General theory of creep flow is proposed which indicates permeability tensor can be represented in terms of on/y four scalar constants
Advmces in Theory of
Figure 7 . Spatially periodic porous media
he problem of flow in porous media is of interest to and scientistsfrom the chemical engineer who quite likely is concerned with multicomponent reacting systems, to the civil engineer who is often faced with a two-phase (air-water) flow process, to the seismologist who must deal with a deformable porous media in a transient state. Even with this vast array of important problems awaiting our attention, this paper will consider only the simplest of all possible cases; the steady, incompressible flow of a Newtonian fluid under conditions such that inertial effects can be neglected. A careful survey of this problem will hopefully provide a sound basis for the study of some of the more important aspects of flow in porous media. T h e ideas presented in this paper are drawn primarily
Tan enormous range of engineers
14
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
from the early work of Hubbert (15) who has given a lucid account of the famous work of Darcy (7) and has presented, on an intuitive basis, a logical route to a theoretical treatment of this problem; from the exhaustive study by Brenner (5) on creep flow in a spatially periodic porous media; from the work of Slattery (30, 37) on the flow of viscoelastic fluids in porous media; and from the author’s own attempt at a theoretical treatment of this problem (38). There are perhaps three fairly distinct, although certainly not mutually exclusive, approaches to the problem of flow in porous media. Since a porous medium is generally not an ordered structure, the idea of developing statistical models is appealing, and a number of attempts have been made in this area [ I , (3, Chap. 6), (28, Sec. 6 . 6 ) , (29)]. Another approach is the method of
FLOW THROUGH POROUS MEDIA SYMPOSIUM
Fluid Motion in Porous Medid STEPHEN WHITAKER
geometric modeling in which one postulates a geometry which hopefully bears some resemblance to the porous media, yet is sufficiently simple to allow the governing differential equations to be solved. Only two geometric models will be referred to in this work; the spatially periodic model of Brenner (Figure l), and a skewed capillary model (Figure 2). I n Brenner’s model the porous media are made up of an infinite array of unit cells (two shown in Figure 1) having boundaries of arbitrary shape and containing one or more solid particles of arbitary shape. So that the flow in a unit cell will be amenable to analysis, it is required that the boundary of the unit cell be a fluid surface. Thus one must accept the existence of rigid supports of negligible hydrodynamic resistance which hold the matrix of unit cells in place.
L
L
-
1
Figure 2. Skcwed capillary model of porous media VOL. 6 1
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The skewed capillary model shown in Figure 2 is a minor variation of the well-known straight capillary model (28, page 114), but it cannot be obtained from that model by a simple transformation of coordinates because the boundary conditions, which may be imposed on the flow, also change under a coordinate transformation. A third approach, which lies somewhere between statistical models and geometric models, is the development of correct averaged forms of the governing differential equations. These equations should be valid for any geometry, thus the results obtained for specific geometric models must satisfy the averaged equations. I n addition, any statistical model should be in accord with the averaged equations. All three methods lead to unspecified parameters which must be determined experimentally, and the primary objective of theoretical work in this general area is to aid in the interpretation of experimental data. I n attacking the problem of incompressible flow in porous media, one is confronted with the fact that the final result is pretty well established-i.e., Darcy’s law gives an accurate description of the flow. Because of this, it is easy to proceed along a variety of approaches, some of which might well be erroneous or wholly intuitive, to the correct final result. We will try to avoid this pitfall in the present study and establish as carefully as possible a logical, correct route to the final result. 1.
The Problem
We are concerned here with the incompressible, creep flow of a constant viscosity Newtonian fluid in a rigid porous media. The equations governing this process are : 1. The continuity equation
T h e velocity, u, and position vector, r, are measured relative to an inertial frame imbedded in the porous media; thus the porous media may move with a constant velocity, u, relative to a frame that is stationary with respect to the fixed stars. 2. The equations of motion 0 =
-vp 4- pg 4-
pv2v
(1-2)
The restriction of constant viscosity is imposed by the form of Equation 1-2. 3. The void volume distribution function a(r) =
1, if r lies in the fluid region 0, if r lies in the solid region
(1-3)
If the function a ( r ) were known it would be possible, in principle, to solve the governing equations and thus determine completely the pressure and velocity fields. However, a ( r ) is never known and we are obviously forced into a different type of analysis-i.e., a derivation of the volume averaged equations. I n this approach we associate with every point in space an averaging 16
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
volume V which contains a volume of fluid V,. volume porosity is given by cI,=
Vr V
The
(1-4)
or more explicitly in terms of the void volume distribution function @ =
1 -J a(y)dV v v
(1-5)
Throughout this work we will require that the averaging volume be constant in time and space; however, the magnitude of the fluid volume V , may vary with position subject to certain restrictions to be discussed later. Before attacking the problem at hand-i.e., obtaining the averaged forms of the governing equations-we must establish some ideas about area and volume averages and what is meant by a “meaningful average,” and we must develop an averaging theorem to relate averages of derivatives to the derivatives of averages. 2.
Averages
In treating problems of flow in porous media, we assume that some microscopic characteristic length, d, exists which is representative of the distance over which significant variations in the point velocity, u, take place. Similarly, we designate the macroscopic characteristic length as L , and assume it is representative of the distance over which significant variations in the volume averaged velocity, (u), take place. I n general, one associates d with the poorly defined but intuitively appealing mean pore diameter, and L with some macroscopic dimension representative of the process under consideration. I n Darcy’s original work one would consider d to be on the order of magnitude of the “diameter” of the sand grains used in the filter, while L would be associated with the diameter of the filter bed. Letting $ be some point quantity (tensor of any order) associated with the fluid (velocity, density), we define the volume average of $ as
One can now ask whether the average ($) is a function that is suitable to use in analyzing flow in porous media. If we were to plot ($) versus Vwe might obtain a function similar to that in Figure 3. Such a curve would be obtained if the point associated with V was in the solid region; thus V , would be zero for small values of V. As V becomes larger, portions of the fluid are contained within V and the average increases from zero going through some fluctuations representative of the variations in the point value of $. For values of V larger than V* the microscopic variations in # are essentially smoothed out, but the value of ($) need not become constant. It should be clear that ($) is a continuous function for any value of V , but for values of V larger than V* the volume average ($) becomes %mooth,” and thus amenable to the type of analysis we have in mind.
The linear term in Equation 2-4 has dropped out since xi is measured from the centroid of V, thus n
JV xfdV = 0 If ($) is a linear function of xd, Equation 2-2 is immediately satisfied; however, we wish to consider a somewhat more general case. An order of magnitude estimate of the first integral in Equation 2-5 can be expressed as,
and the order of magnitude estimate for the second derivative of ($) is given by
Figure 3. Dependence of average on averaging volume
If we let 1 be a characteristic length for the averaging volume, we can impose our first restriction (throughout this paper restrictions will be denoted by R.1, R.2, and assumptions by A.l, A.2, and so forth). d
>= ($)
Neglecting the higher order terms in Equation 2-5, we express the average of the average as
(($>>= ($)
(2-3) For convenience only, we choose the point with which we associate ($) to be the centroid of V and expand ($) in a Taylor series (32, p 228) choosing the centroid as the origin of our coordinate system. (Here x( is the position vector in index notation. Repeated indices are summed from 1 to 3.)
(2-9)
Clearly our analysis must be bound by the restriction 1 o
=
$('pk)dx(t)
(2-1 1)
1/2
where (i), ( j ) , and ( k ) are distinct. If we expand $(',k) in a Taylor series in x(d), we obtain
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where the summation convention does not apply to indices in parentheses. Substituting Equation 2-1 2 into Equation 2-1 1 and evaluating the integrals yield
Estimating the order of magnitude of the second derivative as
(2-14) we can express Equation 2-13 as
Thus if our analysis is to be restricted by I
(5-23)
This result is consistent with that given by Slattery (31) and indicates that the permeability tensor is symmetric. If A ( 1 ) = 0, we obtain the proper form of Darcy's law for the skewed capillary model, and the symmetry requirement is in accord with the spatially periodic model. From the experimentalist's point of view, the problem has been reduced to measuring four scalars @(I), Bp),XI, X2) although the problem could be simplified if a method of making a good guess about the nature of 1 could be developed. Our guess, given by Equation 5-10 is not entirely out of line with the result obtained here. Comparing Equation 5-28 and Equation 4-36, we find 8(1)6i,
(v)* = F(A*, V(P)*)
B
