scaling function for dynamic permeability in porous media

VOLUME 63, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JUL+ 1989

Scaling Function for Dynamic Permeability inPorous Media 0

Two recent Letters, ' one theoretical, Sheng and Zhou(SZ), and the other experimental, Charlaix, Kushnick,and Stokes (CKS), analyzed the frequency dependenceof the dynamic permeability, which is the fluid-flow con-ductance of a porous medium subjected to an oscillatorypressure gradient. Both articles found that the dynamicpermeability of a fairly wide range of porous mediacould be expressed as rc(ro)/xo =f(ro/ro, ), where f(ro) isa nearly universal scaling function, x.o is the dc permea-bility, and m, is a frequency characteristic of the porousmedium. (Oddly, these authors did not demonstrate thatthe experimental and theoretical scaling functions areone and the same. ) The purpose of this Comment is todemonstrate that a simple closed-form expression for thescaling function f(ro) proposed earlier fits both the nu-merical calculations and the experimental measurementsextremely well. This scaling function had been predictedon the basis of certain rigorous properties of K con-sidered as an analytic function of complex co and it hadbeen demonstrated in Ref. 2 that it works well on avariety of random lattices, but Refs. 1 represent the firstsets of hard numbers in realistic porous media.

We had proposed the following scaling function [Eq.(3.4b) of Ref. 3]:

f(co) =[1 —iMro/2] ' —iro

where ro, =t7tt/pftcoa, as defined by SZ, and M—:8atcp/PA ( 8/F2) in terms of quantities defined in Refs. 1

and 2. In Refs. 2 and 3 it was suggested that M is ex-pected to be nearly equal to 1 for most porous media.

SZ ' have calculated the dynamic permeability in threedifferent model porous media: periodic arrays of trun-cated spheres, periodic arrays of truncated octahedra,and tubes with sinusoidally varying radii. Within eachclass, a variety of different parameters was explored.The results of the calculations are reproduced in Fig. l.Superimposed on these results are the values predictedby Eq. (1), where for sake of definiteness I have chosenM=1.

CKS have measured the dynamic permeability in anarrow capillary, a lightly sintered bead pack, and alightly sintered crushed glass pack. ' Their data alsooverlay with the predictions of Eq. (1) but spatial limita-tions do not permit a comparison here.

In porous media for which M differs substantiallyfrom unity, scaling cannot hold; but Eq. (1) is still exact

—4-

(a)

90

75-

60-

45-0)thEO 30-

CL

15—

In m

FIG. l. (a) The magnitude and (b) the phase of scaled dy-namic permeability s(co). The large symbols are the predic-tions of Eq. (1); all other symbols are the calculated values ofSZ (Ref. 1).

in the high- and low-frequency limits and it probablyprovides a reasonable description in the crossover region,for the reasons given in Ref. 2. We view Eq. (1) as therough equivalent of the Debye function for dielectric re-laxation.

David Linton JohnsonSchlurnberger-Doll ResearchOld Quarry RoadRidgefield, Connecticut 06877-4108

Received 5 December 1988PACS numbers: 47.55.Mh, 47. 15.Gf, 47.15.Hg

' P. Sheng and M. -Y. Zhou, Phys. Rev. Lett. 61, 1591(1988); 61, 2391(E) (1988); E. Charlaix, A. P. Kushnick, andJ. P. Stokes 61, 1595 (1988).

2D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid Mech.176, 379 (1987).

D. L. Johnson, J. Koplik, and L. M. Schwartz, Phys. Rev.Lett. 57, 2564 (1986).

580 1989 The American Physical Society