introduction to modeling of transport phenomena in porous media || the porous medium

Chapter·}

The Porous Medium

Phenomena of transport in porous media are encountered in many engineering disciplines. Civil engineering deals, for example, with the flow of water in aquifers, the movement of moisture through and under engineering structures, transport of pollutants in aquifers and the propagation of stresses under foundations of structures. Agricultural engineering deals, for example, with the movement of water and solutes in the root zone in the soil. Heat and mass transport in packed-bed reactor columns and drying processes are encountered in chemical engineering. Reservoir engineers deal with the flow of oil, water and gas in petroleum reservoirs. In all these examples, one or more extensive quantities (i.e., quantities that are additive over volumes, with mass, momentum and energy as examples) are transported through the solid and/or the fluid phases that together occupy a porous medium domain. To solve a problem of transport in such a domain means to determine the spatial and temporal distributions of state variables (e.g., velocity, mass density and pressure of a fluid phase, concentration of a solute, stress in the solid skeleton), that have been selected to describe the state of the material system occupying that domain.

In principle, the equations that describe the various transport phenomena are known and may be written at the microscopic level, where we focus our attention on what happens at a (mathematical) point within a considered phase present in the domain. We may even know the conditions that prevail on the surface that bounds the phase. However, at this level, the equations cannot be solved, since the geometry of the surface that bounds the phase is not observable and/or is too complex to be described. The same is also true for point values of variables within the phase. As a consequence, the description and solution of a transport problem at the microscopic level

3 J. Bear et al., Introduction to Modeling of Transport Phenomena in Porous Media© Kluwer Academic Publishers, Dordrecht, The Netherlands 1990

4 THE POROUS MEDIUM

is impractical and, perhaps, also impossible. Another level of description is, therefore, needed, namely the macroscopic level, at which measurable, continuous and differentiable quantities may be determined and boundaryvalue problems can be stated and solved. Accordingly, the main objective of the present chapter is to introduce the continuum approach that leads to the macroscopic level of describing phenomena of transport in porous media.

We shall begin by defining a porous medium, classifying porous media, and introducing certain geometrical characteristics of porous media. Only certain classes of porous media are treated in this book.

1.1 Definition and Classification of Porous Media

1.1.1 Definition of a porous medium

Soil, sand, fissured rock, cemented sandstone, Karstic limestone, ceramics, foam rubber, bread, lungs and kidneys are just a few examples of a large variety of natural and artificial porous materials encountered in practice. Aquifers from which groundwater is pumped, reservoirs which yield oil and/or gas, sand filters for purifying water, packed-beds in the chemical engineering industry and the root zone in agriculture, may serve as additional examples of porous medium domains. Common to all these examples is the observation that part of the domain is occupied by a persistent solid phase, called the solid matrix. The remaining part, called the void space, is occupied either by a single fluid phase, or by a number of fluid phases, e.g., gas, water and oil. In the latter case, each phase occupies a distinct separate portion of the void space.

For the purpose of this book, a phase is defined as a chemically homogeneous portion of a system under consideration that is separated from other such portions by a definite physical boundary (interface, or interphase boundary). There can be only one gaseous phase in a system, as all gaseous phases are completely miscible and do not maintain a distinct boundary between them. We may, however, have more than one liquid phase in a system. Such liquid phases are referred to as immiscible fluids. Phases can be separated from each other by mechanical means.

A component is part of a phase that is composed of an identifiable homogeneous chemical species, or of an assembly of species (ions, molecules). The number of components in a phase is the minimum number of independent

Definition of Porous Medium 5

chemical species necessary to completely describe the composition of the phase. Thus, each phase may be a molecular mixture of several identifiable components, e.g., ions or molecules of different chemical species in a liquid solution, or in a mixture of gases, or labeled particles of a phase.

Another common characteristic of a porous medium domain is that the solid phase (and, hence, also the void space) is distributed throughout it. What we mean by this is that if we take sufficiently large samples of the porous material at different locations within a porous medium domain, we shall find a solid phase in each of them. At the same time, it is obvious that the size of the samples should be small enough so as to represent a sufficiently close neighborhood around the point (=center) of sampling. For the time being, let us refer to the volume of a sample that satisfies these conditions, as a Representative Elementary Volume (abbreviated REV) of the considered porous medium domain at the given point. In Subs. 1.2.2 we shall discuss this concept in more detail.

The transport of a considered extensive quantity (e.g., mass, momentum and energy) of a phase in a porous medium domain may take place through a single (fluid or solid) phase, through some ofthe phases present in the domain (possibly including the solid phase); or through all of them. In the first case, at least part of the domain occupied by that phase must be connected. In the last two cases, a transfer of the considered extensive quantity may take place across the (microscopic) interphase boundaries separating the phases through which the transport occurs.

With the above considerations in mind, and for the purpose of this book, we now define a porous medium as a multiphase material body characterized by the following features:

(a) A Representative Elementary Volume (REV) can be determined (see Subs. 1.2.2), such that no matter where we place it within a domain occupied by the porous medium, it will always contain both a persistent solid phase and a void space. Similarly, a planar Representative Elementary Area (abbreviated REA; see Subs. 1.2.4) can be determined such that, no matter where we place it within a porous medium domain, it will always contain both a solid phase and a void space. If such REV and REA cannot be found for a given domain, the latter cannot qualify as a porous medium domain.

(b) The size of the REV, or an REA, is such that parameters that represent the distributions of the void space and of the solid matrix within it

6 THE POROUS MEDIUM

Domain boundary - '<

(a) (b)

Figure 1.1.1: Within the indicated domain boundary, the a-subdomain is a single

simply-connected domain in (a) and a single multiply-connected domain in (b). The

,B-subdomain is non-connect'ed in both (a) and (b). Note that the examples here are

limited to two dimensions.

are statistically meaningful. The quantification of this requirement is discussed in detail in Subs. 1.2.3.

In the above discussion, each phase ( gas, liquid or solid) is regarded as a continuum, overlooking its molecular structure. Actually, the methodology presented in Sec. 1.2 for the passage from the microscopic level to the macroscopic one, was originally used for the passage from the molecular level to the microscopic (phase continuum) one, with a Representative Elementary Volume used for averaging the behavior at the molecular level in order to obtain the behavior of the phase as a continuum.

1.1.2 Classification of porous media

Although a variety of transport phenomena is considered in this book, our main interest is in cases where at least one of the transported entities is mass of one fluid phase within the void space. To enable such transport, the configuration of the void space must satisfy certain requirements related to


VOID SPACE SOLID MATRIX

MBe Me Be

MBe

Me PM3 PM4 IP

Be FE

Table 1.1.1: Classification of porous media with respect to void space and solid matrix

connectivity. PM = Porous medium, Me = Single multiply connected domain only, se = Ensemble of simply connected domains only, M se = Combination of Me and se, I P = Isolated pores, and F B = Fluidized bed.

the concept of connectivity. Let us, therefore, start by defining connectivity and then classify porous media according to the type of connectivity of the solid matrix and of the void space.

A domain is said to be connected, or interconnected, if any two points belonging to it can be connected by a curve that lies completely within it (e.g., the a-domain in Fig. 1.1.1).

A connected domain can be either simply- or multiply-connected. It is called simply connected if any closed surface that can be placed within it can be shrunk to a point without leaving it. If it is of a finite extent, it has only a single closed outer boundary (e.g., the a-domain in Fig. 1.1.1a).

A finite connected domain is called multiply-connected if it is also bounded (from the inside) by one or more disjoint closed surfaces (e.g., the a-domain in Fig. 1.1.1b).

A spatial domain is called non-connected (or non-interconnected) if it is composed of a union of disjoint domains, with disjoint boundaries ( e.g., the ,a-domains in Fig. 1.1.1 b).

Table 1.1.1 shows a classification of porous media according to the type of connectivity of the solid matrix and the void space. Other bases for

8 THE POROUS MEDIUM

Solid matrix

'--__ ..... J Void space

Figure 1.1.2: Planar cross-section through REV.

classification, e.g., the void geometry, are also possible. Because of our particular interest in transport phenomena in which ad

vection of fluid mass takes place in the void space, the discussion in this book is related primarily to porous media of type P M4 , i.e., media in which both the solid and the void space are multiply-connected.

1.1.3 Some geometrical characteristics of porous media

Let us now introduce certain geometrical characteristics of the void space within a porous medium, that can be used to quantify the features employed above in the definition of a porous medium. A number of additional characteristics that are relevant to the description of transport phenomena, will be introduced later in this book.

Figure 1.1.2 shows a planar cross-section of an REV of a porous medium domain (see Subs. 1.2.2 for a detailed discussion of the REV). Let U denote the spatial domain of volume U, occupied by a porous medium, and let Uo C U denote the spatial domain of volume Uo of an REV. Likewise, Uos, Uos, Uov and Uov will denote the corresponding sub domains and volumes of the solid matrix (subscript s) and void space (subscript v), respectively.


The locations of U o, Uos and U ov are identified by the position vectors X o, Xos and Xov of their centroids, respectively.

Feature (a)~in Subs. 1.1.1 states that

(1.1.1)

Also

(1.1.2)

where U~v and U: v denote, respectively, the volume of the interconnected void space and the total volume of the isolated simply-connected voids within the REV.

The ratio

(1.1.3)

is defined as the total (volumetric) porosity of the porous medium at point Xo·

From the point of view of fluid flow, U~v is sometime referred to as the volume of the effective void space subdomain. Then

(1.1.4)

is defined as the effective, or interconnected (volumetric) porosity of the porous medium at Xo.

Often, the void ratio, e = Uov/Uos = n/(1- n) is used instead of n.

In what follows, unless otherwise specified, we shall deal only with porous media in which the void space is interconnected. We shall refer to its volume by the symbol Uov , instead of U~v; the symbol n will be used to denote the interconnected porosity.

Let x denote the position vector of a point within Uo• The function ,(x), such that

( ) = {1 for x within U ov , x 0 for x within Uos

(1.1.5)

is called the characteristic (or indicator) function of the void space. For this function, an average, 7, and a deviation from the average, 1', may be defined

10 THE POROUS MEDIUM

by

u1 f ,(x) dU(x) o }Uo(Xo)

1 1 - dUv(x), Uo Uov(Xo)

( 1.1.6)

and

(1.1.7)

where the Xo in -}(XjXo) and in ,(XjXo) indicates that these values correspond to a point x that is located within a domain Uo centered at Xo.

From (1.1.3) and (1.1.6) it follows that

"{(xo) = ~v I = nlxo and 7" = o. o Xo

(1.1.8)

The spatial distribution of the void space within Uo , centered at X o ,

can be described by various geometrical characteristics. One family of such characteristics is composed of spatial averages of products of ,-values taken at different points within the REV. The simplest one is

-}(X)-}(X + h)1 xo,Uo,h

= ~ f [I(x) - "{(xo)][I(x + h) - "{(xo + h)] dU o }uo(xo)

= ~ f ,(xh(x + h) dU - n(xo, Uo)n(xo + h, Uo), o }Uo(xo)

(1.1.9)

where h is an oriented distance between any two points within the REV. The average .y(x)-}(x + h) characterizes the configuration of the void space within Uo . A particular case of this average is obtained for h(= Ihl) = O. Then, (1.1.9) reduces to

(1.1.10)

This parameter represents the spread of the void space about its average density (= n).


Another family of characteristics measures the symmetry of the void space distribution within the REV, through its moments with respect to any plane passing through the centroid, XO' One of them is the first moment

where xov( = -u1 .hu xldU) is the centroid of the void space, Uov , within Uo , ov 0

I is defined by (1.1.5), x == X o , X == x - Xo and * = O. From (1.1.11) we obtain

( ) -00 -,,--

n Xov - Xo = XI = XI, (1.1.12)

valid for any domain Uo •

If the void space is uniformly distributed about Xo within Uo, (1.1.12) reduces to

Xov == XO' (1.1.13)

From the static moments of the REV as a whole, of the void space and of the solid matrix, all with respect to a plane passing through X O , we obtain

(1.1.14)

Equation (1.1.14) implies that the centroids Xos ofthe solid matrix, and Xov of the void space, either coincide, or are always located on different sides of any planar cross-section passing through the centroid, X O , of the REV. In addition, X o , Xos and Xov are located on a common line passing through XO'

Let subscript a denote any phase within the void space of a porous medium, and let Uoo and U oo denote the sub domain and the volume, respectively, occupied by that phase within the REV. Then

N

U ov = L Uoo , Uoo n Uof3 = 0 0:::1

for any a I- /3, (1.1.15)

where N is the number of disjoint fluid phases within Uov • We then have

e _ Uoo 0- Uo ' Leo = n,

( 0)

where eo is called the volumetric fraction of the a-phase within Uo •

(1.1.16)


In analogy to /, we may define a characteristic function /0/ of the a-phase within U, such that

O/(x) = {1 for x within UOO/' / 0 for x outside UOO/.

(1.1.17)

On the interface between the a-phase and all other phases, /0/ is not defined, although its right- and left-side values exist there. Then, similar to (1.1.8), we obtain

(1.1.18)

and, similar to (1.1.12), we obtain

(1.1.19)

Geometrical characteristics of the a-phase configuration within Uo , analogous to (1.1.9) and (1.1.10) can also be introduced.

Let 80 0/ denote the total area of the boundary of the domain UOO/. This boundary is made up of two parts (Fig. 1.1.3) : 80/(3, which is the total area of the interface between the a-phase occupying the domain UOO/ and all other phases (denoted here by 13) within the REV, and 80/0/ which is the area of the portion of the surface 80 bounding Uo which is intersected by the a-phase.

(1.1.20)

Note that we have neglected the possibility of an a - 13 contact on 80 • We may define two corresponding specific areas

(1.1.21)

Since both the entire void space and the solid matrix are interconnected, 80/0/ f:. 0 and 8(3(3 f:. o.

Finally, let us introduce the definition of an areal fraction, (J~, of the a-phase within an REA centered at Xo

A( ) AoO/ I (JO/ Xo = A ' o Xo

(1.1.22)

where a denotes a fluid a-phase within Uov , and nA is the areal porosity of the porous medium. At this point, we should note that both ()~ and nA may depend on the orientation of the considered plane. We shall return to this point in Subs. 1.3.3.


REV, Un

a - phase

Figure 1.1.3: Definition sketch for Sao., S{3{3 and So.{3.

1.1.4 Homogeneity and isotropy of a porous medium

The geometrical characteristics discussed in Subs. 1.1.3 (e.g., porosity and volumetric fraction of a phase) are referred to as macroscopic geometrical parameters. Additional parameters are introduced throughout the book. In this subsection, we consider variations in such porous medium parameters.

A porous medium domain is called homogeneous with respect to a macroscopic geometrical parameter characterizing the configuration of the void space or of any phase within the REV, if that parameter has the same value at all points of the domain. If not, the domain is called heterogeneous, or nonhomogeneous, with respect to that parameter. For example, a porous medium domain, U, is homogeneous with respect to porosity, n, if

Vn=O, in U. (1.1.23)

As a consequence of the microscopic configuration of the void space, or of any phase within the REV, in the neighborhood of a point in a porous medium domain, certain macroscopic porous medium properties at that point


may vary with direction. A porous medium is said to be anisotropic at a point with respect to a property if that property varies with direction at that point. For example, the resistance of a porous material to the transport of various extensive quantities, such as mass or heat, through it, may vary with direction. This behavior reflects the macroscopic effect of the geometrical configuration of the void space (Bachmat,1972). A porous medium is said to be isotropic at a point with respect to a given property, if that property does not vary with direction at that point. A typical porous medium property that exhibits anisotropy is permeability. The reader is referred to Subs. 2.6.5 for a detailed discussion on anisotropy.

1.2 The Continuum Model of a Porous Medium

1.2.1 The need for a continuum approach

As mentioned in the introduction to this chapter, transport problems cannot be formulated and solved at the microscopic level due to the lack of information concerning the microscopic configuration of the interphase boundaries. Moreover, such solution is, usually, of no interest in practice.

In many branches of science, it is convenient to ignore the particulate nature of matter and to adopt the hypothesis that matter is a hypothetical substance that is continuous throughout the spatial domain it occupies and can be described in that domain by a set of variables which are continuous and differentiable functions of the spatial coordinates and of time. This continuum model of matter serves as the fundamental postulate of Continuum Mechanics and Thermodynamics.

The continuum approach, so useful in treating problems related to a single phase (solid, liquid, or gas), can, in principle, and subject to certain modifications, be extended also to a multiphase system such as a porous medium, where the various phases are separated from each other by abrupt interfaces. Accordingly, the real system, consisting of two, or more phases (each of which is already regarded as a continuum, obtained by averaging the behavior of the system at the molecular level, see Bear, 1972, p. 17), that together completely occupy disjoint sub domains within a porous medium domain, is replaced by a model in which each of the phases is assumed to behave as a continuum that fills up the entire domain. We speak of 'overlapping continua', each corresponding to one of the phases present in the

The Continuum Model 15

domain. If the individual phases interact with each other, so will these continua. The space occupied by these overlapping continua will be referred to as the macroscopic space. For each point within this macroscopic space, average values of phase variables are taken over elementary volumes (abbreviated EV), centered at the point,regardless of whether, in the real domain, this point falls within that phase, or outside it. The averaged values are referred to as macroscopic values of the considered variables. By traversing the entire porous medium domain with a moving EV, thus assigning averaged values to every point, we obtain fields of macroscopic variables which are differentiable functions of the space coordinates.

The advantages of the continuum model of a porous medium are:

(a) It circumvents the need to specify the exact configuration of the interphase boundaries, acquiring the knowledge of which is an infeasible task anyway.

(b) It descri1?es processes occurring in porous media in terms of differentiable quantities, thus enabling the solution of problems by employing methods of mathematical analysis.

(c) The macroscopic quantities mentioned in (b) are measurable, and can therefore be useful in solving field problems of practical interest.

Obviously, these advantages are at the expense of the loss of detailed information concerning the microscopic configuration of interphase boundaries and the actual variation of quantities within each phase. However, as we shall see below, the macroscopic effects of these factors are still retained in the form of coefficients, whose structure and relationship to the statistical properties of the void space (or phase) configuration can be analyzed and determined. To express these coefficients in terms of averaged quantities, statistical models of the microscopic variations will, in general, be required. In cases of specific porous media, the numerical values of these coefficients must be determined experimentally, in the laboratory, or in the field.

Altogether, in view of the advantages of the continuum approach, we shall employ it in describing transport phenomena in porous media. In the following subsections, we shall discuss the procedure for passing from the microscopic level to the macroscopic, or continuum, one.

16 THE PORO US MEDIUM

1.2.2 Representative Elementary Volume (REV)

In principle, any Arbitrary Elementary Volume (abbreviated AEV) may be selected as an averaging volume for passing from the microscopic level of description to the macroscopic one. Obviously, different AEV's will yield different averaged values for each quantity of interest, and there is no sense in asking which of them is more 'correct'. The selection of an averaging volume in any particular case depends only on the model's objectives. Also, the size of the 'window' of the instrument that measures an averaged value should correspond to that ofthe selected AEV. In this way, within the range of error introduced by the conceptual model of the process, the predicted and measured averaged values will always be the same. However, the main drawback of this approach is that since every averaged value may strongly depend on the size ofthe selected AEV, it must be 'labelled' (like a yardstick) by the size of the AEV over which it was taken. To circumvent this difficulty, rather than selecting the volume of averaging arbitrarily, we need a universal criterion which is based on measurable characteristics of any porous medium and which determines, for any given porous medium, a range of averaging volumes within which these characteristics remain, more or less, constant. As long as instrument 'windows' are in that range, observed and computed values will be close, within a prescribed level of error. An averaging volume which belongs to that range is considered a Representative Elementary Volume (abbreviated REV).

Thus, the introduction of a Representative Elementary Volume is the first step in passing from the microscopic level, at which we consider phenomena at every point within a phase, to the macroscopic level at which we associate with each point in space volume averaged quantities describing phenomena in its vicinity. This is the volume mentioned in the definition of a porous medium in Subs. 1,1.1.

1.2.3 Selection of REV

The construction of a mathematical continuum model of a porous medium imposes certain restrictions on the size of the REV. Foremost is the requirement that the values of all averaged geometrical characteristics of the microstructure of the porous material at any point in the macrospace of the porous medium domain be single valued functions of the location of that point and of time only, independent of the size of the REV. This statement is a

The Continu urn Model 17

quantification of requirement (b) mentioned in Subs. 1.1.1. Accordingly, we now define a volume U = Uo as a volume of an REV if

(1.2.1)

and a..y(x)..y(x + h)lxo,u,h I = 0,

aU U=Uo

(1.2.2)

where I is the characteristic function of the void space, defined by (1.1.5). In principle, for every point Xo within a given domain, R, one can vi

sualize an experiment consisting of a succession of gradually increasing volumes U1 < U2 < U3 , ••• , all centered at x o , and a concurrent determination of 7(= n) and ..y(x)..y(x + h) for each such volume, hoping that a volume U = Uo , which satisfies both (1.2.1) and (1.2.2) will be found. After repeating this procedure for determining Uo at all points x E R, one can replace the actual porous medium within R, by a model of a fictitious continuum, provided Uo is uniform throughout R. Obviously, this is an impossible task, since it is impractical to observe all points within R.

Instead, let us try to arrive at the size of an REV from its relationships with measurable macroscopic parameters of the microscopic configuration of the void space (Bachmat and Bear, 1986). To this end, let I(X) be regarded as a random function of position, i.e., at any point xp in a porous medium domain, R, the characteristic function, I(Xp ), is a random variable which may attain the values zero or one. Let the probabilities 8 and 1-8 be defined as

P(Ilx = 1) = 81x , p p

Traversing once the domain R,we obtain a nonrandom function I(l)(x). We call I(l)(x) a realization of I(X). Repeating this process N times, we obtain additional realizations, 1(2)(X), 1(3) (x), ... , I(N)( x).

We shall refer to I( x) as a stationary random function in R if:

(a) The expected value of I, given x, is such that

E[{(x)] = 8 = const.

(b) The covariance of I-values at any two points, xp and x q, is such that


for all x E R, where hpq = xp - Xq is the oriented distance between points xp and x q •

(c) Varb] = E{[,(x) - 8]2} = f(O) = const. This is a consequence of (b).

A domain, R, for which (a) and (b) hold, is referred to as macroscopically homogeneous with respect to ,(x).

We shall refer to R as isotropic with respect to ,(x), if

i.e., the correlation function between the values of, at different points within R depends only on the distance, h, between them, and not on their relative orientation.

Our objective is to establish a relationship between moments of the probability distribution of ,(x) and the spatial averages of, over an REV. To this end, we employ the notion of ergodicity of a stationary random function (Yaglom, 1965). A stationary random function is said to be ergodic (or to possess the ergodic property), if any statistical characteristic of the function, taken over a sufficiently large domain of its argument in a single realization, is an unbiased and consistent estimate of the same characteristic over the entire set of possible realizations of the function. An estimate of a population parameter is said to be unbiased if its expected value is equal to the value of the parameter. An estimate of a parameter is said to be a consistent one, if it approaches, probabilistically, the value of the parameter as the sample size increases (Yaglom, 1965). If the function ,(x) possesses the ergodic property within Uo centered at X o , then (Venzel, 1962)

'f(xo) == U1 l,(x)dU = n(xo) ~ ECllx ) = 81x U 0 0

o 0

(1.2.3)

;t(x)..y(x + h)i = ~o Lo ..y(x)..y(x + h) dU Uo,h

(1.2.4)

where Cov")'(h)lxo is the covariance of, in Uo for points spaced an oriented distance h apart, Var")'(xo) = n(1 - n), the symbol T")'(h) denotes the correlation coefficient at x o, between values of, at points spaced an oriented


distance h apart and n( = Uov/Uo) is the porosity at xo, where Uov denotes the volume of voids within Uo•

In fact, the volume Uo of an REV should be sufficiently large, so that the volumetric averages, e.g., those appearing in (1.2.3) and (1.2.4), can be considered as satisfactory estimates of the relevant population parameters of the void space configuration at X o , Le., estimates which are free of errors caused by the size of the sample and its random choice.

In the present case, the sufficient condition for (1.2.3) and (1.2.4) to hold is that I fooo T,y(h) dhl < 00. By definition

As shown by Debye et al. (1957), for an isotropic porous medium and any function T,i h), the relation

(1.2.5)

always holds, where .6. v ( = Uov/ Svs) is the hydraulic radius of the void space (of volume Uov and area of contact, Svs, with the solid). An example of an approximate expression for T"'((h) for an isotropic porous medium with a random distribution of void and solid spaces, is given by Debye et al. (1957), in the form

T"'((h) ~ exp{ - 4.6.v(~ _ n)}' h = Ihl. (1.2.6)

From (1.2.6) it follows that T"'((h) -+ 0 as h -+ 00, ensuring that the sufficiency condition given above holds.

From the above discussion it follows that a necessary condition for obtaining nonrandom estimates (Le., ones that are not subject to sampling errors) of the geometrical characteristics of the void space at any point Xo

which serves as a centroid of a sphere of volume Uo and diameter £, is

(1.2.7)

The magnitude of [min is determined by the chosen accuracy and reliability levels of the parameter estimates. Thus, as a conceptual experiment for estimating the porosity, n, of a porous medium at a point X o , let the volume Uo of a cubical REV centered at that point be split into N disjoint elementary subdomains, each of volume 6U = Uo/N, such that in each of them one may encounter (more or less) either solid or void. The average of


"'( over the N samples is taken as an estimate, n, of the porosity, n, at X o ,

Le.

(1.2.8)

By definition, and by (1.2.3) and (1.2.4), we have

2 1 N N n(l _ n) N N (Tn = N2 E E Cov("'(p,"'(q) = N2 E E T-y(hpq ),

p=lq=l p=lq=l (1.2.9)

where (T~ is the variance of the estimate of n, and hpq = Ixp - Xq I is the distance between points xp and x q.

Employing (1.2.6) we obtain

2 _ n(1 - n) [ ~ ~ { hpq }] (Tn - N2 N + L...J L...J exp - 4~ (1 _ n) .

p=l q=l,p:j:q v

(1.2.10)

From (1.2.10) it follows that if hpq is expressed in units of ~v, we have N = N(n,(T~).

Figure 1.2.1 shows the relationship (T~ = (T~( n, N). For example, for n = 0.4, N = 8000, (T~ = 0.0032.

Hence, for a cubical REV, Uo,min(== .e~l3) = N6U, with each elementary volume, 6U = (C~~v)3, we have

(1.2.11)

where C~ == (.e~~~/ ~v)IN=l is a numerical coefficient. In the above example, this means i::;;in = 20C~~v, where we have added the superscript (n) to emphasize that we have been considering the porosity, n, as the macroscopic geometrical characteristic in determining imino

According to Chebyschev's inequality (e.g., Feller, 1957), the probability that the magnitude of the estimation error exceeds a prescribed level, say {, is bounded from above by

(1.2.12)

Let P* denote a probability such that (TV {2 = P*. Then, N would represent the smallest number of sub domains of Uo which is sufficient to ensure, with a reliability 1 - P*, that the estimation error In - nl will not exceed €. In the above case, this means, for example, that for f = 0.1,


t-- 1111 I r-- ~I- ~ l-

n - 0.2 III

-o-t4-. n::::: 0.1 -n - 0.05 t-- r-.. -r-.r-, ~ \. -?

'" 0

" '7: '\ f\.

\ 1\

Figure 1.2.1: Variance of the estimate, it, of porosity as a function of the number, N,

of elementary subdomains.

P* = 0.32. Obviously, any reduction in f. and in P*, in this example, would require a much larger value of N.

It is of interest to note that in order to determine .e~! for a given porous medium by (1.2.11), for a selected value of (7~ = f.2p*, one has to make use of a preliminary estimate of nand Cb.!:::..v' However, as can be seen from Fig.1.2.1, the effect of differences in n on the value of N and, hence on .e~~ decreases as (7~ decreases.

The requirement of ergodicity also sets an upper bound on the size of the REV. We require .e < .emax , where .emax is the distance between points in the porous medium domain beyond which the domain of averaging ceases to be statistically homogeneous with respect to the moments of ,(x).

In reality, the requirement of homogeneity is seldom satisfied, as the macroscopic parameters of the void space geometry usually vary from point

n

o


n""all:

~~~;;~~~;-~---------4-----

Il(n) :Co - 2" ""all: + Il(n)

:Co 2" ""all:

Figure 1.2.2: Conceptual determination of t~~x by (1.2.15).

to point. However, even for a domain that is heterogeneous with respect to these parameters, one can define around every point, a sufficiently small subdomain, within which these parameters may still be considered uniform, up to a prescribed error level. The size of such a sub domain around a given point serves as the upper bound for the size of the REV at that point.

In order to determine this upper bound, and following the definitions presented above, for a stationary random function, a domain U centered at a point Xo will be called homogeneous with respect to the statistical parameters of ,(x), if

E-y(x) == n(x) = const. = no,

Covl/(x + h), ,(x)] = 1(h),

i.e., a function of h only for all x E U. Then

Var,(x) = 1(0) = const. = no(1 - no).

For a heterogeneous domain, we have n = n(x). We shall refer to the domain U as approximately homogeneous (here, with respect to porosity), if within it

==o<€;:1, (1.2.13)


where nmax , nmin and n are the largest, the smallest and the average values of n, respectively, within U, and C (with 0 < c ~ 1) is an arbitrarily selected small number.

For a sufficiently small domain around X o, any differentiable function, n(x), can be approximated by its linear part (Fig. 1.2.2), i.e.

(1.2.14)

where no = nlxo . Introducing the definition .e~)x == 2Maxlx - xol, and employing (1.2.14),

equation (1.2.13) yields .e(n) = no 8, (1.2.15)

max Igrad nlxo

where 8 = (nmax - nmin) In. The distance .e!:)x (based on porosity) is thus the upper limit for the size of the REV at a point xo, within a porous medium domain at the selected error level.

Altogether, .e(n) has to satisfy the condition

(1.2.16)

at all points,xo, of the given domain. If a non-zero range of .e(n) can be found, which is common to all points

within a given spatial domain, one can adopt the continuum model for the porous medium within that domain.

Finally, we have to relate .e(n) to the size ofthe considered domain. If L* is a characteristic length of the domain, we require that

(1.2.17)

in order to ensure that the boundary region of the domain, which has a width .e(n) , and in which the continuum approach is not applicable (see Subs. 2.7.1), be small compared to the size of the domain itself. The size of the REV in a domain 'R is thus determined by the porosity and the specific surface of the void space in 'R, by prescribed acceptable reliability and error levels in estimating n, by the size of the domain, by the spatial variation of n within the REV and by a prescribed tolerable deviation of n from uniformity within it. If a range satisfying (1.2.16) cannot be found, the domain 'R cannot be represented as a continuum.


So far, the concept and size of the REV have been related to porosity as a geometrical porous medium property. We have indicated this fact by using the superscript (n). Whenever, additional characteristics of the porous medium appear in the macroscopic model that describes a transport problem, e.g., permeability, a range for the REV has to be determined for each of them. If a common REV range can be found, a continuum model of the porous medium can be employed.

One of the requirements for the range of the REV is that an/aU = 0 within it, as defined by (1.2.1). This does not necessarily require that n(x) be uniform within Uo. To illustrate this point, consider the ratio Uv(xo)/U(xo), where U(xo) is the volume of a sphere centered at an arbitrary fixed point, xo, within 'R, and Uv(xo) is the volume of the void space within U(xo).

Figure 1.2.3 shows the variations of the ratio Uv/U as U increases. For very small values of U(xo ), the above ratio is one or zero, depending on whether Xo happens to fall in the void space, or in the solid matrix. As U(xo) increases, we note large fluctuations in the ratio Uv/U. However, as U continues to grow, these fluctuations are gradually attenuated, until, above some value U = Umin , they decay, leaving only small fluctuations around some constant value.

In order to examine the behavior of the function n(xo) == (Uv/U)lxo '

in the domain in which on/aUlxo = 0, consider a domain of averaging in the form of a rectangular prism centered at x o , with edges parallel to the coordinate axes. Its volume is U1xo = ~Xl~X2~X3. By definition

(1.2.18)

where

is the areal average of lover a surface normal to the Xl-axis.


Domain of Domain of

.. !- porous medium -- Domain of (possible) I 'macroscopic I I,~_-------

1.0

I I heterogeneity ------i------------r---

_. Heterogeneous

o

I medium I I I I I I a....- Range for --t I Uo I I I

Homogeneous medium

Volume, U

Figure 1.2.3: Variation of porosity, n, in the neighborhood of a point as a function of

the averaging volume.

Now

on

(1.2.20)


obtain

(1.2.21)

Assuming that -yl (xt, X20, X30) is differentiable with respect to Xl in the domain IXI - xlOl :S 6.xd2, and expanding the terms on the r.h.s. into a power series around x o( XlO, X20, X30), we obtain

Substituting these expressions into (1.2.21), yields

(1.2.23)

In order that "Y(xo ) retain its value for any 6.xl, in the range where an / a( 6.xl) = 0, all terms containing 6.Xl in the last equation must vanish, i.e., I must be a linear function of Xl in that range. This can be written in the form

-yl(Xt, X20,X30) = ao + blXl + f(x2o,x3o). (1.2.24)

However, since -yl(xo) = -y2(xo) = -y3(xo), it follows that

n(x) = ao + b·x, b = const. (1.2.25)

It may thus be concluded that ifthe ratio Uv/U == n(Ulx) has a plateau within a given range of U, then n(xIU) is a linear function of x, including the case n = const., in that range, and vice versa.

If U(xo) is further increased, say beyond some value U = Umax , we may observe a trend in the considered ratio, Uv(xo)/U(xo), due to a systematic variation in the latter.


The Representative Elementary Volume is that volume, Uo(xo), within the range of Umin < U < Umax , that will make the ratio Uv/U independent of U, and hence a single valued function of Xo only. For any U = Uo in that range, the ratio Uv/U represents the porous medium's porosity, n, at Xo.

By (1.2.15), the upper limit of .e(n), for a given 0, is defined by .e0Jx, which, in turn, depends on Umax that indicates the point of deviation from the plateau, produced by the linearity of n(x) in the vicinity of Xo (Fig. 1.2.2).

So far we have investigated the conditions under which a porous medium domain, whether homogeneous, or heterogeneous, can be treated as a continuum. The condition is the existence of a range of volumes for an REV that satisfy certain size constraints.

Turning now to transport and other processes, where the state of each phase is specified by a set of relevant state variables, (e.g., densities), an analogous approach should be undertaken with respect to these variables. A range for REV (= domain of averaging) should be selected for each state variable, following considerations similar to those associated with the geometrical characteristics of the void space. In this way, the size, .e, of the REV corresponding to each state variable will be bounded by .emin that depends on the spatial distribution of the microscopic values of that variable within the phase, and on .emax that depends on the spatial variation of its macroscopic counterpart.

Again, if we have (and usually we do) a number of relevant state variables, the continuum description of the process involving them can be employed only if a common range of REV can be found for all of them. The same range should also be common with that associated with the configuration of the void space.

Two examples of interest may be mentioned. One is pressure wave propagation. In view of the above considerations, if the length of a pressure wave is smaller than .emin of the REV of the porous medium (say, it is of the order of magnitude of the pore size, or less), the process of wave propagation in a porous medium cannot be described by means of the continuum approach.

A second example is the spreading of a component of a phase (e.g., a solute) from a point source. In that part of the domain where the size of the REV associated with the component's spatial concentration distribution is smaller than the lower bound of the size of the REV of the porous medium, the continuum approach to the spreading of the component is not applicable.

28

U v -U

1.0

npm

nJr

THE POROUS MEDIUM

Range for Uo

for fractured porous medium I~ ~I

I I" I ~I I 1 Range for Uo for porous

/\ {I; I medium alone i ;' \ ! II ''\1 /\ I : I ; n Irr:r ~l Rangefor U:-fo~-

/ v I i .. fractures a}one ~I I I I 1 1

O~------~--~---------LI __ ~ Umin,pm Umin,jr Umax,pm Umax,jr

(b)

Figure 1.2.4: Definition of porosity and representative elementary volume for a fractured

porous medium.

If a range Uo can be found which is common to all points within a given spatial domain, both for all relevant geometrical characteristics of the void space and all phases occupying it, and for all relevant state variables, we can define fields of these state variables throughout the domain and treat the latter as a continuum for each of them. The volume Uo satisfying these conditions is the Representative Elementary Volume of the material system within the porous medium domain, introduced in Subs. 1.1.1.

The requirement that across the REV, any macroscopic property (whether one of the void space, or of a state variable) should vary linearly, or approximately so, justifies the assignment of the averaged values taken over the REV to the latter's centroid.


An important consequence of the continuum approach and the introduction of the REV is related to field (and laboratory) measurements of state variables. In order to be comparable with calculated values, based on the continuum model, these measurements should also represent averages taken over the same REV that appears in the (mathematical) description of the considered transport phenomenon. The size of the 'window' of the measuring device should satisfy this requirement.

Sometimes it is convenient to visualize the void space domain of a porous medium as consisting of two or more subdomains, each having a different range of Uo• A fractured porous medium (Subs. 1.5.1), i.e., a porous medium intersected by a network of interconnected fractures, or solution channels, may serve as an example (Fig. 1.2.4). A porous medium domain of this kind may be treated either as a single continuum, or as two overlapping continua that interact with each other (say, in the form of fluid that moves from one subsystem to the other )-one representing the porous medium of the blocks and the other the network of interconnected fractures-within the framework of a common model, only if a range of Uo can be found that is common to the two subsystems throughout it (Fig. 1.2.4b). With subscripts pm and fr representing the porous medium in the blocks and in the network of fractures, respectively, this means that

Umin,fr < Uo < Uma.x,pm, or [min,fr < [ < £ma.x,pm' (1.2.26)

and [min,fr < [ ~ L*. (1.2.27)

Usually (1.2.28)

1.2.4 Representative Elementary Area (REA)

In analogy to the Representative Elementary Volume defined in the previous paragraph, we define a Representative Elementary Area (REA) of a porous medium as a means for transforming quantities associated with areas (e.g., fluxes and stresses). In this case, the transformation is from the microscopic level, at points within a phase, to the macroscopic one, at which we associate with each point in space quantities that describe phenomena in terms of averages taken over a small planar area centered at the point. This is the area mentioned in requirement (a) in Subs. 1.1.1.

30 THE PORO US MEDIUM

The selection of the size of the REA for any process within a given porous medium domain should satisfy requirements which are analogous to those required for the REV (Subs. 1.2.3). We shall employ the symbollA for the length characterizing the REA.

1.3 Macroscopic Values

Having defined the representative elementary volume and area, we now return to the basic idea underlying the continuum approach. For every parameter and state variable of a phase, that is relevant to a considered process that takes place in that phase, within a given porous medium domain, we can define an average (over an REV, or REA) at every mathematical point, denoted by its position vector x, independent of whether or not x falls inside the phase. The considered point, x, serves as the centroid of the REV, or REA. By this procedure, repeated for all points within the considered domain, we replace the actual phase occupying the void space, or part of it, by a fictitious continuum, present everywhere within the entire domain. Throughout the domain, the values of the considered variables, as well as of the various macroscopic characteristics of the phase, are defined at every mathematical point. The values assigned to a point at the continuum (Le., macroscopic) level of description, are averages referred to as macroscopic values.

We consider a porous medium that is a multiphase system. The phases (including the solid one) are denoted by subscripts a = 1,2, 3 .... Each phase occupies at time t a domain UOOi of volume UOOi within Uo• The volumetric fraction of the a-phase within Uo is denoted by BOi (Subs. 1.1.3). Hence

B ( ) _ UOOi(x,t) Oi x, t - Uo ' (1.3.1 )

Note that in Subs. 1.1.3, the symbol a denotes only fluid phases that occupy the void space.

Employing the characteristic function, lOi' defined by (1.1.17), the volume UOOi of the a-phase within Uo may be expressed by

UOOi(x) = r lOi(X + r)dU(r) = r dUOi(r), Juo JUoa

(1.3.2)

where r( = x' - x) denotes the position vector of any point within Uo , with respect to a point x that serves as its centroid (Fig. 1.3.1). The elementary

Macroscopic Values

z

IL-_____ :z:

o

REV

c::J 0: - phase wi thin U

llITTII All other phases

Figure 1.3.1: Definition sketch for volume averaging.

31

volume dU is a (microscopic) element of volume withinUo , while dUa denotes a volume element of Uoa only. Although both x' and x belong to the same coordinate system, we shall employ x' to denote locations of points (at the microscopic level) within Uoa , while x will denote locations of points at the macroscopic level.

1.3.1 Volume and mass averages

Let ea(x', t) = dEal dUa denote the volumetric density (=amount per unit volume) of some extensive quantity E of an a-phase. We have used the position vector x' to emphasize that the field ea (x', t) is at the microscopic level. Mass density and solute concentration (= mass of solute per unit volume of a liquid) may serve as examples of ea. It is assumed that ea is finite, continuous and differentiable everywhere within Uoa •

Two kinds of averages of ea can be defined:

(a) Volumetric intrinsic phase average of ea , taken over the a-phase included in Uo

u t ) f ea(x', tj xha(x', t) dU(x') oa x, t Juo(X)

U ~ ) f ea(x', t;x) dUa(x'), oa x, t Juoa(x,t)

(1.3.3)


where 'Yo: is defined by (1.1.17), the x in (X',tiX) indicates that we consider points x' belonging to an REV centered at x, and we have used the fact that for any function f(x'), we have the identity

f f(x'ho:(x') dU(x') == f f(x') dUo:(x'). Juo JUoa

(1.3.4)

The average ec< C< is a function of the macroscopic space coordinates, x. The symbol a next to the bar in rr indicates that the average is taken over Uoo:.

(b) Volumetric phase average of ec<

eo:(x,t) ~ 1 ec«x', ti xhc«x', t) dU(x') ° Uo

_ ~ f ec«x',ti X ) dUc«x'). o JUoa(x,t)

(1.3.5)

Here the total amount of the extensive quantity of the a-phase is averaged over the entire domain, Uo of the REV. Since we deal with a property of the a-phase only, the integrations in both (1.3.3) and (1.3.5) are over the sub domain Uoo: only.

From (1.3.3) and (1.3.5), it follows that the two kinds of averages are related to each other by

(1.3.6)

where Oc< is defined by (1.1.18). The adjective 'volumetric' in the verbal definition of (1.3.3) and (1.3.5)

will be omitted whenever it is obvious that we deal with volumetric and not with any other kind of average, e.g., an areal one.

When E is an extensive quantity that is defined for all phases present within Uo , e.g., mass, we define the volume average of E by

e = u1 f e dU = u1 L f eo: dU C<

° Juo ° (0:) JUoa

1 IT L Uoc<ec< C< = L Oc<e-;;x = L eo:.

° (C<) (C<) (C<) (1.3.7)

We note that fUoa ec<dUc< represents the total amount of E within Uoc<. The quantity

(1.3.8)

Macroscopic Values 33

defines the deviation of ea at a point x' within an REV centered at x from its intrinsic phase average, over that REV.

By taking the volumetric intrinsic phase average of (1.3.8), we obtain

€a a = o. (1.3.9)

(c) Mass average of e~. Let e~(x',t) = dEa/dmOi denote the spe-cific value of Ea (Le., the quantity of E of an a-phase per unit mass of that phase). Another kind of average, the mass average, < e~ >01, of e~, may then be defined by

-1-1 e~dmOl=-,,()I~ { POl(x',t)e~(x',t)dUOI mOl mOl P 001 JUoa

1 -01 -----;-a e 01 -pe -7]01 a 01 - pOi' (1.3.10)

where pa is the intrinsic phase mass density of the a-phase. The kind of average to be used in each case depends on the way the

averaged quantity is actually measured in the field. For example, if at a point we take a liquid sample out of a porous medium domain, in order to determine the concentration of a solute in it, the latter is an intrinsic phase average, as it is taken only over the liquid phase. We recall that the sample size should be equal to the volume of the liquid within the REV. In general, for a field measurement to be comparable to a quantity appearing in a mathematical model, it should be taken by a device designed such that it indeed reads values that are averages over an REV.

In all the above definitions of averages, we have assigned the average value of a considered extensive quantity to the centroid of the REV. This was justified in Subs. 1.2.3, on the basis of the requirement that the variation of the average values of that quantity within the REV be linear.

Indeed, when considering the ratio Uv/U at some fixed point x (Subs. 1.2.3), the existence of an approximate 'plateau', assumed in drawing Fig. 1.2.3, is based on the assumption that even when the porosity, n, varies considerably within the given domain, its variation within the REV around that point is always approximately linear. In Subs. 1.2.3, we have suggested that the same condition of linearity holds not only for n, but for all other relevant geometrical characteristics of the porous medium, as well as for all relevant macroscopic state variables, for example, a macroscopic quantity eOi as defined by (1.3.5).


1.3.2 Areal averages

Let Ao and Aoa denote the areas of an arbitrarily oriented REA, and the a-area within it, respectively, centered at some point, x, within a porous medium domain (Subs. 1.2.4). We can define two kinds of areal averages for any component of a tensorial a-phase quantity, 1r a associated with area (e.g., flux, stress), such that 1rdAa is additive over Aoa. The first is

1raa(x,t) = A1 1 1ra(x',t;x) dA(x'), oa Aoa(x)

(1.3.11)

called the areal intrinsic phase average of 1ra , taken over the area Aoa.The second is

1ra(x,t) = A1 1 1ra(x',t;x) dA(x'), o Aoa(x)

called the areal average of 1r a, taken over the area Ao. The two averages are related to each other through the

()~( = Aoa/ Ao), i.e., - (}A-a 1ra = a 1ra .

(1.3.12)

areal fraction

(1.3.13)

The areas Ao and Aoa are facing some direction denoted by the unit vector 1I( = Ao/ Ao).

1.3.3 Relationship between volume and areal averages

In the previous subsections, a distinction was made between quantities that are additive over volumes and those that are additive over areas. A relevant case of special interest in the description of transport phenomena is the expression ea yEa (see (2.1.10) for a definition of the velocity VE).

In Continuum Mechanics, the product pvm is the linear momentum density of mass. By analogy, the product ea yEa may be regarded as the linear momentum density of an extensive quantity E of an a-phase. Hence, ea VEadUa is additive over volume, and taking a volume average of it is permissible.

However, ea yEa also represents an E-flux (= amount of E passing through a unit area of the a-phase, normal to V Ea , per unit time). This means that eaVE,,, ·dAa is additive over area. Hence, taking an average of ea yEa over the area Aoa of an REA (which is normal to the direction of V Ea) is also permissible.

A1acroscopic lIalues 35

Let us simplify the notation by introducing the symbol jtEa to denote the total E-flux, ea VEa, at the microscopic level, and examine the conditions

#

under which, at a given point x, the areal average of the flux, ( ea vEa) Ix ==

---jtEa lx, and the volume average, ea yEa lx, of the momentum, are identical. For a point x (with coordinates Xl, X2, X3) which is the centroid ofUo(x, t),

the volume average of ea VIEa (= a component in the direction Xl of ea VEa) is given by

(1.3.14)

We choose an REV in the form of a cylinder of constant cross-sectional area, Ao , equal to the REA normal to the unit vector IXI in the direction of the xl-axis, and length So = (Uo/Ao) in the direction of IXI. We may rewrite (1.3.14) in the form

jiEa(x, t) = - dx~- jiEa dx~ dx;. - 1 1x1+'2° ,X2,X3 1 1 I S!Q A A(' ) III o X1- 2 ,X2,X3 0 oa Xl ,X2 ,X3 X 1 'X2 'X3

(1.3.15) With

(1.3.16)

we obtain from (1.3.15)

(1.3.17)

where 0 80 indicates an average over the length So. In words, (1.3.17) states that the volume average at x is equal to the average over the length So of the areal averages, each taken over cr~sections, Ao.

By developing the areal average jiEa Ixl ~ ~ into a power series about l,w2,w3

the point x, we obtain

(1.3.18)


We now average both sides of (1.3.18) over the length 8 0 of the REV of cross-sectional area Ao. Since (x~ - X1)8o == 0, we obtain

(1.3.19)

Since

(1.3.20)

and (1.3.21)

where L* is a characteristic length of the domain, and O() denotes an order of magnitude (Sec. 3.3), equation (1.3.19) yields

UiEa)80L = jiEalx + OUiEa)lx ((~:) 2). (1.3.22)

Now, since by Subs. 1.2.3, we have Uo "-' £3, and by Subs. 1.2.4, we have Ao "-' (£A)2, if we select £A ~ £, then by (1.2.17), 8 0 == Uo/Ao ~ £ ~ £A ~ L*. Hence, (1.3.22) reduces to

( ·tEa )80 I - ·tEa I )1 -)1' x x

(1.3.23)

up to an accuracy of 0((£/L*)2). In view of (1.3.17), we obtain

jiEa Ix

(1.3.24)

where 0;; (=Aoa/Ao) is the areal fmction of the a-phase. From the definition of the volumetric fraction Oa, it follows that

Higher-Order Averaging 37

By expanding 0:( xi, X2, X3) in a Taylor series around x, and averaging over So, we obtain, in a way similar to the development presented above, and subject to the constraint So ~ L*

Oalx = O:lx' (1.3.26)

regardless of the orientation of A. Hence, from (1.3.24) we obtain

(1.3.27)

This means that subject to a.ll the constraints imposed on the sizes of the REV and REA, the volumetric intrinsic phase average and the areal intrinsic phase average of jtEa (== ea V Ea) at a point are identical.

1.4 Higher-Order Averaging

1.4.1 Smoothing out macroscopic heterogeneity

The macroscopic domain within which transport phenomena take place may be homogeneous, or heterogeneous, with respect to the relevant macroscopic geometrical parameters of the porous medium (e.g., porosity, permeability). When the spatial variations of these parameters are known, it should be possible, in principle, to solve a transport problem on hand at the macroscopic level. However, sometimes we face a situation similar to that which was encountered at the microscopic level, namely, that the detailed information about the spatial variation of the parameters is not available, and/or the parameters suffer discontinuities along unknown surfaces. The way to overcome the lack of information about the heterogeneity at the microscopic level (resulting from discontinuities between void space and solid matrix), was shown to be averaging, employing the concept of a Representative Elementary Volume (REV). One may visualize this averaging as a smoothing operation. The same approach of averaging, or smoothing, may be applied also to heterogeneities that are encountered at the macroscopic level. As a result of this second-order averaging, a new continuum is obtained, which describes the porous medium and phenomena occurring in it at a level called the megascopic level (see, for example, Subs. 2.6.7).

As in the passage from the microscopic level to the macroscopic one, a new Representative Elementary Volume is needed in order to perform the


smoothing. If such a volume cannot be found, it is impossible to pass to the megascopic level. The characteristic size, f*, of this volume, is constrained by

d* ~ f* ~ L, (1.4.1)

where d* is a length characterizing the macroscopic heterogeneity that we wish to smooth out, and L is a length characterizing the considered porous medium domain. In fact, the features of the REV listed in Subs. 1.2.2, as well as the constraints imposed on its size, may, at least in principle, be repeated also here, replacing the terms 'microscopic' and 'macroscopic' by the terms 'macroscopic' and 'megascopic', respectively. Obviously, the length scale of homogeneity at the megascopic level will be much larger than that corresponding to the macroscopic one.

Similar to what happens at the microscopic-to-macroscopic smoothing, here also, the information about the heterogeneity at the macroscopic level appears at the megascopic one in the form of various coefficients that reflect the effect of the actual spatial distribution of the (geometrical) parameters at the macroscopic level on various transport phenomena. An example of transformation from the macroscopic level to the megascopic one is given in the next subsection.

1.4.2 The hydraulic approach

In principle, the flow of a fluid phase, or the transport of any extensive quantity through a porous medium, always takes place in a three-dimensional domain. However, under certain conditions, such a process may be conceptually modelled (always, as an approximation) as one that takes place in a two-dimensional domain. The transformation of the three-dimensional mathematical model into a two-dimensional one, is performed by selecting an appropriate coordinate system and integrating, or averaging the former along a properly selected coordinate axis. In a similar way, a second integration will reduce the model to a one-dimensional one.

Transformations of the kind mentioned above are often employed in Fluid Mechanics, in order to derive one-, or two-dimensional models of flow in channels and pipes. For example, averaging may be performed over the crosssection of a channel, or a pipe, normal to the latter's axis. This procedure is referred to as the hydraulic approach. We shall adopt this term also here, to describe the procedure of reducing models of transport in certain special domains from three to two dimensions.

Higher-Order Averaging 39

Typical porous medium domains to which the hydraulic approach is applicable are aquifers, geothermal reservoirs and oil reservoirs. Usually, the vertical thickness of these porous medium domains is much smaller than their horizontal extent. When various extensive quantities (e.g., mass of a phase or a solute) are transported in such relatively thin domains, the vertical variations of the densities of these quantities are, often, much smaller than the horizontal ones. Under such circumstances, it may be useful to employ the hydraulic approach in order to describe the transport in the aquifer in terms of variables that are averaged over the vertical thickness, obviously, provided this is the kind of information required of the model.

Consider, for example, the macroscopic density ea a(x, y, z, t) of some extensive quantity E of an a-phase. Its average, ea, over the vertical thickness B(x, y, t), of the flow domain in an aquifer, is defined by

ea(x,y,t) = B( 1 ) f eaa(x,y,z,t) dz. x, y, t JB(x,y,t)

(1.4.2)

Thus, ea is a variable that is a function of x, y and t only. In terms of such averaged variables, the description, or model, of the transport of a considered extensive quantity in the aquifer is reduced to a two-dimensional one, in the horizontal, xy, plane.

In addition to the advantage achieved by the reducing the model from three to two dimensions, a two-dimensional transport model also requires less data about the spatial distributions of the various model coefficients that describe the process in the domain. This also means that less field observations are needed in order to evaluate these coefficients. However, it should be emphasized again that the hydraulic approach may be employed only when the vertical variations of the concerned quantities, in comparison with their respective averages, are much smaller (or less important to the modeller) along one axis than along the other ones.

The hydraulic approach is further discussed, with examples, in Chap. 8.

1.4.3 Compartmental models

Sometime, the information required of the model is such that the entire domain of interest may be subdivided into a number of disjoint subdomains, referred to as celis, or compartments. In such a model, the process within each compartment is described in terms of averages taken over the entire


volume of the compartment. The exchange of extensive quantities between adjacent compartments is also described in terms of these averages. This type of model is not considered in this book.

1.5 Multicontinuum Models

In the continuum approach considered so far for the multiphase systema solid phase and one or more fluid phases-called a porous medium, each extensive quantity of a phase, of a component of a phase, or of the porous medium as a whole, was regarded as a single continuum over the entire domain, and all continua overlapped. For practical reasons, a phase may be divided into a number of 'apparent' phases, each regarded as a continuum over the entire domain. Similarly, the solid matrix may be regarded as two or more solid matrix continua. Let us consider two examples of practical interest: the fractured porous medium and the multilayer system. A third example, related to mobile and immobile fluid, is described in Chap. 5.

1.5.1 Fractured porous media

A fractured porous medium is defined in Subs. 1.2.3 as a portion of space in which the void space is composed of two parts (Fig. 1.2.4a): an interconnected network of fractures, and blocks of porous medium, as defined in Subs. 1.1.1. The entire void space is occupied by one fluid, or more. Such a domain can be treated as a single continuum, provided an appropriate REV can be found for it.

However, sometimes the fluid in the fractures behaves differently from that in the porous blocks. For example, when the fractures' widths are large, while the pores in the blocks are very small, practically all flow takes place through the fractures, while most of the fluid storage is provided by the pore space of the blocks, due to their large porosity.

Under such conditions, it is convenient to regard the fluid in the void space as made up of two parts: one in the fractures and the other in the porous blocks, and to regard each part as an apparent phase that behaves as a continuum that occupies the entire domain. Exchange of transported extensive quantities between the two apparent phases (= continua) is assumed possible.

Multicontinuum Models 41

Figure 1.5.1: Multilayer system.

To summarize (see also Subs. 1.2.3), a fractured porous material may be represented by either of the following two models:

• A single continuum, having averaged properties which reflect both the properties of the fractures and those of the void space within the porous blocks .

• A model composed of two overlapping continua, each with its own properties. This model is often called a double porosity model.

The first model is possible if an appropriate REV can be found for the entire fractured porous medium domain. The second model requires a common REV for both subsystems (Subs. 2.4.7 and 6.1.8).


1.5.2 Multilayer systems

Consider a porous medium domain composed of several interconnected layers (Fig. 1.5.1). Each layer is characterized by different transport and storage properties, and all the layers occupy the same domain in the horizontal xy-plane. When the conditions permitting the application of the hydmulic approach prevail (Subs. 1.4.2), e.g., when the thickness of each layer is much smaller than its horizontal extent, and variations of values of state variables across a layer are much smaller than those along it, the transport of extensive quantities within each layer may be assumed to be essentially horizontal. When an exchange of the considered extensive quantities between adjacent layers is possible, say, across relatively thin semipervious layers, the problem of transport in the different layers must be solved simultaneously for all layers.

The set of linked, two-dimensional layers, which replaces the given threedimensional domain, is referred to as a multilayer system. We have to emphasize again that the substitution of the real three-dimensional domain by the multiple layer system is justified only when the conditions underlying the hydraulic approach indeed prevail in each layer. This case is presented here as another example of overlapping continua, where each layer is treated as a continuum and all the layers occupy the same two-dimensional domain.

introduction to modeling of transport phenomena in porous media || the porous medium

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