Comparison of Pore Size as Determined by Mercury Porosimetry and by Miscible Displacement Experiment
A considerable discrepancy between the pore size distribution of porous media as determined by methods of miscible displacement and mercury intrusion porosimetry exists in the literature. This discrepancy is discussed and reconciled by applying the pore structure model proposed by the authors.
In a rarely quoted paper Klinkenberg (1957) developed a computational technique for the calculation of pore size distributions from miscible displacement data. Applying this technique to a sandstone he calculated pore sizes which were considerably smaller, over the entire range of values, than those obtained by mercury intrusion porosimetry. For the first one-third of the saturation range, corresponding to the largest pore radii in the sample, the ratio rmercury poroslmetry/rmlsclble displacement Was equal to about 3. This is a remarkable result because it is well known (Purcell, 1949; Burdine et al., 1950; Dullien and Mehta, 1971/1972), that mercury intrusion porosimetry measures the size of entry pores which may be either smaller than the rest of the pores, or of the same size, but can never be bigger than them. Klinkenberg’s conclusion that “the current concept of pore size distribution is not unequivocally defined”, while accurate, does not go as far as to explain even qualitatively the reason for the particular type of discrepancy found by him. I t is proposed here that the model of pore structure consisting of an assemblage of straight cylindrical capillaries, used by Klinkenberg, is chiefly responsible for the discrepancy. I t is shown below that a more realistic model of pore structure (Dullien, 1975a,b; Azzam and Dullien, 1975) consisting of three-dimensional networks of cylindrical capillaries with step changes in their diameter gives a more consistent account of both the results of mercury intrusion porosimetry and miscible displacement. Klinkenberg considered a single capillary with radius r and length 1, equal to the macroscopic length of the sample, and wrote the following relationship between the volume of displacing liquid which has entered the capillary a t breakthrough (the left-hand term) and the time t it takes for the same volume of liquid to pass through a capillary of radius r under the influence of the pressure gradient Ap/l:
If the capillary has no uniform cross-sectign, but contains “necks” of radius r , and “bulges” of radius rl, then eq 1 is replaced by the following relationship:
where 1, and 11 is the sum of lengths of “necks” and “bulges”, respectively, in a single capillary with step changes in its cross section. The validity of the Hagen-Poiseuille equation has been assumed in each capillary segment. Equation 2 may be rewritten as follows
where 1 , is the effective length of the capillary, y = ls/(ls 111, and x = rs/rI. Klinkenberg combined eq 1 with Darcy’s law
+ (4)
to obtain the following expression for the square pore radi us r 2 8K r2 = (5 €V where V = v/Alt. Combining eq 3 with eq 4 gives
where rS2is the square pore radius calculated from mercury intrusion porosimetry and 8KleV is the value obtained by using Klinkenberg’s model. The ratio expressed by the lefthand term of eq 6 was about 32 = 9 in the case of Klinkenberg’s experiment with a sandstone sample, for the first one-third of saturation range. I t is interesting to compare this result with the values obtained by evaluating the righthand side of eq 6 for a variety of different sandstone samples. According to Dullien’s network model, in agreement with the results of independent treatments (Haring and Greenkorn, 1970; Johnson and Stewart, 1965; Wiggs, 1950) for tightly consolidated porous media (le/1)2 = 3. The quantities y and x have been estimated by Azzam and Dullien (1975) for a number of sandstones, using the pore size distributions of Dullien and Dhawan (1975) and of Batra (1973). The value of the right-hand term in eq 6 has been calculated for twelve different sandstones and the average value of 1 2 was obtained. The lowest value, for Bartlesville sandstone, was 3.4 and the biggest value, for Bandera sandstone, was 31. The value of 9, obtained by Klinkenberg, is well in this range and one may conclude that three-dimensional network models of pore structure, consisting of capillaries with step changes in their cross-section, can be expected t o bring the results of the miscible displacement method in better harmony with the mercury porosimetry pore size distributions. Literature Cited Azzam, M. I. S., Dullien, F. A. L., paper submitted for publication to lnd. Eng. Chem., Fundam. (1975). Batra. V. K., Ph.D. Thesis, University of Waterloo, Waterloo, Ont. Canada, 1973. Burdine, N. T., Gournay, L. S. Reichertz, P. P., Trans. AIM€, 189, 195 (1950). Dullien, F. A. L., AlChEJ., 21, 299 (1975a). Dullien, F. A. L., AlChEJ., 21, 820 (1975b). Dullien, F. A. L., Dhawan, G. K., J. Colloid lnterface Sci., 52, 129 (1975). Dullien. F. A. L., Mehta, P.N., Powder Techno/., 5, 179 (1971/1972). Johnson, M. F. L., Stewart, W. E., J. Catal., 4, 248 (1965). Haring, R. E.. Greenkorn. R. A,, AlChEJ., 16, 477 (1970). Klinkenberg, L. J.. Pet. Trans. A M € , 210, 366 (1957). Purcell. W. R., Trans. AlME, 186, 39 (1949). Wiggs, P. K. C., in “The Structure and Properties of Porous Materials”, p 133, Academic Press, New York, N.Y., 1958.
Chemical Engineering Department University of Waterloo Waterloo, Ontario, Canada
F. A. L. Dullien’ M. I. S. Azzam
Receiued for reuiew October 13, 1975 Accepted F e b r u a r y 16,1976 Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
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