Lilleleht, L. V., Hanratty, T. J., A.I.Ch.E. J.7, 548 (1961). Lockhart, R. W., Martinelli, R. C., Chem. Eng. Progr. 45, 39 ( 1949) . Oliver, D. R., Young Hoon, A., Trans. Inst. Chem. Eng. 46, 106 (1968). Richardson, J. F., Higson, D. J., Trans. Inst. Chem. Eng. 40, 169 (1962). Ross, N. C. J., J. Petrol. Techno!. 10, 57 (1961). Wallis, G. B., “One-Dimensional Two-Phase Flow,” pp 52, 53, McGraw-Hill, New York, N. Y., 1969.
Ward, H. C., Dallavalle, J. M., Chem. Eng. Progr., Sump. Ser. 50 (lo), l(1954). Young Hoon, A., Ph.D. Thesis, University of Birmingham, Birmingham, England, 1967. RECEIVED for review May 21, 1971 ACCEPTED June 12, 1972 Excerpts from this work were presented at the Conference on Engineering in Medicine and Biology, Las Vegas, Nevada, November 1971.
The Dispersion Tensor in Anisotropic Porous Media James A. Guin,’ David P. Kessler,* and Robert A. Greenkorn Purdue University, Lafayette, Indiana 47907
Dispersion in laminar flow through isotropic and anisotropic porous media i s analyzed using a mathematical model based on the random walk. The connection between the random walk model and the continuum model for systems in which the residence time i s a random variable i s presented. Dispersion coefficients are derived in terms of single-step ensemble averages. The analysis results in a new functional dependence for the dispersion tensor Di, upon the average velocity and medium properties. Previous work in predicting the form of the dispersion tensor i s discussed and compared with this new result.
w h e n fluid flows, molecules which are initially in proximity to one another do not necessarily remain so, but can become separated by substantial distances depending on the physical and chemical conditions under which the flow occurs. This mixing process leads to a leveling of gradients in the fluid and is referred to as dispersion. Such mixing processes are accentuated when the fluid flows through a porous medium, that is, some solid matrix through which the fluid passes. It is often a moot point whether such flows are considered more nearly flow through conduits or flow around obstacles; in either case the presence of the solid modifies dispersion. For example, this is why it is difficult to remove a small spot of stain from fabric using solvent-the stain initially spreads because of dispersion. Dispersion in porous media is important in such diverse fields as chromatography, packed beds, the encroachment of salt water into fresh water aquifers during depletion-recharge cycles, the dispersion in the body of chemotherapeutic agents for treatment of disease states in humans and animals, the recovery of crude oil from underground reservoirs, underground waste disposal, dispersion of agricultural chemicals in soils, and flow in filters. There are a number of mechanisms of dispersion (Greenkorn and Kessler, 1969). This paper considers a model for dispeision when the fluid velocity is laminar but large enough that the contribution of molecular diffusion to the total mixing process is small. Dispersion is usually described by a dispersion coefficient which is a second-order tensor. A fundamental question with regard to the dispersion coefficient as usually defined is whether this coefficient is a function only of the properties of the solid matrix or whether in addition it depends on the properties of the flow field. The difference between these two types of Present address, Auburn University, Auburn, Ala. 36830.
dependency has profound implications in the ease with which the dispersion coefficient can be obtained, correlated, scaled, and used. Given a choice, one would rather have a dispersion coefficient which was a function only of properties of the solid matrix much as one would like to have eddy thermal conductivity and diffusivity coefficients independent of the flow field. This will be shown below t b be impossible for the dispersion coefficient other than in some specific cases (much as the eddy coefficients depend on flow field). Methods €or modeling dispersion in porous media may be broadly divided into statistical or continuum methods. Recent review articles by Greenkorn and Kessler (1969) and Nunge and Gill (1969) provide numerous references describing both of these techniques. The objectives of statistical models of porous media are to provide models that can be described statistically in order to study the form of the constitutive relations for parameters in deterministic models (Darcy’s law, volume-averaged continuity equation) ; to help visualize and clarify flow mechanisms; to rationalize dimensional correlations; and to calculate superficial flow fields. None have the direct objective of predicting flow in porous media since the statistical models are idealized, contain too many parameters, and the parameters are not a t the level of flow calculations (i.e., permeability, porosity, and dispersion). The purpose of the present work is to examine the relationship between the dispersion coefficients obtained using a random walk model and those obtained using a continuum treatment. We are not interested here in using these models to predict directly. Rather, since these ideal models, which are real porous media homologs, have pore geometry characteristics that can be exactly described, it is possible t o use them to investigate details of flow in real porous media. This analysis sheds light upon allowable forms for the dispersion tensor, frequently represented as the contracted Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
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have shown that this equality exists only for porous media having special characteristics (e.g., uniform size pores). This investigation is directed primarily toward anisotropic media, whereas previous investigators have directed their efforts toward isotropic porous media. In anisotropic media the situation is complicated since a pressure gradient in a single coordinate direction will give rise to mean velocity components in all three coordinate directions, not just one. I n addition, the general permeability and dispersion tensors have six independent components rather than one or two, respectively, as in isotropic media. The Model Porous Medium
/ Figure 1 . An elemental pore in the idealized porous medium
product of a fourth-order geometrical dispersivity tensor with a second-order tensor composed of velocity components. The random walk method in this work is used as a method of modeling certain microscopic average properties of the porous medium so that the averaged values may serve as parameters in a continuum model. The usual continuum model used for dispersion is the diffusion equation, with a “dispersion Coefficient” replacing the diffusion coefficient, and with a variety of boundary conditions. Although this may seem an unnecessary forcing of a gradient model on a nongradient process, the procedure is not unreasonable considering that (1) a fluid “particle” flows through a porous medium in a series of steps not unlike a random walk, and (2) random walk processes generate diffusion-type equations in their limtinig behavior. The continuum model (Scheidegger, 1954) attempts to describe dispersion in a Cartesian frame with the equation
where the Djk are elements of a symmetric dispersion tensor which by definition accounts for mixing on both the molecular and the convective scale. The velocities U , in eq 1 are average velocity components related to the filter (superficial) velocity by U i = q,/$ where $ is the volumetric porosity. Several investigators have previously used the random walk technique to study dispersion in isotropic porous media. Soheidegger (1954) gave one of the earlier treatments; however, this treatment yields no difference in the longitudinal and transverse dispersion coefficients. It is known from experiments, however, that the dispersion in the axial direction is 3 to 60 times that in the transverse direction (Greenkorn and Kessler, 1969). Josselin De Jong (1958) and Saffman (1959) showed that this effect could be produced by considering a random walk through a network of uniform capillary tubes. Recently, Haring and Greenkorn (1970) have considered a model of nonuniform capillaries and have obtained expressions for the axial and transverse dispersion coefficients in terms of the 6-distribution parameters; however, because of an oversight in computing the filter velocity the expressions so obtained are in error. Similar errors in computing the filter velocity have been made by Scheidegger (1960) and Matheron (1967). The error made by these investigators lies in equating the ensemble average pore velocity t o the spatial average pore velocity, the latter being the correct expression of the filter velocity. Guin, et al. (1971), 478
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A model of a porous medium may be constructed using a pore size probability distribution function f(A,L,a) such that f(A,L,a) dAdLda gives the number of cylindrical pores per unit volume having a cross-sectional area A -+ A dA, length L +L dL, and an orientation a + a d a defined by the angles a1 and a2 as in Figure 1. The mean flow in the porous matrix is assumed to be laminar and to be describable by Darcy’s law
+
+
+
The ith component of the average velocity in an individual capillary is given in steady laminar flow by
(3) c being a dimensionless constant depending on the shape of the individual pore’s cross section. The orientation of each elemental pore in the model to the Cartesian axes may be described by direction cosines (see Figure 1) given by Li = &L.gi. The fluid average velocity vector U in the porous matrix will have direction cosines e t = Vi/ / Uwith / reference to the samk coordinate system. This model has been described in detail by Guin, et al. (1971), and has been shown to have the following macroscopic properties (4)
ALf(A,L,a) dAdLda permeability: k i f = c filter velocity: g ,
=
S, S,
IiljLA2f(A,L,a)dAdLda
(5)
V,f(A,L,a) LAdAdLda
(6)
In all integrals in this paper the domain of integration D is (D:O
