A Discussion About Hydraulic Permeability and Permeability

Petroleum Science and Technology, 28:1740–1749, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 1091-6466 print/1532-2459 online DOI: 10.1080/10916460903261715

A Discussion About Hydraulic Permeability and Permeability M. M. I. AL-DOURY1 1

Environmental Engineering Department, College of Engineering, Tikrit University, Tikrit, Iraq Abstract The unit of permeability is Darcy, md, or unit of area. The symbol used to represent it is k, and the symbol K is used to represent mobility. Many references including some textbooks in soil mechanics contain confusion between these two terms, that is, hydraulic permeability K is used to refer to permeability k. In these books the term hydraulic permeability or coefficient of permeability is used to refer to mobility. Moreover, the symbols k and K are used in a contrary manner. Also, there is an error in the unit of hydraulic permeability. This work is a trial to clarify and correct this error. Keywords coefficient of permeability, Darcy law, hydraulic permeability, mobility, permeability

Introduction I am a head of editorial board of Tikrit Journal for Engineering Science. When I was checking the last volume of Tikrit Journal before publication, I saw research in the field of soil mechanics that contained such confusion. I have read many publications in the field of soil mechanics that contain confusion between permeability and hydraulic permeability in addition to some errors in the units for them. These references also used the term coefficient of permeability to represent hydraulic permeability. Hydraulic permeability is defined as the superficial approach velocity for a gradient of unity (Lambe and Whitman, 1979). They mean by the term gradient a head of water equal to 1 cm per 1 cm length of the sample, that is h=L D 1. This is not right from a purely scientific point of view because this unit does not fall within any scientific system of units. In this work I will use same symbol used by textbooks on soil mechanics.

Darcy’s Equation Darcy (1856) was the first who worked on the flow of water through a sand filter using a pilot apparatus (Figure 1), and he concluded that Q is directly proportional with A and h and inversely proportional with L. Thus: Q D Akh=L

(1)

where Q is flow rate (L3 /T), A is cross-sectional area perpendicular to flow (L2 ), h is difference of head (L), L is length of the sample (L), and k is constant depend on permeability (hydraulic permeability). Address correspondence to Muzher Mahdi Ibrahim Al-Doury, Head of Environmental Engineering Department, College of Engineering, Tikrit University, Tikrit, Iraq. E-mail: [email protected]

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Hydraulic Permeability and Permeability

1741

Figure 1. Apparatus used by Darcy.

His fluid was water. Thus, he did not include the viscosity of flowing fluid, which affects flow rate. Actually, flow is inversely proportional to viscosity. He also stated that k is a constant that depends on the permeability and it is not permeability. However, he gave a length unit for head of water; therefore, the pressure gradient unit is dimensionless. I think that this is not correct, as will be discussed later. This mistake had been transferred to a lot of literature. Darcy’s equation is popular in soil mechanics textbooks and many papers in the form of: Q D AkI

(2)

where I is h=L, head gradient. Taking into account the effect of viscosity and the pressure difference instead of head difference will give the most general form of Darcy’s equation reported in numerous publications (Amiri, 2000; Jin et al., 2004; Huang and Ayoub, 2008; Chapuis and Aubirtin, 2003): Q D AKp=.L/

(3)

where K is permeability (Darcy, 1856), p is pressure difference (atm), and is fluid dynamic viscosity (cp). Seelheim (1880) was the first who stated that the permeability should depend on rock characteristic such as the square of the pore radius. It was also stated that permeability is a property of the substance independent of the flowing fluid. Permeability is the ability of the rock or soil to permit flow of fluids through it (Schneider et al., 1996). It is a characteristic of soil or rock that depends on many parameters such as pore size, pore distribution, and geometry. It does not depend on the fluid that passes through the rock. Brown (2009) stated that k is a coefficient that depends on porous media characteristics such as grain size, pore size, and distribution, and it is not permeability. He stated also that permeability for a media is constant and does not depend on the flowing fluid and its unit is square cm. However, k used in soil mechanics textbooks depends on fluid and soil characteristics.

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M. M. I. Al-Doury

A lot of work concerning permeability had been performed in the fields of civil, mechanical, and petroleum engineering. Only in the field of petroleum engineering there is no confusion between permeability and hydraulic or coefficient of permeability. In petroleum engineering the term hydraulic permeability is referred to as mobility (permeability/) and its unit is L4 /FT or (L2 /dynamic viscosity unit). In the other two fields there is confusion and a mistake in the units of hydraulic permeability.

Units of Permeability Kozeny (1927) stated that the flow of viscous fluid through a tube is similar to flow through porous media. Recalling Hagen–Poiseuille, give the following equation to find the flow through bundle of capillary tubes (Figure 2): QD

nA r 4 p 8L

(4)

where n is number of capillary tube, A is cross-sectional area of one capillary tube, and R is radius of capillary tube. Equating this equation with Darcy’s law .Q D Akh=L/ will give: nA r 4 p D Akh=L 8L

(5)

Thus, the hydraulic permeability k should depend on pore size and dynamic viscosity. Substitution of k D Kg= into Darcy’s equation gives Q D AKP =L. From the definition of porosity () we have D

VP nA r 2 L D D n r 2 Vb AL

(6)

where Vp and Vb are pore and bulk volume, respectively. Then: r 2 KD )r D 8

s

8K

which indicates that the unit of K is square cm.

Figure 2. Simulating porous media as a bundle of capillary tubes.

(7)


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Table 1 Units of permeability

L A p Q K

Darcy

CGS

SI

ft.lb.sec

cm cm2 atm cp cm3 /sec Darcy

cm cm2 dyn/cm2 dyn.sec/cm2 cm3 /sec cm2

m m2 Pa Pa.sec m3 /sec m2

ft ft2 Psf.a lb.sec/ft2 ft3/sec ft2

atm D atmosphere, cm D centimeter, cp D centipose, ft D foot, L D length, lb D pound, m D meter, N D Newton, Pa D pascal (N/m2 ), Psf.a D pound per square foot. Absolute.

Other Units of Permeability Table 1 represents units of permeability in different systems of units. 1darcy D

cm2 cp 1.cm3 =sec/ 1.cp/ D 1.cm/ 1.atm/ atm.sec

1darcy D

1.cm3 =sec/ 1.cp/ 0:01.dyne.sec/cm2 /=.cp/ 1.cm/ 1.atm/ 1013170.dyne/cm2 /=.atm/

D 0:987 10

8

cm2 D 1,000 md

Carman-Kozeny Equation Kozeny (1927) stated that the flow through porous media can be simulated by a flow through a bundle of capillary tubes. He developed the following model: vD

I 3 c ı2

(8)

where is specific weight of fluid (F/L3 ), v is Darcy velocity (L/T), I is pressure gradient (F/L2 /L), is fluid dynamic viscosity (FT/L2 ), is porosity (fraction), C is constant (dimensionless), and ı is specific surface area (L2 /L3 ). It is clear that the unit v in Kozeny’s equation is F /(L2 T), not L/T as he stated. Moreover it is clear that v depends on soil and fluid characteristics. In his studies, Carman (1937, 1938a, 1938b, 1939) proved the validity of Kozeny’s equation and suggested Eq. (9). kD

cg 3 1 1 2 w w S sp:gr .1 /2

(9)

where g is acceleration constant (L/T2 ), w is water density (M/L3 ), S 2 is specific surface area (L2 /M), and sp:gr D s =w is specific gravity of porous media.

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M. M. I. Al-Doury

However, he used a symbol k instead of v and he defined it as the coefficient of permeability or hydraulic permeability. It is clear that the unit k in any system of units is a velocity unit (L/T). This equation is known as the Kozeny-Carman equation and it become popular in use.

Other Forms of Carman-Kozeny Equation Wyllie and Gregory (1955) stated that there are different forms of Eq. (9) such as Eq. (10) (unit for k is L2 ). Robert et al. (2003) also stated that there are many forms of Eq. (9) such as Eq. (11) (unit for k is L/T). Another form is also available, such as Eq. (12) reported by Taylor (1948), Freez and Cherry (1979), and Dominico and Shwarts (1997; unit for k is L/T). kD

3 1 2 cS .1 /2

(10)

kD

cg 3 1 1 w w S 2 sp:gr .1 C /

(11)

k D cd 2

1

w 3 w .1 C /

(12)

where d represents pore diameter (L). Schnieder et al. (1996) used an equation similar to Eq. (10) but they used the symbol K (unit L2 ) instead of k. KD

0:2 m S 2 .1 /2

(13)

Discussion of Confusion and Mistakes In Eqs. (10)–(13), the unit for k is different. It is clear also from Eqs. (9)–(13) that k depends on the characteristics of soil, rock, and fluid. Also, it is obvious that the terms hydraulic permeability and permeability are used interchangeably. Although Lambe and Whitman, in chapter 17 of their book Soil Mechanics SI Version (1979), defined k as the superficial approach velocity for a gradient of unity .k D v=I /, they called it, in different locations in chapters 17 and 19 and in many figures and tables, permeability. Moreover, they gave a conversion chart and conversion constants to convert from k in cm/sec (L/T) to K permeability in square cm (cm2 ). This is not right even it gave right numerical answers because k and K are different identities and they are related by k D K=, as is proved later. If this conversion is right, by similarity one can give a conversion between distance and speed, between density and mass, between work and force, and between any two related variables. They stated that the unit for k is L/T, which is wrong. In spite of their definition for k mentioned previously, they did not include the unit of pressure gradient, which could be psi/ft, Pa/m, mH2 O/m, and so on. For any consistent system of units: k D v.L/T/=Œ.I /.F=L2 =L/ Thus, k will be in [L4 /(FT)].


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Here canceling pressure unit with length unit to give unit of L/T for k is wrong because in any equation, scientific units must be used; that is, the unit for I must be F /(L2 /L) not L/L. On the other hand, using the unit of L/L for I to get the unit L/T for k is also wrong because L for the head must be mH2 O and canceling it with L (unit for length) is wrong. If we assume that this is right, by similarity we can say that the unit of force is equal to cubic meters, unit of radius of capillary tube is N/m2 and so on: F D A.m2 /XP.mH2 O/ D m3 : It must be N. R D 2(N/m)cos=P.mH2 O/ D N/m2 : It must be m. Canceling the units (even if they define the same identity—mass, for example) will give a wrong answer. For example, in environmental engineering the following equation is used to find the mass of bacteria in the aeration unit: XV D YQE.So

Se /c =.1 C kd c /

D (kg bacteria/kg BOD).m3 =day/.kgBOD/m3 /.day/=.day 1 day/ Canceling kg bacteria with kg BOD, even if both are in mass unit, will give the mass of the bacteria in the aeration unit in kg BOD, which must be in kg bacteria. It is very clear that using Lambe and Whitman’s (1979) definition in any system of units will not give the unit of velocity for k but it will be (L4 /FT). Vennard and Street (1982) stated that the unit of the coefficient of permeability is L/T and gave a symbol K instead of k for it. This is an example of a mistake and confusion. Another example of a mistake is the research made by Michaels and Lin (1954). In this research, they show a graph (Figure 19-7) between permeability in square centimeters and 3 =.1 C / using different fluids. It is clear from that figure that the porous media has different permeability for a given porosity depending on the fluid used. Permeability is a characteristic of soil or rock and, like its color or density, is not affected by flowing fluid. Thus, its values must be the same whatever the procedure or fluid used except experimental errors and gas slippage effects. I think that this mistake is due to (1) converting k to K using a conversion factor, (2) not including the viscosity of the fluid used in the experiments, and (3) canceling the unit of head (m of fluid) with unit of length. It is well known that the head expressed in mH2 O is not equal to head expressed in m of any other fluid. They are related by 1 h1 D 1 h2 . Robert (2003) gave these three equations: k D K w =w D Kw g=w

(14)

k D Kw =w

(15)

w D gw

(16)

Equations (14) and (15) are not right because he used Eq. (16) to convert to . It is well known that w D w =w where w is water kinematic viscosity (L2 /T). Moreover, Eq. (14) is wrong because equating Eq. (1) with Eq. (3) will give: Akh=L D AKp=.L/

(17)

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M. M. I. Al-Doury

from which we have k D K=. This relation is completely different from Eq. (14) in which w is introduced to convert the unit for h of Eq. (17) from mH2 O to N/m2 . Moreover, the unit for k in Eq. (14) must be L/T to give the unit for K in square cm. We have seen previously that the unit for k is [L4 /(FT)]. Applying the right unit for k in Eq. (14) will give the unit for K in L5 /F, whereas applying it in Eq. (15) will give K in [(L8 /(M2 )]. Moreover, if we substitute Eq. (14) into Darcy’s equation .Q D AkI / we get .Q D AK I=/. Then K D Q=.A I /. Applying the right units in the last equation will give K in L3 /mH2O as follows: KD

Q.m3 =sec/.N:sec=m2 / D m3 =mH2 O: A.m2 / .N=m3 /I.mH2 O=m/

I think that confusion takes place from canceling pressure units with length unit in the term I of Darcy’s equation .Q D AkI / and then giving k the unit (L/T), which is wrong. This is a basic mistake in the principles. Darcy’s law as it appears in the field of soil mechanics is v D kI , where v is velocity. The range of k for most soils is 10 to 10 10 cm/sec. The unit of permeability Darcy is defined as (1 Darcy can be defined as the permeability of rock that permits flow of a fluid [its viscosity is 1 cp] at a rate of 1 cm3 /sec under a pressure differential of 1 atm when the area of the sample is 1 cm2 and its length is 1 cm). It is clear that this definition of permeability unit cannot be obtained from v D kI because this equation does not include the viscosity that is 1 Darcy D cm2 .cp/atm.sec. while v D kI gives k D v=I which gives unit of velocity (cm/sec) for permeability. All these mistakes are due to not using the basic units in Darcy’s equation. It is well known that the basic units are force (F, Newton), length (L, m), time (t, sec), and temperature (, degree Kelvin) units or mass (M, kg), length (L), time (T), and temperature () because mass and force are related to each other by Newton’s second law F D Ma. Using mH2 O as a unit for pressure is not right basically but it is acceptable for practical applications. Therefore, when applying dimensional analysis for any equation it is not accepted to introduce practical units. This is actually what happens when giving the unit of velocity for k in many publications on soil mechanics and mechanical engineering. In soil mechanics textbooks Eq. (1) is simplified to Q D AkI , where h is in cmH2 O and L is in cm. In determining the units for k, these units are applied giving the unit for k in cm/sec as follows: k D Q=AI D v=I . Thus k D Q.cm 3 =sec:/= fA.cm2 /I.cmH 2 O=cm/g, then canceling cmH2 O with cm will give the unit cm/sec for k. I believe that canceling cmH2 O, which is a pressure unit, with cm length is not right because they are units of different variables as follows: k D Q=AI D v=I . Thus k D Q.cm 3 =sec/=fA.cm2 /I.cmH 2 O=cm/g, then canceling cmH2 O with cm will give the unit cm/sec for k. I believe that canceling cmH2 O, which is a pressure unit, with cm length is not right because they are units of different variables. Thus, the units for k must be written in terms of cm2 /(sec cmH2 O) or any other equivalent unit in other system of units. Equation (1) is a special form of Darcy’s equation (for water flowing through porous media). Application of any system of units will not give the unit for k in L/T, as seen in Table 2. If we consider the unit for k as L/T as is popular in soil mechanics textbooks and apply it in Eq. (1) to find the unit for Q in different systems of units, we find the results in Table 3. The results of Tables 2 and 3 indicate that there is a mistake in the unit used for k (L/T). Thus the unit of k used in soil mechanics textbooks and research must be corrected to be unit of permeability/viscosity .k=/, cm2 /(dyne.sec./cm2 ) D


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Table 2 Units for k (Eq. (1)) for different systems of units

L A h Q k

Popular

CGS

SI

ft.lb.sec

cm cm2 cmH2 O cm3 /sec cm2 /(cmH2 O.sec)

cm cm2 dyn/cm2 cm3 /sec cm4 /(dyn.sec)

m m2 N/m2 m3 /sec m4 /(N.sec)

ft ft2 lb/ft2 ft3 /sec ft4 /(lb.sec)

cm D centimeter, cp D centipoises, ft D foot, lb D pound, m D meter, N D Newton

Table 3 Units for Q as found from Eq. (1)

L A h k Q

Popular

CGS

SI

ft.lb.sec

cm cm2 cmH2 O cm/sec cm2 .cmH2 O/sec

cm cm2 dyn/cm2 cm/sec dyn/sec

m m2 N/m2 m/sec N/sec

ft ft2 lb/ft2 ft/sec lb/sec

cm D centimeter, cp D centipoises, ft D foot, lb D pound, m D meter, N D Newton

cm4 /(dyne.sec.), (L4 /(FT)) in order to avoid confusion otherwise, the general Darcy form Eq. (3) must be used which gives cm2 (L2 ) as a unit for permeability (k). Note that using the units for k listed in the last row of Table 2 in Eq. (1) will give the unit L3 /T for Q. Tables 4 and 5 represent units for K and Q obtained from Eq. (3) for different unit systems.

Table 4 Units for K from Eq. (3)

L A p Q K

CGS

SI

ft.lb.sec

cm cm2 dyn/cm2 dyn.sec/cm2 cm3 /sec cm2

m m2 N/m2 N.sec/m2 m3 /sec m2

ft ft2 lb/ft2 lb.sec/ft2 ft3 /sec ft2

cm D centimeter, sec D second, m D meter, N D Newton, ft D foot, lb D pound

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M. M. I. Al-Doury Table 5 Units for Q from Eq. (3)

L A p K Q

CGS

SI

ft.lb.sec

cm cm2 dyn/cm2 dyn.sec/cm2 cm2 cm3 /sec

m m2 N/m2 N.sec/m2 m2 m3 /sec

ft ft2 lb/ft2 lb.sec/ft2 ft2 ft3 /sec

cm D centimeter, sec D second, m D meter, N D Newton, ft D foot, lb D pound

Conclusion The above discussion clarifies that the special form of Darcy’s equation .Q D AkI / with the unit for k (L/T) used in many textbooks and by a large number of researchers is wrong. Therefore, it is highly recommended that the general form of Darcy’s equation be used in textbooks and then specified for the required field of application. Otherwise, one has to use Eq. (1) with the unit for k as indicated in the last row of Table 2 (L4 /FT).

References Amiri, M. C. (2000). Modified Darcy’s law to predict low Reynolds flow through porous media. Pakistan J. Appl. Sci. 1(1):8–10. Brown, G. (2009). Darcy’s law basics and more. Available at: Biosystems.okstate.edu/Darcy/LaLoi/ basics.htm Carman, P. C. (1937). Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15:150–166. Carman, P. C. (1938a). Determination of specific surface of powders I. Trans. J. Soc. Chem. Ind. 57:225–234. Carman, P. C. (1938b). Fundamental principle of filtration (a critical review of present knowledge) fluid flow through granular beds. Trans. Inst. Chem. Eng. 6:168–188. Carman, P. C. (1939). Permeability of saturated sands, soils and clays. J. Agr. Sci. 29:263–273. Chapuis, R. P., and Aubirtin, M. (2003). Predicting the coefficient of permeability of soil using Carman-Kozeny equation. Can. Geotech. J. 40:616. Darcy, H. (1856). Les Fontaines Publiques de la Ville de Dijon [The public fountation of the city of Dijon, experience and application principles to follow and formulas to be used in the question of the distribution of water]. Paris: Dalmont. Derivation of Darcy’s law. Available at: http://www.answers.com/topic/darcy-s-law#Derivation Dominico, P. H., and Shwartz, F. W. (1997). Physical and Chemical Hydrology, 2nd ed. New York: John Wiley & Sons. Freez, R. A., and Cherry, J. A. (1979). Groundwater. Englewood Cliffs, NJ: Prentice Hall. Jin, G., Patzek, T. W., and Silin, D. B. Direct prediction of the absolute permeability of unconsolidated and consolidated reservoir rocks. Paper no. SPE 90084, SPE ATCE, Houston, TX. Huang, H., and Ayoub, D. Applicability of the Forchheimer equation for non-Darcy flow in porous media. SPEJ 13:112–122. Kozeny, K. (1927). Ueber kapillare leitung des wassers in bodem, sitzungsber alead [Over capillary line of the water in bed, Sitzungsber Alead]. Wiss., Wien 136(29):271–306. Lambe, T. W., and Whitman, R. V. (1979). Soil Mechanics SI Version. New York: John Wiley & Sons.


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Michaels, A. S., and Lin, C. S. (1954). The permeability of kaolinite. Ind. Eng. Chem. 46:1239– 1246. Robert, P., Chapuis, E., and Aubirtin, M. (2003). Predicting the coefficient of permeability of soil using Carman-Kozeny equation. Can. Jeotech. J. 40:616. Schneider, F., Potdevin, J. L., Wolf, S., and Faille, I. (1996). Mechanical and chemical compaction model for sedimentary basin simulators. Tectonophysics 263:307, 317. Seelheim, F. (1880). Methodern zur bestimmung der durchlessigkeit dusbodens [Method for the determination of permeability of the soil]. Zeitschrift Fur Analytische Chemie 19:387–402. Taylor, D. W. (1948). Fundamental of Soil Mechanics. New York: John Wiley & Sons. Vennard, J. K., and Street, R. L. (1982). Elementary Fluid Mechanics, 6th ed. New York: John Wiley & Sons. Wyllie, M. R. J., and Gregory, A. R. (1955). Fluid flow through unconsolidated and consolidated aggregates, effect of porosity and particle shape on Carman-Kozeny constants. Ind. Eng. Chem. 47(7):1379–1388.

Nomenclature A C d g h I K k L N p Q R S sp:gr D S =w Vp ; Vb

cross-sectional area perpendicular to flow, L2 constant pore diameter, L acceleration constant L/T2 difference of head, L h=L, head gradient constant depending on permeability (hydraulic permeability) permeability, Darcy length of the sample, L number of capillary tube pressure difference, atm flow rate, L3 /T radius of capillary tube specific surface area, L2 /M specific gravity of porous media Darcy velocity, L/T pore and bulk volume, respectively

Greek Letters

ı w

specific weight of fluid, F/L3 specific surface area, L2 /L3 fluid dynamic viscosity, FT/L2 water density, M/L3 porosity, fraction