JOSEPH D. PARKER The Institute of Paper Chemistry, Appleton, Wis.
Permeability to W a t e r of
Partially Saturated Glass Fiber Beds Basic understanding - of the liquid relative permeability mechanism in fibrous systems is a foundation for the study of practical processes, such as drying of paper, in which this is a controlling factor
THE
rate at Lvhich partially saturated porous mrdia conduct liquids is generally characterized by a relative prrmeability, the ratio of unsaturated to saturated permeabilitv. In the past, the definition and quaniitative expression of relative permeability have been developed almost entirely from the study of granular structures (7-5, 7, 8, 7 7- 77). In the work described here a quantitative expression was derived to define relative permeability. The results of experiments with a two fluid-phase (water and air) glass fiber system indicated its validity.
Downstream parous plate Upstreom porous plate Experimental fiber bed Compression yokes Gauge peg Differen,liol manometer Mercury manometers
Rotameler
Modified Expression of Relative Permeability
Liquid water is retained in the pores of partially saturated glass fiber beds in accordance with the laws of capillarity. .4n effective mean hydraulic radius of pores, therefore, is given by m = r/Pc
Steady flow through the unsaturated bed is obtained by maintaining a fixed capillary pressure differential across the bed
(1)
assuming a zero contact angle. Based on the capillary flow mechanism ( 5 ) , then, the unsaturated water permeability of these beds is determined from the D'Arcy equation :
By analogy with the Kozeny-Carman expression for the D'Arcy permeability in saturated flow, the corresponding expression of specific unsaturated permeability is readily derived as:
This expression of unsaturated permeability is refined by introducing the pore size distribution. Effective pore size distributions are usually calculated from the desorption phase of the capillary pressure isotherms by means of Equation l . The physical significance of such effective pore size distributions is discussed by Carman ( 3 ) . Actually,
the quantity, y/P,, measures the principal radii of curvature of the liquid menisci in the system. For certain pore shapes, notably cylindrical, this quantity has the same value as the volume to surface ratio of the pore. However, this is not generally true for the irregular shaped pores in real porous media.
Pore size can be defined generally by comparing the void spaces geometrically to small particles. I n small particle technology, the surface to volume ratio of irregularly shaped particles is assumed proportional to the ratio of the second and third moments of some weight average linear measure of the particle
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l.o-~ Ez.945 €=.946 0 €=.943 0 €=.943 A Ez.943 m €=.944 0 L
Ls.82 L a .82 Ls.82 L=.82 L'1.00 La.82
cm. cm.
cm. cm.
em. cm.
01
I I ~ I I I I I
.02
I
0.I
I ii1,1i1 1.0 CAPILLARY
I
i
l U~ i 10
PRESSUR&
pc,
1
I
l
1
1
1
i
1
I
IO0 CM.
Hg.
A Figure 2. Similar capillary pressure curves are exhibited by a variety of incompressible porous media .I O
-
4 I
b ' L I I: I
1'6
CAPILLARY
I
o: pc , CM.
PRESSURE,
jlze (9). The coefficicnt of propoitionality then, is characteristic of the shape and heterogeneity of the particles. By analogy, the surface to volume ratio of a pore space, &,/E, is similarly defined, taking the effective mean hydraulic radius, m, as the linear measure of the pore size. Thus,
Hg.
Figure 1 , With increasing capillary pressure, saturation falls off rapidly and approaches a -residual saturation asymptoticaily '
individual elements over the range of saturation yields the over-all specific permeability:
and the liquid relacive Permeability is
c" m2ds and for a system of uniforrn pores,
- -- -c
so
e
r
(4)
n
Following Wyllie and Spangler (77), porous medium can be considered as comprised of discrete elements, each element containing only the pores of effective size, ?E. Taking that part of the saturation of the over-all medium contributed by a single element as Ar, a specific permeability can be defined for this element in terms of the over-all system parameters similar to Equation 3. Combining with Equation 4, the permei bility is
5umming
the
permeabilities
of
the
Experimental In this initial study, it was deemed advisable to work with the simplest fiber systems possible. Glass fibers provide ideal media as they are geometrically well defined, uniform in size and shape, hydraulically smooth, nonporous, and they form dimensionally stable beds. T o simplify the experiment further, the glass fiber beds were rendered rfiectively incompressible by externally compacting them. The experimental program consisted essentially of determining static capillary pressure curves and steady state water relative permeability curves of
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the partially saturated glass fiber beds. Beds of two sizes of glass fibers were used, having weighted average diameters of 1.41 X lop4 and 4.54 X lop4 cm. These fibers are denoted by their nominal sizes of 1.2 and 3.0 mp. The beds were preformed from low consistency slurries and transferred, fully saturated, to the apparatus for testing. Static Capillary Pressure Measurement. This measurement was made with porous plate equipment similar to that of Christensen and Barkas ( 6 ) . The procedure consisted of applying, stepwise, increasing capillary pressuresi.e., suction-to the initially fully saturated bed. The equilibrium saturation of the bed was determined at each value of capillary pressure. A complete curve of capillary pressure us. saturation could be obtained with a single experimental bed. Steady-State Permeability MeasureExperimental specific permement. abilities at partial saturations were calculated from the measured flow rate and pressure drop through the beds by Equation 2 (see diagram). A differential manometer with a fluid of low specific gravity (1.75) connected across the bed was used to measure the pressure drop. h4ercriry manometers provided for the measurement of capillary pressures at each face of the bed. Water flow rates were determined with a previously calibrated capillary tube flowmeter system. The practical limit of these permeability measurements was a saturation of about 9770. To fix the permeability curve at the high saturations, the permeabilities a t 100% saturation were determined with basically the same equipment developed by Ingmanson (70) in his filtration studies. From rhese experimental data, relative permeability curves were obtained by converting the average capillary pressures to point values of saturation (by interpolation of the appropriate capillary
GLASS FIBER BEDS
-
Grade
-
E .943 ,943 .943 ,943 ,943 ,944 .944 ,944
io-‘
io-$
I
1 .2
1
1
.4
Plate
Fine Fine Fine Fine Fine Fine Medium Medium
.98 .98 .98 .98 .98 1.49
1.00 1.41
1 1 1 1 .6 .8 SATURATION,
I
I 1.0
I
I
1.2
I
j.4
Figure 3. The rapid decline in permeability in the initial stages of desaturation suggests a change from saturated to funicular regime of water retention in the pores
pressure curve) and transforming the specific permeability to relative permeability.
Results and Discussion
As the experimental results for both sizes of fiber were consistent and equally representative of fibrous systems, only the results of the 1.2-mp fiber ale presented here. Static Capillary Pressure Curves. The glass fiber capillary pressure curves (Figure 1) show some slight loss of water before the break-off point. Carman ( 3 ) points out that the breakoff point becomes less well defined when the pores are not uniform in size (Figure 2). In many real porous systems, the slight initial desaturation before the breakoff point is probably due to the opening up of a few largr pores. This relatively small change preceding the break-off point assumes large significance in the analysis of two-phase flow. As seen from the theoretical equation. the capillary pressure measurements enter into the analytical expression of relative permeability in integral form of the pore size distribution, equivalent to the capillary pressure curve. The integral of this curve is very sensitive to the large pore sizes, or low capillary pressures. As Carman reasons, if the mean hydraulic radius is interpreted as the radius of a circular capillary, the liquid retained in an actual pore corresponds to that retained by a group of circular
pores, varying from zero radius upward. This illustrates the continuity of pore size in real systems. It is meaningless in such a case to assume that liquid is held in cylindrical capillaries; and the effective pore size, or mean hydraulic radius, is simply an index of the volume of liquid retained, its value depending on the configuration of the pore space. Steady-State Relative Permeability Curves. Some discrepancies were noted in the experimental relative permeability results (Figure 3) as evidenced by runs 7 and 8. These irregularities were ascribed to the permeability decay of the porous plates and “fiss~reeffects” in the experimental beds. Both of these effects lead to spuriously low calculated permeability values. In view of the consistent permeability results, the permeability decay of the beds was apparently of small significance. Relative permeability curves of the glass fiber beds differ significantly from those of various other media from the literature (Figure 4). Brownscombe and others (2) make the point that as the nonuniformity or complexity of the pore structure increases, the permeability curve deviates more than a straight line. Undoubtedly, this complexity of pore structure was considered to be roughly correlated with the geometrical heterogeneity of the solid particles which comprise the medium. In Figure 4, the glass fiber curves, E and F, fall beyond that of the ungraded alundum core, implying that these beds, which are comprised of very regular and uniform particles, paradoxically have the most complex structure of all. The proper analysis and description of this complexity probably involves more than the heterogeneity of the particles, and porous media of comparatively uniform particles may give rise to exceedingly complex liquid network configurations at partial saturations. Specifically, the rapid fall-off of permeability that accompanies the slight change in saturation of the glass fiber beds at the high saturations was probably due largely to the change from the regime of total pore saturation to the funicular saturation regime in which the liquid exists as contiguous channels throughout the medium. When the saturation regime breaks down to the funicular state. only a relatively small loss of liquid occurs. but the size and shape of the pores which remain filled with liquid may undergo a tremendous alteration. This effect of shape of pores is distinctly different from that of number of pores of a given size, although they cannot be distinguished from the capillary pressure measurements alone, for they manifest themselves in the same way in these measurements. However, their effect on permeability
is quite different. Thus, a volume of pore space contained in a few highvolume pores would conduct water at a greater rate than the same volume of pore space contained in many lowvolume pores? even though both types of pores had the same effective ’.size’’-Le., menisci curvature. In summary, then, it is instructive to describe the change in relative pernieability of two-phase systems as they are desaturated in terms of three major effects. The first effect is based on the well-established concept that as pores increase in size, they contribute proportionally more to permeability than to porosity. The second effect is the increase in tortuosity of the liquid channels as the saturation decreases, and the third is some as yet unidentified effect which, apparently, is related to the shape of the pores. In the absence of more fundamental theory, this effect was attributed to the shape factors associated with the liquid channel system. This is the same argument that was advanced in the development of the Kozeny-Carman equation of relative permeability which led to the introduction of the shape factor, C. As seen from the literature, however, the relative permeability of some granular systems appears to be independent of this shape factor effect. The variation in pore size has been frequently accounted for in the expression of relative permeability as the integral or summation of the product of the pore size squared and the saturation associated with the pore size from the capillary pressure curve or some equivalent modified form of this product. I n this work, the variable pore size term in the derived equation for relative perme-
l
ability, Equation 7, is
rn2ds. Thus,
representing pore size distribution this
.9.8-
>.
.7-
c
Y
A 0
C D E
Sinqle Fmed
CoPilIory elundum
Idbe
[##-)
sure
lqledtdl Irl UnConsolidoled sand FYIId olundum core lunqrodcal Iri 1 . 2 ~ Glass I 8 D e r s
1’61
.e-
SbTURATlON,
I
Figure 4. Compared with granular media curves, the steeper slope of the glass fiber permeability curves indicates a more drastic change in the liquid channel configuration VOL. 52,
NO. 3
MARCH 1960
249
way in the relative permeability expression appears to be generally established. The limitation of this approach is that a knowledge of the distribution of pore sizes corresponding to the residual saturation is not available. However, for most porous systems, the residual saturation represents such a small part of the total pore space that it usually can be ignored. If, as it is hypothesized, the empirical factors introduced in the relative permeability equation are the factors that distinguish between different types of porous media, the values of these factors would be the same for similar media. Values of the composite ratio, k,T2C2/ k0,T,2C,2, were calculated for both sizes of fibers from Equation 7. Within the limits of the experimental measurement, these values were very nearly the same. Thus the difference in the relative permeability curves of the two fiber beds is wholly explained by the variations in the integral term of the pore size distribution,
l
mzds.
The method is at
least internally consistent. AnaIysis of the EmpiricaI Factors. These empirical factor ratios appear to be a complex function of the gore structure and saturation of the media, and it is not possible to separate the composite values into their separate components rigorously without additional information. A possible variation of these factors is suggested by Wyllie and Spangler’s method of analysis (77). They proposed that the hydraulic tortuosity is equivalent to the tortuosity of an electrical path through the unsaturated bed. From electrical resistivity measurements, they defined a resistivity index, I, as T,/ Ts. Experimentally they found that a plot of log Z us. log s, for several granular media, was a straight line of negative slope, following the equation, S-n
(8)
where n is approximately 2.0 for unconsolidated media. Applying this concept of tortuosity to the glass fiber beds, values of the apparent resistivity index were calculated as a function of the saturation from Equation 7 assuming k,C2/k,,CC2 = 1.0. The resulting log 12s. log s curves were nearly identical for both sizes of fibers (Figure 5). The curves approach and become integral with a straight line having a slope of about -1.90, not far different from the value of -2.0 proposed by Wyllie and Spangler for unconsolidated media. Taking this straight line as an assumed tortuosity variation, values of the quantity, k,C2/k,,C,2, were calculated by Equation 7 . A rough speculative estimate of the variation of the individual ratios, k,/k,, and C2/C,2, was obtained from Equations
250
-1
I
T
L m
= thickness of fiber bed, cm. = mean hydraulic radius of capil-
n
= negative slope of log
P,
=
AP, =
Figure 5. Calculated values of the resistivity index suggest a linear variation of unsaturated tortuosity
3 and 4. These are independent equations of the liquid-solid contact surface involving the k, factor in one and the C factor in the other. Assuming a linear distribution of the liquid-solid contact surface area with saturation, values of the respective shape factor ratios were calculated from these equations. The composite products of these calculated ratios are not in close agreement with those calculated from Equation 7. However, they do reach a minimum at the same saturation and follow the same pattern of variation with saturation. Further, the variation of both factors is the same for the two fibers. Nomenclature
A
= cross-sectional area of bed per-
C
=
C,
=
I
=
K
=
K,
=
K,
=
k,
=
k,,
=
INDUSTRIAL AND ENGINEERINGCHEMISTRY
pendicular to direction of flow, sq. cm. factor of proportionality in Equation 4, which applies to total pore space of a medium (waterfilled pores at complete saturation), dimensionless factor corresponding to C for partially saturated media, dimensionless resistivity index = T,/ Ts, dimensionless permeability of saturated porous medium; D’Arcy’s permeability coefficient, sq. cm. wetting-phase specific permeability of unsaturated porous medium, sq. cm. wetting-phase relative permeability of unsaturated porous medium = K,/K, dimensionless channel shape factor in Kozeny constant for saturated porous media, dimensionless wetting-phase channel shape factor for unsaturated porous media. dimensionless
q
=
S,
=
So
=
s
=
T
=
T,
=
y
=
E
=
,u
=
lary or pore (Equation l ) , cm. I us. log s; number of particles, dimensionless capillary pressure or pressure difference across two-fluid phase curved interface in capillary or pore, dynes ‘sq. cm. frictional pressure drop across bed, dynes/sq. cm. volumetric rate of fluid flow, ml. /sec. surface area of liquid-solid contact per unit volume of bed in unsaturated media, sq. cm./ cc. surface area of particles per unit volume of bed, sq. cm. ’cc. liquid phase saturation of porous medium, equal to fractional part of total pore space filled with liquid, dimensionless tortuosity of saturated porous medium, dimensionless liquid phase tortuosity in partially saturated porous medium, dimensionless interfacial tension betlveen t\vo fluid phases, dynes ‘cm. total porosity of porous bed, void volume per bed volume, dimensionless fluid viscosity, dyne-sec. sq. cm.
literature Cited
(1) Baver, L. D., “Soil Physics,” 2nd ed., Wiley, New York, 1948; Soil Sci. Sac. A m . Prod. 3, 52 (1938). (2) Brownscombe, E. R., Slobod, R. L., Caudle, B. H., Oil Gas J . 48, KO.41, 98 119501. ( 3 j Carman, P. C., J . Phys. Chem. 57, 56-64 (1933).
., Kiesling, F. G., Trans. A m . In&. Chem. Engrs. 36,211 (1940). (5) Childs, E. C.. Georqe, N. C., Discussions Faraday Sac. 3, 78 (1948‘ \I . (6) Christensen, G. N., Barkas, W. W.. Trans. Faradav SOC.51. 130-45 (1955) (7) Fatt, I., DGkstra, H., Trans.‘ .4m. Znst. Mining M e t . Engrs. 192, 249 (1951). (8) Gates, J. I., Lietz, W. T., “Drilling and Production Practices,” p. 285, American Petroleum Institute, 1950. (9) Herdan, G., Smith, M. L., “Small Particle Statistics,” p. 50, Elsevier, Houston, 1953. (10) Inemanson. W. L.. Ph.D. dissertation, T h e Instiute of Paper Chemistry, Appleton, Wis., 1951 ; Tappi 35, 439-48 (1952). 1) Martinelli, R. C., Putnam, J. .4., Lockhart, R. W., Trans. A m . Znst. Chem. Engrs. 42, 681 (1946). 2) Miller, E. E., Miller, R. D., J . Appl. Phys. 27, 32 (April 1 9 5 6 ) ; Soil Sci. A m . Proc. 19, 267 (1955). 3) Pearse, J. F., Oliver, T. R., Newitt, D. M., Trans. Znst. Chem. Engrs. (London) 27, 1-18 (1949). 4) Purcell, W. R., Trans. A m . Znst. Mining M e t . Engrs. 186, 39 (19491. 5) Scheidegger, A. E., “The Physics of Flow Through Porous Media,” Macmillan, New York, 1957. 6) Wyckoff, R. D., Botset, H. G.,
Physics 7, 325-45 (1936). 7) Wyllie, M. R. J., Spangler, M. B., Bull. A m . Assoc. Petrol. Geologists 36, 359-403 (February 1952). RECEIVED for review December 3: 1958 ACCEPTED October 26. 1959
