ~ , 'lj;2 = s,

with j1 denoting mass flux of the solute, and j2 denoting entropy flux. By (2.2.109) through (2.2.111)

LP = LH (op"Y /ow"Y) 18 = LH L~~ = LH (op"Y /os)lw'Y tJ tJ(op"Y/OW"Y)18 tJ' tJ tJ(oT/os)lw'Y'

L~~ = L~~ (aT/os )lw'Y = L~~ ( ) tJ tJ (aT/os )lw'Y tJ ' 2.2.126

X} = - ~wp~ I ~WX~ = - n:x~ I, XJ = - ~Ts I :xs, = - ~xT,1 ' u 8 U J U J 8 U w'Y U J U J w'Y

'1 'm'Y LHX~ +LPX~ -LH op"Y I _ L~~ oT I Ji - Ji tJ J tJ J tJ 0 tJ 0 ' Xj 8 Xj w'Y

'2 '8 L~~ X~ + L~~ X~ = -Ln op"Y I - L~~ oT I Ji - Ji tJ J tJ J tJ 0 tJ 0 ' Xj 8 Xj w"f

where L~J XJ, which represents a diffusive mass flux of the solute caused by a temperature gradient, is the thermodiffusion, or Soret effect, and LTJ X}, which represents the flux of entropy caused by a gradient of the chemical potential, is called the Dufour Effect.

It is possible to rewrite these fluxes in terms of more commonly used coefficients, e.g., thermal conductivity and coefficient of molecular diffusion. Thus, in view of (2.2.103), we have

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Microscopic Description 107

12 (OJ.L"! / os )Iw-r pV"! Oij

Lij = (oT / os )Iw-r (OJ.L"! / ow"!)1s' (2.2.127)

where V"! is the coefficient of molecular diffusion o/the ,-component (with respect to the gradient of specific mass of the component), appearing in (2.2.103) and

(2.2.128)

where X is the conductivity of entropy and>" is the thermal conductivity. With (2.2.127) and (2.2.128), the fluxes of mass and entropy take the

forms

a ==

We note that the flux of heat is affected also by the gradient of the ,-component, while the flux of, is also driven by the temperature gradient.

The flux of heat follows from (2.2.51). Accordingly

(2.2.129)

In the examples given above, as a consequence of the interdependence between state variables of a phase, for example, as expressed by (2.2.107), we saw that the flux of a given extensive quantity depends on both the gradient of the corresponding intensive quantity and on that of another one. Actually, the flux of a component, as expressed by (2.2.102), manifests inter­dependence between the concentrations of components in a multicomponent system.

We shall return to coupled fluxes at the macroscopic level in Subs. 2.6.6.

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108 MACROSCOPIC DESCRIPTION

2.2.5 Phase equilibrium

In the previous subsection, we considered the coupling between variables of state of a phase and its effect on transport processes within the phase. In this subsection, we consider coupling between variables of state of differ­ent phases which simultaneoulsy occupy an REV of a porous medium. The phases contact each other along their interphase surfaces, and may interact with each other by exchanging extensive quantities such as mass, momen­tum and/or energy. Although each phase changes its thermodynamic state in the course of time, we shall assume, as a first approximation, that the processes of transport and exchange of extensive quantities, and the rate of transformation of the system from one state to another, are sufficiently slow, so as to allow for spatial variations of some state variables within the REV to smooth out and to bring the phases therein to a state which is close to equilibrium with each other. Under such conditions, the composition of a system is subjected to the relationship

v = 2 + n - r,

referred to as Gibb's phase rule, where v is the number of independent vari­ables which determine the state of the system, r is the number of phases comprising the system, and n is the number of different chemical constituents (thermodynamic components) of which the system is composed.

One should note the difference between the concept of a component intro­duced in Subs. 1.1.1, where the same chemical substance found in different phases represents differrent components, and the notion of a thermodynamic component given here, where the same chemical substance found in different phases represents only one component. Since v ~ 0, the phase rule imposes a restriction on the coexistence of phases and, hence, also on the possibility of phase change. As an example, consider a single chemical compound, say H20, in a single state of aggregation, say water, or water vapor, or ice. In this case n = r = 1, and the state of the system at equilibrium is fully de­termined by 2 + 1 - 1 = 2 independent variables, say p and T, or p and p. Any point in the pressure-temperature plane, determines a possible state of the system. Let the equation of state of a substance be, p = pep, T), and the substance be present, simultaneously in two phases, say, two states of ag­gregation: water and water vapor. In this case, the number of independent state variables reduces to (v = 2 + lcomponent - 2phases = 1), one, say, T. The domain of possible states of this two-phase system in the p - T phase

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Microscopic Description 109

Liquid

Solid+

Vapor

Temperature

Figure 2.2.5: Schematic pressure-temperature diagram for a single component sub­

stance.

diagram lies on a curve p = p(T), which separates the domain where H20 exists only as water from that where it exists only as vapor. Similarly, other p = p(T) curves will represent the domains of simultaneous coexistence of water-ice, or vapor-ice systems.

Finally, there is only one possible state at which the three phases: water, water vapor and ice can coexist at equilibrium, since v = 2 + 1 -3 = O. It is represented by a single point in the p - T diagram.

Figure 2.2.5 shows a typical pressure-temperature diagram that presents the domains of existence of a single component substance in different states of aggregation at equilibrium. The three curves divide the p - T space into three regions, each corresponding to a single state of aggregation, (or phase) of the considered substance. The intersection of the three curves is called the triple point. Each curved segment describes the values of p and T at which the two phases, when placed adjacent to each other, can coexist at equilibrium. Along BC, the p - T conditions are such that a liquid and its vapor can coexist in equilibrium, otherwise, only one phase can exist: above the curve, the considered substance can exist only as a liquid, while below it, it can exist only as a vapor.

Point C, where the p - T curve terminates, is called the critical point. For a single component two-phase system, this point is defined as the highest value of pressure and temperature at which the two phases (= states of a substance) can coexist.

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110 MACROSCOPIC DESCRIPTION

Critical temp.

\ \ ~ ~ T, < T, < To < T3 < T,

T~\' TOT3 T, Vapm

-~ - -Ii ~ Critkal point

L· ·d f I lqUl H...-r---l .......

\ .... VI \ .S/~--~----~'~

8../ \_-Dew point \ '.----~, ~ Z I Liquid + vapor '< ...

::t I ......... J::q'

Specific volume

Figure 2.2.6: Schematic pressure-specific volume-temperature diagram for a single com­

ponent system.

While the p-T diagram shows only the regions of existence of the dif­ferent states of aggregation of a component, a p - v - T diagram, shown schematically in Fig. 2.2.6, where v is the specific volume, presents also the states themselves. Point C is the critical point. The dashed curve encloses the region where the liquid and vapor phases coexist. The solid lines are isotherms, i.e., curves of constant temperatures. Two definitions will help to understand the figure. A bubble point is defined (for a single compo­nent substance) as that state in which the substance is entirely in the liquid phase, and any slight reduction in pressure (or increase in specific volume), at the substance's fixed temperature, produces a vapor phase. Similarly, at a fixed pressure and volume, a slight increase in temperature produces a va­por phase. A dew point is defined, again, for a single component substance, as that set of conditions under which the substance is entirely in the vapor phase; any slight decrease in pressure (or reduction in specific volume), pro­duces a liquid phase at constant temperature; similarly, at a fixed pressure and volume, a slight reduction in temperature produces a liquid phase.

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Microscopic Description 111

A number of isotherms are shown in Fig. 2.2.6. For temperatures T4 and T3 , the curves are entirely in the vapor region as p and v vary. For temperatures Tl and T2, the isotherms are in the vapor region for large values of v. As v is reduced, we move along the isotherms toward the dew point curve. Here, liquid starts to form.

On the other side of the dew point curve, liquid and vapor coexist in equilibrium. As the bubble point curve is crossed, all the vapor condenses and the entire system is in the liquid state. Point C is the critical point.

Before, considering a multicomponent system, let us consider the case in which another component is added to the system for which Fig. 2.2.5 is valid. Let this component be soluble in both the liquid and the vapor of the first component. As an example, we may consider water in equilibrium with its vapor and we add another component, say C02, to the system. The mass of the added component, CO2 in this example, may partially go into solution in the liquid (here water) and partially remain in a gaseous phase (here with the water vapor). According to the phase rule, to be discussed below, the dist~ibution of CO2 between the two phases, at equilibrium, will depend on the pressure and the temperature only.

In order to generalize the discussion, consider a substance A that can dissolve in two immiscible substances, Band C. We define the solubility, RAB , of A in B as

(2.2.130)

where n~ denotes the number of moles of A dissolved in nB moles of B. Similarly

nA R - C

AC - A + ' nc nc (2.2.131)

where n~ denotes the number of moles of A dissolved in nc moles of C. These solubilities are functions of pressure and temperature. The ratio

VA _ RAB "'BC - RAC

is called partitioning factor for A between B and C.

(2.2.132)

When substance C is a liquid and B is its vapor, we use the term equi­librium ratio to denote the ratio

(2.2.133)

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112 MACROSCOPIC DESCRIPTION

where X~ and xt are the mole fmctions of A in the vapor and in the liquid, respectively. A similar equilibrium ratio can be defined for any I-component in a two phase N -component system

where

JC"I _ XJ - Xi'

X"I n~ et - ""N J'

L...Jj=l net

(2.2.134)

a = V,L.

All these equilibrium ratios are functions of pressure and temperature, as well as of the entire composition of the system. The overall composition can be specified by the mole fractions of the system as a whole. Thus,

n"l X 'Y - --=--

- N "I' L:j=l nj

I,j = 1,2, ... , N,

for eac.h component, where n'Y is the number of moles of the I-component in the whole system.

Under certain conditions, e.g., an ideal gas and dilute solutions, the above equilibrium ratios can be computed theoretically. In practice, however, they must be determined empirically.

With XL and Xv denoting the mole fractions of a composite system in the liquid state and in the vapor one, respectively, with

and

and since

we have

X"I tr 1 + Xv(JC"I - 1) = 1, X"IJC"I tr 1 + Xv(JC'Y - 1) = 1,

(2.2.135)

Now, at the dew point, Xv = 1, and the first equation in (2.2.135) gives

(2.2.136)

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Microscopic Description

I

tc Liquid I ~<le I .""c, ~ ___ I_~ 0 .... ).· I

I ~e~ ~-o.'o'o I

~I I

Liquid+ vapor

Dew

I point I I I ·B I I

hi I I

I

I Vapor I I " •• "1

Temperature

113

Figure 2.2.7: Schematic pressure-temperature diagram showing phase-boundary curves.

At the bubble point, Xv = 0, and the second equation in (2.2.135) gives

. (2.2.137)

In both cases, L(-y) X'Y = 1. Figure 2.2.7 presents a pressure-temperature diagram for two states of a

single chemical substance. Examining the T1-isotherm, we note that as the pressure is reduced, at point C we have only liquid, at point B both liquid and vapor coexist, while at A we have only vapor. Along T2-isotherm, we start at point C', at which the system is only in the vapor phase. As the pressure is reduced, the system moves to point B', at which we have both liquid and vapor. A further reduction in pressure will lead to point A', where only vapor exists. This change from vapor to liquid and back to vapor, is called retrograd condensation.

From Fig. 2.2.7, it follows that when a fluid, say, oil, from the high tem­perature and high pressure that prevail in a deep oil reservoir, is brought to the surface, where it is exposed to a different pressure and temperature (at­mospheric pressure and some standard atmospheric temperature), a certain

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114 MACROSCOPIC DESCRIPTION

quantity of vapor would evolve. When the vapor is continuously removed from contact with the remaining liquid, as it is formed, the process is called differential vaporization. When evolving vapor is not removed, the process is referred to as flash evaporation. In either case, when standard atmospheric conditions of pressure and temperature are reached (denoted by SC, or sub­script SC), a certain volume of vapor would result, leaving a certain quantity of residual liquid. We regard the vapor, or gas, that has been produced, as having been dissolved in the volume of liquid at the original (reservoir) pres­sure and temperature.

Denoting the residual liquid, say, oil 0, by Uo , and the gas volume by Ug ,

both measured at atmospheric conditions, we define the gas solubility as

Rg( T) _ Ug,sc o p, - U .

o,sc (2.2.138)

Le., the amount of gas dissolved in oil per unit volume of oil (sometimes denoted by the symbol Rso. This ratio is also called solution gas-oil ratio, and denoted by Rs.

An oil formation volume factor is defined as

( ) Uo(p, T) Bo p, T = Uo(SC) ' (2.2.139)

where Uo(p, T) denotes the volume of oil at the p and T conditions prevail­ing in the reservoir, and Uo(SC) denotes the oil volume under SC conditions. We recall that Uo(p, T) includes in it a certain mass of dissolved gas which will come out of solution when an oil sample pressure is reduced to the atmospheric one. Similarly, in an air water system, Uw(p, T) may include dissolved air. Both the gas solubility, R~(p, T), and the volume formation factor, Bo , are different for the two vaporization processes mentioned above, and so are the equilibrium ratios, defined above. Generally, R~(p, T) de­creases as the oil's density increases (Le., pressure increases). For a system of fixed composition, a pressure is reached at which no more gas can go into solution. This pressure is called saturation, or bubble point pressure, Ps'

Other often used definitions are the formation volume factors for gas and for water

( ) Ug(p,T) Bg p, T = Ug(SC) and ( ) Uw(p, T)

Bw p, T = Uw(SC)' (2.2.140)

We should note that the actual physical composition of a gas is different at different p, T conditions. In the definitions of B o , Bg and B w , the volumes

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A veraging Rules 115

~ Phasel

f&$($88 Phase 2

Phase 3

Figure 2.3.1: An REV occupied by three phases.

Uo , Ug and Uw at p, T, refer to a fixed mass of the component, while the corresponding volumes at SC are those occupied by the same mass at stock tank, or standard conditions.

The state diagrams are essential in conceptualizing the possible existence of a multiphase system in a porous medium.

2.3 Averaging Rules

Figure 2.3.1 shows an REV in a porous medium domain (see also Fig. 1.1.3). Typically, such a system involves either a solid phase and a single liquid phase (e.g., the case of saturated groundwater flow), or a solid phase, a gaseous phase and one or two liquid phases (e.g., the cases of air-water flow in the soil and of solid-gas-oil-water system in an oil reservoir). As stated in Sec. 1.2, our objective is to describe each phase and its behavior as a continuum that occupies the entire domain. In the previous section, we have presented the microscopic balance equations of extensive quantities. In order to transform each of these equations into one that describes the behavior of a considered extensive quantity as a continuum occupying the entire domain, we need certain rules. These rules, often referred to as averaging rules, are presented below.

The discussion will focus on the behavior of an a-phase. Hence, no special symbol will be used to indicate that properties and densities are those of that phase. However, a will remain in such symbols as Uooo SOIOO n Ol

, etc.

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116 MACROSCOPIC DESCRIPTION

2.3.1 Average of a sum

Let G l (x', t) and G2(x', t) be two quantities pertinent to a phase, and G l Ct and a:: be their corresponding intrinsic phase averages, respectively. For example, G l == eI, G2 == e2, (actually, Gi == GiCt == eiCt, i = 1,2). We use x' to denote a point within a phase, while x is used to denote the location of a point at the macroscopic level. Then

U1 f {Gl (x',t)+G2(x',t)}dU(x') oCt Juo(x)

= U1 f Gl(x',t) dU(x') + U1 f G2(x',t) dU(x'), OCt Juoa(X) OCt Juoa(x)

from which it follows that

(2.3.1)

2.3.2 Average of a product

With deviation from the average defined by (1.3.8), we have

and

u1 f Gl(X', t)G2(X', t) dU(x') OCt Juoa(x)

U1 f {GlCt(X,t)+Gl(X',t)}{G2Ct(X,t)+G2(X't)}dU(x') OCt Juoa(x)

U1 {Gl Ct(x, t)~(x, t)} f dU(x) OCt Juoa(x)

+~(x,t)U1 f G1(x',t)dU(x') OCt Juo(x)

+a::(x,t)u1 f G2(x,t)dU(x') OCt Juoa(X)

+ U \ ) f {Gl(x', t)}{G2(x', t)} dU(x'). (2.3.2) 001 x Juoa(x)

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A veraging Rules 117

Hence, employing (1.3.9), we obtain

(2.3.3)

i.e., the average of a product is equal to the sum of the product of the averages and the average of the product of the deviations, all averages being intrinsic phase ones.

2.3.3 A verage of a time derivative

Figure (1.1.3) shows an REV, Uo, of volume Uo, containing a volume Uoa of an a-phase; the symbol f3 denotes the union of all other phases present in Uo. Let e(= ea) denote the density of an extensive quantity E of the a-phase. We assume that e is differentiable with respect to time, and has no discontinuity within Uoa . By regarding Uoa as a material volume with respect to E, and applying the Transport Theorem, (2.1.49), to E within Uoa at time t, we obtain

DE f edU Dt }Uoa(t)

1 oe 1 1 E = - dU + eu·vdB + eV ·V dB, Uoa(t) ot sa,8(t) Saa(t)

(2.3.4)

where Soa = Saa + Sa;3 is the total surface bounding Uoa (see (1.1.20) and Fig. 1.1.3), u is the velocity at which the Sa;3-surface is being displaced and v(= va) denotes the outward (to the a-phase) normal unit vector on Sa;3. We recall that because Soa is regarded here as a material surface, we replace VE·v by u·v on Sa;3

On the other hand, we may focus our attention on the material rate of change of E in the a-phase within Uo as a whole. We may express this rate by

(2.3.5)

where la is defined by (1.1.17). In words, (2.3.5) states that the material rate of change of the total

quantity of E in the a-phase within Uo is equal to the rate of change of E instantaneously within Uo, plus the net efflux of E leaving Uo through its

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118 MACROSCOPIC DESCRIPTION

boundary surface SoC = SO/O/ + S{3{3); v is the outward normal unit vector on So.

Since e vanishes outside the phase, (2.3.5) may be rewritten in the form

~E [ edU=[j[j [ edU+ [ eVE.vdB. t JUoa t JUoa JSaa

(2.3.6)

By comparing (2.3.4) with (2.3.6), we obtain

~ [ e dU = [ [je dU + [ eu·v dB. [jt }Uoa(t) }Uoa(t) [jt }Sa/3(t)

(2.3.7)

Actually, (2.3.7) may be considered as an extension of Leibnitz' rule (of taking the derivative of an integral with respect to a parameter), with the surface integral representing the effect of the rate of displacement of the boundary of integration, SO/{3.

Using the definition (1.3.3) for an intrinsic phase average, we obtain from (2.3.7)

:t(UoO/-ex) = UoO/(~:O/) + ha/3 eu·vdB. (2.3.8)

After dividing by Uo , we obtain

[j()-ex ([je 0/) ~{3 at = ()O/ [jt + eu·v :E0/{3, (2.3.9)

where

~(3 1i eu·v :E0/{3 = U eu·v dB, o sa/3

) dB,

and :E0/{3, defined by (1.1.21), is the specific area of the SO/{3-surface. We recall that () == ()cx and v == vO/.

By (1.3.6), equation (2.3.9) becomes

Oe Fe ~(3 at = at + eu·v :E0/{3. (2.3.10)

Equations (2.3.9) and (2.3.10) relate the time derivative of an average value to the average of the time derivative.

As an example of special interest, let E be the volume, U, of a phase. Then e = 1, ex = 1 and e = (). From (2.3.9), we obtain

[j() 1 1 "......c<{3 -[j = -u u·v dB == U v :E0/{3,

t 0 Sa/3 (2.3.11)

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A veraging Rules

a - phase

dSai -

Figure 2.3.2: Definition sketch for (2.3.15).

119

B

where U/J = U·V is the speed of displacement of a point on Sa(3 with respect to a fixed coordinate system, and :Ea(3 = Sa(3/Uo.

In view of (1.3.8) and (1.3.9), another form of (2.3.9) is

(2.3.12)

When the extensive quantity E is such that we have the densities ea in Uoa and e(3 in Uo(3, we employ (1.3.7) to obtain from (2.3.10)

(2.3.13)

where [e]a(3 == ela - el(3 denotes the jump in e across Sa(3. Note that in (2.3.13), the symbol e does not stand for ea , as up to (2.3.12).

2.3.4 Average of a spatial derivative

Figure 2.3.2 shows an oriented surface element dSa of Sa(3 in a Cartesian coordinate system, Xi (i = 1,2,3), with

(2.3.14)

or, with v == Va

(2.3.15)

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120 MACROSCOPIC DESCRIPTION

By applying Gauss' theorem (2.1.42), valid for any tensorial quantity, Gjkl ... , of an a-phase, to the (interconnected) domain Uoex occupied by that phase, we obtain

aG ex _ UoO jkl ... aXi

f Gjkl ... COS(v, lXi) dS + f Gjkl ... COS(v, lXi) dS, JSa~ JSaa

(2.3.16)

where we recall that the total surface area bounding Uoex is made up of two parts: Sexex and Sex(3 (Fig. 1.1.3).

From the definition (1.3.3) of the intrinsic phase average, we obtain

a 1 a --ex -a Gjkl. .. dU = -a (UoOGjkl...) Xi Uoa Xi

aG j kl... ex --ex ao ( ) uoo a + Gjkl ... Uo-a ' 2.3.17

Xi Xi

However, as shown in Fig. 2.3.3, we have

a 1 - Gjkl... dU aXi Uoa(x)

= lim _1_ ( f Gjkl dU - f Gjkl dU) .6,Xj-+O ~Xi JUoa(X+dX) ... Juoa(X) ...

= .6,lim A 1 {f Gjkl ... dU + f Gjkl... dU

Xj-+O uXi JUoa ,2 JUoa ,3

-(1 Gjkl ... dU + 1 Gjkl ... dU)} Uoa,l Uoa ,2

= }i~ ~1 . (1 Gjkl... dU -1 Gjkl. .. dU) x, 0 x, Uoa ,3 Uoa,l

- ~x·- G· dS· G- dS-1 (1 1 ) - '~Xi (along 4,7,6) Jkl... ,+ (along 4,5,6) Jkl... ,

= f Gjkl ... COS(v, lXi) dS. (2.3.18) JSaa

The replacement of Gjkl. .. dUex by ilxiGjkl. .. dSi is valid only in the limit as ~Xi --+ O.

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A veraging Rules 121

6

2

+ ..... + :r:i

4

Figure 2.3.3: Definition sketch for equa.tion (2.3.18).

One should recall that as we shift our attention from a domain Uo cen­tered at x to the one centered at x + dx, its (say, spherical) shape is main­tained.

From (2.3.16) through (2.3.18), we obtain

a 1 - a (--a aG jkl... a -a Gjkl... dU = -a UoOGjkl...) = UoO a

Xi Uoa Xi Xi

- f Gjkl. .. COS(II, lXi) dB. (2.3.19) JSaf3

Hence, dividing (2.3.19) by Uo , we obtain

OaGjkl ... a _ a 0-'-0' '. v~(3 a - -a GJkl... + GJkl ... t ~O'(3,

Xi Xi (2.3.20)

where ~O'(3 = BO'(3/Uo, Gjkl. .. stands for a tensorial quantity of any rank (of the a-phase), and

r AJi(3 1 1 Gjkl..Yi ~O'(3 = if Gjkl. .. COS(II, lXi) dB.

o Saf3 (2.3.21)

The averaging rule (2.3.20) can also be derived by making use of Gauss' theorem (2.1.44), rewritten here in the form

f a~jkl. .. dU = f Gjkl..YO'i dB + f Gjkl..YO'i dB. JUoa Xi JSaa JSaf3

(2.3.22)

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122

We note that

f Gjkl..Yai dB JSaa

MACROSCOPIC DESCRIPTION

= f Gjkl. .. laVai dB = 88 f Gjkl ... la dU JSo Xi JUo

8(~Uoa) = Uo 8(8a;;;c.a) , (2.3.23) 8Xi 8Xi

where la is defined by (1.1.17). Equation (2.3.20) is obtained by combining (2.3.22) and (2.3.23).

Another form of (2.3.20) is obtained by employing (1.3.6)

(2.3.24)

Equations (2.3.20) and (2.3.24) relate the average of a spatial derivative (or a gradient) to the spatial derivative of an average. Equations (2.3.20) and (2.3.24) may also be written in the symbolic forms

(2.3.25)

and

(2.3.26)

Of special interest is the case E = U, i.e., G = 1, G = 8. Then (2.3.20) yields

or

By combining (2.3.25) with (2.3.27), we obtain

,.-<x(3 -a -- -a 0

8\1G == \1Ga = 8\1G + Gy ~a(3, G == Ga·

(2.3.27)

(2.3.28)

If we replace \1 G by \1·G, where G is a tensor of any rank (~ 1), equations (2.3.25) and (2.3.26) become

-a ~(3 \1·(()G ) + G·y ~a(3,

_ ~(3

\1·Ga + G·y ~a(3.

(2.3.29)

(2.3.30)

Another relationship between the average of a gradient and the gradient of an average is presented in Subs. 2.3.5.

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A veraging Rules 123

Again, like (2.3.7), equation (2.3.26), or any of its equivalent forms, may be considered as an extension of Leibnitz rule, this time for a spatial deriva­tive.

It may be of interest to note that from (2.3.18), it follows that

(2.3.31)

which, upon employing (1.3.6), becomes

(2.3.32)

For G = 1, equation (2.3.31) yields

V() = ~ f 1.1 dS, o lSaa

(2.3.33)

to be compared with (2.3.27), noting that is +s 1.1 dS = o. aa a/3

Another case of special interest is G = Xj. Then, by (2.3.31)

f XjVi dS = [J[J (xoOljUoa,). lSaa Xi

(2.3.34)

Also, by (2.3.33)

(2.3.35)

Hence

1 0 dS [JXoOlj ( ) [JUOOI XjVi = U OOl -£:)-- + XoOlj - Xoj -£:)-,

Saa uXi uXi (2.3.36)

in which, contrary to the usual definition of a deviation by (1.3.8), here x = x - Xo and not x - xoOl.We may also write (2.3.36) in the form

1 1 0 [JxoOlj ( ) 1 [J() -u XjVi dS = ~ + XoOlj - Xoj -()~. 001 Saa uX~ uX~

(2.3.37)

Another form of (2.3.37) is obtained from (1.1.19). Indeed, by (1.1.19)

XoOlj = Xoj + 'YOIXj/(), (2.3.38)

which, upon substitution in (2.3.37), becomes

(2.3.39)

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124 MACROSCOPIC DESCRIPTION

Note that {)xoj/aXi = Oji and recall that v == VOl and (} == (j0i'

For a quantity G such that G == GOI in UoOl,and G == G(3 in Uo(3, with a jump [G]OI,(3(== GIOI - GI(3) at all points of SOl(3, e.g., G is the mass density of an a-phase and of a ,a-phase, (2.3.30) becomes

(2.3.40)

in which G is the volume average of G, as defined by (1.3.7).

2.3.5 Average of a spatial derivative of a scalar satisfying \j2G = 0

The evaluation of the surface integrals in (2.3.25), (2.3.26) and (2.3.28), requires information on the configuration of the SOI(3-surface and on the dis­tribution of G( == GOI ) on it. Often this information is not available. However, sometimes information is available on the derivative of G normal to the SOI(3-surface. To make use of such information, let us develop a second form of the averaging rule for a spatial derivative, this time making use of the known normal derivative at the SOI(3-interphase boundary. We continue omitting the a subscript that indicates that we are considering an a-phase (Le., G == GOI ,

v == VOl' (j == (j0l' etc.). To this end, we make use of Green's vector theorem (2.1.56), for the

domain UOOI surrounded by the surface SOOl' into which we insert aj == Xj and b = G. We obtain (Bachmat and Bear, 1986)

Since aXoj/aXi = 0 within the REV, aXj/aXi == Oji (=Kronecker delta), aXj/aXi = Oji, a2Xj/aXiaXi = aOj;jaxi = 0, and ViOij = Vj, equation (2.3.41) reduces to

f GVj dB = f aaG XjVi dB - f"Va2: Xj dUo JSoa JSoa Xi JUoa ' Xi Xi (2.3.42)

We shall limit the following discussion to a scalar function G = G(x, t) that satisfies the following two conditions:

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A veraging Rules 125

(a) G attains no maximum or minimum value within UOOil i.e., it varies monotonously within the latter. Under this condition (e.g., Morse and Feshbach, 1953)

(b)

within Uocx • (2.3.43)

A further discussion on the conditions that justify (2.3.43) is presented in Subs. 2.6.3.

(2.3.44)

where we have introduced the approximation

(2.3.45)

i.e., we have assumed that the average of the gradient of G on the a - -a portion of the outer surface, So, of the REV is equal to the gradient of the average of G over the domain Uocx of the a-phase within the REV.

With (2.3.43) and (2.3.44), equation (2.3.42) reduces to

(2.3.46)

By applying Gauss' theorem (2.1.42) to the l.h.s. of (2.3.46), we obtain

(2.3.47)

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126 MACROSCOPIC DESCRIPTION

Then, equation (2.3.46) reduces to

aG a aGa 1 aG !lx. = ~T;ij + -V [ Xj~lIi dS, v J vX~ oa JSo.fJ vX~

(2.3.48)

where

T;ij = V1 [ lIiXj dS. (2.3.49) oa Jso.o.

The subscript a was used in T;ij, to emphasize that this coefficient de­pends only on the configuration of the a-phase within Uo•

We have thus achieved the goal of deriving an averaging rule for a spatial derivative that requires information in the form of the second derivative of G on the Sa,B-surface. Equation (2.3.48) is another averaging rule for a spatial derivative, to be compared with (2.3.28). Note that in (2.3.39) we have another interpretation of T;ij.

The coefficient T;ij is a fundamental tensorial property of the configura­tion of the a-phase within the REV. A detailed discussion of this coefficient is given in Subs. 2.3.6. A general discussion on second rank tensors and their properties is presented in Subs. 2.6.5.

We shall consider three types of boundary conditions on the Sa,B-surface that partly surrounds Uoa • The use of the subscript a is resumed as both the a- and the ,8-phases will be considered.

CASE A. The condition on Sa,B is

or aGa ~lIai = 0 vXi

By inserting (2.3.50) into (2.3.48), we obtain

aGa a _ aGa aT* aXj - aXi aij·

on (2.3.50)

(2.3.51)

We have thus related, the average of a gradient to the gradient of an average for this particular case.

CASE A, with Ga representing concentration of a component, corre­sponds, for example, to mass transport by molecular diffusion in a fluid occupying the entire void space, with no diffusion into the solid and without adsorption; a and ,8 denote the fluid and the solid phases, respectively (see Subs. 2.6.3).

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A veraging Rules 127

CASE B. Only two phases, a and (3, say a fluid and a solid, occupy the entire space. The conditions on SOlf3 are:

and

(Af3VGf3'Vf3)I (3 side

of SOIf3

= Gf31 (3 side

of SOIf3

(2.3.52)

(2.3.53)

where AOI and Af3 are constant coefficients that depend on the nature of G and on the nature of the a and (3 phases, respectively, and Gf3 denotes the value of G in Uof3.

This case corresponds, for example, to the case of heat transfer in a porous medium by conduction only, with GOI and Gf3 denoting temperatures and AOI and Af3 denoting the thermal conductivities of the a and (3 phases, respectively.

By applying (2.3.48), first to the a-phase, and multiplying the equation by AOI , then to the (3-phase, and multiplying the equation by Af3, and then adding the two resulting equations, making use of condition (2.3.52), we obtain

Now, by (2.3.49)

[ :tjVOIi dB lso

whence we have a relationship between T~ij and T~ij in the form

801T~ji + 8f3T~ji = Oji.

(2.3.55)

(2.3.56)

Next, we write (2.3.25) twice, once for the a-phase and once for the (3-phase, add the two equations and employ condition (2.3.53). We obtain

~OI --f3 ...",.---cx -f3 801 UGOI + () oGf3 = o801 GOI + o(}f3Gf3

ox . f3 ox· ox . ox . J J J J

(2.3.57)

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128 MACROSCOPIC DESCRIPTION

Finally, multiplying (2.3.57) by A(3 and subtracting the result from (2.3.54), yields

=

(2.3.58)

where we have employed (2.3.56), together with the relationship OOl+O(3 = 1. In the particular case of A(3 = 0, equation (2.3.58) reduces to (2.3.51),

corresponding to CASE A. The same holds when GO/ 0/ = G/. We shall later see that the fact that the averaged G's are equal, means that on the average (but not locally!) there is an exchange between the phases across

-0/ -(3 . SO/(3. For GOI = G(3 ,we obtam from (2.3.57)

{)G {)GOI 0/

{)x· -~' J J

(2.3.59)

where the l.h.s. is the volume average defined by (1.3.7).

CASE C. A single fluid occupies the entire void space. This case is similar to CASE A, except that adsorption of a solute on the solid surface does take place. We assume that the rate of adsorption at a point on SOI(3 is proportional to the difference GOIO/ - GO/Is . Thus, {)GOI/{)xils "10, but

ex{3 ex{3

or on

(2.3.60)

where ~ is a microscopic elementary distance between the a - f3 surface and the interior of the a - phase occupying UOO/' For example, we may view it as proportional to the hydraulic radius ~OI = UoO//SO/(3, such that ~ = ~OI/CO/' where COl is a coefficient that varies with the orientation of elements of the SO/(3 - surface.

By inserting (2.3.60) into (2.3.48), we obtain

(2.3.61)

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A veraging Rules 129

where

(2.3.62)

is another macroscopic coefficient associated with the configuration of the Sa,B-surface within the REV. We note that in (2.3.61), we have introduced

~,B . another macroscopic state variable, viz., Ga ,which is the average of Ga on the Sa,B-surface. Thus, the average of the gradient of Ga depends also on

~,B -a the difference G a - G a

In Subs. 2.6.3, we shall consider adsorption of a component of a fluid phase on a solid surface.

2.3.6 The coefficient T~

The coefficient T~, defined by (2.3.49), expresses the total static moment of the oriented elementary surfaces comprising the Sa,B-surface, with respect to planes passing through the centroid of the REV, per unit volume of the a­phase within Uo• To obtain an estimate of the magnitude of the components T~ij' consider a spherical REV of radius R. Then (2.3.49) can be rewritten in the form

(2.3.63)

where (}~ (= Saa / So) denotes the fraction of the a - a-surface in So and ,---;aa VaWaj represents the average of VaWaj on the Saa-surface.

The term VaWaj is a symmetric second rank tensor. Hence, v;;;v;;a, which is a linear combination of VaWaj, is also symmetric. Therefore, there exists at least one set of three mutually orthogonal planes of symmetry for T;ij, and three principal axes normal to these planes. In the coordinate

system of the principal axes, v;;;v;;a can be expressed in the form

(2.3.64)

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130 MACROSCOPIC DESCRIPTION

a "........-era "........-era where al, a2 and a3 are the principal values of ~ ,i.e., V;l ,V;2 ,

"........-era and V;3 ,respectively. Hence

for i = 1,2,3.

v . t' d" h ,,----.-A a ror an zso ropzc porous me zum, WIt respect to VaWaj ,al = a2

a3 = a, and (2.3.64) reduces to

(2.3.65)

,,---;eta Now, 2:(i) VaWai = a 2:(i) 8ii = 3a. On the other hand, by definition,

,,---;eta 2:(i) VaWai = 1. Hence

(2.3.66)

By inserting this result into (2.3.63), we obtain for an isotropic porous medium

T * O~ 1: aij = Oa Vij· (2.3.67)

Certain porous media satisfy the condition O~ < Oa (Santalo, 1976). Then

o < T~ii < 1 for all i (no summation on i).

However, in the general case of an anisotropic medium and i,j principal directions

aS ,....-aa "........-era "........-era

T~ij = 3 a: (V;l 8li 8lj + V;2 82i82j + V;3 83i83j) . (2.3.68)

Hence, T~ii < 1 for

(2.3.69)

In order to clarify the physical meaning of T~, consider an REV in the form of a cubical box with sides b parallel to the Cartesian x, y, z-axes. Let the void space be made up of tubes, each having a constant cross-section, that connect opposite sides of the box. Each tube may be tortuous, so that its length may be longer than the side b. For such case

( Nx Ny NZ)

O~ = L Axm + L Aym + L Azm /3b2 ,

m=l m=l m=l (2.3.70)

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A veraging Rules 131

(2.3.71)

o~ ~ (L~~l Axm + L~~l Aym + L~~l Azm )b < ~ 00/ 3 L~~l Axmbxm + L~~l Aymbym + L~~l Azmbzm - f. - 3'

(2.3.72)

where Axm , Aym , Azm , are the cross-sectional areas of tubes intersecting each pair of opposite faces normal to the x, y and z axes, respectively, N x , Ny and Nz denote the corresponding numbers of these tubes, bxm , bym , and bzm are the lengths of such tubes, and f. is a small error due to counting intersections more than once. For tortuous tubes, bxm > 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 -;;;;V;;;O/ 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.

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

= [ e dUO/ + [ e\l·yE dUO/ JUoa JUoa

= [ {(e) + e\l.yE} dUO/. JUoa (2.3.74)

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132 MACROSCOPIC DESCRIPTION

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

(Uo~ea) = Uoact" + eV.yEa ),

or uoaia + Uoaea = Uoa(e:" + eV.yEa ). (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 == ea , 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

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Balance Equations 133

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 ae dU = Uo Uoo. at

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

or, in view of (1.3.6)

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

(2.4.2)

(2.4.3)

(2.4.4)

By employing rules (2.3.9) and (2.3.25), we rewrite (2.4.4) in the form

a O..".a ~(3 - e - eu·v E (3 at a

where

or

where:

{)oea at (a)

~(3 1 1 ( .. ) Ea(3 = U ( .. ) dB o sa.{3

-V·O(eV + jEut

(b)

~(3 ·EU " -J ·V LJ a (3

(d) +

and

_---;(){(3

~(V - u)·v Ea(3

(c) (2.4.6)

o rEa p , (e)

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

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134 MACROSCOPIC DESCRIPTION

(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/ + eV 0/ + 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:

--00/

• 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

__ --------~~~&(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.

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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< -00<-0< Oe;O<V·Y - V.(ey +jEU )

,,-{ e-(-Y-_-U-)-+-j-E--U-}-.;(3 ~0<(3 + 0 p-r-EO< . (2.4.8)

in which the l.h.s. expresses the material rate of change of e;, the first term on the r.h.s. expresses the net influx of E through the boundaries of a moving REV by advection (see (2.1.46)), and the second term on the r.h.s. expresses the net influx by dispersion and diffusion.

Let us develop macroscopic balance equations for the particular extensive quantities for which microscopic equations are presented in Subs. 2.2.2.

We shall continue to omit the subscript that denotes the a-phase.

2.4.2 Mass balance of a phase

Let E represent the mass, m, of an a phase, (see Subs. 2.2.2(a)). Then, yE == ym, e == pm == p, rm = 0 (because of the principle of mass conservation) and jm( == jmU) = p(ym _ Y) is the diffusive mass flux of the phase (= molecular diffusion).

By averaging (2.2.24), or from (2.4.7), we obtain the differential macro­scopic mass balance equation

BOP -0< ----oQ' -0< ,""" _______ ;0<(3 at = - V·O(py + pY + jm ) - {p(Y - u) + jm}·v ~0<(3. (2.4.9)

By (2.3.3)

we may rewrite (2.4.9), in terms of ym, in the form

(2.4.10)

If there is no transfer of mass across So<(3, the So<(3-surface is a material surface for the mass of the phase, and the surface integrals on the r.h.s. of (2.4.9) and (2.4.10) vanish.

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136 MACROSCOPIC DESCRIPTION

If we assume that the advective flux is much larger than the sum of the dispersive flux and the diffusive one, i.e.

(see discussion in Subs. 4.1.1), and if Sex/3 is a material surface with respect to the phase, (2.4.9) reduces to

where

80pex _ex--ex -ex -- = -\l·Op Y = -\l.p q, 8t

(2.4.11)

(2.4.12)

is the specific discharge of the a-phase. Equation (2.4.11) is commonly used as the basic (macroscopic) differential mass balance equation of an a-phase. For a single fluid phase that fills the entire void space, we replace the symbol a by f and 0 by the porosity, n, obtaining

8np! !--! ! ---at = -\l·np Y = -\l.p q, (2.4.13)

where q = nY! is the specific discharge of a fluid that fills the entire void space.

By neglecting the dispersive mass flux in (2.4.10), and assuming no mass transfer across the Sex/3-surface, the mass balance equation takes another fundamental form

80pex = _ \l.o~ymex 8t

(2.4.14)

This form is preferable to (2.4.11), whenever the mass-weighted specific discharge, qm (== oym ex ), is used. Only by neglecting the molecular diffusion of the mass, we obtain qm = q.

The problem of mass transport is further discussed in Chaps. 4 and 5.

2.4.3 Volume balance of a phase

For E = U, e = 1, yE = Y, jUU = (Y - Y) = 0, the source function is given by pru == \l·Y, obtained from (2.4.1), with e = 1. Then, (2.4.7) becomes

80 ~ r rCY./3 --ex - = -\l·OY - (Y - u)·v ~ /3 + O\l·Y m ex, (2.4.15)

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Balance Equations 137

or, in view of (2.3.11)

_(){ __ (){ ,-<x{3 '\7.(}V (== V·q) = OV·V - V·v E(){{3. (2.4.16)

For a flow of a phase that is microscopically isochoric, and an S(){p-surface that is a material surface for the phase, (2.4.15) reduces to

ao -(){ at = -V·OV == -V·q, (2.4.17)

which is commonly used to describe the flow of a homogeneous fluid phase in multiphase flow in a porous medium. It is often referred to as the continuity equation of a phase. We note that (2.4.17) can be obtained from (2.4.11) by assuming lP = const., or pOI = o. If, in addition, the microscopic boundary of the phase is such that ao fat = 0, equation (2.4.17) reduces to

or V·q = o. (2.4.18)

It is of interest to note that V·V = 0 does not necessarily imply that V·V = O.

Equation (2.4.18) is commonly used as the continuity equation for the flow of a single component, incompressible, homogeneous fluid phase that occupies the entire void space of a rigid porous medium. The exact condi­tions that lead to (2.4.18) are discussed in Subs. 3.3.2 and 4.1.2. A flow that satisfies (2.4.18) is referred to as macroscopically isochoric flow.

2.4.4 Mass balance equation for a component of a phase

Let E be the mass m"l of a ,-component of a phase. In this case, e == pm,,! = p"l,jE =jm"! ==j"l. By averaging (2.2.56), or directly from (2.4.7), we obtain

aop at ______________ A{3

r- ~

{p"l(V - u) + P}·v E(){{3 + (}prm "! • (2.4.19)

The last term on the r.h.s. of (2.4.19) expresses the rate of production of the ,-component in the phase, per unit volume of porous medium. Equation (2.4.19) is called the macroscopic differential mass balance of a component of a phase in a multiphase system; it is historically also called the equation

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138 MACROSCOPIC DESCRIPTION

of hydrodynamic dispersion. It takes into account the production of the con­sidered component within the phase (last term on the r.h.s.) and transport (by advection and molecular diffusion) of the component across the (possi-

~

bly moving) interphase boundary. The term plV expresses the dispersive flux of the mass of the ,-component in the a-phase. It is further discussed in Subs. 2.6.4. We note that the total flux of the ,-component is made up

-----cOl of an advective flux (= pIOl'\F), a dispersive flux (= plV ) and a diffusive one (= ?). We also note that E(;)? "=I O. By adding (2.4.19) over all ,-components, we obtain (2.4.9).

The problem of hydrodynamic dispersion is considered in Chap. 6.

2.4.5 Balance equation for the linear momentum of a phase

Let E be the linear momentum, M = m vm, of a phase. For this case, e = pvm. The diffusive flux of momentum, expressed by (2.1.38), is given by jMm == pvm(vM - ym) = -17, where 17 is the stress. The rate of production of momentum per unit mass of the phase is rM = F, i.e., the intensity of the external body force acting on the phase. The microscopic equation to be averaged is (2.2.26), rewritten for an a-phase. By averaging this equation, or from (2.4.7), we obtain

{)(Jpym Ol

----'-:::-- = {)t

V.(J{pymOlvmOl + (;Vmrvm a _ qot} ~--------------~~p {pym(vm - U) - 17 }.V ~olP + (JpFOl . (2.4.20)

Since

(2.4.21)

by (2.3.3) (2.4.22)

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Balance Equations 139

equation (2.4.20) can also be written in the form

(2.4.23)

When combined with the mass balance equation (2.4.10), equation (2.4.23) becomes

()~ avm a + ~() fJym a = _ V.() {fJym a Vm a + ~ym ym a at at

+fJymyma} _ ()(~vma + fJym al.vvma

-----------------~~ _{pym(vm - u) - u}·11 ~a~ + v·()ua + ()~, (2.4.24)

where ( .. f denotes the transpose of ( .. ). In indicial notation, the last equa­tion takes the form

(2.4.25)

In the particular case of p=const., an Sa~-surface that is a material sur­face with respect to the a-phase, and a flow that is macroscopically uniform, Le., avra jaXj = 0, equation (2.4.24) reduces to

We note that the first term on the r.h.s. of (2.4.26) is nonlinear with respect to the velocity vector.

By assuming that the mass and momentum dispersive fluxes appearing in (2.4.24) are much smaller than their advective counterparts (see Subs.

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140 MACROSCOPIC DESCRIPTION

3.3.3), i.e.

and assuming that the SOl/3-surface is a material surface with respect to the mass of the a-phase, i.e., p(ym - u)·" == 0, equation (2.4.24) reduces to

aym Ol 8/?,-- = at

8."Q1-0I ~ ~/3 P ym . vym + cr'" 1:01/3

+ V ·8ua + 8""jj'P ,

which can also be written in the form

fJ7j0l Dm ym 01 ~/3-0I P Dt = V ·8crOl + cr·" 1:01/3 + 8pF

(2.4.27)

(2.4.28)

2.4.6 Heat balance for a phase and for a saturated porous medium

We shall deal here only with the case of a single fluid phase that occupies the entire void space of a porous medium domain (see Chap. 7 for further discussion of heat transport). For such fluid phase, the heat balance equation at the microscopic level is (2.2.44). In it, we shall neglect the term T : vym, which expresses the rate of internal energy increase (per unit volume) by viscous dissipation, as it is usually much smaller than the other terms. The ( dimensionless) Brinkman number, defined by

Br _ JLcCVc/ d c )2 - Ac( dT)e/ d~

(2.4.29)

gives the ratio between the heat produced by viscous dissipation and that supplied by conduction. In it, de is a length characterizing the void space. The characteristic hydraulic radius, (d, )e, or the term Jke/ne may be used here for de (see discussion in Subs. 3.3.3 for the interpretation of the char­acteristic values indicated by subscript c).

Here we shall assume Br < 1. This approximation will be applied when rewriting (2.2.44) both for the fluid phase and for the solid one. For the sake of brevity, we shall omit the heat source terms in both phases. If necessary, they may be introduced at a later stage.

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Balance Equations 141

With the above simplifications, the microscopic heat balance equation (2.2.44) for the fluid phase takes the form

V·(PfCVTf Vj + j7)

aPfl m DrCv Tf aTf vf V·Vf + TfPf----nt, (2.4.30)

where subscript f denotes the fluid that fills the entire void space, and all other symbols are defined in Subs. 2.2.2( c). The last term on the r.h.s., with Cv denoting the specific heat of the fluid at constant volume, vanishes when Cv = 0, or when Cv = const. The last case underlies the development in the present subsection. Although assumed here that Cv is a constant, we have written the fluid's balance equation in the form (2.4.30) to emphasize the interpretation of PfCVTf as the heat density.

By taking a phase average of (2.4.30), with Cv = const., we obtain

V·(PfCvTfVj +j7)

U1 [ PfCVTf(Vj - u)·v f dB o lSsf

Since the (microscopic) solid-fluid interface, Ss" is a material surface with respect to the fluid's mass, the second term on the r.h.s. of (2.4.31) vanishes.

At this point, we introduce the following simplifying assumptions (see Subs. 3.3.3):

IPiTt' I ~ Ip/T/I, --f f-f so that PfTf ~ Pf Tf . (2.4.32)

--~~-----f ---f T apf I v·vm '" r faPf I v.vm f

f aTf Vf f - f aTf Vf f· (2.4.33)

PfTfV.Vjf ~ p/T/V.Vjf, (2.4.34)

which is based on the assumption that the absolute value of the average of a product of two or three deviations is much smaller than that of the product of the corresponding averages.

(2.4.35)

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142 MACROSCOPIC DESCRIPTION

i.e., the diffusive mass flux (= molecular diffusion), and hence its contribu­tion to the total heat flux discussed below, is negligible, and

-f ______ f s

IV·nVf 1 ~1Vj"lIf I~fs.

With these and earlier assumptions, (2.4.31) reduces to

where, following (2.2.74) and (2.2.76)

and f3r = --.!.. 8Pf I Pf 8Tf PI

(2.4.36)

(2.4.37)

are, respectively, the coefficients of fluid compressibility at constant tem­perature, and of thermal (volumetric) expansion at constant pressure, and

o 0 f (PfCVTf)V f expresses the flux of heat due to thermal dispersion. We note that the total heat flux is made up of an advective, a dispersive and a diffu­sive (= conductive) heat fluxes.

For a thermo-elastic isotropic solid phase, denoted by the subscript s, the microscopic heat balance equation is given by

8psCsTs ( .H) 8cs 8t = -v· PsCsTsVs +Js - Ts'l} 8t ' (2.4.38)

where 'I} is a coefficient defined for an isotropic solid by

see (2.2.98), Cs is the solid's specific heat at constant solid strain, 0' s is the stress in the solid, and Cs is the solid's dilatation. Both Cs and 'I} are assumed to be constant.

By averaging (2.4.38), and introducing the following simplifying assump­tions (see Subs. 3.3.3):

o s

Ip/CsT/1 ~ IPs(CsTs) I,

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Balance Equations

--s ( )

-S 8cs 1- n Ts TJ 8t '

8(1 - n)£:;s 8t

and

143

noting that Ssj is a material surface with respect to the solid's mass, i.e., (Vs - u)·vs == 0, we obtain

8 ( )-S -s _ ( ){-SC -s-V s -Hs } 8t 1-nps CsTs --V·1-n Ps sTs s +Js

_~ L -H. dB _ T s 8(1- n)£:;s U Js Vs s TJ 8t

o Sfs (2.4.39)

We note that in (2.4.37) and (2.4.39), we have

d -H -H an Js 'Vs = -Jj 'Vj

at every point on Ssj. Equations (2.4.37) and (2.4.39) are the two macroscopic heat balance

equations for the fluid and for the solid phases, respectively. In order to express these equations in terms of average temperatures as the only state variables, we have to introduce appropriate expressions for the macroscopic

conductive fluxes, jI/j and j¥"s, as well as for the surface integrals that express the (average) rate of exchange of heat between the phases. These topics are discussed in Subs. 2.6.3.

At the microscopic level, we assume, on the basis of thermodynamic con­siderations, a condition of no-jump in the temperatures of the fluid and the solid at their common boundary, i.e., [Tjj,s = O. Actually, this assumption is not always justified; under certain conditions, the interpretation of the be­havior at a microscopic boundary is as if there exists a 'skin' or a 'boundary layer' to the transfer of heat, such that even at the microscopic level, we may not employ the no-jump condition for the temperature.

Even if we do invoke a condition of no-jump in temperatures at the microscopic interfaces, this does not necessarily imply the equality of the macroscopic temperatures of the two phases at a point. Thus, when Ts s f= T/, the two balance equations have to be solved simultaneously because they are linked by the terms that express the exchange of heat between them. This is, for example, the case when large solid blocks are surrounded

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144 MACROSCOPIC DESCRIPTION

by relatively narrow fluid filled fractures. Actually, we also have p/, vmf f

and rsS as additional variables, which means that we need more balance equations, (see, for example, the discussion in Chap. 4 on deformable porous media).

Under the condition of T/ =f T/, heat is transferred from the phase having a higher temperature to the other phase. This exchange is expressed by the surface integrals in (2.4.37) and (2.4.39). Very often, this rate of transfer is expressed as

(2.4.40)

where a~ is referred to as a heat transfer coefficient The concept of a heat transfer coefficient that is a constant is question­

able, as it may depend on the rate at which phase temperatures at the microscopic level deviate from their average values. In a fluid phase, this may also strongly depend on the fluid's velocity.

Often, because solid grains, or blocks, are relatively small and the velocity of the fluid in the void space is small, the two phases maintain thermal equilibrium, or approximately so, i.e., T/ = T/ = T. Then, by adding (2.4.37) and (2.4.39), we obtain

a {)t {np/Cv + (1- n)p/Cs}T

f -f -s 0 of = -V·{npf CVTVf + (1- n)psSCsTVs + n(pfCvT)Vf }

"Hf "Hs {f3Tf -f a(l- n)c/} -V.{nJf + (1- n)Js } - f3p V·nVf + 'fJ at T,

(2.4.41)

where we have used the equality of heat flux at points on the fluid-solid boundary, Sf s'

For this case, of T/ = T/ = T, the total heat flux njljf + (1- n)j¥S is expressed in terms of T by (2.6.122).

The topic of heat transport is further discussed in Chap. 7.

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Balance Equations 145

2.4.7 Mass balance in a fractured porous medium

A fractured porous medium has already been introduced in Subs. 1.2.3 (and Fig. 1.2.4) and 1.5.1. In the latter, we have also introduced the so-called double porosity model, in which we visualize the void space in the fractures as a continuum (occupied by one or more fluids), while the void space within the blocks is regarded as another contimuum that is occupied by the same fluid, or fluids. Each of the three overlapping continua: the solid matrix, the void space of the fractures and that of the blocks, occupies the entire considered domain. The two void-space continua may exchange fluid (or fluids) mass between them at every (macroscopic) point within the considered domain. The transport of other extensive quantities, e.g., heat, may also take place within each of the three continua, with possible exchange between them. We often refer to the fluid in the fractures and to the (same) fluid in the void space of the blocks as two 'apparent phases'.

In the present subsection, we shall consider the mass balance of a single fluid that occupies the void space of both the fractures and the porous blocks, as well as the balance of the mass of a component present in that fluid. We shall employ subscripts s, fr and pb to denote the solid, the fluid in the fractures and that within the porous blocks, respectively. Needless to add that we assume that a common REV can be found for the network of fractures and for the porous medium in the blocks.

The microscopic mass balance equation is (2.2.24). Let us rewrite this equation, once for the fluid in the fractures (subscript fr) and once for the fluid in the void space of the porous blocks (subscript pb), in the form

aPjr \7 (V .m) at = - . Pjr jr + Jjr , (2.4.42)

aPpb _ \7 (V .m) 7ft - - . Ppb pb + Jpb . (2.4.43)

Since Sjr,pb = Sjr,! + Sjr,s, the averaged mass balance for the fluid within the fractures is given by

_jr--jr 0 0 jr -. -jr \7.0jr(Pjr Vjr +PjrVjr +Jjr )

U1 [ {Pjr(Vjr-u)+jir}·VjrdS o 18fT ,f

U1 r {Pjr(Vjr-u)+jjr}.Vjr dS, (2.4.44)

o 18fT ,s

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146 MACROSCOPIC DESCRIPTION

where jm == jmU, 0fr( = Uofr/Uo) denotes the volumetric fraction of volume of fractures within 0 the REV, u denotes the speed of displacement of an Sfr,s-surface, and ( ) denotes here the deviation of ( )fr from its intrinsic phase average over the REV. The last term in (2.4.44) vanishes because the fracture-solid interface, Sfr,s is a material surface with respect to the fluid's mass. The remaining surface integral, over the fracture-fluid surface in the void space, Sfr,f expresses the rate at which fluid mass is transported, by advection and diffusion, from the fractures to the porous blocks. Within the fracture network, we have mass transport by advection, dispersion and diffusion.

The averaged mass balance for the fluid within the blocks is given by

aOpbPpbPb (_pb~b 0 0 pb ..-pb) at v ·Opb Ppb V pb + Ppb V pb + Jpb

u1 { {ppb(Vpb - u) + j;b}·"'pb dS, (2.4.45) o lSj,jr

where Opb( = Uopb/Uo) is the volumetric fraction of the void space within the blocks (= n - 0fr), ()Pb denotes deviation of ( .. )pb from its intrinsic phase average, and we have already taken into account the fact that Spb,s is a material surface with respect to the fluid's mass. In (2.4.45), the surface integral expresses the rate at which fluid mass moves from the void space in the blocks to the fractures, per unit volume of porous medium. Denoting this rate by 1m , we may rewrite the two balance equations in the form

aOfrPf/r =

at 00 (-fr-V fr 0 VO fr omfr) 1m

- v· fr Pfr fr + Pfr fr + Jfr + , (2.4.46)

o 0 (_pb-y-Pb 0 VO pb -r;nPb) 1m - v· pb Ppb pb + Ppb pb + Jpb - ,

(2.4.47)

where (2.4.48)

We may apply the same procedure also to the mass of a ,-component of the fluid phase. The two balance equations would then be

-fr a8frP}r (-;yfr--fr 0"( 0 fr om"Yfr) m"Y

at = -V·Ofr Pfr Vfr + PfrVfr +Jfr + I , (2.4.49)

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Balance Equations 147

{)() ~b pbPpb __ '("7.() (~b~b 0,,/ V pb .m,,!pb) 1m"! {)t - v pb Ppb pb + Ppb pb + Jpb - ,

(2.4.50)

where P}r and P;b denote the component's mass density (=concentration) in the fluid occupying the fractures and the blocks, respectively, and 1m "!

denotes the rate of transfer of the mass of the ,-component from the blocks to the fractures, per unit volume of porous medium.

Often, the rate of component transfer, 1m "!, is assumed proportional to the difference between the average concentrations in the two continua

m"! _ * (~b --;:ylr) 1 - air Ppb - P Ir , (2.4.51)

where ajr is a transfer coefficient, often regarded as a constant proportional to the coefficient of molecular diffusivity in the porous blocks, to the area of contact between the two continua, and inversely proportional to some characteristic distance between the fractures and the interior of the blocks. However, the assumption that this coefficient is a constant is rather ques­tionable, especially in the case of large blocks. This observation stems from the fact that the characteristic distance to the point within a block to which the average value may be attributed, varies continuously, as the component diffuses into the block, or out of it, especially in large blocks.

An analogous model can be written for the transport of heat in a frac­tured porous medium

The transport of a component of a fluid phase in a fractured porous medium, is further discussed in Subs. 6.1.8.

2.4.8 Megascopic balance equation

In many cases of practical interest, the macroscopic domain, within which transport phenomena are considered, may be heterogeneous with respect to relevant macroscopic coefficients. Reference to this fact was already made in Subs. 1.4.1. As was indicated there, the detailed information about the spatial variation of macroscopic coefficients of transport, may either be lack­ing, or of no interest to the user of the required model output. In such a case, the transport phenomena are chosen to be described in terms of spatial averages, taken over a domain which is orders of magnitude larger than that of an REV, or an REA, introduced in Chap. 1.

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148 MACROSCOPIC DESCRIPTION

In Subs. 1.4.1, we referred to this procedure as higher order averaging, and to the level of description resulting from it as the megascopic level. This is, for example, the level of description used in the hydraulic approach (Subs. 1.4.2).

The general differential megascopic balance equation for any extensive quantity, E, is derived by averaging the general macroscopic balance equa­tion (2.4.7). The latter is valid at any point in a porous medium domain regarded as a continuum.

With GOt denoting the macroscopic value of GOt, as defined by (1.3.5), we introduce the definition of the megascopic value of GOt, defined by

GOt(x,t) = 1 ~ GOt(x',x,t)dUo(x'), Uo Jlio

(2.4.52)

where Uo is a spatial domain, of volume U 0, centered at point x, and x' is a point inside U o.

The domain U 0 will be referred to as representative macrovolume (ab­breviated RMV) of the porous medium at point x, if its length scale, £*, satisfies, in principle, conditions which are analogous to those imposed on an REV, viz.

(a)

(b)

(2.4.53)

where d* is the length scale of the macroscopic heterogeneity, which the averaging smoothes out, and L represents a characteristic length of the problem domain. An example of d* is the vertical length of the domain in the hydraulic approach.

(iGa(x, t) I = o. aU u=fJo

(2.4.54)

All specific properties of the RMV, and the conditions for defining a differ­entiable field, GOt(x, t), are analogous to those specified for the REV in Subs. 1.2.3.

The spatial domain within which GOt is continuous and differentiable, will be referred to as the megascopic space. The coordinates in this space will be referred to as megascopic coordinates.

By averaging«2.4.7) over an RMV, at any point within the megascopic space, we obtain

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Stress in a Porous Medium

8eO/ 8t

v. (eO/ v:=a + ~~O/ + ~ €O/ V 0/ 0/ + OO/€O/ ~ 0/ 0/)

r-----------~~---~~ + Jfiu - {eO/(V 0/ - u) + J~u}.vO/ ~O/~

149

(2.4.55)

where a bar symbol over a macroscopic value indicates an average over an RMV, with

and

(.) = rr - (··t, or (.) = n - ( .. ) is the deviation of a macroscopic value of a quantity at any point within an RMV, from its average over the RMV.

As could have been expected, the megascopic balance equation (2.4.55) contains new additional dispersive fluxes which result from the variability

of the relevant macroscopic quantities. One of these fluxes is ;0/ V ~ 0/, which will be referred to as the macrodispersive flux of E. As in the case of the macroscopic level of description, also at the megascopic level, one has to express the various dispersive fluxes in terms of megascopic quantities. This is done by introducing auxiliary models ofthe distribution ofthe macroscopic values of these quantities within an RMV.

A megascopic expression for the macro dispersive flux is derived in Subs. 2.6.7.

2.5 Stress and Strain in a Porous Medium

When dealing with strain in a porous medium domain, or with the transport of extensive quantities in a deformable porous medium, we have to refer to the stress in the porous medium system as a whole. In this section we shall consider this topic, with special emphasis on the stress that produces porous medium strain.

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150 MACROSCOPIC DESCRIPTION

2.5.1 Total stress

Here, we deal with stresses in multiphase flow in a deformable porous medium (Bear and Pinder, 1978). As an example, let us assume that the void space contains two fluids (e.g., oil and water). We shall refer to one fluid (subscript w) as the wetting fluid, and to the other one (subscript n) as the nonwetting fluid (see further explanation ofthese terms in Subs. 5.1.1). By rewriting the averaged momentum balance equation, (2.4.28) for each of the three phases, we obtain

--n o _nDmvmn nPn Dt V·OnU n n + OnPnFnlt + ~ [ un·vndS,

o }Sns+Snw

(2.5.1)

V ·OwU w w + OwPw F w w + ~ [ U w·vwdS, o }Sws+Swn

--s o -=-IJDm Vms sPs Dt

where s denotes the solid.

(2.5.2)

(2.5.3)

We note that the outward normals are such that Vn == -Vs on Sns,

Vs == -Vw on Ssw and Vn == -Vw on Snw. The symbol Dm( )/Dt denotes the material derivative from the point of view of an observer traveling at the average mass weighted velocity of the considered phase.

By summing the three equations, we obtain --ex

"" o.."..-a Dm VmOl L...t exPex Dt

(ex=n,w,s)

where, by (1.3.7)

U = ~ 1 udU = ~ I: 1 UexdUex o Uo 0 (ex=s,n,w) Uo

I: Uex == Us + Un + U w ' (ex:::s,n,w)

(2.5.5)

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Stress in a Porous Medium 151

defines the volume averaged stress, or total stress, and

defines the total body force per unit volume of porous medium at a (macro­scopic) point.

Using (2.2.82), the total stress is also given by

0' = Us + Tn + T w - Pn I - Pw I. (2.5.6)

We have used the symbol

[(··)h,2 == (··)Iside 1 - (··)Iside 2

to denote the jump from side 1 to side 2 across the microscopic interface. By examining (2.5.4), we note that the interaction between the phases is accounted for by the three surface integrals.

The first integral on the r.h.s. of (2.5.4) describes the interaction across the (microscopic) interface between the nonwetting and wetting fluid phases. In principle, similar interactions also occur at the solid-fluid interfaces, i.e., the second and third integrals. On the other hand, we usually assume con­tinuity of traction, i.e., [O']w,s·vw = 0 and [O']s,n·vs = 0, and neglect surface tension phenomena at fluid-solid interfaces. Hence, the last two surface inte­grals in (2.5.4) vanish. A detailed discussion on surface tension is presented in (Subs. 5.1.1).

Back to the first integral, actually, at every point on the microscopic interface between two immiscible fluids (here the wetting and the nonwetting ones), overlooking their molecular structure, and regarding them as two continua separated by a sharp interface, the following equation expresses continuity of momentum transfer (Landau and Lifschitz, 1960)

(1 1) 8,wn r' + r" /wnvi + 8Xi

2 8,wn () */wnVi + -8-' 2.5.7 r Xi

where /wn is the magnitude of the surface-tension between the wetting and the nonwetting fluids. This is a concept that introduces the molecular level effects between the two fluids in the form of a force (per unit length) that is tangent to the interface, and r* is the mean radius of curvature of the latter, with r' and r" denoting its principal radii of curvature. The l.h.s. of

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152 MACROSCOPIC DESCRIPTION

(2.5.7) expresses a jump in the component in the ith direction of the total momentum flux. The r.h.s. may be interpreted as the rate of production of linear momentum per unit area of the interface. In this equation, /nw may be nonuniform, e.g., because of impurities and temperature variations.

Note that because /wn exists only in the interphase surface, the gradient of /wn in (2.5.7) should be interpreted as

in which tk denotes a coordinate in the interphase surface. When the n - w-interface is a material surface with respect to fluid mass,

the advective momentum flux vanishes, i.e., [pym(ym - u)]n,w·Vn = O. For a stationary fluid, the viscous stress, r( = u + pI), vanishes and the jump, -[u]n,w·vn, reduces to [p]n,wvn. The same conclusion can be obtained if we assume, as an approximation, that the component of the shear force normal to the interface is much smaller than the force due to pressure, i.e.

If also V/wn = 0, equation (2.5.7) reduces to

Pn - Pw = (:' + r:' )/wn' (2.5.8)

known as the Laplace formula. Since (Pn - Pw) > 0, the pressure is greater in the non wetting fluid for which the surface is convex. The difference Pn - Pw is called the capillary pressure. In Sec. 5.1, we shall introduce the difference Pc = p:;;n - P:;:w = Pe( Ow) called macroscopic capillary pressure.

Neglecting inertial forces, (2.5.4) reduces to

(2.5.9)

where

(2.5.10)

A better understanding of (2.5.7) can be obtained by rewriting it in a local coordinate system at a point on the interface. With tI, t2 denoting the unit vectors in the plane tangent to the interface, and recalling that /wn

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Stress in a Porous Medium 153

= 1'wn(tt, t2), 81'wn/8sv = 0, we write a balance along the normal to the plane, in the form

(2.5.11)

in which -TijVWj expresses a force component normal to the interface, per unit area of the latter, due to shear. A balance in the tangential plane can also be written

(2.5.12)

Since tWi = 0, the balance in the tangential plane, takes the form

(2.5.13)

In the case of a single fluid phase, or neglecting Fe, as an approximation, in the case of multiphase flow, (2.5.9) reduces to

(2.5.14)

known as the equilibrium equation (see Biot, 1941, and Verruijt, 1969).

2.5.2 Effective stress

In the case of two fluid phases, the total stress at a point within a porous medium domain is given by (2.5.6). Consider the particular case of a single fluid phase (subscript J) that fully occupies the void space. Then, neglecting the shear stress in the fluid, or when the system is at rest, we have, from (2.5.6)

- - (1 )-s -11 0" - - n O"s - nPI . (2.5.15)

Our objective is to determine the stress that produces the strain of the solid matrix in a porous medium domain. The knowledge of this strain is required in problems of flow through deformable porous media.

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154 MACROSCOPIC DESCRIPTION

The total force due to the stress acting on the solid matrix within an REV, per unit volume, is given by

~ [ Us·Vs dB = ~ [ Us·Vs dB + ~ [ Us·Vs dB. ok_ ok.. ok,. (2.5.16)

By (2.3.30), and assuming that [ulf,s·vs = 0, this equation can be rewrit­ten in the form

(2.5.17)

The 1.h.s. represents the resultant force (per unit volume) that produces the strain of the solid matrix within an REV. We note that it is composed of two parts: the effect of Us, which expresses the effects of stresses in the solid matrix surrounding the REV (through Sss), and the effect of the fluid within the REV. Thus, Us constitutes only part of the stress that produces the strain of the solid matrix.

Terzaghi (1925) introduced the concept of effective stress, or intergranu­lar stress in soil mechanics (see any text on soil mechanics). Essentially, this concept assumes that as the solid comprising the solid matrix (e.g., a gran­ular material) is (almost) completely surrounded by an ambient fluid, the latter's pressure (actually stress, unless we neglect T f) acting on the solid­fluid interface produces a stress of equal magnitude in the solid (say, within each individual grain), without contributing to the solid matrix' deforma­tion which is produced mainly by forces that are transmitted (in a granular material) from grain to grain at contact points. Thus, the strain producing stress, or intergranular stress, is obtained by subtracting the pressure in the fluid from the stress in the solid, where both pressure and stress are average values, and we recall that in this book stress is positive for tension, while pressure is positive for compression.

Essentially, the concept of effective stress stems from the observation that the deformation of a granular material is much larger than can be explained by the compression of the material itself. This suggests that deformation is produced mainly by the rearrangement of grains, with localized slipping and rolling, implying that deformation is governed by the transmission of localized normal and shear forces at contact points. As these forces are not affected by the pressure in the water, a change in fluid pressure, accompanied

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Stress in a Porous Medium 155

by an equal change in total stress produces no deformation and, hence, should produce no change in effective stress.

Accordingly, when the shear stress, T f, is neglected, we write (2.5.15) in the form

U (1- n)u/ - np/I

(1- n){u/ + p/I} - (1- n)p/I - np/I,

or u = u~ - p/I, (2.5.18)

where PI' is the (average) pressure in the fluid, and the effective stress is expressed by

(2.5.19)

Written in this form, it is evident that the effective stress is made up of two parts: one is an average stress (positive for tension) within the solid matrix, and the other is an average pressure (positive for compression) in the water filling up the void space. The stresses u and u~ are forces per unit area of porous medium cross-section. In soil mechanics, the minus sign in (2.5.18) is usually replaced by a plus sign, i.e., both u and u~ are positive for compression.

When T f within the fluid cannot be neglected, we obtain

u - u' + Uf! - S , (2.5.20)

with (2.5.21)

Returning now to the case of two fluids (wand n) that together fill the void space, with negligible shear stress within both fluids, let us define an average pressure of the two fluids (subscript and superscript v), by

-v 1 -w -n Pv = -( OwPw + OnPn ),

n

with u = u~ - ~I. Another form of (2.5.22) is

-v 1_n -n-w 1_w Pv = -{nPn - Ow(Pn - Pw )} = -(npw + OnPc),

n n

where Pc = pnn - Pww

(2.5.22)

(2.5.23)

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156 MACROSCOPIC DESCRIPTION

is the (macroscopic) capillary pressure.

We can now rewrite (2.5.18) in the form

where

(1 - n )0" s s - OwPw WI - OnPn nI

(1- n){O"/ + PvVI} - (1- n)pvVI

O -wI 0 =--7tI - -, Ow-wI On-nI - wPw - nPn - 0" s - -Pw - -Pn , n n

(2.5.24)

(2.5.25)

is the effective stress for this case. In unsaturated flow (= air-water flow), the nonwetting fluid is air while the wetting one is water. When we take the air pressure to be atmospheric, and with Pa denoting the pressure in excess of the atmospheric one, Paa == 0, and (2.5.24) reduces to

0" -_ 0"' Ow -WI --Pw . 8 n (2.5.26)

In determining the average void pressure, P;u, we have taken a volume average of the pressure in the various fluids occupying the void space. Other weights in determining P;u would lead to equations which are different from (2.5.24) and (2.5.26). For example, some authors (e.g., Aitchison and Don­ald, 1956) use for air - water flow (with Paa = 0)

(2.5.27)

where X is some function of the moisture content, Ow. Bear et al. (1984) used (2.5.27) with X(O) = Ow/no

An average pressure for the case of three fluid phases that together oc­cupy the void space is discussed in Subs. 5.1.3.

Verruijt (1984) distinguishes, in a granular solid matrix, between the in­tergranular stress, O"~, related to the pressure in the fluid, defined by (2.5.18) and the effective stress, O"~.

For his effective stress, O"~, Verruijt (1984) suggests modifying (2.5.18) to the form

(2.5.28)

where b should be such that the deformation of the granular skeleton (say, soil) will be fully determined by the effective stress, O"~. For a compressible fluid, a change in volume, fj.Uj, is related to a change in pressure, fj.p/, by

fj.Uj = -f3jfj.p/Uj, (2.5.29)

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Stress in a Porous Medium 157

where U, is the initial volume of fluid, and 13, is the fluid's compressibility. The solid's volume change is related to changes in fluid pressure, !::..p/, and to changes in the isotropic component, (7~(= ~ I:(k)(7~)kk)' of (7~, by

(2.5.30)

where Us is the initial volume of solid and 13; and f3~ are compressibilities related to changes in the (all around) fluid pressure and in the intergranular stress, respectively. They both depend on the elasticity of the solid material.

A bulk, or total porous medium compressibility, 13, can be defined by

!::..U = f3U !::..O', (2.5.31)

where 0' is the isotropic component of the total stress (= ~ I:(k) O'kk), and U denotes the initial bulk volume. This expression corresponds to what is known in soil mechanics as dry compression test.

In a compression test of a saturated sample, with drainage being pre­vented by an impermeable membrane surrounding the sample, we obtain

or

(2.5.32)

Since there is no drainage, we have

(2.5.33)

Since!::..O' = !::..(7~ - !::..p/, we obtain from (2.5.29) through (2.5.33)

!::..U U

f3(!::..O' + !::..p/) - 13: !::..p/

-f3,n!::..p/ - 13:(1- n)!::..p/

+f3~(l - n)(!::..O' + !::..p/)

!::..p/ 1 - !::..O' = B = 1 + n(f3, - f3n/{f3 - f3~(l - n)}'

(2.5.34)

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158 MACROSCOPIC DESCRIPTION

The coefficient B(> 0) is called Skempton's (1954) coefficient. For 13: = f3~ = 0, i.e., incompressible solid, B reduces to

Back, now, to the definition of effective stress in (2.5.28), with

Verruijt (1984) requires that

or f).U = f3U{f).O' + (1- b)f).p/}

and that, instead of (2.5.30), we shall have

-f).Us = -f3:Usf).p/ = -f3:U(l - n)f).p/,

or, with (2.5.31)

(2.5.35)

(2.5.36)

(2.5.37)

By comparing (2.5.36) with (2.5.37), Verruijt reaches the conclusion that

1

1 + n(f3j/f3 - b)' (2.5.38)

i.e., identical to (2.5.34) when f3~ = 0, and bin (2.5.28) is expressed by

(2.5.39)

In most natural soils, this ratio is very small, so that the effective stress is identical to the intergranular stress. We shall henceforth use u~ to denote both stresses. .

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2.5.3 Forces Acting on the Solid Matrix

We consider a single isochorically moving fluid that occupies the entire void space. The total force exerted on the solid matrix by the fluid, across their common interface surface, Sis, per unit volume of porous medium, is ex­pressed by

(2.5.40)

where we have made use of the relation U I = T 1- Pil. By (2.6.46)

1 1 Tf' '1/1' dB - _TIl CI "'''q171: -u I)) - r A 2 '-<I) r)" o Sfs ul

(2.5.41)

By (2.3.30) and (2.6.14)

Substituting (2.5.41) and (2.5.42) into (2.5.40), yields

n (Opf + 7/ g OZ ) (T'!', _ (h) _ pf an ax j ax j )1) OXi

_ICI m + J.t 6.2 (Xijqrj' I

(2.5.43)

Expressing the last term in (2.5.43) by making use of (2.6.48), we obtain another form of (2.5.43)

(2.5.44)

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160 MACROSCOPIC DESCRIPTION

If both the inertial force and the internal friction within the fluid can be neglected, Le., we invoke (2.6.51) and (2.6.52), the last equation reduces to

( &Pi i fJZ) i fJn Fi = -n fJxi + Pi 9 fJxi - P fJxi

-(%: + Pig %:J, (2.5.45)

or, by employing (2.6.54)

T. _ -ik-1 m -i fJn ~~ - nil ij qrj - P fJxi' (2.5.46)

where the second term on the r.h.s. expresses the effect of heterogeneity in porosity.

We should keep in mind that the development of the force :F as presented above is based on the assumption that the fluid's motion is microscopically isochoric.

The total force, Ftotal,i, acting on the solid matrix within an REV, is given by the r.h.s. of (2.5.3). Making use of (2.5.21), we can write this force in the form

where we have made use of the fact that -asijVsj surface. Expressing Fi by (2.5.45), we obtain

D TTmi fJ-i -i m Vi Tij np Dt + fJXj

fJpi fJz fJa~ji fJXi - Ptolalg fJXi + fJx j ,

where Ptotal = (1- n)psS + np/.

(2.5.47)

(2.5.48)

The first two terms on the r.h.s. of (2.5.48) vanish if inertia and shear in the fluid are neglected. Under such conditions, the total force acting on the solid may also be written in the form

( &pi -i fJZ) fJz fJa~ji Ftotal,i = - -fJ + Pi g-fJ - P.ubmergedg-fJ + -fJ-'

Xi Xi Xi Xj (2.5.49)

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Stress in a Porous Medium 161

where

P - P -f - (1 )(-S -f) submerged - total - Pf - - n Ps - Pf ,

and Psubmergedg is the buoyancy weight of a saturated porous medium. In soil mechanics, the sum -(\7P/ +pf g\7 z) is called seepage force, while

- \7p/ is called boundary neutral force. For a pf = const., the seepage force

(per unit volume of porous medium) is expressed as _pf g\7<pf, where </> is the piezometric head of the fluid. Thus, the seepage force acts in the direction of the hydraulic gradient (Subs. 2.6.1). We note that the seepage force is independent of the medium's permeability. Under the conditions for which the motion equation (2.6.54) is valid, the seepage force is equal to J.Lk-1qr, where k, defined by (2.6.55), is the porous medium's permeability.

Thus, the total force acting on the solid is made up of the seepage force, the force of buoyancy, taking into account the solid's submerged density, and the gradient of the effective stress in the solid matrix.

Finally, we may express the force, :Ftota!, in terms of the total stress, 0',

in the form

aO" ·i az ;:':otal,i = -a J - Ptotalg-a '

Xj Xi (2.5.50)

or in terms of qr

:F. . = -f k-:-.l "171: _ az aO"~ji total,t J.L tJ qrJ Psubmergedg-a + -a--

Xi Xj (2.5.51)

By adding (2.5.2) for the fluid, and (2.5.3) for the solid, we obtain

(2.5.52)

where the second term on the l.h.s. represents ;:':otal,i.

Equation (2.5.52) is the starting point for treating cases in which inertial effects cannot be neglected.

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162 MACROSCOPIC DESCRIPTION

2.6 Macroscopic Fluxes

Equation (2.4.7) is the general macroscopic differential balance equation for any extensive quantity, E. In order to solve it for the macroscopic variables O(x, t) and ea, we need additional relationships for the source functions, prE, and for the fluxes (e.g., for Va (or Vma, had we employed vm instead of

--a ~~ ~~ both V and jE in (2.4.1)), eV , jEa, ~(V - u)·lI ~a~ and jE. lI ~a~). These additional relations depend on the specific nature of each relevant phase and each extensive quantity.

In the present section, we consider the advective, diffusive and dispersive fluxes. Whenever it is clear that we refer to a single, (say, a)-phase only, the subscript a will be omitted, except in the rT symbol.

2.6.1 Advective flux of a single Newtonian fluid

The general macroscopic balance equation (2.4.7) contains the expression oeava , called the (macroscopic) advective flux of E, Le., the quantity of E advected by the average volume weighted velocity, Va, of a fluid phase, (per unit area of a porous medium). Likewise, the expression oeaVma is the (macroscopic) advective flux of E, carried by the average mass weighted velocity, Vma, of a fluid phase.

For fluid volume (e a = 1), we define two kinds of fluxes. One is

called the mass weighted specific discharge. The other IS

q == ovct,

(2.6.1)

(2.6.2)

called specific discharge. In practice, the choice between using qm and q depends on the measured quantity, viz., mass or volume.

In principle, an expression for the advective mass flux, e.g., Opavm a, can be obtained by solving for Vm a any of the macroscopic momentum balance equations presented in Subs. 2.4.5, e.g., (2.4.24), or (2.4.27), supplemented by macroscopic constitutive equations that describe the behavior of the con­sidered fluid phase at the macroscopic level. Specifically, we need expressions which relate the stress, qa, and the various dispersive fluxes, to the average velocity, Vma. However, under certain conditions, the momentum balance equation, say (2.4.27), can be reduced to simpler forms. For example, we

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Macroscopic Fluxes 163

may assume that certain terms are much smaller than others and may, there­fore, be deleted from the equation. In what follows we shall introduce a number of simplifying, albeit approximate, assumptions in order to derive simplified expressions for the velocity Vm ex .

For a fluid that occupies the entire void space, we replace () by n. How­ever, it is possible to visualize that part of the fluid is stagnant, or practically so, say due to the presence of dead-end pores, or due to the strong adher­ence of the (rather thin) film of fluid (e.g., water) to the solid. We may then define an effective porosity, ne , with respect to the flow, and use it to replace () in (2.4.27). However, in the present Subsection, we shall assume that the entire void space is occupied only by a single mobile fluid. In Subs. 2.6.2, we shall consider a multiple fluid phase system. We shall return to the concept of immobile fluid in Subs. 5.3.6 and 6.4.3.

As a point of departure for determining the macroscopic velocity of a fluid phase, that occupies the entire void space, (Le., () = n), let us start from the macroscopic momentum balance equation (2.4.28), which, in view of (2.3.30), can be rewritten here in the form

-j _jDmVr

np Dt

--j OCT" j n~ + npFi (2.6.3)

OXj

where, the symbol f indicates the fluid phase. We recall that underlying (2.6.3) are the assumptions that the fluid-solid interface is a material surface with respect to the fluid's mass, and that all the momentum dispersive fluxes have been neglected.

In terms of the viscous stress, Tij, and the pressure, p, with CTij = Tij -

p6ij (see (2.2.82)), and for a body force that is due to gravity, viz., Fi = -gOZ/OXi, with the z-coordinate directed upward, equation (2.6.3) takes the form

-j -j --j --j _j Dm Vr op oz OTij

np = -n- - npg- + n-o . Dt OXi OXi x j

(2.6.4)

The first term on the r.h.s. represents the total force due to pressure. Since

(2.6.5)

this term includes the force of pressure exerted on the fluid across the fluid­solid interface and across the fluid portion of So. Note that v == Vj == -VS'

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164 MACROSCOPIC DESCRIPTION

The second term represents the force due to gravity. The third term represents the force due to the viscous resistance to flow.

Let us further develop the terms on the r.h.s. of (2.6.4), that still involve microscopic variables, in order to express them in terms of macroscopic ones.

As shown in Subs. 2.3.5, under certain conditions, the relationship be­tween the average of a gradient and the gradient of an average, can be presented in a modified form. Accordingly, if we assume here that the fluid pressure within the void space of an REV varies monotonously, i.e., with no maximum or minimum, then

in Uov • (2.6.6)

Under this condition, (2.3.48) is applicable, and we may use it to write

[)p 1 ()P1 * 1 1 0 [)p -[) . = -[) . Tji + -u Xi-[). Vj dS, (2.6.7) X~ xJ n 0 Sfs xJ

where the coefficient Tli , == Tjji' defined by (2.3.49), takes the form

TiJ = _1_ f XWj dS. nUO}Sff

(2.6.8)

In order to express the surface integral in (2.6.7) in terms of macroscopic variables, let us focus our attention on the normal derivative, ([)p/[)Xj)Vj,

on the S 1 s-surface, that appears in that integral. Assuming that in the vicinity of the (possibly moving) fluid-solid surface, the components normal to the latter, of both the inertial force and the viscous resistance to the flow, are negligible, relative to the normal components of pressure gradient and gravity, i.e.

I (D;;m - ~~: )Vjl ~ I (:~ + pg ::Jv+ we obtain, say from (2.6.4), written at the microscopic level

Hence

[)p '" [)z [)x. Vj = -pg [)x. Vj = -pg83jvj on Sis.

J J

_pi g83 j_1_ f XWj dB nUO}Sfs

_p1 g( 83i - T;i).

(2.6.9)

(2.6.10)

(2.6.11)

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Macroscopic Fluxes 165

In writing (2.6.11), we have made the approximation p < pI within Uol' and

We turn now to the gravity term in (2.6.4), and express it by

--1 oz _ -1 .

pg-o - P g83~' Xi

(2.6.12)

(2.6.13)

By combining (2.6.7), (2.6.11) and (2.6.13), we obtain for the first two terms on the r.h.s. of (2.6.4)

(2.6.14)

noting that T;i = 83 jTJi = (oz/oXj)Tk The r.h.s. of (2.6.14) represents another form of the resultant force acting on the fluid, per unit volume of porous medium, due to pressure and gravity.

Our next objective is to express the average viscous resistance force,

n( OTij / ox j)l, appearing in (2.6.4), in terms of the macroscopic velocity. As we shall see below, this force is made up of two parts: one within the fluid and the other at the fluid-solid interface. Before proceeding with the de­velopment of the required expressions, we recall that (2.6.4) is valid for any fluid phase. In order to derive an equation for the macroscopic advective flux of a particular fluid, an appropriate constitutive equation (Subs. 2.2.3), representing the relationship between the components Tij and the velocity components, Vim, for that particular fluid, must be introduced. For this purpose, let us consider the particular case of a Newtonian fluid (i.e., lin­early viscous fluid), for which the constitutive equation is given by (2.2.86), repeated here for convenience in the form

( OVim OVt) "oVr Tij = J.L -0- + -0- + oX -o-8ij.

Xj Xi Xk (2.6.15)

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166 MACROSCOPIC DESCRIPTION

Hence

OTij

ox' J

(2.6.16)

Following the methodology of deleting nondominant effects, presented in Sec. 3.3, we compare the first and third terms on the r.h.s. of (2.6.16), seeking the condition under which

(2.6.17)

We obtain

(2.6.18)

Since the terms denoted by an asterisk are of order one, the required condition is

(2.6.19)

where LiJ1.) and LiV ) are lengths characterizing the spatial variations in dy­namic viscosity and in velocity, respectively, within the fluid phase. Practi­cally, this condition is always satisfied and, hence, we may delete the first term on the r.h.s. of (2.6.16). Similar considerations will also hold for the second term. Thus, we may replace (2.6.16) by

(2.6.20)

By averaging (2.6.20), and assuming

1<1' +"Aul{o:l);:] }fl <: 1<1>+ N'lf (0;:;/1. (2.6.21)

and

(2.6.22)

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Macroscopic Fluxes 167

we obtain

-8 . .f 82TTm I 8 (8V:m) I r~J _-I Yi (-I \/11) J - -f.L +f.L +'" ---8xj 8xj8xj 8Xi 8xj

(2.6.23)

By employing (2.3.20) twice, we obtain

(2.6.24)

and

(2.6.25)

Hence

Another form of (2.6.26) is obtained by making use of (2.4.16). By taking a derivative of the latter equation, we obtain

(2.6.27)

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168 MACROSCOPIC DESCRIPTION

With this equation, (2.6.26) can be written as

-j OTij

n-­ax-J

(2.6.28)

where we note that the last term on the r.h.s. vanishes for microscopically isochoric flow, i.e., when aVr /aXj == o.

Our next objective is to express the surface integrals appearing in (2.6.26) in terms of macroscopic variables.

At this point we introduce the important assumption that the fluid ad­heres to the solid walls, i.e.

(2.6.29)

with Vs denoting the solid's velocity. This assumption is often referred to as the 'no-slip' condition. Under this condition

-s an c.! v. - si -a Xj

an-Vs-/ o-Vs-/ ---:::-- + n--OXj OXj ,

(2.6.30)

where we have introduced the approximation Vsils ~ Vsi s, and made use Js

of (2.3.27). Similarly

f -s-s ~V:n _ s ~ __ an Vsj aVsj

J vJ L..J j s - a + no· Xj Xj (2.6.31)

By making use of (2.4.15), recalling that Sjs is a material surface with respect to the fluid's mass, and that, approximately, q ~ qm, i.e., neglecting the diffusive flux of the fluid's mass, we obtain

~ __ ds

'oVr aVr j an 1 (an aqj ) an aXj Vi I:js ~ - aXj aXi = -;:;, at + aXj aXi·

(2.6.32)

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Macroscopic Fluxes 169

At any point in the close proximity ofthe Sjs-surface, the fluid's velocity vector, V m , can be expressed as

vm Vllmll + vt~t' + vt%"t" V:m V "1I + V:mt'-t' + V:mt'!t" JJ JJ JJ' (2.6.33)

where II, t' and t" are the unit vector normal to Ssi (= the principal normaQ and the two tangential ones to Sis, in the 'local' Cartesian coordinates at a point of Sis; SII, St' and St" are lengths along these vectors, respectively. These three mutually orthogonal unit vectors are obtained as the intersec­tions of the osculating plane, the normal plane and the rectifying plane.

A similar expression can be written for Vsls . By taking the difference 's vm - Vsls ,we obtain 's

(2.6.34)

Since Sis is a material surface with respect to the fluid's mass, we have in its immediate neighborhood

(2.6.35)

Then recalling that VWj + t~tj + t~'tJ = bij, (2.6.36)

we obtain for the velocity components in the tangential plane

Vi*m - Va1ls,s = CVr - Vsjlss,)(t~tj + t~'tJ) = (vT - Vsjls,)(bij- VWj), (2.6.37)

where we have introduced the star symbol (*) to emphasize that the com­ponents Vi*ffi here are in the plane tangential to the solid. At any internal point, the transformation ~m = Vimbij holds. We have thus related the ve­locity components in the close neighborhood ofthe Sis-surface to the latter's configuration.

___ ----ds With (2.6.37), we can now express the surface integral 8Vim jas il L.is,

appearing in (2.6.26), in the form

...--is aVim -a L.is-

SII

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170 MACROSCOPIC DESCRIPTION

where Vim 1.0. denotes the velocity at a point within the fluid, located at an elementary distance ~ from 818, and the characteristic value, ~c, of ~, is defined by (2.6.38).

In deriving (2.6.38), we have made use of the 'no-slip' condition on 818, "..-'18

and the approximation l'sj ~ l'sj 8, with the approximation sign chang-ing into an equality one when the solid (not the solid matrix) is rigid, or approximated as such.

The distance ~c appearing in the approximation of the velocity gradient in (2.6.38), is a characteristic distance from the solid walls to the interior of the fluid phase. It is some measure of the pore size. For example, it can be taken as proportional to the hydraulic radius, ~v == ~ 1, defined as the ratio of void space volume, Uov , to the interface surface area, 881 , i.e.

where C 1 is a macroscopic dimensionless shape factor, and 'E 1 8 (= 818/ Uo )

is the specific area of the f -s-interface within Uo , per unit volume of Uo •

With the above definitions, (2.6.38) takes the form

(2.6.39)

where (2.6.40)

is another coefficient that characterizes the configuration of 818 , and

(2.6.41)

is the relative mass weighted specific discharge (i.e., relative to the solid).

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Macroscopic Fluxes 171

By inserting (2.6.30), (2.6.31), (2.6.32), and (2.6.39) into (2.6.26), we obtain

--j OTij

n--ox' J

(2.6.42)

When we assume that the flow at the microscopic level is isochoric, i.e., v·vm = 0, in view of (2.6.39) and (2.6.31), the total viscous resistance reduces to

-j OTij n-

ox' J

where we note the effect of solid deformation.

(2.6.43)

Finally, by assuming, still for microscopically isochoric fluid motion, that

I oq;!l I I OVsi SI ox' ~ n ox' ' J J

(2.6.44)

Henceforth, this equation will be employed as the expression for the total resistance to the flow.

It should be interesting to examine the term nOTij/oX/ that appears in (2.6.4). As stated above, this term expresses the viscous resistance to the flow. By making use of (2.3.20), we can express this resistance as

8~jj on~jj 1 1 n-o ' = -0-'- + -u TijVj dB,

xJ x J 0 Sfs (2.6.45)

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172 MACROSCOPIC DESCRIPTION

where all terms express resistance forces per unit volume of porous medium. We note here that the total viscous resistance is made up of two parts. The first is a resistance resulting from the internal friction in the fluid. The second one results from friction at the fluid-solid interface.

By comparing (2.6.45) with (2.6.44), we may conclude that only the last term on the r.h.s. of the latter expresses the drag at the fluid-solid interface, i.e.

u1 f TijVj dB = -p/ ~~ Ctijq;j. (2.6.46) o lSls f

The remaining term, p?82q:;il8xj8xj, represents the resistance to flow due to the internal friction in the fluid.

Since the fluid-solid interface is a material surface, (2.4.17) holds when the flow is assumed to be microscopically isochoric. Under such conditions, the l.h.s. of (2.6.3) can be rewritten in the form

-f Dm v;n-f _ -f{ 8qr 8 (qrqj)} np D = P -8 + -8 -- . t t Xj n (2.6.4 7)

We have thus achieved our goal of expressing all the terms that appear in the averaged balance equation (2.6.4) in terms of macroscopic variables.

By inserting (2.6.14), (2.6.44) and (2.6.47) into the averaged momentum balance equation (2.6.4), we obtain

pf{8qr + ~(qrqj)}=_n(8pf +pfg8Z)T~ 8t 8xj n 8xj 8xj J

-f 82q;!i -f .. Cf m (2648) + f1 8x .8x' - f1 Ct~J ~2 qrj' ..

J J f

where T/j and Ctij(== Oij - vwisf ) are two tensorial properties of the config­uration of the Sfs-surface in saturated, single phase flow. Both coefficients are second rank tensors that constitute macroscopic representations of the microscopic configuration of the fluid-solid interface within the REV. The first, Tji, transforms the local body force into a macroscopic one. The sec­ond, Ctij, introduces the effect of the configuration of the solid-fluid surface in the term that transforms part of the force resisting the flow at a point to an averaged resistance force at the fluid-solid interface.

If, in addition, Vs = 0, then, by (2.4.16), written for j3 == s, we have V·q = 0, and q == qn so that (2.6.48) further reduces to

pf(8qi + q.8(qdn)) = _n(8pf +pfg 8z )T~ 8t J 8xj 8xj 8xj J

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Macroscopic Fluxes 173

(2.6.49)

Equation (2.6.48) represents an approximation of the macroscopic mo­mentum balance equation for a fluid phase that fully occupies the void space of a porous medium domain. All the variables appearing in it, viz. q::i, q'f, pf, n, are macroscopic ones.

In (2.6.48), the term on the l.h.s. represents the inertial force acting on the fluid, per unit volume of porous medium. Another form of it is

(2.6.50)

-f 2 -f-f 1 8Vm ! 8Vm ! where(Vm ) =~m ~m andwji(="2( 8~j -~))representsthemacro-

scopic vorticity tensor of the fluid. This vorticity vanishes when the (macro­scopic) flow is irrotational, i.e., when the (average) velocity field possesses a potential (see Subs. 2.1.12).

The first term on the r.h.s. of (2.6.48), as explained following (2.6.14), represents the resultant force acting on the fluid, due to gravity and to pressure gradient, per unit volume of porous medium.

The second term on the r.h.s. of (2.6.48) represents the force acting on the fluid, due to the viscous resistance to its flow inside the fluid, per unit volume of porous medium. Actually, this force is exerted on the fluid within the REV, across the fluid-fluid portion of the surface bounding the REV.

The last term on the r.h.s. of (2.6.48), expresses the viscous resistance, or viscous drag force exerted by the solid phase on the flowing fluid at their contact surfaces within the REV, per unit volume of porous medium.

Often, in a given problem, one of these forces is much smaller with respect to the remaining ones and, therefore, may be deleted from the momentum balance equation. This may be so for the entire flow domain, or only for a part of it, on which our attention is focused. Hence, following the discussion on the deletion of non dominant effects presented in Sec. 3.3, we may now proceed to consider simplified cases of (2.6.48).

CASE A. The flow in a given domain is such that the viscous resistance force, due to the momentum transfer at the solid-fluid interface, is much larger than both the inertial force and the viscous resistance to the flow

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174 MACROSCOPIC DESCRIPTION

inside the fluid, i.e.

(2.6.51)

and

I-I .. CI ml I-I o2q::l I 11 at) ~J qrj > 11 oXjoXj . (2.6.52)

In Sec. 3.3, it is shown that conditions (2.6.51) and (2.6.52) prevail when the Reynolds number, Re, defined by (3.3.13), the Darcy number, Da, defined by (3.3.14) and the Strouhal number, St, defined by (3.3.17), are such that

St ::; 1 and 1

ReDa2' ~ 1. (2.6.53)

In most regional groundwater flow problems, Da~ ~ 1 and St ::; 1, and, therefore, (2.6.51) and (2.6.52) are valid even for Re equal to several tens.

Under such conditions, the momentum balance (or motion) equation (2.6.48) reduces to

where

m _ ("Tnl -s) _ kj/ {opl -I oZ} qrj = n Vj - Vsj - - p;f ox/ + P g ox/ '

n~2 n3

kj/ = C/(aji)-lTii = 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/

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Macroscopic Fluxes 175

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 = coTn3 / 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 _ pf cp - z + -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

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176 MACROSCOPIC DESCRIPTION

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

-f 1p! dpf i.p* = z + Po 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) m + Jlf ox j + pf 9 ox j + qrk = o. (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 ReDa2 <: 1. (2.6.62)

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Macroscopic Fluxes 177

For an isotropic porous medium, equation (2.6.61) reduces to

(2.6.63)

where the scalars T* and k are the tortuosity and permeability ofthe isotropic porous medium, respectively (see Subs. 2.6.5).

Equation, (2.6.63), with qm ~ q, but without the coefficient l/nT* ap­pearing in the first term, and without the gravity term, was proposed by Brinkman (1948) and is known as Brinkman's equation. The coefficient l/nT* introduces the effect of the geometry of the microscopic fluid-solid interface at the macroscopic level.

CASE C. We may encounter situations, especially at the onset of flow and in oscillatory flows, in which the Strouhal number may be large, such that the local acceleration, o~ml / at, may not be neglected in (2.6.48). Then, (2.6.48) reduces to

-1 -pI oV:m k·· (Op-l OZ) * -1 . J ~J -1 m _ p;1 (Til) klJ ---at + p;1 {]x j + P 9 ax j + qri - o. (2.6.64)

CASE D. Let us consider the situation during a period, for example, immediately following the onset of flow from rest, in which the inertial effects are much larger than the viscous ones, as manifested by the second and third terms on the r.h.s. of (2.6.48). Employing the methodology outlined in Sec. 3.3, we compare pairs of terms, appearing in (2.6.48), with each term representing a certain force. From the discussion that follows (3.3.15), we conclude that when

1 ReDa2" ~ 1, (2.6.65)

the contribution of the convective acceleration to the inertial force is much larger than the force of viscous resistance exerted by the solid matrix on the fluid.

Similarly, let us compare the force resulting from the local acceleration, with that acting on the fluid, due to the viscous resistance to the flow inside the fluid. We seek the conditions leading to

l am I I 02 m I pI ~ ~ p;1 qri. at OX·Ox· J J

(2.6.66)

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178 MACROSCOPIC DESCRIPTION

Following the procedure outlined in Sec. 3.3, we obtain

Ipioqilotl Pcqc(L~q))2 Ip*(oqilot)*1 ITii 02q;:'i loxjoxjl - (!).t)~q) J1.cqc I J1.* 02 (q;:'i )* loxjoxjl

(L~q))2 Ip*(oqi IOt)*1 (!).t)~q) Vc 1J1.*02( q~)* loxjoxjl '

(2.6.67)

where v( = J1.1 p) is the kinematic viscosity of the fluid, subscript c repre­

sents characteristic values, (!).t)~q) = (!).q)max/(oqi IOt)max represents the characteristic time interval during which the fluid's specific discharge, at a point, undergoes a significant change, and L~q) = (!).q)max/(oqi loxj)max is a characteristic distance over which that change in specific discharge takes place.

The inertial term will dominate when (!)'t)(q)

( ) c ~ 1 or Fov ~ 1, (2.6.68) (Leq )2lve

where

(2.6.69)

is the Fourier number, associated with the fluid's viscosity, that expresses, at a point, the ratio between the time interval during which a significant change in velocity (= momentum per unit mass) occurs, and the time re­quired for smoothing out spatial velocity differences by molecular transfer of momentum (see further discussion in Sec. 3.3).

Under conditions (2.6.65) and (2.6.68), which eliminate the effects of viscosity, both in the form of drag on the solid and internal friction within the fluid, the motion equation (2.6.48) reduces to

to

oqi a (qiqj) _ ( 1 opi OZ )T* --+- -- --n --+g- 'i. at ax 'n -pi ox' ax ' ) ) ))

(2.6.70)

In view of (2.6.50), for macroscopically irrotational flow, (2.6.70) reduces

~ oVri + (_l_&Pi + OZ )T~' + ~ (Vmi)2 = 0, 9 at pi 9 OXj OXj Jl OXi 2g

(2.6.71)

in which each term expresses a force per unit weight of fluid.

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Macroscopic Fluxes 179

2.6.2 Advective fluxes in a multiphase system

So far, we have assumed that a single fluid phase occupies the entire void space. When two immiscible fluids occupy disjoint (microscopic) sub do­mains that together fill up the entire void space, because of surface tension phenomena, one of the fluids, called wetting fluid, denoted by subscript w, tends to adhere to the solid wall of the void space, while the other, called nonwetting fluid, subscript n, tends to stay away from the solid wall (see Subs. 5.1.1). This means that a wetting fluid always coats the entire solid surface. When a porous medium, initially saturated by a wetting fluid, is drained, a small amount of the latter will always remain on the solid wall in the form of a very thin film, with a thickness of some tens of molecules, that adheres to the wall by strong molecular forces and cannot be displaced. Even if initially a void is occupied by a nonwetting fluid, a wetting fluid that invades the void space will tend to spread on the solid wall by imbibition, gradually displacing the nonwetting fluid. This film behaves practically like a solid, transmitting momentum across it (almost) without changing.

According to this picture, the total surface surrounding the wetting fluid is made up of a wetting fluid-solid part (not including the thin film part) and a wetting fluid-nonwetting fluid one. In addition, there exist wetting fluid-wetting fluid, and nonwetting fluid-nonwetting fluid interfaces on the external boundary of an REV. Similarly, the nonwetting fluid is in contact with both the solid (across the film) and the wetting fluid, through portions of the total surface surrounding it.

The momentum that can be exchanged between the two fluids across their common boundary must be taken into account in the derivation of an averaged momentum balance equation for each of them. Thus, for the wetting fluid, the momentum transfer must be expressed by the sum of two integrals of the same integrand: one over the Sws-surface, between the wetting fluid and the solid, and the other, over the Swn-surface, between the wetting fluid and the nonwetting one. In this way, two expressions will be derived, one for each surface integral, in terms of averaged velocities and coefficients that represent the configuration of the phase within the REV. This conceptual model serves as a basis for the derivation of flux expressions that will exhibit coupling between adjacent immiscible fluids, due to momentum transfer across the microscopic interfaces that separate them. A as a result, the pressure gradient in one fluid will also cause movement in the other fluid. In a case with phase change, the interface is not a material

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180 MACROSCOPIC DESCRIPTION

surface, and momentum transfer accompanies mass transfer. As a point of departure for the derivation of an average momentum bal­

ance equation for a phase, taking into account the exchange of momentum across Swn, we shall use (2.6.4). When written for an a-fluid phase, with a = W, or n, this equation takes the form

__ (} apa a _ (} az a (} aT aij a - a !:I aPa9!:1 + a !:I

UXi uXi UXj (2.6.72)

Noting the assumptions that underlie it, we assume that (2.6.9) is valid for both the fluid-fluid and fluid-solid portions of the surface that surrounds each phase. Then, (2.6.14) is applicable to each of the two fluids, and we can write

(2.6.73)

Similarly, (2.6.26) is applicable to each of the two fluids separately, if we replace in it the average over the SJs-surface by the average over the sum of the partial surfaces, Sws and Swn. Thus, we have for the wetting fluid

(} [hwij W _ -w ( a 2 q:i w - J.Lw

aXj aXjaxj

(2.6.74)

According to (2.6.27), the last term in (2.6.74) vanishes for an incom­pressible fluid, or when the motion is isochoric.

We continue to adopt the no-slip condition on the Sas-surface, i.e.

vml =V I a SOlS - s SOlS' a = w,n. (2.6.75)

Under this condition, (2.6.30) is valid, with subscript f replaced by a and porosity, n, replaced by (}a.

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Macroscopic Fluxes 181

As a consequence, we obtain for the wetting phase

{)2qm, {) (, IWs+wn ) {) it + -{) V;IVwj ~ws+wn Xj Xj Xj

2 -s {) q~i {) (0 {)Vsi ) roJ +___

- {)X ,{)X ' {)X' w {)X' J J J J

{) (, .AJJn) + {)Xj (V;I - Vs/)Vwj ~wn. (2.6.76)

In the same way, we obtain for the same wetting phase

{)2qm, {) ( .AJJs+wn ) {)2 qm , {) ( {)"fT':'S) WJ ' m rWJ YSJ {) {) + -{) VwJ'Vwj ~ws+wn c:= {) {) + -{) OW-{)--Xi Xj xi Xi Xj Xi Xj

{) (' IWn) + {)Xi (V;} - Vsj S)Vwj ~wn. (2.6.77)

Similar equations can be written for the nonwetting phase. By analogy to (2.6.38), we can now express the surface integral for the

,,--___ lWs+wn

wetting phase, 8 V;I / {) Sv ~ws+wn, in the form

where -u;n is the velocity of the (microscopic) fluid-fluid interface, with Uj

c:= V,;} I Swn' an d

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182 MACROSCOPIC DESCRIPTION

(ws) _ J: ".---IWB Ciij = Vij - VwiVwj ,

(wn) _ J.: ".---IWn Ciij = Vij - VwiVwj ,

where C:U, defined by this expression, is a shape factor associated with the wn-surface area, on the w side of this surface.

By analogy to (2.6.32), we may write

By inserting (2.6.76), (2.6.77), (2.6.78) and (2.6.79) into (2.6.74), we obtain

() OTwij W

wax-3

An analogous equation can be written for the nonwetting fluid. The l.h.s. of (2.6.72) can be rewritten for the wetting fluid, in the form

() ~Dm~ wpw Dt

An analogous equation can be written for the non wetting phase.

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Macroscopic Fluxes 183

Finally, by inserting (2.6.81), (2.6.80) and (2.6.73) into (2.6.72), we ob­tain for the wetting phase

(2.6.82)

As in the case of advective flux of a single fluid phase, we shall refer to the particular case in which the effects of inertia and internal friction in the fluid can be neglected, in comparison with the viscous resistance force at the fluid-solid and fluid-fluid interface surfaces. In such a case, (2.6.82) reduces to

C (ws) G' (wn) -w Waij I;ws m -w waij I;wn m ,-/um J.Lw 6.w Ow qrwj + J.Lw 6.w Ow (qwj - Ow Uj )

= -Ow (8:;; + Pw Wg ::JT~ji. (2.6.83)

An analogous equation can be written for the nonwetting phase. To eliminate -u;n from (2.6.83), we turn to the microscopic jump con­

dition (2.5.13), of the shear stress in the plane tangent to the nw-interface. Here we rewrite this condition in terms of fluid velocities, in the form

-tni {J.Ln (8Vni + 8Vn}) + A~ 8Vnk Oij _ J.Lw (8V': + 8V;}) 8xj 8Xi 8Xk 8xj 8Xi

\" 8V:k l: } 8'Ywn -Aw-8--Uij l/nj = -8--tni. Xk Xi

(2.6.84)

Since tnjl/nj = 0, equation (2.6.84) reduces to

{ ( 8V.'11?- 8V.11?- ) (8V.'17\ 8V.m. ) } -tni J.Ln 8:; + 8:: -J.Lw 8x:t + 8x~J l/nj

(2.6.85)

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184 MACROSCOPIC DESCRIPTION

For the nonwetting fluid, we have

( {)Vni ()Vnj ()Vnj) tni -()- + -{)-VnWnj + -{)-tnWnj

Snl/ Snl/ Snt

{)V:~ {)Vnj = ~tni+--V'

{)snl/ {)snt nJ (2.6.86)

where dSnt is an elementary length in the direction It in the tangential plane, with

{)( .. ) {)( .. ) -- = --tni· {)snt {)Xi

Similarly, for the wetting fluid

t ( {)V;i ()V;}) ni {)x j + {)Xi Vnj

( {)V;i {)V;} ()Vwj ()V;})

= tni -{)-Vwj + -() twj + -{)-Vwi + -() twi Vnj SWI/ Swt SWI/ Swt

{)V~ {)V;} = --() tni - -{)-Vnj (2.6.87)

SWI/ Swt

By substituting (2.6.86) and (2.6.87) into (2.6.85), we obtain

( {)Vni ()V~). ({)Vnj ()V;}). _ (),wn . f..tn-{)- + f..tw-{)- tm + f..tn-{) + f..tw-{)- VnJ - --() . t m ·

Snl/ SWI/ Snt Swt Xt

(2.6.88)

Assuming that the dominant fluid velocity variation in the close vicinity of the wn-interface, is that of its tangential component along the normal to the interface, equation (2.6.88) reduces to the form

{)Vni {)V;i {),wn (2 6 89) f..tn- + f..tw- ~ --. . . {)snl/ {)SWI/ {)Xi

By taking an average of (2.6.89) over the totality of the microscopic interface surfaces within an REV, proceeding in a way similar to that used in (2.6.78), we obtain

(2.6.90)

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Macroscopic Fluxes 185

Hence, the average speed of displacement of the nw- interface is given by

"...----'Wn

,,-Mn U· J

,...-'Wn where we have assumed ItOt ~ p:;:x, II = w, n.

{),wn

{)x· J C' C' , .,....-n n +..--w =

Itn An Itw Aw

(2.6.91 )

By inserting (2.6.91) into (2.6.83), we obtain for the wetting phase

-w Cw 'Ews (ws) m Itw .6.w Ow llij qrwj

(2.6.92)

An analogous equation can be written for the nonwetting phase. The two equations, one for the wetting phase and one for the nonwetting

phase, constitute a set of two equations in q~j and q~j. By solving this set for q;:wj, we obtain

where a (n w) = ---,,----_'E.....:n.....:w::...-,..--_

A + Aw ' H rf'c' ..-wc' ,.....n n ,...,w w

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186 MACROSCOPIC DESCRIPTION

and

(2.6.94)

can be regarded as the permeability of the w- phase in a two phase system. This definition conforms to that given by (2.6.55) for a single fluid phase that occupies the entire void space, with (}w = n, and n/'£W8 = Uow/ SW8

=dw •

An analogous equation, and a corresponding definition for the perme­ability, can written for the nonwetting phase.

The motion equation for the wetting phase can now be rewritten in the form

(2.6.95)

Similarly, the motion equation for the nonwetting phase, is given by

m nn ({)p;;n -n {)Z) qrnf. = - "'f.j {)Xj + pn 9 {)Xj

~w

(nw) (8p:;W +.,,-w {)Z) !:nw {)/wn "'f.J' -{)-- Pw 9 -{) - "'f.J' -{)-

Xj Xj Xj (2.6.96)

In these equations

"'ww = Afi [(I + ::~~a(nw>'T:-1'kn )-T:V(}w] .. , f.j J-Ln n '3

[ (nw) ] ",wn w a (nw) *-1 * f.j = Af.i ....-n(}2 a ·Tn .kn,Tn(}n .. ,

J-Ln n ')

[0~ ] ",nn = An, 1 + a a(nw),T*-l.kw f.j f., r(}2 w .. '

w w ')

(nw) [ (nw) ] n a (nw) *-1 * "'f.j = Af.i n-w(}2 a ·Tw ·kw·Tw(}w .. '

J-Lw w I)

",!:wn = A~ [a(nW)a(nw>.( d n 1 + '£nw a(nw),T*-l.k )] f.j f.1 re' r(}2 n n ,,' n n n n I)

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Macroscopic Fluxes 187

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

Assuming that

~_--::-_A11n

'O'Ywn oCw ----oCw OXi

~_-::-..... A11n 'O'Ywn oT

+ -o-T- -OX-i

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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 _ e (~ vmS) _ kwij ({)PWW -w {)Z) qrwi - w Vwi - si - - ~ -{) . + Pw g-{) . '

J.lw XJ XJ (2.6.98)

where

(2.6.99)

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Macroscopic Fluxes 189

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 assump­

tion 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.

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190 MACROSCOPIC DESCRIPTION

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),

and, for a binary system

(2.6.104)

(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 J -- vJ.l, (2.6.106)

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

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Macroscopic Fluxes 191

(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)

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192 MACROSCOPIC DESCRIPTION

For diffusion within the pore space, we may use the hydraulic radius ~ 1 as the characteristic length, Le. Thus, when the condition

( ~t)(P"Y) FoZ> = e > 1

- ~}/1)'Y (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 Uov , 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 Ga == p'Y. Hence, by averaging (2.6.103), making use of (2.3.51), we obtain

-;;:y _ V'Y {)p'Y _ V'YT* {)p'Y1 _ (1)*'1) .. {)p'Y1 J. -- ---n ··----n Jt--, J {)x j Jt {)Xi {)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

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Macroscopic Fluxes

(2.6.112) becomes

- ap"l p. = -1)"1-CtJ ax.

J

193

(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

-j I f)p"l p"l - p"l S J""Iv· - -1)"I-v· - 1) Is ·z- 1- "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 Uo. 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)

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194 MACROSCOPIC DESCRIPTION

,--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 FoA 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

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Macroscopic Fluxes 195

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 -s jlfm = nj7 + (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

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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

-0 ex --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 ea 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-

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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

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198

--v --

Abrupt front

at t = 0

-c=l

I .... ~------ L = V t

MACROSCOPIC DESCRIPTION

···········1 1

iMil t/j ·1

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

-v

-

o

Tracer injected at t = 0

Time t=o

(a)

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.

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Macroscopic Fluxes 199

1.0

~ ~

I~Actual C~ith dispersion)

/! I I.-- Without dispersion

0.5

~ o 1 2 3 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

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200

(a)

MACROSCOPIC DESCRIPTION

Direction of average flow ~

(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.

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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

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202 MACROSCOPIC DESCRIPTION

(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.

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Macroscopic Fluxes 203

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

~: + V.eVE == ~: + VE·Ve + eV·VE = prE. (2.6.125)

The corresponding averaged equation is given by

8ex 11 0 a a ~ - - - eu·v dB + VE·Ve + eV·VE = prE . 8t Uoa Saf)

(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 0 ~ 0 0 0 0 a 8t VE·Vea - eV·VE + (prE) - VE·(Ve) + VE·(Ve)

(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=to = x - xo(to) = xE(to). 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, VE , 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) I t=to+.:lt 0

= - Vea.VElxE(t) dr to

ft=to+.:lt 0 ~ + lto {(prE) - eV·VE + F}lxE(t),T dr

= -(xE(t) - xE(to))·Vealto+e.:lt ft=to+.:lt 0 ~

+ lto {(prE) - eV·VE + F}lxE(to),T dr,

(2.6.128)

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204 MACROSCOPIC DESCRIPTION

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.VEa !: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

or, in indicial notation D ,E _ Vb E oEa

ij - i Xj •

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

a 2 a

(2.6.132)

(2.6.133)

According to

Dxf 0E 1 D (xf) for i = j, -.. { otXi "2 Dt VEx.? =

t J (2.6.134)

oED 4>j oE a CE)2D 4>/'

Xi Dt Xi Xi Dt for i f; j,

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Macroscopic Fluxes

X' 1

T xlP

1

I-

Figure 2.6.4: Physical meaning of D'E.

205

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 oEOI

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)

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206 MACROSCOPIC DESCRIPTION

Le., DE represents a macroscopic rate of strain only. o Ci

Our next task is to express the components ViE xf ,in terms of compo-nents 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

o E 0 E Ci I 0 E 0 E Ci I 0 E 0 E Ci Vi Xj t = Vi ltj /Llt)o ~ Vi ltj (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 .eE PeE

Llt=(Llt)o=V;X+;~/LlCi -V:Ci 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.

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Macroscopic Fluxes 207

;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 veloc­ity, va, in that REV. With va and V in the directions indicated by the unit vectors 1s and 1s', respectively (Fig. 2.6.5), we may express the above transformation in the form

v: - (3*~ - (3*;-;C¥V .1 ' - (3*~V dXi dXi s' - I's' - S - , ds ds'

(2.6.142)

where (3* is a coefficient of proportionality. In the Xi - system, the above relation takes the form

V . - (3*.,.....--av. dx j J - 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 o 0 ~

and Vj = Tajl VI . (2.6.144)

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208 MACROSCOPIC DESCRIPTION

Hence

o 0 a_Q'~

TOIikTOIj/ Vk Vi . (2.6.145)

For heat transport, we replace V~ by >"01/ POlCVOI' so that

PeH = VaOl ~OI •

01 >"01/ POICVOI

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

IE 0 Eo EOI """"Q""""""""OO .E 0 .E 0 01

Dij - Vi Xj = {ViVj + (Ji /e)(Jj Ie) }(~t)o -01-01 E E E

f ETo TO 01 Vk Vi PeOi ( .!E/oe)( .!E/oe)OI fOi PeOi •

01 OIik OIjl VOl 1 + Pe~ + J t JJ VOl 1 + Pe~ (2.6.146)

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

V ·VE V·(VE - V) + V·V

= _V~V.(~e) _ ~ +V:V.(V;), (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 {D~ I(. '["7)} Pe~ f~ 1 + V·V (~t)o "" 1 + ~~ + p, v P Pe~ + 1 VOl

E cE ( 6.2 ) PeOi + 1 + -s; 1 + I ~ Pefl + 1

(2.6.148)

where (V) .01

I == I(p, V p) = V'[:V· : - ~ (2.6.149)

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

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Macroscopic Fluxes 209

(2.6.150)

-et where we have deleted the terms e ,and assumed

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, prE = 0, (2.6.151)

and (2.6.150) reduces to

(2.6.152)

where the coefficient

E Vk etvi'" PeE aetiklm et

vet Pe~ + 1 + fg' / ~et

E Wvi'" E E aetiklm vet f(Peet ,fet / ~et) (2.6.153)

is called the coefficient of mechanical dispersion (dims. L2T-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)

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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,

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Macroscopic Fluxes 211

(JO/ 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.

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

where

A' - {)x~ {)x~ {)x~ {)x~ A .. rs ... uv -!l !l • "!l!l 'J ... lm , ~ UXi UXj UXl uXm ~

n indices '- v ~ n indices n derivatives

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

(2.6.158)

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.

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212 MACROSCOPIC DESCRIPTION

A second rank tensor A, with components Aij, is said to be symmetric, if

for all i:f j. (2.6.159)

Thus, for example

* 3()~~a 3()~~a * Taij = TVaWaj = -() VajVai = T aji ,

a a (2.6.160)

defined by (2.3.49), and

~(3 ~(3 aij == Dij - VaWaj = Dji - VajVai = aji, (2.6.161)

defined by (2.6.40), are symmetric second rank tensors, because VaWaj == VajVai. The permeability of a fluid a-phase in a porous medium, is defined by (2.6.55) as

k ()a~~ ( )-IT * ajf. = c;;- aji aib (2.6.162)

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 im­

posed 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)

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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

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214 MACROSCOPIC DESCRIPTION

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

A·e=C, (2.6.169)

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 == IXb we obtain

from (2.6.168) (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)

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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.

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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 ()a~; ( )-IT * ajl = ----c;- aji ail· (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 = ~ ()~ ~2

a 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).

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Macroscopic Fluxes 217

(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)

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218 MACROSCOPIC DESCRIPTION

A comparison of (2.6.188) with (2.6.189), yields

E _ E _ _ (J~ E~ aO/L = aO/iiii - 2Q1 + Q2 - (JCX.e (Tii) , i = 1,2,3,

(no summation on i), (2.6.190)

E _ E (J~ E-o -2cx acxT = acxijij = Q2 = (Jcx.e (Tij) , i,j = 1,2,3,

(no summation on i or j),

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.

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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

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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)

In a Cartesian coordinate system with e = lXI, we have

and

Hence

Ttl = (Tl - T';)6I i6U + T';6il ,

aji = (aL-aT)6I j 6li+ aT6ji,

(ajitl = (a'Ll - arl )6lj6li + arl 6ji'

(2.6.200)

(2.6.201)

(2.6.202)

(2.6.203)

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Macroscopic Fluxes 221

(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

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222 MACROSCOPIC DESCRIPTION

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·C)(B·D), (A·D)(B·C),

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

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

(A.C)(B·e)(D·e), (A·D)(B·e)(C·e).

By a procedure similar to that employed above, we obtain

Qikj.e = Qlbikbj.e + Q2bijbk.e + Q3bi.ebkj + Q4eiekbj.e

+Qsbikeje.e + Q6eiejbk.e + Q7eibkje.e + QSeiekeje.e

+Q9bijeke.e + QIObi.eekej, (2.6.208)

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) + Q2bijbk.e + Q4(eiekbj.e + eie.ebkj)

+Qs(bikeje.e + bi.eekej) + Q6eiejbk.e + Qs(eiekeje.e)

+Q9bijeke.e, (2.6.209)

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

we obtain

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Macroscopic Fluxes

and (2.6.209) reduces to

Qikj£ = QI(OikOj£ + Oi£Okj) + Q20ijOk£

+Q4( Oikeje£ + Oif.ekej + Ojkeief. + OJ£eiek)

+Q60k£eiej + Q8eiekejee + Q90ijeke£,

Finally, symmetry in the indices i, k and j, f, i.e.

yields Q6 = Qg,

and (2.6.210) takes on its simplest form

Qikj£ = QI(OikOjf. + Oif.Okj) + Q20ijOkf.

+Q4(oikeje£ + Oi£ekej + Ojkeie£ + OJ£eiek)

+Q6(ok£eiej + Oijekef.) + Q8eiekeje£,

which contains only five scalar coefficients. By substituting

o 0 a Qikj£ = TO/ikTO/jf. ,

223

(2.6.210)

(2.6.211)

(2.6.212)

(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

-0 -20/ (Tn) - Qll11 = 2QI + Q2 + 2(Q4 + Qs) + Q6 + Q8 + Qg,

-0 -20/ -0 -20/ (TI2 ) = (TI3) -0 -20/ -0 -20/ (T22) = (T33) -0 -20/ -0 -20/ (T23) = (T32) -0 -20/ -0 -20/ (T2I) = (T3I ) ooa 000'

- QI2I2 = QI3I3 = Q2 + Q6,

= Q2222 = Q3333 = 2QI + Q2,

- Q2323 = Q3232 = Q2,

- Q212I = Q3I3I = Q2 + Qg,

TllT22 = TllT33 - Q1l22 = Q1133 = QI + Q4, boO! 000'

T22Tll = T33Tll - Q2211 = Q3311 = Ql + Qs.

From -o-o-a 0 0 ex TllT22 = T22Tll ,

(2.6.214)

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224 MACROSCOPIC DESCRIPTION

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 0 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

where

o 0 c; * E E{ *( ) = Tc;ikTc;j.e Tc;jmfc; = fc; Q1T2 Oik0.em + 0i.e0km

+Q2T;Ok.eOim + [Ql(Tt - Tn + Q4TnOik€.e€m

+[Q2(Tt - Tn + Q6TnOk.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}

= aNN ~ aNN' TN (Oik0.em + Oi.eOkm) + (aNN,TN )Ok.eOim

( * aNN - aNN' *) (l: 0 ) + CLNTL - 2 TN uik€.e€m + i.e€k€m

+(aLNTi - aNN' TN )Ok.e€i€m

( aNN - aNN') * ( ) + CLN - 2 TN Okm€i€.e + Om.e€i€k

+(aNL - aNN,)TNOim€k€.e

+[(aLL - 2CLN - aLN)Ti (2.6.216)

(2.6.217)

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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 2QlT2f~ = (aNN - aNN/)TN

a2 {Q2 + Q9cos2(e, ya)}T2f~ = {aNNI + (aNL - aNN/)cos2(e, YO<)}TN a3 2{Ql(Ti - Tn + Q4Ti}f~cos(e, y a )

{2CLNTZ - (aNN - aNN/)TN}cos(e, yo<)

a4 {Q2(Ti - Tn + Q6Ti}f~ +{(2Q4 + Q8 + Q6)(Ti - T2) + Q8T2}f~cos2(e, yo<)

aLNTZ - aNN,TN +{(aLL - 2CLN - aLN)TZ + (aNN - 2CNL - aNL)TN}cos2(e, yo<)

a5 2Q4T2f~cos(e, yo<) = {2CLN - (aNN - aNN' )}TNcos(e, va). (2.6.220)

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226 MACROSCOPIC DESCRIPTION

In the case of an isotropic dispersivity, all terms in (2.6.219) which con­tain components of e vanish and the coefficient of dispersion, D~m' reduces to the form given by (2.6.194), with Nand N' replaced by L and T.

2.6.6 Coupled fluxes

In discussing the diffusive mass flux in Subs. 2.6.3, we have neglected the phenomenon of coupling between microscopic fluxes, that stems from the interdependence, shown in Subs. 2.2.4, between the various state variables of a phase.

A more rigorous approach to the derivation of the macroscopic diffusive mass flux, and that of other extensive quantities, requires the averaging of the general form of the microscopic flux presented by (2.2.111), rewritten here in the form

)'? = -Lr: 8<lF R .. £:I , r=1,2, ... q, ... , , 2,)=1,2,3,

~ ~J UXj (2.6.221)

where q is the symbol that denotes an extensive quantity, ~r is the intensive quantity corresponding to the r extensive quantity (including r = q) of R coupled ones, and the L1; is the phenomenological coefficient that expresses the contribution of the gradient of a ~r intensive quantity to the flux of the q extensive quantity

By averaging j? over an a-phase, employing (2.3.51), i.e., neglecting the (or in the absence of) transfer of q across the interphase boundaries surrounding the a-phase within an REV, we obtain

where

-:q0l __ rqr 8~rOl R " 1 2 3 Ji - '-"k!:) , r=1,2, ... q, ... " 2,)=",

~ UXk

rqr _ LqrT* '-'ik - ij OIjk

(2.6.222)

(2.6.223)

is the macroscopic coefficient of the diffusive flux of q, stemming from the gradient of ~rOl.

Thus, it may be concluded that the solution of a macroscopic transport problem of any extensive quantity, always requires the simultaneous solution of the balance equations of that quantity and of all extensive quantities that are coupled to it at the microscopic level. '

Let us use this opportunity to comment on the use here of the term 'coupling' .

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Macroscopic Fluxes 227

In Subs. 2.2.4, the term 'coupled' (microscopic) fluxes of extensive quan­tities of a phase, was introduced to indicate the interdependence between gradients of state variables that correspond to different extensive quantities at the microscopic level. In the first part of the present subsection, we have shown the macroscopic counterparts of these fluxes. These should be em­ployed when writing the macroscopic balance equations for the corresponding extensive quantities of a phase in a porous medium domain.

However, the term 'coupling' is often used for additional situations.

• Consider a single phase that occupies the entire void space. Since, now at the macroscopic level, p = p(p, c, T), balance equations have to be written and solved simultaneously, for the mass of a phase, mass of a component and heat. We speak of 'coupling' between the macroscopic transport equations of the various extensive quantities. This coupling remains even if we neglect the coupling between the fluxes of these quantities, as developed above.

• When the concentration of a certain component in a multicomponent phase, is the state variable for which a solution is sought, coupling with the other components may be due to the fact that a source term that appears in the balance equation of one component, in addition to its dependence on the concentration of that component, is a function also of the concentrations of other ones, e.g., due to chemical reactions among them.

• Consider a number of fluid phases that together occupy the entire void space, or a solid and a number of fluid phases that together occupy a porous medium domain. Let us consider an extensive quantity that can be transferred across interphase boundaries, say, as a result of differ­ences in the corresponding intensive quantity. This exchange between the phases, taking place at the microscopic level, is expressed by the surface integral that appears in the macroscopic balance equation. The balance equations that describe the rate of change of the considered extensive quantities in the individual phases, are then linked, or cou­pled, to each other because of the interphase transfer terms. Therefore, they have to be solved simultaneously. In the absence of interphase exchanges, the problems are decoupled.

As an example, consider the case in which two, or three phases occupy the entire void space. For the sake of simplicity, let us assume as in the case discussed in Subs. 2.6.2 and in Chap. 5, that the flow is under

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228 MACROSCOPIC DESCRIPTION

isothermal conditions, that each phase is a single component one and that interphase boundaries are material surfaces with respect to mass transfer. The only extensive quantity that is being transferred across interphase boundaries is momentum. In Subs. 2.6.2, we show that macroscopic fluxes of two microscopically adjacent phases are coupled to each other, due to the transfer across the interphase boundaries of the tangential component of the momentum flux. In addition, even when the latter transfer is neglected, the transfer of the momentum due to the normal component of the momentum flux, leads to coupling between the two phases, in the form of a relationship between the difference in pressure between them (= capillary pressure) and their saturations. This results from the fact that we consider here a flux which is a second rank tensor .

• Consider the transport of an extensive quantity in two macroscopic domains with a common boundary (say, when a jump exists in the values of macroscopic transport coefficients along the latter). As we shall see in Sec. 2.7, the condition on such boundary is the equality of the normal component of the total flux of the considered extensive quantity, on both sides ofthe boundary. However, the value ofthe state variable for which a solution is sought, say, pressure, on the boundary (Le., in both domains as the boundary is approached) is, a-priori, not known. Under such conditions, the transport problems in the two domains have to be solved simultaneously. Again, we often speak of 'coupling' between the transport problems in the two domains.

2.6.7 Macrodispersive flux

In Subs. 2.4.8, we have encountered the megascopic flux

A _A Ct

eCt V Ct ,

referred to as the macrodispersive flux. Note that we have omitted here the subscript a that denotes the phase, except in the average symbol.

In order to express this flux in terms of megascopic values of e and V, We make use of the appropriate macroscopiC expression for the velocity of the considered phase. As an example, let us consider the isochoric flow of a Newtonian fluid phase for which the specific discharge is expressed by

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Macroscopic Fluxes 229

(2.6.98). For the particular case of an isotropic permeability and a rigid, stationary solid matrix, we may rewrite this equation in the form

After some algebraic manipulations, and neglecting the effects of varia­tions in density, we obtain

As to i, we may follow a procedure similar to that employed in developing the dispersive flux, viz., starting from a model in the form of the balance equation (see (2.4.7))

~ = _v.(e-vEa) at a,

and leading to the expression

(2.6.226)

where (.6.t)o denotes a representative travel time of an ea-particle through the RMV. Hence, we may define

(2.6.227)

where f*E is a characteristic radius of the RMV. Noting that

(2.6.228)

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230 MACROSCOPIC DESCRIPTION

we obtain from (2.6.226) and (2.6.227)

__ ~~a

~~a *E V V - -E-eV = -£ ·Ve = -D ·Ve

Va ' (2.6.229)

where, employing (2.6.225), and deleting variations of VIP, JI!', and 7f" within the RMV

(2.6.230)

where

(2.6.231)

represents the scalar macrodispersivity of the isotropic porous medium with respect to E, while DE is the coefficient of mechanical macrodispersion of E in the isotropic porous medium. Both coefficients are at a point in the megascopic space. It is of interest to note that while the dispersivity of a macroscopically isotropic porous medium is determined by two scalers, namely longitudinal, aL, and transversal, aT, dispersivities, with respect to the macroscopic velocity vector, the macrodispersivity in the same case is represented by a single scalar, A. As a consequence, there is no mechani­cal macrodispersion across macroscopic streamlines. However, the effect of diffusion does still exist as a result of averaging VE a over the REV.

2.7 Macroscopic Boundary Conditions

Macroscopic differential balance equations for extensive quantities such as mass, mass of a component, linear momentum and energy, were developed and presented in Sec. 2.4. In Subs. 2.2.3, we saw that balance equations have to be supplemented by constitutive equations that provide information on the behavior of the materials involved in any particular case of transport. Functions that represent the rate of production of extensive quantities that are relevant to the considered process, must be provided. Obviously, we also need information on the numerical values of all the parameters pertinent to the porous medium domain within which the transport phenomena of inter­est take place and to the phases contained in it. When all this information is put together, we obtain a closed set of equations, the solution of which

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Macroscopic Boundary Conditions 231

yields the values of state variables that describe the future distributions of the considered extensive quantities.

However, the above closed set of equations has an infinite number of possible solutions. To obtain from this multitude of possible solutions, a unique one, that corresponds to a particular case of interest, it is necessary to provide supplementary information, namely

(a) the configuration of the boundaries of the domain within which the phenomena under consideration take place,

(b) a description of the initial state of the considered system (= initial conditions), and

(c) a description of the interaction ofthe system under consideration with its environment, i.e., conditions on the boundaries specified in (a). These conditions are referred to as boundary conditions.

Different boundary conditions result in different solutions. Hence, the importance of stating them in a way that reflects the actual conditions of the problem on hand, for example, in terms of known fluxes, or known values of state variables that are imposed on the domain by its environment.

Altogether, to solve a problem of transport in a specified domain, means to determine the spatial and temporal distributions of certain dependent variables that satisfy the given set of equations and initial conditions at all points within the considered domain, as well as the conditions specified on its boundary.

As we shall see below, and more specifically in Part B, the various bound­ary conditions take the form of equalities between either the values of the dependent variables, or of fluxes, on 'both sides' of a considered boundary. In such equalities, the information related to the external side must be known. It is obtained from the known conditions that are planned to be maintained in the environment, or by assuming, on the basis of past experience (or even by guessing, subject to a-posteriori verification) the future situation that will prevail on the external side of the boundaries.

It is usually implicitly assumed that the interaction between the external world and the investigated domain is such that the former imposes known conditions on the latter, but is not affected by any processes that take place in it. However, sometimes the interaction between the two domains across the common boundary is such that the behavior in one domain is affected by what happens in the other, so that the conditions that will prevail on the

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232 MACROSCOPIC DESCRIPTION

common boundary are unknown a-priori. Nevertheless, as we shall see below, no matter what values will these conditions take on in the future, they must always satisfy certain functional relations. Under such conditions, the lack of information related to the external side of a considered domain, requires a simultaneous solution for both the considered domain and for the one that is external to it.

A typical example of a boundary ofthis kind occurs between two adjacent porous medium domains of different solid matrix properties (Subs. 2.7.3 and Subs. 2.7.4).

Since we are interested in the statement of problems at the macroscopic level, the boundary conditions must also be stated at that level. In the present section, we develop the general forms of these conditions.

2.7.1 Macroscopic boundary

In previous sections, we have often referred to transport phenomena that take place in a specific porous medium domain. Such a domain is bounded by a closed surface, possibly including segments at infinity. A considered do­main may be, for example, a subdomain of a larger porous medium domain, separated from the former by an arbitrarily chosen (mathematical) surface, possibly coinciding with a surface of discontinuity in any macroscopic pa­rameter characterizing the solid matrix. Another example is the surface that separates a porous medium domain from its environment, when the latter may be devoid of solid matrix (i.e., n = 1), or it may be devoid of any pore space (i.e., impervious, with n = 0).

Figure 2.7.1 shows four regions of different media: a region with no void space, two regions containing solid matrices of different porosities and a region with no solid matrix. By employing the methodology of averaging over an REV in order to determine the porosity, we note that the latter varies gradually as we move along the x-axis. Although we observe rather steep changes in porosity (shown by the S-shaped dashed lines), no abrupt change occurs. Thus, in the continuum sense, sharp boundaries that delineate the different media at the macroscopic level, do not exist anywhere along the x-axis. However, we recall that in defining a porous medium in Sec. 1.1 and in Subs. 1.2.3, we required that the variation of any macroscopic quantity (here porosity) over the REV be linear. If this condition is not satisfied in the region of transition from one porosity to the other, as shown, for example, in Fig. 2.7.1, the actual variation in porosity in every region of transition,

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Macroscopic Boundary Conditions

n Idealized boundary

Idealized boundary

Transition zone in which macroscopization conclitions are not satisfied

"

233

Continuous variation in n accorcling to the continuum approach

A-- --+~ -- - \d'JPb- B

Domain with no void space n=O

Porosity nl > n2

Porosity n2

.External domain without a Bolid matrix n = 1.0

Figure 2.7.1: Introduction of abrupt macroscopic boundaries.

must be replaced by an idealized boundary in the form of a surface across which an abrupt change in porosity takes place.

The boundary surfaces introduced in this way, divide the entire domain into sub domains separated from each other by sharp boundary surfaces. The continuum approach is applicable to each subdomain. Across the boundaries, we assume the existence of a jump in porosity and in other macroscopic ma­trix properties. On the two sides of each such boundary, the values of these properties are obtained by extrapolating the spatial linear trend in property values, as the boundary is approached from within each sub domain. The sharp boundary may be arbitrarily located at any point within the tran­sition region. For convenience, however, we usually locate it at the point corresponding to the mean porosity between the two adjacent regions (Fig. 2.7.1). In this way we hypothesize the existence of regular continuum do­mains for all phases present in the system up to the boundary surface on

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234

Fluid

MACROSCOPIC DESCRIPTION

Fluid

Fluid

REA (At)

Macroscopic boundary

Figure 2.7.2: Schematic microscopic interpretation of a jump in areal porosity.

both its sides. At the same time, discontinuities are introduced across such boundaries. Under such conditions, a given transport problem must be for­mulated separately for each region, with appropriate conditions specified on the common boundaries. These conditions describe the interactions between adjacent regions across the common boundaries.

As shown in Subs. 1.3.3, subject to conditions of linear variations of macroscopic quantities across an REV, the (volumetric) porosity is equal to the areal one. Hence, a jump in volumetric porosity implies also a jump in the areal porosity. Figure 2.7.2 illustrates this jump in the macroscopic sense, in the case of a single fluid that occupies the entire void space. We note the presence of fluid - fluid, fluid - solid, solid-fluid and solid-solid segments along the boundary. Similar considerations apply to a number of fluid phases that occupy the void space. Using the nomenclature of Fig. 2.7.2, this means that

and

One may argue that because the probability of a fluid - solid interface coinciding with the boundary is negligibly smail, the schematic diagram shown in Fig. 2.7.2 is unrealistic. Indeed, in the real (microscopic) mul­tiphase porous medium, no jump in any of the phases occurs across the (macroscopic) boundary. Also, in the strict sense of a continuum model, we cannot have two areal porosities (or two values of any other solid matrix

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Macroscopic Boundary Conditions 235

property) at the same point. However, as illustrated earlier in the discus­sion that led to the introduction of abrupt changes in porosity, the proposed conceptual model does not attempt to describe the real medium (in the mi­croscopic sense). Instead, our conceptual model assumes an abrupt change in the continuum properties across the boundary surface. In doing so, we hope that the resulting model will not deviate significantly from the real world. This assumption should be verified by experiments as part of the process of model validation.

So far we have considered boundaries which are either some arbitrary surfaces, or surfaces of (hypothetical) discontinuities in solid matrix proper­ties. However, a sharp boundary may be introduced as an approximation in three additional cases.

(a) A fictitious abrupt boundary between two miscible flu­ids. Here the 'two miscible fluids' may, actually, be the same fluid, but with different concentrations of certain components. The transport of these com­ponents is consi'dered in detail in Subs. 2.4.4, 2.6.2 and 2.6.3, and in Chap. 6. At this point, it is sufficient to note that, usually, a transition zone is created between two adjacent domains with different concentrations of the consid­ered component (or components) of the fluid occupying the void space. The concentration varies gradually across this transition zone. When the latter is narrow, relative to the two domains of interest, we may approximate it as a sharp boundary between two fluids, across which the concentration changes abruptly from that of one fluid to that of the other. The interface between fresh water and salt water in a coastal aquifer (Subs. 8.4.4) may serve as an example of such a fictitious sharp boundary between two miscible fluids.

(b) A boundary between two immiscible fluids. Here the satura­tion of each fluid, due to capillary effects (Sees. 5.4 and 8.5), varies gradually across a transition zone. If this zone is narrow relative to the domains of interest on its two sides, it may be approximated as a sharp boundary across which we stipulate a jump in the saturation of the considered fluids. The phreatic surface (Subs. 4.2.6 and Chap. 8) may serve as an example; the two fluids are air and water, and we assume that only water is present in the void space below this surface, while only air plus water at the irreducible water saturation is present in the void space above it.

(c) A boundary between different states of aggregation of the same material. Under certain conditions, the material in the void space

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236 MACROSCOPIC DESCRIPTION

undergoes a change of phase (=change of state of aggregation). Evaporation, condensation, freezing, thawing and melting, may serve as examples. When this phenomenon takes place within a relatively narrow zone in a porous medium domain, we introduce, as an approximation, a macroscopic bound­ary surface across which the phase change is assumed to take place. We assume that the void space on each side of such a boundary is completely occupied by a different state aggregation of the same material. The interface boundary may move as a result of the change of phase. Transport problems having such a boundary are called Stefan problems.

Because of the approximation involved in the introduction of sharp bound­aries to replace the transition zones that exist in reality in all the cases mentioned above, measurements within the transition zones should not be expected to compare with predictions obtained by solving the mathematical models which include such (hypothetical sharp) boundaries.

In general, a boundary surface may be stationary or moving. It may also be material or non-material with respect to any considered phase (see Subs. 2.1.5). Equations (2.1.23) through (2.1.25) are valid, with F(x, t) = 0 denoting the equation of a boundary surface, u denoting the velocity of points on it, and 1.1 denoting the outward normal unit vector on it.

2.7.2 The general boundary condition

The general boundary condition for the extensive quantities mass, mass of a constituent, momentum and energy, all of a phase in a multiphase contin­uum, arises from the balance of that quantity as it is transported across the boundary. Intuitively, we could state that in the absence of sources and sinks of a considered extensive quantity on the boundary, the total amount of that quantity, transferred by all phases present in the porous medium domain, must be conserved as it is being transported across the boundary. However, usually, we need information regarding the transport of the various extensive quantities by each phase separately. Had we assumed that conservation of an extensive quantity is maintained separately for each phase, then the quantity that is transported across the boundary in any considered phase should have entered only that phase on the other side of the boundary. However, this as­sumption violates our conceptualization of the boundary as shown in Fig. 2.7.2, where the jump in areal porosity requires that the phases comprising the medium cannot be completely continuous across the hypothesized con­tinuum boundary. This implies that in addition to a portion of the boundary

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Macroscopic Boundary Conditions 237

5 = 51 + 8 2

U = U1 + U2

= -UV2

Figure 2.7.3: Definition sketch for developing the general boundary condition.

across which phases remain continuous, surfaces of contact must also exist on the boundary between any given phase and all other phases. Across such surfaces of contact, extensive quantities may be transported from one phase to another that faces it on the other side of the boundary.

To obtain the balance of an extensive quantity across a boundary for an individual phase in a multiphase system, let us consider first a porous medium domain with only two phases - a and (3. Figure 2.7.3 shows a porous medium domain, U(t), that is composed of two parts: UI(t) and U2(t), separated from each other by a common macroscopic surface S*(t). The domain U(t) is bounded by the surface Set) (= SI(t) +S2(t». The outward normal unit vector to Uf., .e = 1,2, on S*, is denoted by Vf., with VI

= -V2 == v. The outward normal unit vector on S is denoted by N. Each surface may be moving at a velocity u f::. yE.

The macroscopic global balance of an extensive quantity, E, having a density, e, within Uf., .e = 1,2, is obtained by writing the macroscopic equiv­alent of (2.2.8), employing (2.1.50) and Gauss' theorem (2.1.45)

f (Be + V.eyE _ prE) dU lUl(t) Bt

= f e(yE - u).v dB + f e(yE - u)·N dB, (2.7.1) ls;(t) lSl(t)

where the overbar indicates the volume average defined by (1.3.7).

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238 MACROSCOPIC DESCRIPTION

The macroscopic global balance equation of E over U (= U1 + U2 ),

bounded by 8 (= 8 1 + 8 2) is given by

f (ae + V.eVE _ prE) dU }U(t) at

= f rSE dB + f e(VE - urN dB, (2.7.2) }s*(t) }Sl (t)+S2 (t)

where r SE denotes the sources of E on 8*, when such sources exist. By writing (2.7.1), once for .e = 1 and then for .e = 2, adding the two

equations, and comparing the sum with (2.7.2), we obtain the condition

(2.7.3)

Since (2.7.3) is valid for any size of 8*, the integrand must vanish at every point of 8*. Hence, the condition

(2.7.4)

must be satisfied at every point of the boundary 8*. By making use of (1.3.7), we rewrite (2.7.4) in the form

L:[8aea(V~ - uth,2'V + L: 8ar~Ea = 0, (2.7.5) (a) (a)

or

L:[8a{ ea a(v a a - u) + J*E + jEUa} h,2'V + L: 8ar~Ea = 0, (2.7.6) (a) (a)

---o---a where L(a) 8a = 1, J*E (== ea Va) denotes the dispersive flux of E (see

Subs. 2.6.4) and jEUa denotes the averaged diffusive flux of E. Equation (2.7.6) represents the general macroscopic boundary condition

for any extensive quantity, E, in a porous medium. We note that it expresses the notion that E does not accumulate on the boundary. The difference in flux between the two sides is balanced by the (possible) production of Eon the boundary. Obviously, (2.7.6) can be modified to include the possibility of accumulation of E on the boundary. In the absence of sources on the boundary, (Le., rSE == 0), equation (2.7.6) reduces to the continuity of the total flux across the boundary

L: [8a{ ea a(v a a - u) + J:E + j~Ua} J1,2'V = 0, (2.7.7) (a)

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Macroscopic Boundary Conditions 239

where we note that the total flux is made up of advection relative to the (possibly moving) boundary, dispersion and diffusion.

In the following subsections, we shall consider particular cases of E and boundary types.

In practice, usually, the boundary conditions used in models describing transport phenomena in porous media, do not take the no-jump forms de­veloped above. Instead, they take the form of specification of the values of variables, or of their derivatives, on the boundary. Since, however, these conditions are supposed to represent the true physical aspects of the trans­port phenomena, they must satisfy the no-jump condition developed above, as the latter expresses nothing but a statement of continuity of the total flux of the considered extensive quantities across the boundary.

In Part B, the general boundary condition developed here is applied to cases of transport of specific extensive quantities. We shall see there how the general condition - expressing flux continuity - is reduced to the more familiar and simple forms of boundary conditions, specified in terms of variables and/or their derivatives. In particular, we shall examine the assumptions underlying the reduction of the general no-jump condition to these simplified forms.

Boundary conditions for specific extensive quantities are presented below. The intrinsic phase average symbol is omitted, as the entire discussion will be at the macroscopic level.

2.7.3 Boundary conditions between two porous media in single phase flow

Consider a (macroscopic) boundary across which there exists a discontinuity in the volumetric porosity and hence also in the areal one. Let the entire void space be occupied by a single fluid phase.

We shall consider the transport of mass, mass of a component, momen­tum and energy, in the absence of sources of these extensive quantities on the boundary. The general boundary condition for any of the extensive quanti­ties is given by (2.7.7), in which Q represents both a solid (8), and a fluid (f).

Whenever the considered extensive quantity is such that any portion of the boundary that is an interface between two phases, i.e., a solid-fluid, or fluid-fluid interface, is material with respect to that quantity, the no-jump boundary condition can be written separately for each phase.

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240 MACROSCOPIC DESCRIPTION

(a) Conditions for mass of a phase

In the absence of phase change (see Subs. 2.7.7), the fluid-solid portion of a boundary is a material surface with respect to (advective, dispersive and diffusive) mass (transport). We shall express this in the form of an assumption

[A2.1] Mass is not transported across the solid-·fluid portions of the (macro­scopic) boundary.

Under this assumption, with ea in (2.7.7) replaced by the mass density, Pa, with a = s for the solid and a = f for the fluid that occupies the entire void space, i.e., Of == n, and Os == 1 - n, the general (no-jump) boundary condition for mass of a phase, can be written separately for each of the two phases, in the form

a = f, s, (2.7.8)

where J~ and J~m denote the diffusive and the dispersive mass fluxes, re­spectively, of the a-phase. No diffusive mass flux of the solid exists. We recall that all symbols in (2.7.8), and in what follows, denote intrinsic phase averages.

Let us q:msider a number of particular cases of (2.7.8). We shall assume that

[A2.2] The dispersive flux of any extensive quantity in the solid phase may be neglected.

Therefore, for the solid phase, (2.7.8) reduces to

(2.7.9)

Let us further limit the discussion, here and henceforth in this section, by assuming that

[A2.3] The (macroscopic) boundary is a material surface with respect to the mass of the solid phase, i.e., despite deformation and movement of the boundary, no solid mass is transported across it.

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Macroscopic Boundary Conditions 241

Hence

(2.7.10)

Vsl1'v = V sl2 'v = u·v.

Assumptions [A2.2] and [A2.3] thus reduce (2.7.9) to an expression that provides no information with respect to [Psh,2' The mass density of the solid phase on each side of the boundary, may take on a different value. Assump­tion [A2.3] also implies that there is no jump in the normal component of the solid phase displacement, w, i.e.

[Wh,2'V = 0. (2.7.11)

If we now add the assumption that

[A2.4] The solid's displacement in the direction tangent to the boundary is the same in both porous medium domains, i.e., the domains do not slip with respect to each other.

we obtain [Wh,2 = 0, (2.7.12)

which is a more stringent condition than (2.7.11). The velocity ofthe bound­ary, defined by u·v = d(w·v)/dt, should be used in all the expressions in which u appears. From (2.7.12) it follows that [Vsh,2 = 0, with u = Vsll = (dw/dt)lm' £ = 1,2.

For the fluid phase (a = /), equation (2.7.8) takes the form

[OfPf(Vf - u) + Of(Jj + Jjm)h,2'V = 0,

or, in view of (2.7.11)

where qrf(= Of(Vf - Vs)) is the fluid's specific discharge relative to the solid; it can be expressed by an appropriate motion equation (Subs. 2.6.1).

Following (2.6.112) and (6.1.1), both written for Ph we can expressed the sum of the diffusive and dispersive mass fluxes by

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242 MACROSCOPIC DESCRIPTION

Then, (2.7.8) takes the form

[p,qr, - O,(Djh.Vp/)h,2·Y = 0, (2.7.13)

This is the most general boundary condition that can be stated for the fluid's mass. However, it may be further simplified by the introduction of a number of assumptions.

To begin with, we recall that in the continuum model employed here, a discontinuity in solid and fluid properties (including, for example, density) is possible. Although in the fictitious continuum model employed here, a discontinuity in all state variables is acceptable at a macroscopic boundary, we shall assume that

[A2.5] At every point on a macroscopic boundary, there exists no disconti­nuity in the (intrinsic phase averages of) scalar intensive quantities.

Under this assumption, we must have [p/h,2 = 0, as otherwise, the in­finite density gradient associated with a discontinuity in p, will instanta­neously equalize the densities on both sides of the boundary through the process of diffusion.

We note that from this assumption, it follows that the continuity in the flux of mass across a boundary, implies also a continuity in volumetric flux. The error introduced as a consequence of this assumption is imbedded as an error in the velocity.

With this assumption, (2.7.13), reduces to

(2.7.14)

Subject to assumption [A2.5], equation (2.7.14) is the most general bound­ary condition for mass flux across a boundary between two porous media, in the case of a single fluid phase that occupies the entire void space.

Let us further assume that (see assumption [A4.1])

[A2.6] The sum of the fluid's diffusive and dispersive mass fluxes is negligible, with respect to the advective one.

Then, (2.7.14), combined with [P,h,2 = 0, reduces to

[8,V']1,2'Y == [q,h,2'Y = [O,h,2 U ' Y '

(2.7.15)

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Macroscopic Boundary Conditions 243

Since [O/h,2 =1= 0, equation (2.7.15) indicates that the normal components of the volume weighted velocity of the fluid, on both sides of the boundary, are not the same.

With [A2.6]' but without [A2.5], the general boundary condition (2.7.13) reduces to

(2.7.16)

(b) Condition for mass of a component of a phase

For the mass flux of a component of a fluid phase, we also invoke as­sumption [A2.5], which in this case means

(2.7.17)

where p'Y is the concentration of the ,-component in the fluid phase. Actually, we have to be careful in writing the no-jump condition (2.7.17),

as this condition may be valid only for the chemical potential of the compo­nent. Nevertheless, we shall continue to use here the notation p'Y, with the understanding that whenever necessary, it will be replaced by an appropriate measure of the concentration of the component.

When the fluid-solid portion of the boundary is also material with respect to the mass of the ,-component, considerations similar to those that lead to (2.7.14), yield here, for the fluid alone

p'Y[qr/h,2·V - [OID]h·V p'Yh,2·V = 0, (2.7.18)

where we have made use of (2.6.112) and (2.6.152) to express the sum of the diffusive and dispersive fluxes of the ,-component, in terms of V p'Y.

By summing (2.7.18) for all components, with E("!) p'Y = Pj, we obtain (2.7.14). Equation (2.7.18) is the most general condition for the mass flux of a ,-component across a boundary, in terms of p'Y, provided we accept [A2.5], i.e., that [PI h,2 = 0.

If we also invoke assumption [A2.6], which leads to (2.7.15), then (2.7.18) reduces to

(2.7.19)

(c) Condition for the momentum of a phase

Unlike the two extensive quantities considered above, momentum can cross (by diffusion) from one phase to another through a portion of the boundary that is common to both.

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244 MACROSCOPIC DESCRIPTION

As can be seen in the first term on the r.h.s. of (2.4.20), the total mo­mentum flux of any phase is composed of advective, dispersive and diffusive fluxes. Accordingly, the general no-jump condition, (2.7.7), is written for the momentum flux of the porous medium as a whole, in the form

I:[Oa{Pa V:(V: - u) + J~M - 0' a} h,2'V = 0, a = j, s, (2.7.20) (a)

where J~M denotes the dispersive momentum flux, and we have neglected the dispersive mass flux.

Note that (2.7.20) implies that continuity is required for the force acting on the boundary whose normal is v. Because the flux of momentum is a second rank tensor, the no-jump condition is required not only for the normal component of the force, but also for the tangential one. Momentum is transferred also (through shear) by the latter component.

Making use of [A2.2] and [A2.3], equation (2.7.20) reduces to

[OJ{PJ Vj(Vj - u) + JjM - 0' J} - Os 0' sh,2'V = 0, (2.7.21)

which states a condition of no-jump, or continuity, in the total momentum flux across the boundary in both phases.

One can adopt the assumption that

[A2.7] The sum of advective and dispersive momentum fluxes of the fluid across the boundary is much smaller than the diffusive one, expressed by -O'J.

Then, (2.7.21) reduces to

[O'h,2'V == [OJO'J + OsO'sh,2'V = 0, (2.7.22)

where 0'(== 0') denotes the total stress in the porous medium. As we have seen in Subs. 2.5.2, deformation in a porous medium is not

produced by the phase average stress in the solid, 0' s, alone, but by the effective stress, O'~, that takes into account the effect of the .pressure in the fluid enveloping the solid phase. Accordingly, making use of (2.5.21), another way of writing (2.7.22) is

(2.7.23)

where O'~ = Os(O's - O'J) is the effective stress that produces solid matrix deformation in a saturated porous medium, (with O'~ == O'~, O's == O'ss,

O'J == 0'/. As is often done in flow through porous media, we may assume that

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Macroscopic Boundary Conditions 245

[A2.8] The (macroscopic) viscous part ofthe fluid stress vector acting on the boundary is negligible, when compared with the force due to pressure, i.e.

Then, (2.7.23) reduces to

[O'~ - pIh,2'v = O.

If [A2.5] can be adopted, then

[Pfh,2 = o. and (2.7.24) reduces to the condition

[0'~h,2'V = O.

(2.7.24)

(2.7.25)

(2.7.26)

We wish to emphasize again that (2.7.25) was obtained from [A2.5] and not from the boundary condition (2.7.20), which expresses the continuity of momentum flux.

( d) Condition for energy

Energy can be transported across portions of the boundary that are common to two phases. We can, therefore write a no-jump condition only for the porous medium as a whole, and not for the individual phases, separately.

The total energy flux in any a-phase, is given by

where we have neglected the effects of velocity fluctuations within the REV. Employing [A2.3] and (2.7.11), and neglecting the dispersive heat flux in

the solid, J;H, the no-jump condition, (2.7.7), for the energy of the porous medium as a whole, takes the form

[OfPf(If + HVr)2)(Vj - u) - OfO'f"Vj + Of(Jlj + JjH)h,2'V

+ [-OsO's·V:n + OsJHh,2'V = o. (2.7.27)

Let us consider the expression

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246 MACROSCOPIC DESCRIPTION

appearing in (2.7.27). We can rewrite this expression in the form

or

[OJ(Vj - U).Uj]1,2·'" + [OjUj]1,2 U.'"

+[Os(V~ - U).Uslt,2·'" + [OsUsh,2U·"',

[OJ(Vj - u).Uj + Os(V~ - u)·Ush,2·'" + u.[OjUj + OsUsh,2·"'.

In view of equations (2.7.11) and (2.7.15), the first term reduces to OJ(Vj - u)[Ujh,2·"'. However, in view of assumption [A2.5], [Ph2 = 0, [rh2 = 0, and hence we have [u j h,2 = o. The second term vanishes in view of (2.7.11). The last term vanishes in view of (2.7.22). Hence, (2.7.27) reduces to

[Ojpj(Ij + HVt)2)(Vj - u) + OJ(J7 + JjH)h,2·'"

+[OsJ~h,2·'" = O. (2.7.28)

As stated above, the fluid-solid portion of the boundary is not a material surface with respect to (the conductive part of the) energy transport. Hence, as in the case of momentum, energy may be exchanged between the fluid phase and the solid one across their common portion of the boundary, and energy is not conserved within any of the phases alone. Equation (2.7.28) includes the possible exchange between the two phases, and it is impossible to separate this equation into two equations, one for each phase.

Let us add the assumption that

[A2.9] The fluid and solid phases are everywhere, locally, in thermal equi­librium.

i.e., Ts = Tj == T. Then only one energy balance equation is required to describe the thermodynamic state of the fluid-solid system as a whole. For convenience, we select the equation of balance (2.4.41) for the combined fluid-solid system. This means that boundary conditions should also refer to the combined fluxes through both the fluid and the solid phases. The exchange between the phases need no longer be considered.

Note that even if we choose to construct a model for Ts f:. T" we have a condition only on the combined fluxes in the two phases.

One should also note the difference between [A2.9] and [A2.5]. While the former stipulates local equilibrium at the macroscopic level, between two

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Macroscopic Boundary Conditions 247

phases, the latter states that local thermodynamic equilibrium exists within each phase, considered as a continuum, as well as at points on the interface between phases.

Let us further assume that

[A2.1O] The flux of kinetic energy is negligible with respect to the thermal one.

Then, (2.7.28) reduces to

{PfQrf[If h,2 + [ L: BaJ !! + Bf JjH h.2}·V = O. (2.7.29) (a=f,s)

At this point, we introduce specific (to the considered phases) expressions for the internal energy, I" and for the diffusive and dispersive heat fluxes appearing in (2.7.29). For example

with [Cvfh,2 = 0, (2.7.30)

and

'"' B (JH + J*H) - _A*H·VT L..J aa a - , (2.7.31) (a=f,s)

with CVf denoting the fluid's specific heat at constant volume and A*H denoting the sum of the thermal conductivity of the combined fluid-solid continuum and the coefficient of thermal dispersion in the fluid (noting that all along we have neglected the dispersive heat flux in the solid).

Then, (2.7.29) can be rewritten in the form

(2.7.32)

Since, by [A2.5], we have

[Th,2 = 0, (2.7.33)

equation (2.7.32) reduces to the condition

(2.7.34)

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248 MACROSCOPIC DESCRIPTION

2.7.4 Boundary conditions between two porous media in multiphase flow

Here, two or more fluid phases occupy the void space on both sides of the boundary. To generalize the discussion, we shall assume that each phase is composed of a number of components, some of which are capable of diffusing in more than one of the fluid phases. However, we shall assume that these diffusing components constitute but a small fraction of the mass of these phases, so that the latter may still be regarded as immiscible fluids. The discussion will be limited to two fluid phases only: a wetting fluid, denoted by subscript w, and a nonwetting one, denoted by subscript n.

We shall consider a number of extensive quantities: mass of a phase, mass of a component of a phase, momentum and energy. The general no-jump condition for any extensive quantity, E, is given by (2.7.7).

The general no-jump condition (2.7.7) reduces to

L [Oaea(Va - u) + Oa(J~ + J:E )h,2·V

(a=w,n)

+[OsPs(Vs - u) + Os(J~ + J:E)h,2·V = o. (2.7.35)

As we shall see below, whenever the considered extensive quantity cannot CluSS (microscopic) interphase boundaries, the no-jump boundary condition can be written separately for each phase. Otherwise, because of the pos­sible transfer from one phase to another across their common portion of the boundary, only one no-jump condition can be written for all the phases together.

(a) Mass of a fluid phase

In this case, ea = pa, 0: = w, n, and we assume that

[A2.11] Any interface between phases is a material surfaces with respect to the mass of the phases.

In this case, the general no-jump condition (2.7.7) is

L [OaPa(Va - u) + Oa(J~ + J:m)h,2·V

(a=w,n)

+[OsPs(Vs - u) + Os(J: + J:m)h,2·V = o. (2.7.36)

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Macroscopic Boundary Conditions 249

By employing [A2.1], [A2.3] and [A2.11], the no-jump condition (2.7.36) can be decomposed into separate no-jump conditions: one for the mass of each of the fluid phases. The term for the mass of the solid phase vanishes. The condition for each of the fluids is the same as the one written for the case of a single fluid phase that occupies the entire void space, namely (2.7.8).

(b) Mass of a component of a phase

Here, ea = p~. We recall that a considered ,-component may diffuse from one fluid to another across their common portion of the boundary.

Because, by [A2.1], no mass of a component crosses the fluid-solid portion of the boundary, (2.7.36) reduces to

L [Oap~(Va - u) + Oa(J~ + J~'Y)h,2·V = o. (2.7.37) (a=n,w)

In view of assumptions [A2.5] and [A2.6], written for Pa, we may write

a=w,n. (2.7.38)

Also, by invoking [A2.5], we may write

[P~h,2 = 0, [p~h,2 = o. (2.7.39)

Note the comment that follows (2.7.17). Hence, in view of (2.7.39) and (2.7.38), equation (2.7.37) reduces to

L [Oa(J~ + J~'Y)h,2·V = 0, (2.7.40) (a=w,n)

or L [OaD:h "v7 P~h,2·V = o. (2.7.41)

(a=w,n)

We note that this boundary condition expresses the continuity of the sum of diffusive and dispersive fluxes in both fluid phases. Because, upon crossing the boundary, the component can diffuse from one fluid phase into the other, it is impossible (and perhaps not necessary) to write a separate condition for the flux of the component in each phase.

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(c) Momentum of a phase

For momentum, e = pVm • Using [A2.1] and [A2.3], in the form

L [8aPa V:(V: - U)h,2'11 - [8sO's + 8aO' ah,2'11 = O. (2.7.42) (a=w,n)

By invoking [A2.7]' we obtain

[O'h,2'1I =: [8sO's+ L 8aO'a] '11=0, (a=w,n) 1,2

(2.7.43)

to be compared with (2.7.22), i.e., no jump takes place in the component of the total stress normal to the boundary.

We note the coupling that appears in the boundary conditions for mo­mentum, (2.7.42) and (2.7.43), between the fluid and the solid phases because of the (hypothesized) contact on the boundary between the solid phase and each of the fluid phases.

Following the discussion on effective stress in Subs. 2.5.2, and the ideas leading to (2.7.23) for a single fluid phase that occupies the entire void space, we express the effective stress, O'~, for the multiphase case, by (2.5.25), or, more generally, by

0'~=8s(O's-~ L 8aO'a) =8s(O's- L SaO'a) , (a=w,n) (a=w,n)

(2.7.44)

where the second term in the parentheses represents the average effect of the fluids that 'surround' the solid matrix. In determining the average pressure of the fluids that occupy the void space, the weight given to each of the fluids is its saturation

Sa( = 8a/n), with L Sa = 1. (a=w,n)

Other weights may also be introduced. With (2.7.44), equation (2.7.43) becomes

[O'h,2'11 =: [O'~ + L SaO'a] '11 = O. (a=w,n) 1,2

(2.7.45)

For the sake of simplicity, let us assume that [A2.8] is valid separately for each fluid, and that [A2.5] can be applied to the pressure of each of the fluid phases, i.e.

[Pnh,2 = 0, [Pwh,2 = O. (2.7.46)

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Macroscopic Boundary Conditions 251

W€ then obtain from (2.7.45)

[q~h,2'V - [Snh,2PnI·v - [Swh,2PwI·v = 0. (2.7.47)

Since Sw + Sn = 1, another form of (2.7.47) is

(2.7.48)

where the difference Pn - Pw (both average values) is recognized as the cap­illary pressure, Pc(Sw), discussed in Subs. 2.5.1 and 5.5.1.

Any attempt to further simplify (2.7.48), e.g., by decomposing it into separate conditions on [q~h,2, or on [Swh,2' is not possible. From (2.7.46) and the discussion to be presented in Sec. 5.1, it follows that on a boundary between two porous media in multi phase flow, we must have [Pc]1,2 = 0, and hence we must also have [Swh,2 =J 0. If we assume that [q~h,2'V = 0, as in (2.7.26), then we shall obtain either Pn - Pw = 0, or [Swh,2 = 0, on the boundary. Both conclusions are unacceptable, as Pc = ° only for Sw = 1.

From this discussion we must conclude that in the case of multiphase flow, (2.7.26) and (2.7.46) can be used simultaneously, only as an approximation of the single condition (2.7.48).

2.7.5 Boundary between two fluids

Here we consider the condition on an (assumed) abrupt boundary (=inter­face) between two fluids that occupy sepamte domains in a porous medium. In Subs. 2.7.1, this interface was introduced as an approximate transition from one fluid to another, whether these fluids are miscible, or immiscible. The solid matrix itself undergoes no change in porosity (nor in any other property) across this kind of boundary, so that [Oah,2 == [nh,2 = 0, where a denotes a fluid phase.

(a) Mass of a phase Since in this case, the boundary contains no fluid-solid part, the condi­

tions for the mass of the fluid and of the solid, can be separated. Hence, for the fluid, we obtain from (2.7.7)

(2.7.49)

We note that no information on [Pfh,2 can be derived from (2.7.49), and, in general, we may have [Pfh,2 =J 0.

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No condition is required for the mass of the solid. Note that in this case, the interface is not a material surface with respect to the solid.

Since

[A2.12] The (possibly moving) fluid-fluid interface is a material surface with respect to the mass of each fluid.

and recalling that we have a different fluid on each side of the boundary, equation (2.7.49) reduces to a separate condition for each fluid, viz.

f = 1,2, (2.7.50)

where we have expressed the sum of diffusive and dispersive mass fluxes Jj and Jjm, respectively, in terms of V' Pj' If conditions permit the application of [A2.6] to the considered problem, then for the fluid on each side of the interface, (2.7.50) reduces to

(Vj - u)lrv = 0, f = 1,2, (2.7.51)

and

(2.7.52)

Otherwise, (2.7.50) remains the boundary condition for each of the fluids.

(b) Mass of a component Although we consider here a hypothetical abrupt interface between either

miscible or immiscible fluids, let us discuss the case where

[A2.13] Each fluid phase is composed of a number of components, some, but not all of which are capable of diffusing through the interface, into an adjacent fluid. However, the diffusing components contribute only a very small fraction of the fluid's density.

In this way, the interface remains a material surface with respect to the total mass of each of the fluids, but not necessarily with respect to the mass of a considered I-component. The no-jump condition for the normal mass flux of a I-component is expressed by

(2.7.53)

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Macroscopic Boundary Conditions 253

where p} denotes the concentration of the i-component in the fluid phase. We note that in (2.7.53), a different fluid is present on each side of the boundary i.e., f has a different meaning for £ = 1 and £ = 2.

In view of (2.7.51), equation (2.7.53) reduces to

(2.7.54)

Equation (2.7.54) implies nothing with respect to [p}h,2' unless we invoke [A2.5] to yield (2.7.17), i.e., lP}h,2 = O. If [A2.5] is applicable, but [A2.6] is not, then (2.7.53) becomes

(2.7.55)

For a component that does not diffuse across the boundary, when neither [A2.5] nor [A2.6] is applicable, the condition is

{P}(Vf - u) - D}h:~7p}n~·v = 0,

When [A2.6] is applicable, the condition is

(c) Momentum

£ = 1,2. (2.7.56)

£ = 1,2. (2.7.57)

In view of (2.7.51), condition (2.7.7), applied to momentum, becomes

(2.7.58)

Because there exists no solid-fluid portion of the boundary, we may write the no-jump condition separately for each phase. Hence, we write

[ush,rv [ufh,2'V

0,

-[Pf ]I·v + [r f h,2'V = O. (2.7.59)

(2.7.60)

At this point, we invoke [A2.8], and [A2.5], which leads to (2.7.25), i.e., to

Then, (2.7.60) reduces to [U~h,2'V = o. (2.7.61)

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254 MACROSCOPIC DESCRIPTION

(d) Energy

From I, = Cv,T" equation (2.7.31), and the conditions [Psh.2 = 0, and [Th,2 = O,'based on assumption [A2.5], it follows that the no-jump condition for the total energy flux (through both the solid and the fluid) takes the form

TO,[p,Cv,(V, - U)h.2·V - [A*H:VTh.2·V = 0, (2.7.62)

where A*H (= O,A, + OsAs + O,Dlj), defined by (2.7.31), is different on each side of the boundary ..

If [A2.6] is also applicable, (2.7.62) is further reduced to

[A*H.VTh.2·V = 0. (2.7.63)

Shape of the interface

The interface velocity, u, or its normal component, U·V, appears in some of the equations developed above. Since U is a-priori unknown, these equa­tions, in their present form, cannot be used as boundary conditions. In fact the interface equation, F(x, t) = 0, may be regarded as an additional un­known variable for which a solution is sought. It is, therefore, necessary to express F(x, t) in terms of the state variables of the problem. Once F(x, t) is known, (2.1.25) may be used to determine u·v. An interface of this kind is often referred to as a free surface.

In principle, the explicit functional form of F(x, t) can be derived from any of the no-jump conditions of a considered given problem. For example, if condition (2.7.63) is valid, we may use (2.1.25) to write

F(x, t) == A*H.VTll·V F - A*H.VTI2·V F = 0. (2.7.64)

However, this equation expresses F(x, t) in terms of V F. It is much more convenient to derive F(x, t) from a condition of no-jump in the value of the state variable of the problem. Denoting the state variable, e.g., pressure, concentration, or temperature, by O(x, t), the no-jump condition [Oh,2 = 0 on the boundary can be used to define the shape of the interface in the form

F(x, t) == O(x, t)ll - O(x, t)1z = 0. (2.7.65)

Actually, the problem is non-linear, as the values of O(x, t) in both do­mains are not known a-priori. However, once these values have been deter­mined (say, by some iterative technique), (2.7.65) can be used to obtain an explicit expression for the shape of the interface.

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Macroscopic Boundary Conditions

A well mixed zone

I I I I I Io.-Li I I I I I I .

·1- Porous medium

Figure 2.7.4: Nomenclature for a well mixed zone.

2.7.6 Boundary with a 'well mixed' domain

255

To simplify the presentation in this subsection, we shall refer, as an exam­ple, to the concentration, p''I, of a ,-component of a fluid, say, water, that occupies the entire void space.

Consider a saturated porous medium domain in hydraulic contact with a body of fluid (e.g., a river, or lake), and let the fluid contain a ,-component which will serve as the considered extensive quantity (Fig. 2.7.4). No sub­script will be used to denote this fluid. We assume that

[A2.14] The external body of fluid is 'well mixed' i.e., it is everywhere at a uniform value of the considered state variable along the normal to the boundary at every point of the latter.

In the present example, the condition of no-jump in the total flux of the considered component, takes the form of (2.7.53), in which Of is replaced by

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256 MACROSCOPIC DESCRIPTION

the porosity, n. This condition may also be written in the form

{p"(V - u)}lfb·v - {p"Y(q - nu) + n(J"Y + J*'Y)}lpm·v = 0,

(2.7.66)

where q = nV is the specific discharge of fluid in the porous medium, sub­scripts fb and pm denote the fluid body and the porous medium, respec­tively, Blfb = 1, and p" denotes the (assumed constant) concentration of the ,-component in the 'well mixed' fluid body. Equation (2.7.66) expresses the continuity of the mass flux across the boundary, of the ,-component in the water, by advection, diffusion and dispersion. Since the fluid body is assumed to be at a uniform concentration at the boundary, only advection takes place in it. Assumption [A2.1] underlies (2.7.66). In view of (2.7.11), we could replace (q - nu) in (2.7.66) by qr.

To simplify the presentation, let us assume that the boundary is station­ary, i.e., u = 0. Then, (2.7.66) reduces to

(2.7.67)

where the sum J"Y + J*"Y is expressed in terms of V p"Y . Consequently, when no advection takes place across the boundary, Vlfb·V

== q·v = 0, J*"Y = 0, and (2.7.67) reduces to

(2.7.68)

This implies that no transport of mass by molecular diffusion takes place across such a boundary, even when p"Ylpm =I p". This conclusion is unac­ceptable, as under the physical conditions of this example, we should expect transport of the ,-component to take place between the porous medium do­main and the adjacent fluid body by molecular diffusion, as this remains the only possible mode of transport.

The error in the conclusion expressed by (2.7.68) stems from the assump­tion that a 'well mixed' zone exists on the external side ofthe boundary. This assumption, in the absence of advection, combined with the sharp boundary approximation, must yield no mass flux by diffusion across it. In order to reinstate the diffusive-dispersive flux, which, as we know should take place in reality, we introduce the concept of a transition zone, or buffer zone, at the boundary (Fig. 2.7.4). We may associate the width of this transition zone, ~, with the magnitude of an REV, assuming that the abrupt bound­ary passes through its mid-point. Instead of the boundary between the body

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Macroscopic Boundary Conditions 257

of fluid and the porous medium, we now consider the boundary between the latter and the transition zone. Assuming that the sum of dispersive and diffusive fluxes through the transition zone is proportional to the average concentration gradient, and that the latter is proportional to the concentra­tion difference p'Y - p", we express the condition of continuity of flux at the boundary by

(2.7.69)

where a* is a coefficient, often referred to as a tmnsfer coefficient, such that a*(p" - p'Y) represents the sum of diffusive and dispersive fluxes through the transition zone.

Since, with [Pfh,2 = 0, we have Vlfb·V = q·v, equation (2.7.69) reduces to

(2.7.70)

which now serves as the boundary condition. In the absence of advection, or when Iq·vl ~ a*, equation (2.7.70) re­

duces to a*(p" - p'Ylpm) = -nDl·p'Ylpm·v. (2.7.71)

We note that if we accept (2.7.70), then P"lfb =I p'Ylpm on the bound­ary, i.e., a jump in concentration takes place on the boundary. This is a consequence of introducing the transition zone and the 'well mixed zone' approximation.

When Iq·vl ~ a*, equation (2.7.70) reduces to

(2.7.72)

which is identical to (2.7.67), yet is based on different reasoning. The entire development presented above can also be applied to cases

where temperature and pressure serve as state variables, as well as to cases where the external domain is a solid body. The 'well mixed' zone may be either a solid body, or a fluid body.

2.7.7 Boundary with fluid phase change

The assumed sharp boundary is between two states of aggregation of the same substance within the void space. Across such a (possibly moving)

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258 MACROSCOPIC DESCRIPTION

boundary, a change of state (e.g., freezing, thawing, evaporation, condensa­tion) of a phase may take place. As indicated in Subs. 2.2.5, within a given range of pressure and temperature, the considered substance may be either in a solid, a liquid,or a gaseous state. The solid matrix remains unchanged, although it is possible to consider cases in which the solid matrix melts, or dissolves. We assume that

[A2.15] Only a single state (rather than a mixture of states) of a fluid, is present in the void space on each side of the boundary.

Problems with such a boundary are often referred to as Stefan type prob­lems.

As in the previous sections, all boundary conditions are developed from the condition of no-jump in the total flux of a considered extensive quantity in crossing the boundary. Since in this case, [Osh,2 = 0, the sum in (2.7.6) and (2.7.7) does not include the solid phase. We shall consider here only conditions related to the fluid. Hence, no subscript will be used to indicate the fluid.

( a) Conditions for Mass

From (2.7.7), we obtain

(2.7.73)

recalling that a different state of aggregation (gas, or liquid) of the considered substance occupies the entire void space on each side of the boundary. Since [Oh,2 == [nh,2 = 0, and [Uh,2·V = 0, equation (2.7.73) takes the form

(2.7.74)

Equation (2.7.74) cannot be further reduced, because, in general, [ph,2 :f: 0, and [Vh,2 :f: 0.

The first inequality stems from the nature of the phase change, except at the critical point, where the densities of the two states of the considered substance are identical (e.g., Amyx et al. 1960, pp.212-217). The jump in the normal component of the velocity arises from the change in the density, or specific volume, of the substance upon the change of state. For example, if on one side of the boundary, the substance is at rest, the change in specific volume that occurs as a result of phase change, induces a velocity on the

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Macroscopic Boundary Conditions 259

other side. For a change from solid to liquid, or vice versa, this effect may be negligible. However, changes from liquid to vapor, and vice versa, are associated with significant changes in the specific volume of the considered substances.

Since the mass of a considered substance does cross it, the boundary of phase change considered here is not a material surface, so that

(V - u)lrll f:. 0, f = 1,2.

However, we recognize that in crossing the boundary, the mass assumes a different state of the same substance.

(b) Conditions for Momentum

Adopting assumption [A2.7], the conditions for momentum are identical to those presented in Subs. 2. 7.4( c).

In particular cases, an appropriate constitutive relation must be intro­duced for each state of the considered substance.

(c) Energy

Using assumptions [A2.10] and [A2.11], together with [Uh,2·11 = 0, and solid phase properties, with [8h,2 = 0, the condition of no-jump in the total energy flux in the direction normal to the boundary, reduces to

(2.7.75)

Although with [A2.5] and [A2.10], we have [Th,2 = 0, the jump in in­ternal energy, [lh,2 == [pCvTh,2 (f:. 0), in (2.7.75), expresses the jump in the energetic state of the substance in the void space on both sides of the boundary.

When we consider a change of phase from solid to liquid, or from liquid to vapor, the jump in energy represents the additional energy required to produce a more disordered state of the molecular structure, i.e., the energy required to further separate the molecules from each other. The jump in the energetic state of a substance is manifested by the fact that for the thermal flux, we have [-A*H·VTh,2 f:. OJ this means that part of the sum of dispersive and diffusive heat fluxes entering, or leaving, the boundary is

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260 MACROSCOPIC DESCRIPTION

compensating for the energy consumed by the phase change. This is an example of a sink, rSE, on the boundary.

Alternative forms of (2.7.75) are obtained by expressing the internal en­ergy in terms of the enthalpy, h, viz.

[(ph - P)Vh,2·1I - [ph - ph,2U.1I - [A*H.V'Th,2·1I = 0,

(2.7.76)

where h = I + pip, with the quantity pip expressing the energy associated with the volume per unit mass. It is often assumed that

[A2.16] Only a small part of the energy required to produce a change of phase is derived from the change in volume (Denbigh, 1971).

Then, (2.7.76) is reduced to

(2.7.77)

where L 1,2( = [Phh,2) is the latent heat of phase change, defined per unit volume. It represents the energy required to produce a change in the state of a unit volume of substance. It may also be defined with respect to the density of one of the states of a considered fluid phase, e.g., in the form

where L1 and L2 are the latent heat per unit mass of states 1 and 2, present on sides 1 and 2, of a boundary, respectively.

In (2.7.77), the quantity [phVh,2·1I, is associated with the change in volume of the fluid in the void space, due to phase change, and the resulting advective energy flux that is induced across the boundary. Some authors (e.g., Roberts et al. 1979) neglect this effect.

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Macroscopic Boundary Conditions 261

(d) Boundary shape

In the case of a solid-liquid (= liquified solid), or a liquid-vapor (= gas containing the liquid's vapor), boundary, the shape of the boundary, can be derived (Subs. 2.7.5) from the condition [Th,2 = 0, viz.

F(x, t) = T(x, t)ll - T(x, t)b = O. (2.7.78)

The comment that follows (2.7.65) is also valid here. The relationship expressed by (2.7.78) is valid also for changes from a solid state to a liquid one, and vice versa (of the material that occupies the void space).

2.7.8 Boundary between a porous medium and an overlying body of flowing fluid

Only the case of momentum is considered here. When a fluid continuum is bounded by a solid one, we usually assume

that the fluid adheres to the solid. For a stationary solid, this means zero velocity at the fluid-solid boundary.

When a moving (viscous) fluid is bounded from below by a porous medium domain, the tangential stress (=momentum flux) at the bound­ary between the two domains affects the velocity of the fluid just below the (assumed abrupt) boundary. We recall (Subs. 2.7.1) that within half an REV from this boundary, the behavior of the fluid in the porous medium should not be expected to be similar to that encountered farther away. N everthe­less, we assume that the latter kind of behavior can be extended up to the boundary.

Consider the case of flow of an incompressible Newtonian fluid at a spe­cific discharge qo (with the possibility of qo = 0), in a homogeneous isotropic nondeformable, stationary porous medium domain (side 1), z < O. This domain is overlain by a moving fluid (side 2), with 08 12 = 0, 011 2 = 1. The discussion in Subs. 2. 7.3( c) is applicable, for example, by starting from (2.7.23). Since in the present case O'~ll'v = 0'~12'v = 0, and pil = p12' equation (2.7.23) reduces to

(2.7.79)

In order to express (2.7.79) in terms of fluid velocities, we have to use an appropriate constitutive relation. For T 1 b, i.e., in the fluid continuum,

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262 MACROSCOPIC DESCRIPTION

we use (2.2.88), with vm == Vi' For 'Till (== 'T/11)' we use an approximate macroscopic constitutive relation obtained by averaging (2.2.88), with q~ ~q.

Let us demonstrate (2.7.79) by considering uniform flow of a fluid con­tinuum occupying the entire void space in the domain z > O. In that domain Vxb = V, Vyb = 0, Vzl 2 = O.

Continuity of mass flux across the boundary, requires that qzl1 = O. For this case, (2.7.79) becomes

(2.7.80)

Hence, at least near the boundary, we cannot have the pre - assumed uniform flow at the specific discharge of qx = qo = const. Instead, a transition zone exists, across which the qx varies. An approximation of (2.7.80), would be (see Subs. 2.7.6 and (2.6.38))

1 M BVI -a (qxl1 - qo) = £l , n uZ 2

(2.7.81)

where aM is a transfer coefficient (dims. L -1) for momentum transfer, that depends only on porous medium properties, such as permeability and poros­ity. Beavers and Joseph (1967) proposed aM == eM n/.../k, where eM is a dimensionless constant that depends only on nand k.

We note that in the rather narrow zone of transition from qo to qx 11 , the characteristic length for velocity variations is small and, following the discussion in Sec. 3.3, we may have to use the Brinkman equation (2.6.64).

By substituting (2.7.81) into (2.7.80), we obtain the boundary condition

(2.7.82)

which is a mixed type boundary condition in terms of qx' The fluid velocity, qxI1/n, in the vicinity of boundary between the two

domains, is often referred to as slip velocity. Usually, qxl1 ~ qo.