Transport in Porous Media 4 (1989), 9%104. 97 �9 1989 by Kluwer Academic Publishers.

Short Communication:

An Inflow-Outflow Characterization of

lnhomogeneous Permeable Beds

MIHIR SEN and K. T. YANG Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556 U.S.A.

(Received: 2 June 1987; revised: 26 June 1988)

Abstract. It is proposed that the inflow to and outflow from a permeable bed be related by an integral. For vanishing fluid velocities, it is linear and its kernel is a material property characterizing the bed. Some of the properties of the kernel are considered here. The approach is particularly useful for tilted, fractured and other inhomogeneous beds, since no continuum characteristics need be assumed. The effects of nonlinearity are also discussed.

Key words. Integral representation, heterogeneous beds.

1. Introduction

Permeable materials considered in chemical engineering applications or hydro- logical studies include a wide diversity of possible beds such as capillary tube

bundles or packed sand. There is need for a kinematic characterization of these

beds independent of the dynamic laws which govern the flow within it. The

object ive here is to advance this point of view by defining a relevant quantity

which can parametrize the bed and which would have practical usefulness. Thus,

the purpose is not to discuss the dynamics of the flow, nor the response of the bed to different pressure gradients, gravity, capillary or other forces which drive flow in porous media. Rather, the goal is to consider the kinematics of flow which results from a given distribution of the flow passages within the bed, and to

characterize the bed without any a priori knowledge of its nature. Thus, a wide variety of possibilities used in different branches of engineering should be

included. For simplicity, however, we will only consider the flow of incom- pressible single-phase fluids.

For porous beds, a geometrical proper ty called tortuosity has been proposed,

sometimes defined as the average ratio of the length of the flow path and the straight-line distance between inflow and outflow (Scheidegger, 1974). This is, however, very difficult to measure directly. Others (Reiss, 1980) have used approaches such as fracture porosity to deal with inhomogeneous formations. They are, however, based on cont inuum-type principles which are suitable only if we look at geometr ical scales much larger than the inhomogeneit ies that are

present. Furthermore, the definition of local characteristics must assume some

98 MIHIR SEN AND K. T. YANG

form of the expected microstructure even before measurements are made; for example, the same measuring technique will not be used for filtration through a layer of sand as for flow along a set of capillary tubes.

Practical considerations prohibit the direct measurement of point-wise proper- ties since experiments have to be made with beds of finite depth. Normally one has to make a homogeneity or similar assumption to deduce the local charac- teristics of the bed. These local properties can then be employed to calculate the flow through the finite bed. It appears then that, for some cases at least, it is preferable to cut short this procedure and seek to characterize the bed from the point of view of its overall behavior. Thus, we should look for quantities for which we do not have need to know the details of the flow field within the

material, nor to assume a material continuum. This is especially true in the presence of inhomogeneities for which local properties could have discontinuities or vary considerably within the bed. A spatial average of local properties would not be very useful either. Such is the case for fissured rocks, for instance, the transport properties of which are almost entirely controlled by the fissures which are several orders of magnitude more permeable than the solid rock (Streltsova, 1976). Characterization through local properties becomes difficult if not im- possible in such cases.

It is necessary then to introduce a definition by which any finite bed can be characterized. It should be closely related to the way in which the information will be used in order that the assumptions involved in the whole process be kept to a minimum. This way it would be possible to encompass all kinds of permeable beds without regard for their internal structure. Since flow distribution is one of the quantities of practical importance, we examine here a way of using it in an input-output relation for an entire bed. This relation makes possible a charac- terization of the bed which can be used later to determine flow distributions for the same bed under different flow conditions. Though the characterization depends indirectly on the details of the internal structure of the bed, no homogenei ty or other similar assumption will be made; we will not be able to say much about the local material or flow properties, but will avoid the question altogether. A similar transfer function approach to the t ime-dependent tracer movement with respect to depth was proposed by Jury (1982) in which lateral

dispersion was neglected.

2. Integral Representation

Although we can set up the defining relations for a permeable material of arbitrary shape with arbitrary inflow and outflow surfaces, we will for simplicity consider such a bed confined between plane, parallel inflow and outflow surfaces S and S' respectively, shown in Figure 1. The inflow and the outflow velocities, u(x) and u'(x') respectively, are assumed parallel and normal to both surfaces, x

AN I N F L O W - O U T F L O W C H A R A C T E R I Z A T I O N

X J S

99

x' -~u '

Fig. 1. Schematic of a permeable bed.

,

and x' are two-dimensional vectors on the surfaces S and S' such that ]x-x '] is the lateral distance between them.

The contributions from the entire inflow surface can be summed to give the total outflow at a specific location. They are thus related by

Is A.(x, x') u(x) dx, (1) U'(X')

where Au is the kernel of the integral. If necessary for experimental or com- putational purposes, this integral can be approximated by a discrete form. The inflow and outflow surfaces are divided into N and N' numbers of small enough elements to give

N

U) = Z Aijui for 1 ~< j ~< N', (2) i = 1

where Aij is a rectangular N x N' matrix operator depending on the fluid velocity U.

3. Properties of A. (x, x')

On physical grounds, the kernel A~ should be bounded, and should vanish as the lateral distance between the inflow and outflow points I x - x'l ~ ~. In fact, for a bed which is of finite lateral extent, A , = 0 if Ix -x ' l is greater than the lateral extension of the bed. Another restriction is imposed by mass conservation, which for an incompressible fluid is

Is u(x) dx = Is, u'(x') dx',

100

giving

Is A.(x, x') dx' = 1. (3) t

Furthermore, if the inflow is a Dirac delta distribution at Xo given by u(x)= a(x-xo) , Equation (1) becomes

u'(x') = A . (x0, x'). (4)

In principle then, A~,(x, x') can be determined by injecting a unit flow velocity at different positions represented by x = Xo, and determining the form of the outflow for each inflow position.

If differentiability with respect to velocity is assumed, then for small velocities the function Au can be considered to have an expansion of the form

Au = Ao + [@A,],=oU + ' " ", (5)

where Ao = Ao(x, x'). The second term on the right is the Fr6chet derivative [@A~],=o (of Au with respect to u at u = 0) operating on u. A linearized version of A, can thus be defined as

MIHIR SEN AND K. T. YANG

Ao = lim (A. ) , (6) u - + 0

In some physical situations, Au may indeed be independent of u for all u. This will be so, for instance, when the flow is directed along specific channnels within the material such that there is no lateral dispersion. This is the case for flow through ducts and pipes. Then, Au = Ao in Equation (5) and the integral transform in Equation (1) is strictly linear. The flow distribution is determined entirely by the geometry of the flow passages. In general though, the local fluid velocity within the material depends not only on the geometry of the flow passages within the permeable material but also on local pressure gradients which affect lateral dispersion. However, in the limit of vanishing pressure gradient and velocity, only the former determines the local velocity. Then Ao as defined by Equation (6) becomes independent of velocity,

3.1. EXTERNAL HOMOGENEITY AND SYMMETRY

A bed will appear to be externally homogeneous if the relation between inflow and outflow at two points is determined only by the difference in lateral coordinates. In other words, the locations of the inflow and outflow velocities do not matter, only the difference in their positions does. Then

Ao(x, x') = Ao(Ix- x'l). (7)

As a special case, if the inflow U were uniform we can use Equation (3) to obtain

AN INFLOW-OUTFLOW CHARACTERIZATION 101

u'(x') = UIs Ao(lX-X']) dx

= u , (8)

signifying that the outflow would also be uniform. Internal homogeneity, on the other hand, has to be defined on the basis of a microscopic filtration velocity or similar local quantity. External homogenei ty does not restrict the material to internal homogenei ty and is in that sense more general. Though an internally homogeneous bed will also be homogeneous externally, the converse is not always true. The property of symmetry with respect to the origin can also be defined for a bed if A0(x, x') = Ao(x', x). Thus, an externally homogeneous bed is symmetric with respect to any origin.

3.2. COMPOSITE BEDS

The beds can be stacked upon each other in the form of n parallel layers, each one with a possibly different A , . Their overall transfer function A*(x, x') is related to that of each one of the constituent layers by

n * ( x ' x ' ) = fI '" f fi el(x, 1 ,x i )dx ldX2 . . . d x n -1 , i ~ l

$1 . . . S n - i

where the integrat ions are over each one of the intermediate

$1, $2 . . . . , S , -1 . xi is a plane vec to r on the intermediate surface Si.

(9)

surfaces

4. l l lus trat ive E x a m p l e s

The function A,(x, x') can be constructed for certain simple cases to give an idea of some of the results that can be experimentally expected and their relation to the physical nature of the bed. These illustrations could find applications in chemical reactors or hydrological processes. It must be emphasized that examples

of this type help in the interpretation of the experimentally determined A , , but are not really intended for modelling. In fact, two beds with entirely different internal structures could conceivably give similar A, 's. Such beds would then be similar with respect to their external flow characteristics.

4.1. TILTED LAYERS

It is possible to encounter beds of the form of tilted layers where the flow occurs much easier along the inclined layers than across them. If no cross-flow is allowed

Au(x, x') = ~ ( x - x' + A), (10)

where A is the shift in lateral position of the outflow with respect to the inflow.

102 MIHIR SEN AND K. T. YANG

From Equation (1) we get

u'(x') = u(x' - A). (11)

The inflow and outflow are identical except for a constant shift in position and so the bed is externally homogeneous.

4.2. FRACTURES

The characterization discussed here will be most useful for beds with in-

homogeneities. Consider, for example, an otherwise externally homogeneous bed with a fissure running from Xl on S to x2 on S'. An inflow U at x = xl will result in the same outflow U at x ' = x2. If the fissure does not interfere with the homo- geneity of the rest of the bed, one can write Ao = AN + A~o, where Ao H and A~o are due to the homogeneous and inhomogeneous parts, respectively. However , it is more likely that there will be interference between the two and superposition cannot be assumed. At any rate, the combined Ao can be determined as usual.

5. N o n l i n e a r A s p e c t s

In general, lateral dispersion within a permeable bed is a function of the velocity

distribution, which in turn affects A~. As a consequence, for nonvanishing velocities A , is dependent on u and the integral transform represented by Equation (1) is nonlinear. However , if A~ varies smoothly with u, the expansion of Equation (5) exists and so does A0. The nonlinearity in this case is due to the dynamics of the flow. Though the general case is nonlinear, the range of applicability of linearity is of considerable interest for a given bed.

Interactions between the flow and the matrix of the permeable~bed are also a sou rce of nonlinearity. Even though the physics may be quite complex, it is evident tbat - thel tow within the impermeable material may cause a change ' in its internal structure. Ho'wever, it is not necessary that this be reflected in a change in A , . As an example, the cfiange could be localized enough so as not to affect the overall infli3w-outflow relation. Another possibility is that the overall charac-

teristics of the bed after being affected by the flow are in some sense similar to those before the change, so that A , is not affected.

This flow-structure interaction brought about by a change in u can be of different kinds. The simplest would be a smooth and reversible change which we can call linearly or nonlinearly elastic, depending on its dependence on velocity. For reversibility, the permeable bed structure deforms continuously as the velocity u changes due to fluid dynamic forces and returns by the same path as the change is reversed. It is also possible to have a smooth but irreversible change in A , which would be identified as hysteresis. This would be so, for instance, in the presence of adhesion to wetting surfaces, as in the experimental observations of Levec et al. (1986).

The kernel A , is a useful quantity even in cases where it changes dis-

AN INFLOW-OUTFLOW CHARACTERIZATION 103

continuously with u, implying the nonexistence of @Au at this point. This might happen if there is a strong internal change of the matrix within the permeable bed such as fracture, dislocation or other velocity induced discontinuity. It is difficult to imagine a situation in which this process could be reversible, although that must be kept in mind as a theoretical possibility. It could not occur, however, at vanishingly small velocities so that we can be assured that A0 defined in Equation (6) always exists. It must also be pointed out that u(x) is a function and it is possible to induce discontinuities by altering its functional nature and not just its magnitude. Such would be the case if a given mass flux were concentrated over a small area to produce hydraulic fracture. In some applications, this would be quite desirable.

6. Discussion

It is to be remembered that our purpose has been to introduce a new charac- terization of permeable beds, not to model them. The principal feature of this characterization is that it specifies the hydrodynamics of the flow. Once the flow is thus determined, transport of scalar quantities can be computed. This is most effective if the transport is convective and nondiffusive which, of course, is an idealization. The characterization is not based on continuum assumptions and is thus especially useful for the study of inhomogeneous and nonlinear permeable beds. This approach can be useful for a wide variety of applications, including, for example, the passage of solutes in beds which cannot conventionally be called porous. The unification of the characterization of a wide diversity of permeable media is one of the advantages. The simplicity of the properties of the charac- terization parameters makes it particularly useful for applications.

Another important aspect of the characterization discussed here is that it is measurable. Unlike some other descriptions of permeable beds, it can be applied to a great variety of beds regardless of their microstructures. Though the result obtained will be related in some way to the details of the internal flow passages in a manner similar to tortuosity, its determination is much more direct from an experimental point of view. However, careful experimentation is needed to reveal the range of validity of a linear approximation and use of a constant A0. This is similar to the procedure generally used in mechanics wherein it is important to determine the range over which a material parameter such as coefficient of elasticity or thermal conductivity can be considered approximately constant. The important point, however, is that the basic definition introduced here is valid under all circumstances.

References

Jury, W. A., 1982, Simulation of solute transport using a transfer function model, Water Resour. Res. 18, 363-368.

104 MIHIR SEN AND K. T. YANG

Levec, J., S~ez, A. E., and Carbonell, R. G., 1986, The hydrodynamics of trickling flow in packed beds, Part II: Experimental observations, AIChE J. 32, 369-380.

Reiss, L. H., 1980, The Reservoir Engineering Aspects of Fractured Formations, Gulf Publ. Co., Houston.

Scheidegger, A. E., 1974, The Physics of I~7ow Through Porous Media, 3rd Edn., University of Toronto Press, Toronto.

Streltsova, T. D., 1976, Advances and uncertainties in the study of groundwater flow in fissured rocks, in Z. A. Saleem (ed.), Advances in Groundwater Hydrology, AWRA, Minneapolis, pp. 48-56.