Electrical Conductivity of Partially Saturated Porous Solids The typical decrease in electrical conductivity as electrolyte i s displaced from a porous solid by a gas can be accounted for as a random breaking of conducting paths in a resistance network in which the cavities play the part of resistances.
T h e resistance of a porous body completely filled with a conducting liquid has been calculated from a model in which cavities are arranged in a simple cubic array and each cavity is connected to it's sis nearest neighbors (Owen, 1952). The relative resistance of a porous body that is only partially filled with electrolyte has been calculated from a model in which capillaries intersect at' random (Wyllie and Gardner, 1958), and simulated on a computer for a model of a network of tubes (Fatt, 1956). The pressures for displacement of a wetting liquid or for mercury penetrat'ion were calculated for a model which is similar to Owen's model in t,hat each cavity is connected to six others and, like Fatt's model, it is a network (Iczkowski, 1967, 1968). It is of interest to determine the relative resistance of models like these as a function of the saturation of conducting liquid. The void space can be visualized by considering a unit cube of a simple cubic lattice as in Figure 1, A . h sphere centered at each corner of the cube cuts out' sections like those shown in Figure I, B . When these spherical sections are removed, the result is an individual cavity of the void space (Figure 1, C). Each cavit'g, like the cube from which it is derived, is joined to six others which hurround it. The total void space is a network of such cavities joined together as in Figure 2. The distribut'ioii of liquid inside the network of connected cavit'ies can be visualized in Figure 3, which repre,sents a cross section of the porous solid and shows some cavities filled with liquid (represented by light squares) and the remaining cavities filled with gas (represented by dark squares). According to Bruggeman's (1935) method, only a differential volume of the porous solid needs to be Considered as having structure, and the rest of the porous solid may be considered homogeneous, but having the same properties as the structured part. I n the present treatment, only a single cell is considered to have structure and the remaining cells are considered to be continuous with regard to the electrical properties. With regard to saturation properties, the entire pore space is considered to be made of cells, which may be filled or empty. Figure 4 shows a magnified view of the structured part' which is drawn as an assembly of cubes for simplicity rather than the more complex shapes of Figure 1, C. The st'ructured part is located in a t'hin cross-sectional slab oriented perpendicular t o the direction of current flow. The central cavity of the structure is in parallel with t'he other cavities in the cross section and is in series with the front and back sections of the porous packing. Calculation
If the central cell and all of the six surrounding cells are filled with electrolyte, the resulting circuit is like that shown in Figure 3, A . Since all the other cells of the porous solid are regarded as continuous, the potential halfway through the thickness of the cross section is the same as the potential a t the four openings of the central cell which are connected to the cross section. Therefore, there is no flow of current from the cell to the center of the cross section. The resistance 674 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
of the cross-sect'ional slab alone is given in Figure 5, B , and a t t'he top of Table I. If the central cell is filled and only five of the surrounding cells are filled, t,here are two cases, depending on whether the empty cell is in the cross-sectional element or in the front or back sections. I n the former case, conduction occurs through the cell from the front to the back sections and the corresponding circuit is the same as t h a t for six filled surrounding cells. I n the latter case, conduction cannot occur directly through the cell from front to back, but a potential difference still exists between the front (or back) section and the center of the cross-sectional element which is regarded as continuous. Therefore, some conduction occurs from the front (or back) through the four faces leading to the center of the cross-sectional element as shown in Figure 5 , C. The corresponding circuit is shown in Figure 5, D , and as t8he third circuit in Table I. If only four of the surrounding cells are filled, there are three possibilities. Both empty cells may be in the cross section; the corresponding circuit is like that of Figure 5, B . One empty cell may be in t h e cross section and the other empty cell may be in the front or back; the corresponding circuit is like that of Figure 5, D. One empty cell may be in front and the other may be in back of the central cell; no conduction occurs through the cell and the corresponding circuit' does not contain a contribution from the central cell, as is shown in Figure 5, E . The remaining equivalent circuit,s for three or fewer surrounding cells filled with liquid are summarized in Table I. Before the central cell is emptied, the resistance configurations are like those of Figure 5, B , D , or E . hfter the central cell is emptied, all the configurations revert t o that of Figure 5, E , which has resistance R , the resistance of the crosssectional element wit,hout the additional central cell. The resistance of configuration 5, B , is R, R / ( R o R ) , where Ro is the resistance of the additional central cell. The resistance change due to emptying the central cell is the difference between the resistances of configurations 5, B and E:
+
AR = R - R,R/(R, f R ) since R, >> R.
=
R2/(R,
+ R ) x R2/Ro +
+
The resistance of configuration 5, D , is (R/2) R [ p ( k - 1) ~ ] / [ 2 p 2 ( k - 1) T ( k - 1) R ] . The resistance change due to emptying the central cell in t'his configuration is the difference between the resistances of configurations 5, D and E:
+
AR
=
since
p
(IC
- 1) R2/[4p+ 4 (IC
and r
+
- 1) r + 2 ( k - l ) R ] ( k - 1)R2/4[p+ (IC - 1)rI
>> R.
I n Table I, the singly connected cent'ral cells or central cells which are not connected to both the front and back sections have configurat'ions like that' of Figure 5, E , and do not contribute resistance changes when emptied. Isolated cells cannot be empt'ied.
Table I Liquid Total ConNo. of Connecnec- Gas Forms tions to tion Con. (6!k) Front or No. of k 6-k Back TyDeS
List of TYDeS
Corresponding Equivalent Circuit
Figure 3. Liquid (open squares) and gas (cross-hatched squares) in cavities
f+Yrrff
Figure 4.
Structured part of porous solid Resistance of Cells
Rear P a r t
Front Section
w Resistance of the Central Cell (’3.)
A
Figure 1.
B
Resistance of Cells in the Cross Section f R )
r
C
Individual cavity of the void space
Resistance o f ,
the Central Cell (R,)
R / 2 R/2
In order t o determine p arid r in the above equations, a powdered carbon-\vas composite was molded between spheres to form a model of the cavity shown in Figure 1, E . T h e plane faces of the model cavity were coated with a thin layer of powdered silver-was composite to eliminate contact resistance and nietal foil strips were attached to the silver composite. 13y measuring the resistance across opposite faces, the resistance between adjacent faces, and the resistance between a face and its opposite and adjacent faces joined together, R,, r , and p were evaluated for the carbon composite model cavity to yield p = 3.371 R, and r = 0.572 R,. The probability that a cavity will be surrounded b y li other liquid-filled cavities is (i)sk(l- s)~-’, k = 0 , . , , , 6, where is defined as m!/n! ( m - n)! for n 5 m arid (;) = 0 if n > m. The saturation, s, is also the probability t h a t any given cavity will be filled with electrolyte. The factor (;) can be split into a sum of contributions for configurations like t h a t
(T)
R (E)
Figure 5. Resistance configuration
of 5, B , D , or E . The probability of configuration 5, B , is (k42)s k ( l - s)6-lc, k = 2, . . . , 6; the probability of configurations like that of 5, D , is 2 ( k ! l ) sk (1 - s)~-’> k = 2, . . , , 5 ; and the remainder are like configuration 5 , E . The resistance change, dR, \Then the central cell is emptied, -dn, is equal to the probability of each configuration times Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
675
The relative conductivity, K , = R ( l ) / R ( s ) , is plotted in Figure 6, A , as a function of saturation, s. The corresponding log-log plot shonm in Figure 6, B , is nearly linear. L
Discussion
0
0.1
0.2
0.9
0.4
0.5
0.6
Saturation
0.1
0.8
0.9
1.0
I
hlthough K , is a polynomial in s, it behaves like a simple power of s, with K , = P.The present treatment s h o m analytically that this behavior, which is typical of many porous solids, can be accounted for b y random breaking of conducting paths in a resistance network in which the cavities play the part of the resist,ances.X similar result \yas demonstrated b y Fatt, (1956) using computer simulation of a net'work of tubes. Different porous media show different slopes of the log-log plot which vary from 1.4 for certain sandstones (Ryllie and Spangler, 1952) to 3 for sintered electrodes (Ksenzhek et al., 1964) or mercury penetration (Goddard et al., 1962). The present treatment can be varied to apply to different porous media by considering cavities having different numbers of faces (Fatt, 1956). The asymmetry between r p and Ro in the present treatment also increases the slope of the log-log plot above that found by F a t t . Further improvements might involve using a structured part containing more cavities, averaging over all orientations of the unit cubes and over all the distorted cubes (Iczkowski, 1967, 1968) rather than just using a perfect cube. The present t'reatment applies to displacement of electrolyte by air or to mercury penetrat'ion, in which the conducting fluid is uniformly distributed across the thickness of the porous solid, and not to imbibition of electrolyte.
+
-0.14
-1.2
-1.0
-0.5
-0.6
-0.4
-0.2
0
Log Saturation
Figure 6.
Relative conductivity as a function of saturation Literature Cited
the resistance change for that configuration, summed over all configurations:
Substituting d n = N d s , where 1Y is the total number of cells in the cross section, rearranging, and integrating yields
Bruggeman, D. A . G., Ann. Physik 5,636 (1935). Fatt, I., Trans. A.Z.M.E.207, 144, 160, 164 (1956). Goddard, R. R., Gardner, G. H. F., U'yllie, 11. R. J., "Interactions between Fluids and Particles," p. 326, Institution of Chemical Engineers, London, 1962. Iczkowski, R. P., IND.ENG.CHEM.FUNDAM. 6, 263 (1967); 7, 572 (1968). Ksenzhek, 0. S., Kalinovski, E. A., Baskin, E. L., Zh. Prikl. Khim. 37,1045 (1964). Owen, J. E., Trans. S.Z.M.E. 195, 169 (1952). Wyllie, hl. R. J., Gardner, G. 11. F., World Oil 146,210 (1958). Wyllie, hI. R. J., Spangler, 11.B., BulZ. Amer. Assoc. Petrol. Geol. 36,359 (1952). RAYMOND P. ICZKORSKI
A llis-Chalmers Greendale, W i s . 65120 RECEIVED for review December 15, 1969 ACCEPTED July 22, 1970
676 Ind. Eng. Cham. Fundam., Vol. 9, No. 4,1970
