Application of the Kozeny-Carman Equation to

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Application of the Kozeny-Carman Equation to Permeability Determination for a Glacial Outwash Aquifer, Using Grain-size Analysis Hilmi S. Salem To cite this article: Hilmi S. Salem (2001) Application of the Kozeny-Carman Equation to Permeability Determination for a Glacial Outwash Aquifer, Using Grain-size Analysis, Energy Sources, 23:5, 461-473, DOI: 10.1080/009083101300058480 To link to this article: http://dx.doi.org/10.1080/009083101300058480

Published online: 29 Oct 2001.

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Energy Sources, 23:461 ± 473, 2001 Copyright # 2001 Taylor & Francis 0090-8312 /01 $12.00 ‡ .00

Application of the Kozeny-Carman Equation to Permeability Determination for a Glacial Outwash Aquifer, Using Grain-size Analysis HILMI S. SALEM Atlantic Geo-Technology Halifax, Nova Scotia, Canada Permeability (hydraulic conductivity) is a measure of the ability of a porous medium to convey a single Xuid or multiphase Xuids through its body under a certain hydraulic gradient. In this study, the permeability was determined for the grain-size fractions and the layers of a glacial outwash aquifer and its aeration zone (northern Germany) using grain-size analysis and the Kozeny-Carman equation. The aquifer and the aeration zone are composed of unconsolidated sediments characterized by lateral and vertical heterogeneities. The sediments consist, primarily, of silts, sands and gravels, with a majority of sands and a variety of grain sizes. The permeability of the aquifer and the aeration zone ranges from 2:85 £ 10 ¡4 to 4:60 £ 10¡1 m/s, corresponding to the grain-size fractions, and from 4:02 £ 10¡6 to 3:35 £ 10¡2 m/s, corresponding to the layers. These ranges of permeability agree well with those obtained experimentally from pump tests and laboratory measurements. Several empirical equations correlating between various parameters, with coeYcients of correlation ranging from 0.65 to 1.0, were obtained. Keywords glacial aquifers, grain-size distribution, Kozeny-Carman equation, permeability, porosity, speci® c surface area

Numerous fresh-water aquifers of glacial origin are present in diŒerent 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. 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 melt-water streams and moraines in front of the glaciers. In their studies on glacial aquifers, McDonald and Wantland (1961), Frohlich (1973), Kelly (1977), Urish (1981) , and Frohlich and Kelly (1985) pointed out that glacial aquifers are greatly heterogeneous, laterally and vertically. The area of investigation, Segeberger forest, forms about 10% of the total area of the province of Schleswig-Holstein (S-H), northern Germany. The sediments of the area are 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 Received 31 March 2000; accepted 11 May 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.com

461

462

H. S. Salem

level. The annual precipitation on the area is about 800 mm/yr, distributed as follows: 500 mm evaporation, 60 mm runoŒ, and 240 mm in® ltration recharging the aquifer system. The aquifer ranges in thickness from 30 to 70 m, with a water table about 5 to 10 m deep and about 20 m underneath the areas of higher altitude. The aquifer is underlain by an aquiclude composed of boulder clays, known as ``Geschiebemergel.’’ Fluid ¯ ow through porous media may be quantitativel y evaluated by knowledge of several physical parameters and conditions, as well as several lithological attributes. These include velocity of ¯ ow; gravity acceleration; pressure drop; hydraulic gradient; permeability; porosity; speci® c surface area; density, viscosity, and temperature of ¯ uid; pore-water salinity; geometry of pores and pore channels (tortuosity); shape and size of pores and pore channels; and size, shape, type, packing, sorting, distribution, and orientation of grains (anisotropy). In the present study, emphasis is placed on determination of permeability, with respect to the porosity and the speci® c surface area, for the grain-size fractions and the layers of the S-H aquifer and the aeration zone, using grain-size analysis and the Kozeny-Carman equation.

Theory For permeability …k† determination, several equations have been theoretically, experimentally, and empirically developed (e.g., Slichter, 1899; Van Terzaghi, 1923; Krumbein and Monk, 1942; Wyllie and Rose, 1950; Loudon, 1952; Rose, 1957; Harr, 1962; Chilingar et al., 1963; Davis and De Wiest, 1966; Berg, 1970; Todd, 1980; De Marsily, 1986; Katz and Thompson, 1986; Johnson et al., 1987; Salem, 1994; Salem and Chilingarian, 1999). Those equations are dependent, one way or another, on three fundamental equations: Darcy’s equation (Darcy’s law), Poiseuille’s equation, and the Kozeny-Carman equation. The temporal rate of change of the position of a ¯ uid particle is represented by means of the particle velocity. Fluid velocity is one of the major factors in the mechanism of ¯ uid ¯ ow through a medium because it is aŒected by the dynamic and kinematic relationships among the ¯ uid, medium and ¯ ow. The apparent velocity of ¯ uid ¯ ow …¾ap †, in cm/s, also known as ® lter (microscopic) velocity, is de® ned as the mean velocity of a ¯ uid, ¯ owing with a rate q, in cm3 /s, through the entire space of a porous medium, having a cross-sectional area A, in cm2 , and permeability k, in cm/s, i.e., ¾ap ˆ

q A

…1†

Equation (1) is one of the simplest forms of Darcy’s law. The actual velocity of ¯ uid …¾ac †, in cm/s, also known as distance (macroscopic) velocity is obtained by dividing ¾ap by the fractional porosity …¿† of the medium. For a ¯ uid with viscosity · (in cP) ¯ owing with a rate q (in cm3 /s) under a pressure gradient ¢P (in atm/cm), normal to cross-sectional area A (in cm2 ) of a medium with permeability k (in Darcy), the apparent velocity …¾ap †, in cm/s, is (Darcy’s law): ³ ´ q k …2† ¾ap ˆ ˆ ¢P A ·

The Kozeny-Carman Equation to Permeability Determination

463

Using the same units given in Equation (2), k (in Darcy) for the same medium, having length L (in cm), is kˆ

q·L A¢P

…3†

For a ¯ uid with viscosity · (in Poise) ¯ owing through a number of capillary tubes nc , each with capillary length Lc and capillary radius rc (both in cm), where ¢P is in dyne/cm 2 , the general form of the Poiseuille’s equation for capillary ¯ ow is !³ ´ nc ºr4c ¢P qˆ …4† 8· Lc For linear ¯ ow and capillary ¯ ow, the Darcy’s law and the Poiseuille’ s equation are quite similar (Tiab and Donaldson, 1996). Using the same units given in Equations (2)± (4), q of a ¯ uid passing through capillary tubes with a total cross-sectional area At (in cm 2 ) is ³ ´³ ´ kAt ¢P qˆ …5† Lc · Based on the equations of Darcy and Poiseuille, Kozeny (1927) derived an equation that was later modi® ed by Carman (1937, 1938) and was thus known as the Kozeny-Carman equation. The Kozeny-Carman equation is a good-working hypothesis for determination of permeability or speci® c surface area (Loudon, 1952; Wyllie and Splanger, 1952; Rose, 1957; Kezdi, 1974; De Marsily, 1986; Salem and Chilingarian, 1999). The Kozeny-Carman equation and its derivatives can be expressed as follows: ¾ap ˆ where ¾ap 5 ¢P 5 g 5 ¿ 5 Kcc 5 ss 5 · 5 Lc 5

¢Pg¿3 Kcc s2s ·Lc

…6†

apparent velocity of ¯ uid ¯ ow (cm/s); pressure gradient in the direction of ¯ uid ¯ ow (g/cm2 ); gravity acceleration (ˆ 981 cm/s 2 ); porosity (fractional); Kozeny-Carman coe cient (dimensionless); speci® c surface area (cm ¡1 ); viscosity of ¯ uid (Poise ˆ g/cm.s); length of capillary tubes (cm).

Substituting · by f¸»g, where ¸ is ¯ uid’s kinematic viscosity (in cm2 /s) and » is ¯ uid’s density (in g/cm3 ), and then substituting f¢P=»Lc g by the hydraulic gradient, I, dimensionless, Equation (6) is ¾ap ˆ

Ig¿3 ¸Kcc s2s

…7†

It is important to mention that the symbol I is used for the hydraulic gradient instead of i because i is used later for the grain-size fractions. Substituting f¿ap =Ig by k, then k (in cm/s) is

464

H. S. Salem g¿3 ¸Kcc s2s

kˆ

…8†

Carman (1937, 1938) substituted ¿3 by f¿3 =…1 ¡ ¿†2 g, dimensionless, known as porosity factor, which is a measure of texture of the medium, relating k (in cm/s) to grain size; consequently, Equation (8) is kˆ

(

¿3 …1 ¡ ¿†2

)» ³

g …¸Kcc s2s

´¼

…9†

For k, in cm 2 , the Kozeny-Carman equation can be simply expressed as (Wyllie and Gregory, 1955; Chilingar et al., 1963; Pfannkuch, 1969; Salem and Chilingarian, 1999) kˆ

¿

…10†

Kcc s2s

The Kozeny-Carman coe cient, Kcc , is de® ned as the product of tortuosity …½† and shape factor …Shf †, both are dimensionless, i.e., fKcc ˆ ½Shf g. 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 re¯ ect the geometry of the cross-sectional area of the pore channels normal to the ¯ ow direction. For unconsolidated sediments, Carman (1937, 1938) assigned a value of 2.0 for ½ and a value of 2.5 for Shf , which both results in a value of 5.0 for Kcc . The speci® c surface area, ss , is variably de® ned as the interstitial surface area of the pores and pore channels for each unit of bulk volume, grain volume, pore volume, in cm2 /cm 3 (ˆ cm¡1 ), or for a unit of weight of a material, in cm2 /g. For a medium, with grain diameter D, in cm, and porosity ¿, in fraction, Carman (1937) gave the following equation for ss : ss ˆ

6…1 ¡ ¿† D

…11†

If porosity and grain size are known, then Equation (11) can be used to obtain the speci® c surface area of fractions of sediments …ssf , in cm¡1 †. For a layer consisting of diŒerent fractions (X i to X n), with various grain sizes and various speci® c surface areas, the speci® c surface area of all sediment fractions in that layer (ssl , in cm ¡1 ) can be obtained as (Loudon, 1952) ssl ˆ L

(

n X iˆ1

X i ssi

)

…12†

where L 5 dimensionless coe cient ranging from 1.1 (for rounded grains) to 1.4 (for angular grains); ssi 5 speci® c surface area of fraction i, in cm¡1 (ˆ 6=Dmi †; Dmi 5 mean diameter of the grains in fraction i (in cm).


465

Methodology The permeability, k, was determined for the grain-size fractions and the layers of the S-H aquifer and its aeration zone with respect to the porosity, ¿, and the speci® c surface area, ss . That was achieved by using grain-size analysis (GSA) of the sediments obtained from 6 wells at depth (Z) ranging from 1 to 32 m and by applying the Kozeny-Carman equation. The fractions of silts, ® ne, medium, and coarse sands and gravels, as well as a small amount of clays were recognized at various depths. The weight percentage (WP) and the grain-size distribution (GSD), both obtained from GSA, were plotted on semi-log paper to construct the accumulative curves that represent the various fractions of the sediments. The accumulative curves were then used to obtain the grain size (Gs , in mm) as D10 , D50 , and D90 , which are, respectively, de® ned as the size of the grains (in terms of diameter D) regressed at WP of 10, 50, and 90% . The uniformity coe cient …Uc †, dimensionless, was obtained as the ratio of D regressed at 90% to D regressed at 10% , i.e., fUc ˆ D90 =D10 g. For each accumulative curve, the values of Uc and D50 (in mm) were introduced in the following empirical equations, given by Urish (1981) for similar heterogeneous glacial aquifers, to obtain the porosity, in % , as maximum …¿max † and minimum …¿min †, from which the average porosity …¿av † was obtained as f…¿max ‡ ¿min †=2g log ¿max ˆ 1:62563 ¡ 0:08653 log D50 ¡ 0:03636 log U c

…13†

log ¿min ˆ 1:53902 ¡ 0:18968 log D50 ¡ 0:08201 log Uc

…14†

The speci® c surface area of the grain-size predominant fractions of the grain sizes of the sediments in each layer (ssf , in cm ¡1 ) was obtained in accord with Equation (11). The speci® c surface area of all the fractions in each layer (ssl , in cm ¡1 ) was obtained in accord with Equation (12). For this purpose, one requires Gs regressed at D10 , D50 , and D90 and the corresponding WP (both read from the accumulative curves), as well as a value for L, which was assumed as 1.3, because of the variations in the grain shape of the sediments. The values of porosity and speci® c surface area were then used in the KozenyCarman equation (Equation 9) to determine the permeability …kf † and …k1 †, in cm/s, for the fractions and the layers, respectively. A value of 0.01 cm 2 /s was used for ¸ and a value of 5.0 (dimensionless) was used for Kcc . The values of D10 , D50 , and D90 (in mm), U c (dimensionless), ¿av (in % ), ssf and ssl (in cm ¡1 ), kf and k1 (in m/s), corresponding to the various layers penetrated by the six wells at diŒerent depths …Z†, are given in Table 1. Empirical equations [Equations (15)± (32)], along with their coe cients of correlation …Rc †, ranging from 0.65 to 1.0, are given in Table 2. These 18 equations represent relationships correlating among the various parameters determined in this study. Examples of the relationships are given in Figures 1± 4.

Results and Discussion The grain-size analysis showed that the sediments exhibit a range of Gs from µ 63 mm (clays and silts) to ¶ 2 £ 104 mm (coarse gravels). The analysis also showed that the amount of clays is very small, agreeing with the results obtained by Schroeter (1983), who indicated that the clay fraction does not exceed 1.5% of the

466

1± 5 77 722

3.5± 4 76.2 77

5.5± 6.5 710.5

1± 6 78 714 722

1± 6 712.5

1± 9 710 719 721 732

B

C

D

E

F

Z (m)

A

WELL

85 20 102 83 95

230 160

16 145 180 130

12 215

15 10 80

100 600 125

D10 (mm)

440 925 390 290 500

600 380

220 1,800 3,500 600

110 800

200 35 440

590 3,200 525

D50 (mm)

6,750 30,000 2,200 775 7,000

3,800 1,400

1,550 4,600 4,400 4,600

575 17,500

525 182 2,600

3,800 5,400 2,700

D90 (mm)

79.4 1,500 21.6 9.3 73.7

16.5 8.8

96.9 31.7 24.4 35.4

47.9 81.4

35.0 18.2 32.5

38.0 9.0 21.6

Uc

33.5 26.0 36.6 39.9 33.1

35.1 38.6

39.5 28.1 25.0 33.7

45.0 29.6

42.3 50.8 35.2

33.3 29.3 35.2

¿av (% )

91 48 98 125 81

65 97

166 24 13 67

300 53

173 844 89

68 13 75

ssf (cm ¡1 †

153 1,133 212 431 148

226 153

1,896 284 173 575

2,878 158

2,285 7,100 190

172 49 218

ssl (cm¡1 )

1.20 £ 10¡2 3.00 £ 10¡2 1.19 £ 10¡2 8.42 £ 10¡3 1.51 £ 10¡2

2.52 £ 10¡2 1.31 £ 10¡2

4.71 £ 10¡3 1.33 £ 10¡1 3.93 £ 10¡1 2.27 £ 10¡2

1.78 £ 10¡3 2.96 £ 10¡3

4.22 £ 10¡3 5.37 £ 10¡5 2.53 £ 10¡3 7.02 £ 10¡4 4.44 £ 10¡3

2.08 £ 10¡3 5.27 £ 10¡3

3.57 £ 10¡5 9.49 £ 10¡4 2.18 £ 10¡3 3.02 £ 10¡4

1.94 £ 10¡5 3.30 £ 10¡3

2.76 £ 10¡5 4.02 £ 10¡6 2.98 £ 10¡3

3.33 £ 10¡3 3.35 £ 10¡2 2.23 £ 10¡3

2.13 £ 10¡2 4.60 £ 10¡1 1.94 £ 10¡2 4.80 £ 10¡3 2.85 £ 10¡4 1.36 £ 10¡2

k1 (m/s)

kf (m/s)

Table 1 Grain size of sediments (diameter of grains; D10 , D50 , and D90 ), in mm, regressed at weight of 10, 50, and 90% ; uniformity coe cient …Uc †, dimensionless; average porosity …¿av †, in % ; speci® c surface area of the predominant grain-size fractions of sediments in a layer …ssf †, in cm¡1 ; speci® c surface area of all grain-size fractions of sediments in a layer …ssl †, in cm¡1 ; permeability of the predominant grain-size fractions of sediments in a layer …kf †, in m/s; and permeability of all grain-size fractions of sediments in a layer …k1 †, in m/s. The parameters were determined from grain-size analysis for samples taken from 6 wells penetrating the aeration zone and the aquifer (S-H, northern Germany) at depth (Z) ranging from 1 to 32 m


467

Table 2 Empirical equations [Equations (15)± (32); listed as indicated in the text] with coe cients of correlation (Rc , ranging from 0.65 to 1.0) correlating among grain size of sediments (diameter of grains; D10 , D50 , and D90 ), in mm, regressed at weight of 10, 50, and 90% ; average porosity …¿av †, in% ; speci® c surface area of the predominant grain-size fractions of sediments in a layer …ssf †, in cm¡1 ; speci® c surface area of all grain-size fractions of sediments in a layer …ssl †, in cm¡1 ; permeability of grain-size fractions …kf †, in m/s; and permeability of layers …k1 †, in m/s. These equations are based on grain-size analysis for samples obtained from 6 wells at depth (Z) ranging from 1 to 32 m, penetrating the aeration zone and the aquifer (S-H northern Germany). The equations represent grain size ranging from µ 63 mm to ¶ 2 £ 104 mm, which correspond to clays and silts (µ 63), ® ne sands (63± 200), medium sands (200± 630), coarse sands (630± 2000), ® ne gravels (2000± 6300), medium gravels …6:3 £ 103 –2 £ 104 †, and coarse gravels …¶ 2 £ 104 †. Equations (21), (26), (28) and (32) correspond, respectively, to Figures 1± 4 Eq. # (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

Equation …¡0:095548 † ¿av ˆ 52:525 D10

…¡0:157640 † ¿av ˆ 92:052 D50 …¡0:130970 † ¿av ˆ 97:993 D90 …¡1:0499† ssl ˆ 36299 D10 † ssl ˆ 1:03 £ 105 D…¡0:90457 50 …¡0:63162 † ssl ˆ 57009 D90 …4:9434† ssf ˆ 1:9334 £ 10¡6 ¿av …4:4436† ssl ˆ 5:4557 £ 10 ¡5 ¿av …1:1724† kf ˆ 1:04 £ 10¡4 D10 kf ˆ 1:03 £ 10¡6 D…1:5673† 50 …1:9423† k1 ˆ 1:589 £ 10¡7 D10 …¡0:096014 † ¿av ˆ 23:436 kf † ssf ˆ 10:155 k…¡0:28482 1 …¡0:57796 † ssl ˆ 35:937 kf kf ˆ 32:143 s…¡1:7268† sf …¡1:8536† k1 ˆ 44:284 ssl ssl ˆ 4:9474 s…0:99285† sf …1:0031† k1 ˆ 4:3827 £ 10¡2 kf

Nature of Correlation

Rc

Inverse Correlation

0.65

Inverse Correlation

0.95

Inverse Correlation

0.92

Inverse Correlation

0.90

Inverse Correlation

0.96

Inverse Correlation

0.91

Direct Correlation

0.96

Direct Correlation

0.92

Direct Correlation

0.77

Direct Correlation

0.99

Direct Correlation

0.97

Inverse Correlation

0.92

Inverse Correlation

0.85

Inverse Correlation

0.96

Inverse Correlation

1.00

Inverse Correlation

0.99

Direct Correlation

0.96

Direct Correlation

0.72

Figure

Fig. 1

Fig. 2 Fig. 3

Fig. 4

sediments. The D10 (Table 1) covers a Gs range of 10± 600 mm (median ˆ 100 mm), corresponding to silts, ® ne and medium sands, and the small clay portion. The D50 (Table 1) covers a Gs range of 35± 3500 mm (median ˆ 500 mm), corresponding to silts, ® ne, medium, and coarse sands, and ® ne gravels. The D90 (Table 1) covers a Gs range from 182 to 3 £ 104 mm (median ˆ 3800 mm), corresponding to ® ne, medium, and

468

H. S. Salem

coarse-grained sands and gravels. The median values (100, 500, and 3800 mm) correspond, respectively, to ® ne sands, medium sands, and ® ne gravels. The sediments within the aquifer exhibit a general range of Gs from 100 to 6300 mm, corresponding to ® ne, medium, and coarse sands, as well as ® ne gravels. The Gs range of the aquifer coincides with the median values of Gs (100, 500, 3800). These results indicate that the values of Gs (regressed at D10 , D50 , and D90 ) cover together the whole range of the size of the grains present in the aquifer and the aeration zone. The obtained physical parameters (Table 1) have the following ranges and median values (given in brackets): ¿av ˆ 25± 51% (35.1% ), corresponding to a range of ¿max of 33± 55% and a range of ¿min of 16± 42% ; ssf 5 13± 844 cm¡1 (81 cm¡1 ); ssl ˆ 49± 7100 cm¡1 (218 cm¡1 ); kf ˆ 2:85 £ 10¡4 –4:6 £ 10¡1 m/s (1:15 £ 10¡2 m/s); k1 ˆ 4:02£ 10¡6 –3:35 £ 10¡2 m/s (2:18 £ 10¡3 m/s). The wide ranges of these parameters can be contributed to the high degree of lateral and vertical heterogeneitie s characterizing the sediments of the aquifer and the aeration zone. It is important to mention that the median values are used instead of the mean values, because the median values are more representative of the wide variations of the parameters obtained than the mean values. In addition, a median value (representing the value of the middle variable of a class of data and corresponding to the 50% point on an accumulative frequency range of values) is not sensitive to extreme values, as in the case of the mean value. Regarding the porosity in particular, the diŒerence between the mean value (35.3% ) and the median value (35.1% ) is negligible because ¿ has no extreme values, similar to the other parameters. The interrelationships among the various parameters determined for the fractions and the layers [Table 2; Equations (15)± (32); Figures 1± 4] show the presence of a strong interconnection between Gs (represented by D10 , D50 , and D90 ), ¿, ss , and k. Table 2 shows that an increase in ¿ is associated with a decrease in Gs [Equations (15)± (17)]; an increase in ss is associated with a decrease in Gs (Equations (18)± (20) and with an increase in ¿ [Equations (21)± (22); Figure 1]; and an increase in k is associated with an increase in Gs [Equations (23)± (25)] and with a decrease in ¿ [Equation (26); Figure 2], as well as with a decrease in ss [Equations (27)± (30);

Figure 1. Average porosity …¿av †, in % , versus speci® c surface area of the predominant grainsize fractions of sediments in a layer …ssf †, in cm¡1 ; 19 readings representing 6 wells penetrating the aeration zone and the aquifer (S-H, northern Germany).


469

Figure 2. Permeability of the predominant grain-size fractions of sediments in a layer …kf †, in m/s, versus average porosity …¿av †, in % ; 19 readings representing 6 wells penetrating the aeration zone and the aquifer (S-H, northern Germany).

Figure 3. Permeability of the predominant grain-size fractions of sediments in a layer …kf †, in m/s, versus speci® c surface area of all grain-size fractions of sediments in a layer …ssl †, in cm¡1 ; 19 readings representing 6 wells penetrating the aeration zone and the aquifer (S-H, northern Germany).

Figure 3]. The speci® c surface area of the predominant grain-size fractions …ssf † and that of all the fractions together in a layer …ssl † show a progressive increase with each other [Equation 31]. Also, the permeability of fractions …kf † and that of layers …k1 † increase progressively with each other [Equation (32); Figure 4]. The fact that ssf < ssl and kf > k1 (Table 1) suggests that an increase in the number of fractions in a layer, particularly those of a ® ner grain size, leads to an increase in the speci® c surface area of the sediments, which results in a lower permeability of that layer. The uniformity coe cient …Uc ˆ D90 =D10 † does not show strong correlations with the various parameters obtained because it re¯ ects a wide range of grain sizes, indicating lack of uniformity of the sediments. The range of Uc [º 9–97, with a single reading of 1500 (median ˆ 32.5); Table 1] agrees well with the ranges of heterogeneous glacial deposits given by other researchers.

470

H. S. Salem

Figure 4. Permeability of the predominant grain-size fractions of sediments in a layer …kf †, in m/s, versus permeability of all grain-size fractions of sediments in a layer …k1 †, in m/s; 19 readings representing 6 wells penetrating the aeration zone and the aquifer (S-H, northern Germany).

The speci® c surface area is a sensitive parameter to the variations of Gs . The results indicate that an increase in the value of ss is related to an increase in the amount of ® ne-grained sediments (associated with an increase in ¿ and a decrease in k). Also, a decrease in the value of ss is related to an increase in the amount of coarse-grained sediments (associated with a decrease in ¿ and an increase in k). It is important to mention that if clays were present in a greater amount, the values of ss would be much higher than those obtained, and thus the values of k would be much lower than those obtained. Davis and De Wiest (1966) presented k values for clays, ranging from 10¡10 to 10¡8 m/s (0.01± 1 mD). For hydrocarbon reservoirs, composed of shaly (clayey) sandstones, Salem and Chilingarian (1999) obtained values of ss of up to º 4:8 £ 104 cm ¡1 …k ˆ 1:9mD†, corresponding to clays …Gs ˆ 1:2 mm†. The variations of ss aŒect strongly the variations of k, because ss contributes to the nature of the grain-to-grai n contact and the size of the pores, which both play a considerable role in the mechanism of ¯ uid ¯ ow through the pore channels (capillaries).

Conclusions Permeability of aquifers can be determined empirically, theoretically or experimentally from laboratory measurements, tracer tests, or pump tests. For the glacial aquifer in S-H (northern Germany), the permeability, along with the speci® c surface area and the porosity, was determined from grain-size analysis using the KozenyCarman equation. The aquifer is composed of unconsolidated, heterogeneous sediments that consist of a variety of grain sizes and diŒerent mineralogical compositions. Variations of the parameters that control the hydraulic ¯ ow through the aquifer are strongly aŒected by the grain-size distribution of the sediments. The nature of the relationships between the various parameters is primarily dependent on the grain-size distribution and variations of the grain size. A decrease in the grain size leads to an increase in the speci® c surface area, which results in a decrease in the


471

permeability. The high degree of heterogeneity of the aquifer contributes to the wide variations of the physical properties of the sediments, re¯ ected in the water-¯ ow nonuniformity. The results obtained agree well with those obtained experimentally for the same aquifer or other aquifers composed of glacial deposits. The obtained empirical equations, correlating among several parameters, can be successfully applied to similar deposits.

Nomenclature A At D Dm D10 D50 D90 Gs I Kcc L Lc Rc Shf Uc X i¡n X Y Z a g k kf kl nc q rc ss ssi–n ssf ssl L · ¸ ¿ ¿av ¿max ¿min

Cross-sectional area of a porous medium, normal to ¯ uid ¯ ow (cm2 ) Total cross-sectional area of capillary tubes in a porous medium (cm2 ) Diameter of grains (cm, mm, mm) Mean diameter of grains (cm, mm, mm) Size of grains regressed at 10% of weight (cm, mm, mm) Size of grains regressed at 50% of weight (cm, mm, mm) Size of grains regressed at 90% of weight (cm, mm, mm) Grain size (cm, mm, mm) Hydraulic gradient (dimensionless) Kozeny-Carman coe cient (dimensionless) Length of medium (cm) Length of capillary (cm) Correlation coe cient (dimensionless) Shape factor of grains, pores, and pore channels (dimensionless) Uniformity coe cient (dimensionless) Weight percentage of grain-size fractions (i± n) in a layer (% ) Variable of the X-axis in the empirical equations of Figures 1± 4 (various dimensions) Variable of the Y-axis in the empirical equations of Figures 1± 4 (various dimensions) Depth (m) Empirical-equation coe cient in Figure 1 (dimensionless) Gravity acceleration (ˆ 981 cm/s2 ) Permeability (cm/s, m/s, mD, Darcy, cm2 ) Permeability of the predominant grain-size fractions in a layer (m/s) Permeability of a layer (m/s) Number of capillaries in a medium Rate of ¯ uid ¯ ow (cm3 /s) Radius of capillary (cm) Speci® c surface area (cm¡1 , cm 2 /g) Speci® c surface area of grain-size fractions (i± n) in a layer (cm ¡1 ) Speci® c surface area of the predominant grain-size fractions in a layer (cm ¡1 ) Speci® c surface area of all grain-size fractions of sediments in a layer (cm ¡1 ) Coe cient of roundness and angularity of grains (dimensionless) Viscosity of ¯ uid (Centipoise ``cP,’’ Poise, g/cm.s) Kinematic viscosity of ¯ uid (cm2 /s) Porosity (fraction or % ) Average porosity (% ) Maximum porosity (% ) Minimum porosity (% )

472 » ½ ¾ac ¾ap ¢P GSA GSD S-H WP

H. S. Salem Density of ¯ uid (g/m3 ) Tortuosity (dimensionless) Actual velocity of ¯ uid (cm/s) Apparent velocity of ¯ uid (cm/s) Pressure drop (atm/cm, dyne/cm2 , g/cm 2 ) Grain-size analysis Grain-size distribution Schleswig-Holstein Weight percentage (% )

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