The Influence of Clay Conductivity on Electric

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The Influence of Clay Conductivity on Electric Measurements of Glacial Aquifers Hilmi S. Salem Published online: 29 Oct 2010.

To cite this article: Hilmi S. Salem (2001): The Influence of Clay Conductivity on Electric Measurements of Glacial Aquifers, Energy Sources, 23:3, 225-234 To link to this article: http://dx.doi.org/10.1080/00908310151133906

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The Influence of Clay Conductivity on Electric Measurements of Glacial Aquifers HILMI S. SALEM

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Atlantic Geo-Technology Halifax, Nova Scotia, Canada The conductivities of pore water and clays are important mechanisms in the process of electric-current conduction through porous media. The clay conductivity contributes effectively to the process of electric-current conduction, particularly when the medium is saturated with fresh water. In the present study, the conductivity of clays was investigated in relation to the formation resistivity factor and specific surface area. The formation resistivity factor is an important parameter in defining variations of the formation and porewater resistivities. The apparent and intrinsic formation resistivity factors, along with other petrophysical and hydrophysical parameters, were determined from surface electric measurements and analyses of sediments and water samples for a glacial aquifer (northern Germany). The aquifer is saturated with fresh water and composed of unconsolidated sediments that consist primarily of silts, sands, and gravels, with a majority of sands and a small amount of clays. The sediments of the aquifer and the overlying layers are characterized by a variety of grain sizes and lateral and vertical heterogeneities. The ratio of the intrinsic formation resistivity factor to the apparent formation resistivity factor is a good measure in interpreting the nature of relationships among several parameters that govern the electric-current conduction and fluid flow through porous media. Empirical equations, with coefficients of correlation ranging from 0.92 to 1.0, were obtained. Also, an equation that can be used to calculate clay resistivity from pure-water resistivity and formation resistivity factor (apparent and intrinsic) was obtained. Keywords

formation resistivity factor (apparent and intrinsic); fresh water; bulk, pore-water, and clay resistivities; glacial aquifers; ionic double layer; specific surface area

Many fresh-water aquifers of glacial origin are present in different regions of the world, such as the northern central part of the United States of America, the southern part of Canada, and the northern part of Europe. The glacial aquifers generally consist of silts, sands and gravels, with variable amounts of clays. The sediments of glacial aquifers have a wide range of grain size, grain shape, and grain type (mineralogy). Large quantities of the sands and gravels were deposited as outwash material, swept out from the melting glaciers by the forces of the meltwater streams and moraines in front of the glaciers. In their studies on glacial aquifers, McDonald and Wantland (1961), Frohlich (1973), Kelly (1977), Urish (1981 ), Frohlich and Kelly (1985), and Mazac et al. (1985) pointed out that glacial aquifers are characterized by a high degree of heterogeneity, laterally and vertically. Received 7 December 1999; accepted 31 January 2000. Address correspondence to Hilmi S. Salem, Atlantic Geo-Technology, 26 Alton Drive, Suite 307, Halifax, Nova Scotia, B3N 1L9, Canada. E-mail: hilmisalem @canada.co m

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The area of investigation, Segeberger forest, forms about 10% of the total area of the province of Schleswig-Holstein (S-H), northern Germany. The area is of Pleistocene age, similar to all glacial deposits in northern Europe (Einsele and Schulz, 1973). The altitude of the area ranges from 25 to 65 m above sea level. The annual precipitation in the area is about 800 mm/yr, distributed as 500 mm evaporation, 60 mm runoff, and 240 mm infiltration recharging the aquifer system. The aquifer is about 30–70 m thick, with a water table about 5–10 m deep underneath the surface and about 20 m underneath the areas of higher altitude. The aquifer is underlain by an aquiclude composed of boulder clays, known as “Geschiebemergel.” This study investigates the influence of clay conductivity on the conduction of electric current through the S-H aquifer. To achieve this goal, analyses of surface electric measurements and water and sediment samples were performed.

The Ionic Double Layer and Clay Conductivity Clay minerals have the property of adsorbing certain ions at their surfaces, which enables them to produce electric fields around themselves. This phenomenon is known as clays polarity. If, however, the distribuion of unbalanced charges on the clay surfaces is heterogeneous, or if the ions in the solution are of different valences, the available ions will not be adsorbed. As a result, a so-called ionic double layer will be developed at the clay surfaces (Winsauer and McCardell, 1953 ). The double layer can be measured as the difference in electric potential between the solid and fluid elements. As the grains and pores become smaller and smaller, the overall influence of the double layer, with respect to the electric-current conduction, becomes increasingly important. The double layer consists of a rigidly bound (or fixed) layer and a diffusive layer of less rigidly bound ions. The thickness of the double layer is a function of salinity of the pore water; i.e., as salinity increases (leading to an increase in the pore-water conductivity), the double layer will be thinner. Sheriff (1984) defined the double layer as the layer of the ions at the solid-fluid interface, with a thickness of < 100 Å (Å “Angstrom” = 10–10 m). The influence of clays on the mechanism of electric-current conduction through porous media is due to the double-layer effect, associated with the charged surfaces of clay minerals. The surface conductance of clay minerals is a function of pore-water conductivity (salinity), pore-water acidity-alkalinity indicator (pH), specific surface area, ionic composition, and ionic exchange capability of the matrix minerals. The amount and type of clays are important factors affecting the whole process of electric-current conduction and the magnitude of the overall resistivity (or conductivity) of a medium. McNeil (1980 ) pointed out that the presence of various fractions of kaolinite and/or montmorillonite, for instance, yields different readings of electric resistivity. These are examples of clay resistivity given by various researchers: 15–50 ½ .m (Archie, 1942); £ 20 ½ .m (Frohlich, 1973); 5 ½ .m (Henriet, 1976); and 10 ½ .m (Mazac et al., 1985). In comparison to the resistivity of clays, Archie (1942), for example, gave values ranging from 600 to 1530 ½ .m, corresponding to resistivity of sands and gravels.

Theory The formation resistivity factor (F) is widely used in aquifer and reservoir investigations, as well as in engineering-site investigations. It is a well-known parameter in interpreting electric measurements (surface and borehole). Archie (1942 ) defined F as the ratio of the resistivity of a water-saturated formation (Rb; bulk resistivity) to the resisitvity of pore-water (RW ), i.e.,

F = Rb /RW.

(1)


The Influence of Clay Conductivity on Electric Measurements

227

The formation resistivity factor is a function of several physcial parameters and lithological attributes, which include, in addition to the formation and pore-water resistivities, temperature, viscosity, and degree of saturation of pore water; clay content (type and amount of clays); mechanism of charge fixation at the fluid-solid interface (represented by specific surface area and surface conductance); intricate geometry of the pores and pore channels (represented by tortuosity); size and type of pores (represented by porosity); formation ability to transmit fluid (represented by permeability); cation-exchange capacity; and size, shape, type, packing, sorting and distribution of grains. It is also a function of degrees of cementation, consolidation and compaction (diagenesis) of sediments, particularly when the sediments are affected by high overburden pressure, as in the case of hydrocarbon reservoirs. The formation resistivity factor is classified into apparent (Fa) and intrinsic (or true, Fi ). The Fi describes the sediments if the solid constituents were nonconductive and if these constituents exert no other influences on the conduction process of electric current, such as ion exchange and surface conductance. Otherwise, F is appparent (Fa ), which accounts for the other influences produced by the clay content. Patnode and Wyllie (1950) observed that the presence of clays in a porous medium results in a considerable conductance affecting the real value of F. Thus they proposed one of the earliest models, which accounts for Fa and Fi, in relation to the pore-water resistivity (RW ) and clay resistivity (Rcl ):

1 5 Fa

1 Fi 1

Rw . Rcl

(2)

Equation (2) indicates that Fa approaches Fi if the expression RW /Rcl is very small. This expression becomes very small (or negligible) when RW is low (high pore-water conductivity) in comparison to Rcl (low clay conductivity). Naturally, RW can be high (low conductivity), as in the case of fresh water, or it can be low (high conductivity), as in the case of saline water. The clays existing in a saturated formation always have high conductivity. When the pores are saturated with highly saline water (brine), the conductivity of clays becomes insignificant, because most, if not all, the electric current is conducted via the saline water (Alger and Harrison, 1989; Salem, 1994; Salem and Chilingarian, 1999). When the pores are saturated with fresh water, the clay conductivity becomes more important in the mechanism of electric-current conduction. Thus Fa ¹ Fi when the medium is saturated with fresh water, and Fa = Fi when the medium is saturated with saline water, even in the presence of clays. Winsauer and McCardell (1953) introduced a model in which Fi is defined in terms of Fa (= Rb/RW ) and clay conductivity ( Ccl = 1/Rcl):

Fi 5

Fa 1

CclRb 5

Rb

1 1 Rw

1 Rcl

.

(3)

By substituting ( Rb/RW ) by Fa, one obtains

1 5 Fa

1 Fi 1

CclRw . Fi

(4)

Equation (4) suggests that Fa approaches Fi when the expression CclRW (= Rw/Rcl ) is very small. It also suggests that an increase in Ccl leads to a decrease in Fa, indicating a decrease in Rb. The Winsauer and McCardell model [Equation (4)] is similar to the Patnode and Wyllie model [Equation (2)], with the exception that the Winsauer and McCardell model is valid for

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a range of RW of between 0.04 and 8 ½ .m and the Patnode and Wyllie model is valid for a wider range of RW , i.e., from 0.12 to 3215 ½ .m. Hill and Milburn (1956 ) adopted a quantitative approach by postulating the following equation that relates the measured (apparent) formation resistivity factor, Fa, to the intrinsic formation resistivity factor at very low electrolyte resistivity of 0.01 ½ .m ( Fi = F0.01 ) by taking into account the effect of clays within the matrix, represented by the coefficient b:

Fa = F0.01 (100 RW )b log (100 R ).

(5)


w

For clean sediments (which have no clays), b is zero; consequently, Equation (5) becomes Fa = F0.01 = Fi. Otherwise, b has a range from –0.22 to zero. Equation (5) indicates that when the pore water is saline (highly conductive), the whole current will be conducted via the saline water and thus any conduction via the clays will be prevented. Pfannkuch (1969) developed a model for unconsolidated porous media, which accounts for various parameters that affect the conduction of electric current and the flow of fluid. The Pfannkuch’s model relates Fi to Fa with respect to porosity ( u ), tortuosity (r ), pore-water conductivity (CW ), clay conductivity ( Ccl), surface conductivity occurring at the fluid-solid interface ( CS ), and specific surface area (ss):

Fi 5

Ccl r Cw

Fa 1 1

12

2

u 1

u

ss

Cs Cw

.

(6)

The expression Ccl/CW is the ratio of clay conductivity to pore-water conductivity, and the expression Cs/CW is the ratio of surface conductivity to pore-water conductivity. If CW is high (saline water), then both expressions will approach zero and thus Fi = Fa. If CW is low (fresh water), then both expressions will have a considerable role and thus Fi will be different from Fa. If the sediments primarily consist of nonconductive grains, such as sands and gravels with a small fraction of clays, the expression Ccl/CW will be insignificant. In this case, the expression Cs/CW will become more effective, as the conduction of electric current takes place via the interconnection between the clay surfaces and the pore water. Consequently, Equation (6) will be

Fi 5

Fa 1 1

ss

Cs Cw

.

(7)

The specific surface area, ss, is variably defined as the interstitial surface area of the pores and pore channels for each unit of bulk volume, grain volume, and pore volume in cm 2/cm 3 (= cm –1 ), or for a unit of weight of a material, in cm 2/g. The specific surface area (ss), in cm –1, can be obtained as (Wyllie and Gregory, 1955; Chilingar et al., 1963; Pfannkuch, 1969; Salem and Chilingarian, 1999)

ss 5 u

k Kcc

1 2

.

(8)

Equation (8), known as the Kozeny–Carman equation (Kozeny, 1927; Carman, 1937, 1938 ), takes into account the influences of permeability ( k ), in cm 2, porosity ( u ), in fraction, and Kozeny–Carman coefficient (Kcc ), dimensionless. The Kozeny–Carman coefficient is defined as the product of tortuosity, r , and shape factor, Shf, both dimensionless, i.e., { Kcc = r Shf}. The shape factor is a measure of the shape of the grains, pores, and pore channels in a porous medium. The tortuosity and shape factor reflect the geometry of the cross-sectional


229

area of the pore channels normal to the direction of fluid flow. For unconsolidated sediments, Carman assigned a value of 2.0 for r and a value of 2.5 for Shf, which result in a value of 5.0 for Kcc. By introducing the definition of ss [given in Equation (8)] into Equation (7), then Fi can be obtained as

Fi 5

Cs Cw

Fa 1 1

1 2

u kKcc

.

(9)

By rearranging terms and defining parameters, Fa is (Pfannkuch, 1969)


Fa 5

Fi AssR0.5 w

11

.

(10)

where A is the surface conductance factor (mho), RW is the pore-water resistivity (½ .cm), and ss is the specific surface area (1/cm). The surface conductance factor, A, indicating the double-layer effect, is a function of porewater conductivity (which is directly proportional to pore-water salinity), surface conductance, and specific surface area. Using the dimensions given in Equation (10), A can be obtained as {4.0 ´ 10 -8 RW 0.5 } (Pfannkuch, 1969; Bikerman, 1970; Urish, 1981). Worthington (1977, 1982 ) gave the following equation based on the models of Patnode and Wyllie (Equation (2)) and Winsauer and McCardell (Equation (4)):

1 5 Fa

1 Fi 1

Rw . B

(11)

Worthington pointed out that the factor B describes the matrix effective resistivity. This factor is large for clean sand formations and thus Fi approaches Fa. Lower values of B and higher values of Fi/Fa suggest that electric current is partially conducted via the clays.

Methodology Profiles of vertical electric soundings (using Schlumberger configuration), along with grain size and water analyses for samples obtained from 6 wells penetrating the S-H aquifer, were performed. Along with the analysis of the electric measurements, accumulative curves that represent the fractions of the sediments were constructed on semilog paper between the weight percentage and the grain-size distribution. The accumulative curves were then used to obtain the grain size, for which the specific surface area was obtained in accordance with the Kozeny–Carman equation. The bulk resistivity was obtained from the electric measurements, and the pore-water resistivity was obtained from the water analysis. The bulk and pore-water resistivities were used to obtain the apparent formation resistivity factor, in accordance with Archie’s equation. The apparent formation resistivity factor was then used, along with the surface conductance factor and the specific surface area, to determine the intrinsic formation resistivity factor, in accordance with an equation obtained in this study [discussed below; Equation (15)]. Empirical equations, with coefficients of correlation (Rc ) ranging from 0.92 to 1.0, were obtained. These equations correlate among the apparent formation resistivity factor (Fa ), intrinsic formation resistivity factor ( Fi ), pore-water resistivity (RW ), surface conductance factor (A ), and the formation resistivity factors’ ratio ( Fi/Fa).

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Results and Discussion The fractions of silts, fine, medium, and course sands and gravels, as well as a small amount of clays were recognized at various depths within the aeration zone and the aquifer. The results show that the amount of clays is very small, agreeing with the results obtained by Schroeter (1983), who pointed out that the clay fraction is about 1.5%. The wide variations of the grain size, reflected in wide variations of the various parameters, indicate that the sediments are greatly heterogeneous, laterally and vertically. The pore-water resistivity of the top soil and the aeration zone, which increases sharply with depth down to about 5 m, exhibits a range of 43–250 ½ .m (average = 131 ½ .m). The sharp increase of RW with depth for the sediments above the water table is due to the leaching of ions, which resulted from the infiltration process. The RW within the aquifer has a range of 13–83 ½ .m (average = 39 ½ .m). In contradiction to the overlying sediments, RW of the aquifer shows a slight decrease with depth, associated with a general decrease in the bulk resistivity that ranges from 80 to 3395 ½ .m (average = 789 ½ .m; median = 450 ½ .m). The slight decrease of RW (increase of water conductivity) with depth within the aquifer may be attributed to the presence of a higher amount of total dissolved solids (TDS) in the lower zones of the aquifer as a result of the water flow and circulation. The decrease of resistivity (bulk and pore water) with depth may also be attributed to the degradaton of the sediments towards a finer scale. The fine-grained sediments have a higher conductivity than the coarse-grained sediments because of the great adsorptive capacity and the greater specific surface area of the fine-grained sediments. The specific surface area has a range of 60–180 cm –1 (average = 118 cm –1 ). The apparent formation resistivity factor, Fa, which represents variations in the bulk and pore-water resistivities and reflects the influence of clay conductivity on the current conduction, ranges from 2.8 to 16.6 (average = 8.0). The apparent formation resistivity factor decreases with a decrease in the grain size (lower bulk resistivity) because of the greater effect of the surface conductance produced by the finer grains (clays). The intrinsic (true) formation resistivity factor, Fi, ranges from 2.83 to 17 (average = 8.4). The difference between Fi and Fa has a range of 0.02–1.5, and the formation resistivity factors’ ratio ( Fi/Fa ) has a range of 1.01–1.13 (average = 1.05). Both parameters (Fi and Fa ) show a direct correlation with each other [(Figure 1; Equation (12); Rc = 1.0)]:

Fi 5

2

0.030966 1

1.0557 Fa .

(12)

For the sediments within the aquifer, the formation resistivity factors’ ratio tends to decrease, which suggests that Fi becomes closer to Fa. The closer the value of Fi to Fa, the lesser is the effect of clay conductivity on the electri-current conduction. This observation may suggest that the electric current conducted through the aquifer takes place primarily via the pore water, as a result of increasing the pore-water salinity (higher TDS). This phenomenon is supported by the fact that the pore-water resistivity of the aquifer is lower (higher conductivity) than that of the overlying sediments. The relationship between Fi/Fa and RW shows a direct correlation [(Figure 2; Equation (13); Rc = 0.92)].

Fi Fa 5

1.0002 1

5.5065 3

102

4

Rw .

(13)

Equation (13) indicates that by decreasing RW (increase in pore-water conductivity), the formation resistivity factors’ ratio tends to decrease ( Fi becomes closer to Fa). This suggests that the clay conductivity becomes less effective, which may be attributed to a lesser amount of clays or to the presence of different types of clays.



231

Figure 1. Apparent formation resistivity factor ( Fa ), dimensionless , versus intrinsic formation resistivity factor ( Fi ), dimensionless , for the Schleswig–Holstein aquifer, northern Germany.

Figure 2. Pore-water resistivity ( RW ), in ½ .m, versus the ratio of intrinsic formation resistivity factor to apparent formation resistivity factor ( Fi /Fa ), dimensionless , for the Schleswig–Holstein aquifer, northern Germany.

The formation resistivity factors’ ratio is also correlated to the surface conductance factor, A, which has a range from 2.12 ´ 10–6 to 6.32 ´ 10–6 mho (average = 3.46 ´ 10–6 mho) [(Figure 3; Equation (14); Rc = 0.92)]:

Fi/Fa = 0.9512 + 27788 A.

(14)


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Figure 3. Surface conductance factor ( A ), in mho, versus the ratio of intrinsic formation resistivity factor to apparent formation resistivity factor ( Fi/ Fa ), dimensionless , for the Schleswig–Holstein aquifer, northern Germany.

Equation (14) suggests that as Fi becomes closer to Fa (lower Fi/Fa ), the pore water becomes more effective in conducting the current and thus A becomes smaller, indicating a lesser role of the clay conductivity. The clay resistivity (Rcl ), determined in accord with the following equation [(Eq. (15), obtained by trial and error)], has a range of between 1.8 and 14.3 ½ .m (average = 6 ½ .m), which agrees well with the values given by other researchers mentioned earlier:

Rcl 5

RW Fi Fa . Fi 1 Fa

(15)

Conclusions The influence of clay conductivity on electric current transmitted through a porous medium becomes more evident when the medium is saturated with fresh water. The clay-conductivity influence was investigated for the Schleswig–Holstein aquifer (northern Germany) using surface electric measurements and water and sediment analyses. The aquifer is saturated with fresh water and composed of glacial deposits consisting of silts, sands, and gravels, and a small amount of clays, with a variety of grain sizes. The sediments of the aquifer and the overlying layers exhibit a range of bulk resistivity from 80 to 3395 ½ .m and a range of pore-water resistivity from 28 to 250 ½ .m, meanwhile the clays exhibit a range of resistivity from about 2 to 14 ½ .m. The wide ranges of resistivity, reflected in wide ranges of petrophysical and hydrophysical parameters that govern the electric-current conduction and the fluid flow through the aquifer, are due to the lateral and vertical heterogeneitics of the sediments. The clay influence on the electric-current conduction, indicated by differences between the appar-


233

ent and intrinsic formation resistivity factors, seems to be effective, although the amount of clays is so small. The clay conductivity is attributed to the surfaces of clays interfaced with the pore water. The ratio of the intrinsic to the apparent formation resistivity factors proves to be a good measure in interpretation of electric measurements. The equations obtained in this study can be successfully applied to similar areas.


References Alger, R. P., and C. W. Harrison. 1989. Improved fresh water assessment in sand aquifers utilizing geophysical well logs. The Log Analyst 30 (Jan.-Feb.):31–44. Archie, G. E. 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. AIME 146:54– 62. Bikerman, J. J. 1970. Physical Surfaces. New York: Academic Press, 476 pp. Carman, P. C. 1937. Fluid flow through granular beds. Trans. Inst. Chem. Engs. 15:150–156. Carman, P. C. 1938. The determination of the specific surface of powders I. J. Soc. Chem. Indus. 57:225–234. Chilingar, G. V., R. Main, and A. Sinnokrot. 1963. Relationship between porosity, permeability and surface areas of sediments. J. Sed. Petrol. 33:759–765. Einsele, G., and H. D. Schulz. 1973. Ueber den Grundwasserhaushalt im norddeautschen Flachland. Teil I: Grundwasserneubildung bewaldeter und waldfreier Sanderflaechen Schleswig-Hostein. Bes Mitt. Dt. Gewaesserkdl. Jb. 36:72 pp. Frohlich, R. K. 1973. Detection of fresh water aquifers in the glacial deposits of North-western Missouri by geoelectrical method. Water Resour. Res. 9:723–734. Frohlich, R. K. and W. E. Kelly. 1985. The relation between hydraulic transmissivity and transverse resistance in a complicated aquifer of glacial outwash deposits. J. Hydrol. 79:215–229. Henriet, J. P. 1976. Direct application of Dar Zarrouk parameters in groundwater surveys. Geophys. Prosp. 24:344–353. Hill, H. J., and J. D. Milburn. 1956. Effect of clay and water salinity on electrochemical behavior of reservoir rocks. Trans. AIME 207:65– 72. Kelly, W. E. 1977. Geoelectric sounding for estimating aquifer hydraulic conductivity. Ground Water 15:420–425. Kozeny, J. 1927. Ueber kapillare Leitung des Wassers im Boden. Sitzungsber. Akad. Wiss., Wien 136a:271–306. Mazac, O., W. E. Kelly, and I. Landa. 1985. A hydrogeological model for relations between electrical and hydraulic properties of aquifers. J. Hydrol. 79:1–19. McDonald, H. R., and D. Wantland. 1961. Geophysical procedures in groundwater study. Trans. Am. Soc. Civ. Engs. 126:122 –135. McNeil, J. D. 1980. Electrical Conductivity of Soils and Rocks. Geonics Ltd., Tech. Note, TN5, 22 pp. Patnode, H. W., and M. R. J. Wyllie. 1950. The presence of conductive solids in reservoir rocks as a factor in electric log interpretation. Trans. AIME 189:47– 52. Pfannkuch, H. O. 1969. On the correlation of electrical conductivity properties of porous systems with viscous flow transport coefficients. Intern. Assoc. Hydrau. Res., 1st Intern. Symp. Found. Trans. Phen. Porous Media 23:496–505. Salem, H. S. 1994. The electric and hydraulic anisotropic behavior of the Jeanne d’Arc Basin reservoirs. J. Pet. Sci. Eng. 12:49–66. Salem, H. S., and G. V. Chilingarian. 1999. Determination of specific surface area and mean grain size from well-log data and their influence on the physical behavior of offshore reservoirs. J. Pet. Sci. Eng. 22:241–252.

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Schroeter, J. 1983. Der Einfluss von Textur- und Struktureigenschaften poroeser Medien auf die Dispersivitaet. Ph.D. dissertation, Kiel University, Germany, 232 pp. Sheriff, R. M. 1984. Encyclopedic Dictionary of Exploration Geophysics. SEG:Tulsa, OK, 266 pp. Urish, D. W. 1981. Electric resistivity-hydraulic conductivity relationships in glacial outwash aquifers. Water Resour. Res. 17:1401–1408. Winsauer, W. O., and W. M. McCardell, 1953. Ionic double-layer conductivity in reservoir rocks. Trans. AIME 198:129–134. Worthington, P. F. 1977. Influence of matrix conduction upon hydrogeophysical relationships in arenaceous aquifers. Water Resour. Res. 13:87–92. Worthington , P. F. 1982. The influence of shale effects upon the electrical resistivity of reservoir rocks. Geophys. Prosp. 30:673–687. Wyllie, M. R. J., and A. R. Gregory. 1955. Fluid flow through unconsolidated porous aggregates, effects of porosity and particle shape on Kozeny-Carman constant. Indus. Eng. Chem. 47:1379–1388.

Nomenclature A B Ccl Cs CW F Fa Fi Kcc Rc Rb Rcl RW Shf X Y

b k ss u r pH S-H TDS

Surface conductance factor (mho) Matrix resistivity factor (½ .m) Clay conductivity (1/½ .m) Surface conductivity at fluid-solid interface (1/½ .m) Pore-water conductivity (1/½ .m) Formation resistivity factor (dimensionless ) Apparent (measured) formation resistivity factor (dimensionless ) Intrinsic (true) formation resistivity factor (dimensionless ) Kozeny–Carman coefficient (dimensionless ) Correlation coefficient (dimensionless ) Bulk resistivity (½ .m) Clay resistivity (½ .m) Pore-water resistivity (½ .m, ½ .cm) Shape factor of grains, pores, and pore channels (dimensionless ) Variable of the X-axis in the empirical equations of Figures 1–3 (various dimensions) Variable of the Y-axis in the empirical equations of Figures 1–3 (various dimensions) Measure of clay conductivity (dimensionless ) Permeability (cm 2 ) Specific surface area (cm –1, cm2/g) Porosity (fractional) Tortuosity (dimensionless ) Acidity-alkalinity indicator (dimensionless ) Schleswig–Holstein Total dissolved solids (ppm)