Fluid-Flow Characterization of Porous Media (for the ...

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Fluid-Flow Characterization of Porous Media (for the Example of the Jeanne d?Arc Basin Consolidated Reservoirs) Hilmi S. Salem Version of record first published: 29 Oct 2010.

To cite this article: Hilmi S. Salem (2000): Fluid-Flow Characterization of Porous Media (for the Example of the Jeanne d?Arc Basin Consolidated Reservoirs), Energy Sources, 22:6, 557-572 To link to this article: http://dx.doi.org/10.1080/00908310050013776

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Fluid-Flow Characterization of Porous Media (for the Example of the Jeanne d’Arc Basin Consolidated Reservoirs ) HILMI S. SALEM

Downloaded by [Mr Prof. Hilmi S. Salem] at 23:53 30 December 2012

Atlantic Geo-Technology Halifax, Nova Scotia, Canada Flow of fluids in porous m edia is a com plicated process go v erned by sev eral physical parameters and properties of the solid particles and fluid m olecules, as well as the interactions between them . The Kozeny-Carman equation is one of the powerful equations that pro vides a better understanding of fluid flow. The equation can be successfully applied to viscous flow and laminar flow through unconsolidated and consolidated porous m edia if som e of its elements are known . One of these elem ents is the Kozeny-Carm an coefficient (K cc ), which has a specific influence on the m echanism of fluid flow in porous media. In this study, K c c , along with other param eters, was obtained num erically from well-log m easurem ents for several wells penetrating the Canadian offshore Jeanne d’Arc Basin reserv oirs, which consist of consolidated and heterogeneous rocks characterized by a complex network of pores and pore channels. It is shown that K c c is not a constant but a v ariable depending on se veral physical param eters and lithological attributes and strongly related to fluid flow as well as to electric-current conduction in porous media. The Kozeny-Carm an coefficient, obtained as tortuosity (t ) tim es shape factor (S h f ) agrees well with that obtained as t tim es Archie cem entation factor (m ). This observation suggests that S h f and m are analogous to each other. An a v erage v alue of 7.5 was obtained for K c c , which can be used for sim ilar media instead of the value of 5.0 that is inaccurately used for consolidated porous m edia. Also, an a v erage v alue of 3.3 was obtained for t and an a v erage v alue of 2.28 was obtained for S h f (analogous to m ). Em pirical equations linking K cc and a v ariety of petrophysical parameters were also obtained. Keywords

consolidated porous media, fluid flow, Kozeny-Carman coefficient

The Hibernia and Terra Nova reservoirs of the Jeanne d’Arc Basin, offshore Newfoundland , and the Sable Island reservoirs, offshore Nova Scotia, Canada, are composed primarily of shalestones, sandstone s, siltstones, and carbonates, and enriched with clays. They are complex mosaics characterized by fine to medium size of grains, high specific surface area, and high degrees of mineralogical and lithological heterogeneities and hydraulic and electric anisotropies. They are satu-

Received 25 March 1999; accepted 12 June 1999. Sincere thanks are extended to F. Elsheik for his critical review of the manuscript. Address correspondence to Dr. Hilmi S. Salem, Atlantic Geo-Technology, 26 Alton Drive , Suite 307 , Halifax, N.S., B3N 1L9, Canada. E-mail: [email protected]

557


558

H. S. Salem

rate d with oil , gas , and brine and affe cte d by high ove rburde n pre ssure . Furthe r de tails and re fere nce s are give n by Sale m (1994 , 2000a, 2000 b ) and Sale m and Chilingarian (1999 , 2000a, 2000 b ). De termination of pe trophysical param e te rs , such as porosity, pe rme ability, tortuosity, form ation resistivity factor , wate r saturation , spe cific surface are a , pore ge ome try factor , Archie ce me ntation factor , shape factor , and Kozeny-C arm an coe fficie nt, along with knowle dge of the grain and pore characte ristics , is e sse ntial for e valuation of the hydraulic , e le ctric , and e lastic be havior of porous me dia. The Kozeny-C arm an ``constant’ ’ ( K cc ) , as it is wide ly known , of the Koze nyC arm an e quation (Kozeny, 1927; C arm an , 1937 , 1938 ) is de fine d as tortuosity (t ) time s shape factor ( S hf ). Tiab and Donaldson (1996 ) pointed out that K cc varie s be twe e n flow units , but is constant within a give n unit. The Kozeny-C arm an ``constant’’ e xhibits wide variations in the sam e me dium and from me dium to me dium , due to variations in se ve ral prope rtie s and param e te rs , including size , type , and shape of grains (de gre e of sphe ricity-angularity of grains ) ; size and shape of pores and pore channe ls; tortuosity; anisotropy; and de gre e s of compaction , consolidation , and ce me ntation. Thus , it is more accurate to call K cc the ``Koze nyC arm an coe fficie nt.’’ Atte mpts to re nde r the Koze ny-Carm an e quation applicable to consolidate d porous me dia have ce ntere d on me thods that e nable one to de te rmine an appropriate value for K cc . A succe ssful application of the Kozeny-Carm an e quation in de te rmining e ithe r pe rme ability ( k ) or spe cific surface are a ( s s ) for porous m e dia is gre atly de pe nde nt on advance knowle dge of the value of K cc . This e quation c an be succe ssfully used , not only for unconsolidate d porous me dia, but also for consolidate d porous me dia , if adjuste d value s of K cc , porosity ( f ) , and k (in the c ase of s s de te rmination ) , or K cc , f , and s s (in the case of k de te rmination ) , are known (Sullivan & He rte l , 1942; W yllie & Rose , 1950; W yllie & Splange r , 1952 ; Corne ll & Katz , 1953; W yllie & Gre gory, 1955; Sale m , 1992 ). Muskat and B otse t (1931 ) obtaine d , e xpe rime ntally, a value of 4.65 for K cc , corre sponding to flow of air through be ds of glass sphe res. Sullivan and He rte l (1942 ) conducte d e xpe rime nts on aggre gate s of glass fibe rs and showe d that K cc e xhibits a range of betwe e n 3.07 (for flow paralle l to the axe s of glass fibe rs ) and 6.04 (for flow norm al to the axe s of the fibe rs ). The me asure me nts of Muskat and B otse t (1931 ) and Sullivan and He rte l (1942 ) indicate d that K cc be have s anisotropically. It incre ase s whe n flow is transve rse to the be dding plane s or orie ntation of the fibe rs or grains , and de cre ase s whe n flow is paralle l to the be dding plane s or orie ntation of the fibe rs or grains. Rose and Bruce (1949 ) use d a value of 2.5 for the shape factor ( S hf ) and obtaine d value s of up to 400 for K cc , refle cting nonide al syste ms. The y pointed out that K cc incre ase s with incre asing clay content and with de cre asing pe rm e ability. For a range of tortuosity (t ) from 1.0 to 2.45 , corre sponding to diffe re nt dire ctions of flow in porous me dia, W yllie and Spangle r (1952 ) assum e d value s ranging from 2.47 to 3.21 for S hf , and obtaine d value s ranging from 4.05 to 7.48 for K cc . In the ir e xpe rime nts on fluid flow through consolidate d sam ple s of sandstone s , lime stone s , and dolomite s , Corne ll and Katz (1953 ) assume d a value of 2.5 for S hf and obtaine d value s for K cc ranging from 3.40 to 8.15 that corre spond to a range of t from 1.36 to 3.26. W yllie and Gregory (1955 ) conducte d e xpe rime nts on diffe re nt groups of aggre gate s (cylinde rs , disks , cube s , prism s , and sphe res ) , with f ranging from 12% to 52% , and obse rve d that K cc inve rse ly varie s from 17.2 to 3.0 with f , and dire ctly with s s .

Fluid-Flow Characterization (Jeanne d’A rc Basin Reserv oirs)

559


Theory Fluid flow in porous me dia is a com plicate d proce ss gove rne d by seve ral physical param e te rs and lithological prope rtie s. Two m ajor kinds of flow in porous m e dia are re cognize d for sm ooth channe ls: viscous flow and turbule nt flow. For viscous flow (also known as lam inar flow ) , fluid flow is along stre am line s paralle l to the walls of channe ls of the me dium. For turbule nt flow , the stre am line patte rn doe s not e xist, be cause fluid passe s through the me dium in an irre gular m anne r. V iscous flow is confronted with the internal friction force s of fluid , me anwhile the turbule nt flow is confronte d with the ine rtial force s. V iscous flow de pe nds on the ve locity of fluid relative to the boundarie s of channe ls , and turbule nt flow de pe nds on the ve locity of fluid re lative to the ine rtial force s. B oth kinds of flow have bee n give n gre at atte ntion by rese arche rs working on fluid dyn am ics and othe r re late d are as.

Darcy’s Law, Hagen-Poiseuille’s Law, and the Kozeny-Carman Equation M any e quations have bee n used for re lating the flow of fluid to the characte riz ation of porous me dia. O ne of the most fundam e ntal e quations that has be e n wide ly use d for almost one -and-a-half ce nturie s is D arcy’ s law , which was obtaine d e xpe rime ntally by Darcy (1856 ) for flow of wate r through sands and sandstone s. D arcy’ s law has be e n m odifie d by m any re se arche rs to be use d for various porous me dia saturate d with a single-phase fluid or multiphase fluids. D arcy’ s law can be e xpre sse d in the following three e quations: D

kA

P

( )( )

qs

m

L

(1 )

which yie lds

ks

qm L

(2 )

D PA

and v ap s

q A

s

k

D

P

( )( ) m

L

(3 )

whe re

q s volum e rate of fluid flow , cm 3 r s k s pe rme ability (D arcy s 9.87 = 10 y 9 cm 2 s 0.987 m m 2 ) A s cross-se ctional are a , cm 2 m s viscosity of fluid , cP D P s pre ssure drop in the dire ction of flow , atm

L s thickne ss of m e dium , or le ngth of flow path , cm v ap s appare nt (m acroscopic ) ve locity of fluid flow , cm r s , from which actual ve locity ( v ac ) can be obtaine d , i.e . , ( v ac s v ap r fractional f ).

560

H. S. Salem

The H age n-Poise uille ’s law for viscous flow ( q ) , in cm 3 r s , of fluid with viscosity m , in Poise , passing through circular capillarie s (channe ls ) , with radius rc and le ngth L c , both in cm , unde r pre ssure drop D P , in dyne s r cm 2 ( s 1.0133 = 10 6 atm ) , is

qs

p

D

rc4

P

( )( ) 8m

Lc

(4 )


If the conducting channe ls of the fluid we re represe nte d by a numbe r of paralle l capillary tube s of various radii and le ngths , the n the flow rate , q , through the syste m is

qs

N

p

8

p

nrc4i

is 1

D

P

( ) m L ci

(5 )

whe re

N s numbe r of c apillary tube s (groups of channe ls ) n s numbe r of channe ls in a group rc i s radius of channe l i , cm L c i s le ngth of channe l i , cm If m and D P we re e xpre sse d , re spe ctive ly, in cP and atm winste ad of Poise and dyne r cm 2 ; E qs. (4 ) and (5 ) x , the n q (in cm 3 r s ) can be obtaine d according to D arcy’ s law (whe re k in D arcy) as give n in E q. (1 ). Similarity betwe e n D arcy’ s law wE q. (1 ) x and the H age n-Poise uille ’s law wE q. (4 ) x raise s a comparison be twe e n turbule nt flow through porous m e dia and viscous (laminar ) flow through circular , straight channe ls. Darcy’s law can be compare d to the H age n-Poise uille ’ s law by using the Re ynolds numbe r ( R n ) and the friction factor ( f f ). The Re ynolds numbe r is de fine d as the ratio of the ine rtial force s to frictional force s , which can be e xpre sse d as

Rn s

r

() m

v ap ¥

(6 )

whe re

R n s Re ynolds number , dim e nsionle ss 3 r s fluid de nsity, g r cm

m s fluid viscosity, Poise s g r cm r s v ap s appare nt ve locity of fluid , cm r s

¥ s characte ristic line ar dime nsion indicating ge ome try; in the case of channe ls , ¥ s rc s channe l radius , in cm The Re ynolds numbe r has the value of 0.0 whe n the fluid is not moving (flow is ze ro ). W he n the fluid is moving along sm ooth stre am line s paralle l to the channe l walls , R n re ache s value s of up to 2 ,000. W he n the ine rtial force s be come some wh at e ffe ctive (so that both laminar flow and turbule nt flow occur ) , the value of R n


561

incre ase s to 4 ,000. W he n the flow be come s fully turbule nt (so that fluid ve locity varie s randomly with time ) , R n re ache s critic al value s of gre ate r than 4 ,000. The friction factor is a function of R n and the re lative roughne ss of the channe ls. For a bundle of channe ls , with radius rc we quivale nt to ¥ of E q. (6 ) x and le ngth L c , both in cm , the friction factor is D P grc

( )

ff s c

(7 )

2 2 r L c v ap

whe re

f f s friction factor , dime nsionle ss


D

D

c

s coe fficie nt that de pe nds on the type of the ge ome try of the channe l and ge ne rally range s from f 0.6 to 2 , dime nsionle ss

P s pressure drop , g r cm 2 g s gravity acce le ration , cm r s 2 The following re lationship wE q. (8 ) x be twe e n f f and the inve rse of R n can be applie d to D arcy’s law for porous me dia and the H age n-Poise uille ’s law for c apillarie s whe n the flow is viscous , and it no longe r holds whe n the flow is no longe r viscous:

ff s c

D P grc

( ) 2r L

2 c v ap

y1

s Rn

s

m

1

( )( ) r

rc v ap

(8 )

which yie lds v ap s

c grc2

D

P

( )( ) 2m

Lc

(9 )

For straight channe ls of noncircular cross se ction , the channe l radius , rc , c an no longe r be used for the characte ristic line ar dime nsion associate d with the channe l cross se ction , and thus , rc is re place d by the me an hydraulic radius ( rh ) use d as characte ristic line ar dime nsion. The me an hydraulic radius is de fine d as the cross-se ctional are a norm al to the flow divide d by the pe rime ter available to the fluid. By re placing rc with rh and the te rm c r 2 with the inve rse of the y dim e nsionle ss shape factor of the channe ls ( S hf ) , i.e . , S hf1 s c r 2 , or S hf s 2 r c , the n E q. (9 ) c an be writte n as

v ap s

grh2

D

P

( )( ) S hf m

Lc

(10 )

The shape factor of channe ls ge ne rally range s from about 1 to 3.5 , with an ave rage value of about 2.5. It incre ase s with incre asing the de gre e of irre gularity of the channe ls (de cre ase s with incre asing the de gre e of re gularity ). The shape factor ( c de cre ase s with incre asing the de gre e of be have s in an opposite way to c irre gularity of the channe ls , and vice ve rsa ).

562

H. S. Salem

The me an hydraulic radius can be , alte rnative ly, de fine d as the volume of fluid wor volume of void that is de fine d in term s of porosity, i.e . , void volume s f r 1 y f x in the channe l syste m divide d by the spe cific surface are a ( s s ) available to the fluid. The spe cific surface are a is variously de fine d as the surface are a of the pores and pore channe ls pe r unit of bulk volume , sb , pe r unit of grain volume , s g , pe r unit of pore volume , s p , in 1 r cm , or pe r unit of we ight , s wt , in cm 2 r g. B y taking into account the alte rnative de finition of the me an hydraulic radius , i.e . , rh s f r s s (1 y f ) , and conside ring the actual rate of fluid flow ( q ) , i.e . , q s v ap A f , the relationship for viscous flow of an incompressible fluid in porous me dia is


qs

gA

1

m s s2

S hf

D

P

( )( )( ) Lc

f

3

(1 y f )

2

(11 )

Comparing the viscous flow through porous m e dia to the laminar flow through straight channe ls raise s the point that flow stre am line s though porous me dia are not straight but repre sent tortuous paths through the interspace s of the m e dia. The re fore , tortuosity (t ) , dime nsionle ss , must be take n into conside ration. Tortuosity is de fine d as the ratio (or the square of the ratio ) of the e ffe ctive le ngth of flow path of fluid ( L e ) to the le ngth paralle l to the ove rall dire ction of the pore channe ls in a porous me dium ( L ) , i.e ., t s L e r L . The re fore , t is a corre ction factor applie d to take care of the e xtra le ngth of the sinuous conne cte d channe ls (conduits ). Conse que ntly, the rate of fluid flow , q , must be multiplie d by t , and thus , E q. (11 ) is

qs

gA

1

D P

m ss

t S hf

Lc

( )( )( ) 2

f

3

(1 y f )

2

(12 )

whe re

q s rate of fluid flow , cm 3 r s g s gravity acce le ration s 980 cm r s 2 A s cross-se ctional are a of channe ls , cm 2 m s viscosity of fluid , Poise s g r cm s y ss s spe cific surface are a , cm 1

t s tortuosity , dime nsionle ss

S hf s shape factor of channe ls , dime nsionle ss P s pressure drop in the dire ction of flow , g r cm 2 D

L c s le ngth of channe ls s thickne ss of be d , cm f

s porosity , fractional

E quation (12 ) , give n by Koze ny (1927 ) and modifie d by Carm an (1937 , 1938 ) , is known as the Koze ny-C arm an e quation. It can be use d succe ssfully for viscous flow through porous me dia (D arcy’ s law ) and laminar flow through straight channe ls (H age n-Poise uille ’s law ).



563

The stre am line through a porous me dium can adopt an ave rage angle of 45 8 with the horizontal , and he nce , t s 1 r cos 45 8 f 2. The angle be twe e n the flow line and the horizontal cannot, howe ve r , have a constant value ; it varie s from point to point along the path of flow , and thus , t will vary corre spondingly. C arm an (1937 ) r conside red the corre ction for the channe l le ngth as (2 ) 1 2 s 1.414 , which re sults in 2 a value of 2 for t , i.e ., t s (1.414 ) . By assigning a value of 2.5 for S hf and a value of 2 for t , Carm an (1937 ) obtaine d the value of 5.0 for the Kozeny-C arm an coe fficie nt ( K cc ). The Koze ny-Carm an coe fficie nt is actually a combination of se parate and appare ntly inde pe nde nt quantitie s (tortuosity and shape factor ) , i.e ., K cc s t S hf . Tortuosity is a me asure of the comple xity of the tortuous passage s of fluid flow and the orie ntation of pore channe ls relative to the ove rall dire ction of flow. The shape factor is a me asure of the shape of pore channe ls , pores , and grains. The le ngth of the colle ctive pore channe ls (represe nte d by t ) and the shape of pore channe ls , pore s , and grains (re prese nte d by S hf ) vary conside rably from me dium to me dium , and the re fore , K cc shows conside rable variations. Carm an (1937 ) , howe ve r , pointe d out that the value of 5.0 for K cc is valid for unconsolidate d porous me dia composed of sphe ric al grains with uniform size , and note d that it is unknown whe the r or not the value of 5.0 can be use d for nonsphe rical grains and mixtures of grains with diffe rent size s. R ose and Bruce (1949 ) , W yllie and Spangle r (1952 ) , and W yllie and Gre gory (1955 ) pointe d out that a value of gre ate r than 5.0 should be use d for K cc for consolidate d porous m e dia , which can m ake the Kozeny-Carm an e quation wE q. (12 ) x applic able to consolidate d porous me dia saturate d with a single -phase fluid or multiphase fluids. Substituting K cc , dime nsionle ss , for t S h f in E q. (12 ) , the pe rme ability ( k ) c an y be obtaine d (in cm 2 ) by using s s (in cm 1 ) and f (fractional ) as

ks

f

3

K cc s s2 (1 y f )

2

(13 )

Carm an (1937 ) pointe d out that the te rm f 3 r (1 y f ) 2 , known as the porosity factor , affords sufficie nt corre ction for change s in f . To obtain a simple r form of the Koze ny-C arm an e quation , the porosity factor can be re place d with f , and thus , E q. (13 ) is

ks

f

K cc s s2

(14 )

The Shape Factor of the Kozeny-Carman Equation and the Archie Cementation Factor of the Archie-Winsauer Equation The shape factor ( S hf s K cc r t ) is the only factor in the Koze ny-Carm an e quation that cannot be me asure d dire ctly. For a porous m e dium , consisting of uniform grains and pore s (such as sphe res and be ad packs ) , the variation of t is conside rable and the constancy in the value of S hf is probably fairly accurate ; but for a nonuniform porous me dium , the constancy in S hf se e ms to be a little pre sum ptuous (Chilingar e t al., 1963 ). A tkins and Smith (1961 ) pointe d out that S hf of the Koze ny-Carm an e quation is an alogous to the Archie ce me ntation factor ( m ) of the

564

H. S. Salem

Archie-W insaue r e quation (Archie , 1942; W insaue r e t al. , 1952 ) , which can be e xpre sse d as

m s y

(log F y log a ) log f

(15 )

whe re

m s Archie ce me ntation factor , dime nsionle ss F s form ation resistivity factor , dime nsionle ss


a s pore ge ome try factor , dime nsionle ss f

s porosity , fractional

Both param e te rs ( S hf , m ) are functions of the sam e influe nces (e .g., shape of pore channe ls , pore s , and grains; mine ralogy and size of grains; tortuosity; pe rm eability; spe cific surface are a; anisotropy; type of porosity; me chanisms of diage ne sis; e tc. ). This sugge sts that m can be use d for de te rmination of K cc of the Koze ny-Carm an e quation wE qs. (12 ) , or its de rivative s x , and S hf can replace m in the Archie-W insaue r e quation wE q. (15 ) x.

Methodology Diffe re nt digital and an alog logs for seve ral we lls , pe ne trating the Hibe rnia and Te rra Nova rese rvoirs of the Je anne d’Arc B asin , we re an alyze d at sam pling-de pth inte rvals ( D Z ) of 0.2 m. E mphasis is place d on two H ibe rnia we lls (B-27 and K-18 ) and two Te rra Nova we lls (C-09 and E-79 ). The Koze ny-Carm an coe fficie nt was obtaine d as K cc1 , i.e ., K cc1 s t S hf , whe re S hf was assigne d a constant value of 2.5 , and as K cc 2 , i.e ., K cc2 s t m , whe re m is a variable obtaine d in accord with E q. (15 ) which re quire s F , f , and a. The form ation re sistivity factor was obtaine d in accord with the formula of Archie (1942 ) , i.e ., F s R b r R w , whe re R b is the bulk re sistivity, in V m , obtaine d from the de e p induction resistivity log , and R w , in V m , is the pore -wate r (brine ) re sistivity, obtaine d from the spontane ous pote ntial log. The porosity was obtaine d from a combination of various logs , and a was obtaine d from the inte rse ction of the best-fitting line of the F ] f relationship at r the porosity value of 100% . Tortuosity was obtaine d as ( Ff )1 2 , whe re F is dim e nsionle ss and f is fractional. T able 1 shows e xample s of the results obtaine d at D Z of 10 m for the Hibe rnia K-18 we ll and the Te rra Nova C-09 we ll , including t , m , R w , K cc1 , and K cc2 . To unde rstand the influe nce of K cc (obtaine d as K cc1 and K cc2 ) on the fluid flow and e le ctric-curre nt conduction in the rese rvoirs , K cc was corre late d with F (Figure s 1 and 2 ) for the Te rra Nova E-79 we ll , and with s s (Figure 3 ) and k (Figure 4 ) for the Hibe rnia B-27 we ll. For both we lls (Te rra Nova E-79 and Hibe rnia B-27 ) , 360 and 210 re adings , respe ctive ly, we re obtaine d at D Z of 1 m and plotte d in Figure s 1 ] 4. The spe cific surface are a ( s s ) was obtaine d using the e quation of Chilingar e t al. (1963 ) , which re quire s F , f , and k . The pe rme ability ( k ) was obtaine d using the e quation of Timur (1968 ) , which re quire s f and wate r saturation ( S w ). The wate r saturation was obtaine d using the e quation of Sim andoux (1963 ) , which re quire s seve ral param e ters. Coe fficie nts of corre lation ( R c ) ,


565

ranging from 0.90 to 0.95 , we re obtaine d for the four relationships (Figure s 1 ] 4 ). It is important to me ntion that the symbol ``K cc ’’ use d in the re m aining part of this study indicate s K cc 1 and K cc2 , unle ss othe rwise indicate d.


Results and Discussion The Koze ny-C arm an coe fficie nt ( K cc ) for the Hibe rnia and Te rra Nova re servoirs varie s conside rably from we ll to we ll and rese rvoir to re servoir due to the e ffe cts of de pth , lithology , wate r saturation , clay content , anisotropy, e tc. For seve ral we lls inve stigate d at D Z of 0.2 m , approxim ate ly 2.5 = 10 4 re adings we re obtaine d for e ach param e ter an alyze d , including K cc . An ove rall-ave rage value of 7.5 was obtaine d for K cc , which repre sents a ge ne ral ave rage range of be twe e n 4.1 and 10.4. To the author’ s knowle dge , since Carm an (1937 ) assigne d the value of 5.0 to K cc in the Kozeny-Carm an e quation for unconsolidate d porous me dia , no othe r value was use d , ne ithe r for unconsolidate d nor for consolidate d porous m e dia. B ase d on the pre se nt re sults , the K cc value of 7.5 (gre ate r than that re porte d by C arm an ) can be use d for similar form ations. The results obtaine d at D Z of 0.2 m showe d that t , m , and K cc are m aximize d for the Hibe rnia re servoir and minimize d for the Te rra Nova rese rvoir. This obse rvation m ay be attributed to the influe nce of de pth (the Hibe rnia rese rvoir is de e pe r than the Te rra Nova re servoir; Table 1 ). The incre ase in de pth , accompanie d by incre ase in ove rburde n pre ssure (gre ate r com paction , consolidation , and ce me ntation ) , results in change s in te xture and mine ralogy. The pore -wate r re sistivity has a wide r range for the Te rra Nova re se rvoir than the Hibe rnia re servoir (Table 1 ) , which m ay be attributed to the prese nce of m arlstone and conglome rate in the Te rra Nova rese rvoir. For both rese rvoirs , K cc 1 ( s t S hf ) range s from 4.46 to 42.92 , whe re as K cc2 ( s t m ) range s from 3.88 to 39.25 (Table 1 ). At close r sam pling inte rvals , K cc e xhibits value s of lowe r than those give n in Table 1 , accom panie d by value s of lowe r than 1.0 for t and lowe r than 1.2 for m , which m ay indicate pre se nce of microfractures and pe rfe ct sphe rical grains , and r or lack of grains of irre gular shape . The se crite ria he lp to e ase the flow of fluid and the conduction of e le ctric curre nt in the rese rvoirs. The relative ly high value s of K cc , accompanie d by re lative ly high value s of t and m (T able 1 ) , m ay indicate the pre sence of complicate d tortuous passage s , whe re the flow of fluid and the conduction of e le ctric curre nt face a gre ate r re sistance . Also , K cc e xhibits some value s of up to 150 obtaine d at D Z of 0.2 m , which m ay re fle ct ve ry complicate d ne twork of the pores and pore channe ls and e nrichme nt of plate-like clay m ine rals. Be cause K cc and t are functions of anisotropy, both param e te rs ( K cc and t ) incre ase whe n the flow is norm al to the bedding plane s (more resistance to the flow ) and de cre ase whe n the flow is paralle l to the be dding plane s (le ss resistance to the flow ). The re lationships be twe e n K cc 1 ( s t S h f ) and F , and K cc2 ( s t m ) and F for the Te rra Nova E-79 we ll wFigures 1 and 2; E qs. (16 ) and (17 ) x show value s of R c of 0.95 and 0.91 , re spe ctive ly:

K cc 1 s y 1 .1244 q 4 .3684 log F

(16 )

K cc 2 s y 1 .1723 q 3 .9726 log F

(17 )

566

H. S. Salem Table 1

Tortuosity (t ) , dime nsionle ss; Archie cem e ntation factor ( m ) , dime nsionle ss; pore-wate r re sistivity ( R w ) , V m ; and Kozeny-C arm an coe fficie nt w K cc1 s t S hf ( S hf is shape factor s 2.5 ) and K cc2 s t m x , dime nsionle ss , at sam pling-de pth intervals (D Z ) of 10 m for the H ibe rnia K-18 we ll (4 ,055 ] 4 ,545 m ; 490 m ) and the Te rra Nova C-09 we ll (3 ,188 ] 3 ,548 m ; 360 m ).a Hibe rnia K-18 well (4 ,055 ] 4 ,545 m ; 490 m )

Z


(m ) 4 ,055 4 ,065 4 ,075 4 ,085 4 ,095 4 ,105 4 ,115 4 ,125 4 ,135 4 ,145 4 ,155 4 ,165 4 ,175 4 ,185 4 ,195 4 ,205 4 ,215 4 ,225 4 ,235 4 ,245 4 ,255 4 ,265 4 ,275 4 ,285 4 ,295 4 ,305 4 ,315 4 ,325 4 ,335 4 ,345 4 ,355 4 ,365 4 ,375 4 ,385 4 ,395 4 ,405 4 ,415 4 ,425 4 ,435 4 ,445 4 ,455

t

m

6.51 7.78 4.93 4.41 3.66 5.59 4.97 3.71 2.83 3.34 3.16 3.03 2.92 4.01 3.49 4.12 3.09 5.48 5.08 4.50 4.82 2.54 6.39 3.59 2.62 2.43 2.64 3.03 5.38 4.21 5.54 4.53 3.13 2.35 2.81 2.28 16.10 3.17 17.17 4.96 4.22

2.53 2.26 2.29 2.25 2.20 2.19 2.60 2.15 2.17 2.21 2.25 2.30 2.18 2.19 2.17 2.17 2.16 1.84 2.47 2.66 2.54 2.20 2.34 2.19 2.22 2.25 2.29 2.30 2.56 2.19 2.68 2.28 2.54 2.25 2.26 2.34 2.36 2.40 2.29 2.32 2.25

V

Te rra Nova C-09 well (3 ,188 ] 3 ,548 m ; 360 m )

Rw m

0.011 0.003 0.020 0.011 0.011 0.003 0.010 0.010 0.020 0.011 0.014 0.021 0.018 0.008 0.011 0.007 0.011 0.001 0.012 0.021 0.009 0.020 0.003 0.010 0.029 0.029 0.023 0.022 0.009 0.008 0.008 0.006 0.038 0.052 0.025 0.065 0.001 0.107 0.001 0.014 0.006

Z K cc1

K cc2

(m )

t

m

16.26 19.45 12.33 11.03 9.16 13.97 12.43 9.27 7.06 8.36 7.89 7.57 7.31 10.02 8.72 10.30 7.72 13.70 12.71 11.25 12.05 6.34 15.98 8.98 6.55 6.08 6.60 7.57 13.46 10.53 13.86 11.33 7.83 5.88 7.01 5.69 40.25 7.91 42.92 12.40 10.55

16.44 17.60 11.31 9.94 8.06 12.27 12.93 7.98 6.14 7.38 7.10 6.96 6.38 8.76 7.58 8.93 6.68 10.08 12.55 11.97 12.25 5.58 14.94 7.87 5.82 5.47 6.05 6.97 13.76 9.22 14.84 10.32 7.94 5.29 6.34 5.33 37.94 7.61 39.25 11.50 9.47

3 ,188 3 ,198 3 ,208 3 ,218 3 ,228 3 ,238 3 ,248 3 ,258 3 ,268 3 ,278 3 ,288 3 ,298 3 ,308 3 ,318 3 ,328 3 ,338 3 ,348 3 ,358 3 ,368 3 ,378 3 ,388 3 ,398 3 ,408 3 ,418 3 ,428 3 ,438 3 ,448 3 ,458 3 ,468 3 ,478 3 ,488 3 ,498 3 ,508 3 ,518 3 ,528 3 ,538 3 ,548

1.81 3.82 2.18 2.68 2.24 2.33 3.43 1.99 1.93 2.67 5.69 3.49 3.12 3.66 4.74 3.80 7.02 2.51 2.23 1.93 6.69 5.61 5.51 3.51 2.72 3.49 1.86 1.79 2.23 2.06 2.03 4.55 3.83 2.34 2.35 2.20 2.29

2.15 2.19 2.22 2.25 2.22 2.25 2.31 2.22 2.21 2.18 2.27 2.22 2.22 2.19 2.18 2.16 2.11 2.14 2.28 2.01 2.11 2.19 2.16 2.18 2.49 2.15 2.27 2.25 2.15 2.20 2.06 2.25 2.21 2.20 2.23 2.18 2.17

V

Rw m

K cc1

K cc2

0.059 0.011 0.038 0.024 0.037 0.034 0.024 0.043 0.050 0.027 0.011 0.033 0.019 0.014 0.005 0.030 0.001 0.592 2.135 2.184 0.001 0.003 0.001 0.011 0.632 0.025 3.236 5.072 1.695 0.510 0.784 0.021 0.018 0.077 0.084 0.064 0.060

4.53 9.55 5.45 6.70 5.59 5.82 8.57 4.98 4.83 6.69 14.23 8.73 7.80 9.16 11.86 9.50 17.55 6.28 5.58 4.83 16.73 14.02 13.78 8.78 6.80 8.72 4.66 4.46 5.56 5.14 5.07 11.38 9.58 5.86 5.87 5.50 5.71

3.90 8.35 4.82 6.02 4.96 5.23 7.91 4.42 4.27 5.83 12.91 7.74 6.91 8.04 10.35 8.20 14.79 5.38 5.08 3.88 14.14 12.30 11.89 7.67 6.78 7.49 4.23 4.02 4.78 4.52 4.17 10.22 8.47 5.16 5.24 4.81 4.95


567

Table 1 Tortuosity (t ) , dime nsionle ss; Archie cem e ntation factor ( m ) , dime nsionle ss; pore-wate r re sistivity ( R w ) , V m ; and Kozeny-C arm an coe fficie nt w K cc1 s t S hf ( S hf is shape factor s 2.5 ) and K cc2 s t m x , dime nsionle ss , at sam pling-de pth intervals (D Z ) of 10 m for the H ibe rnia K-18 we ll (4 ,055 ] 4 ,545 m ; 490 m ) and the Te rra Nova C-09 we ll (3 ,188 ] 3 ,548 m ; 360 m ). a (Continued ) Hibernia K-18 we ll (4,055 ] 4,545 m ; 490 m )

Z


(m ) 4 ,465 4 ,475 4 ,485 4 ,495 4 ,505 4 ,515 4 ,525 4 ,535 4 ,545 a

t

m

4.05 3.46 2.35 2.79 2.65 3.84 6.12 4.34 7.27

2.12 2.20 2.15 2.21 2.18 2.28 2.26 2.33 2.31

V

Te rra Nova C-09 we ll (3 ,188 ] 3 ,548 m ; 360 m )

Rw m

0.016 0.043 0.063 0.018 0.021 0.010 0.005 0.017 0.003

The general range s of the Hibe rnia well are : t s 2.28 ] 5.69 ] 42.92 , K cc2 s 5.05 ] 39.25; 2.01 ] 2.49, R w s 0.001 ] 5.072 V

Z K cc1

K cc2

10.12 8.66 5.87 6.98 6.63 9.60 15.30 10.86 18.18

8.59 7.63 5.05 6.17 5.78 8.76 13.82 10.13 16.79

(m )

t

m

V

Rw m

K cc1

K cc2

parameters (give n in the Table at D Z s 10 m ) for the 17.17 , m s 1.84 ] 2.68 , R w s 0.001 ] 0.107 V m , K cc1 s and for the Te rra Nova well are : t s 1.79 ] 7.07 , m s m , K cc1 s 4.46 ] 17.55, K cc2 s 3.88 ] 14.79.

Figure 1. Form ation resistivity factor ( F ) , dime nsionle ss, ve rsus Kozeny-C arm an coe fficie nt ( K cc1 ) , dimensionless ( K cc1 s t S h f , where t is tortuosity and S h f is shape factor ) , for the Te rra Nova E-79 we ll (3 ,125 ] 3 ,485 m ) , re prese nting 360 re adings at sampling-depth intervals ( D Z ) of 1 m.


568

H. S. Salem

Figure 2. Form ation resistivity factor ( F ) , dime nsionle ss, ve rsus Kozeny-C arm an coe fficie nt ( K cc2 ) , dime nsionless ( K cc2 s t m , whe re t is tortuosity and m is Archie ce me ntation factor ) , for the Te rra Nova E-79 we ll (3,125 ] 3,485 m ) , re pre se nting 360 re adings at sampling-depth intervals ( D Z ) of 1 m.

Figure 3. Specific surface are a ( s s ) , in cm y 1 , versus Kozeny-C arman coe fficie nt ( K cc1 ) , dimensionless ( K cc1 s t S h f , whe re t is tortuosity and S h f is shape factor ) , for the Hibernia B-27 well (4,160 ] 4,370 m ) , repre se nting 210 re adings at sampling-de pth inte rvals ( D Z ) of 1 m.



569

Figure 4. Pe rme ability ( k ) , in mD , ve rsus Kozeny-C arman coe fficie nt ( K cc1 ) , dime nsionless ( K cc1 s t S h f , where t is tortuosity and S h f is shape factor ) , for the Hibernia B-27 well (4,160 ] 4,370 m ) , re prese nting 210 re adings at sampling-depth intervals (D Z ) of 1 m.

The ge ne ral range of F (f 5 ] 2 ,000 ) , with a m ajority of re adings of be twe e n f 7 and 300 , corre sponds positive ly to a ge ne ral range of K cc of be twe e n f 2 and 20 , with a m ajority of re adings of be twe e n f 2.5 and 10. Figures 1 and 2 indicate that if F incre ase s by f 100-fold , K cc will incre ase by f 4-fold. Using E qs. (16 ) and (17 ) , the F value of 10 , for e xample , corre sponds to the K cc 1 value of 3.24 and to the K cc 2 value of 2.80 , and the F value of 25 corre sponds to the K cc1 value of 4.98 and to the K cc 2 value of 4.38. This indicate s that the diffe rence s be twe e n the value s of K cc 1 ( s t S hf ) and K cc2 ( s t m ) are sm all. This obse rvation , supporte d by the sm all diffe re nce be twe e n the range s of K cc 1 and K cc2 , sugge sts that S hf and m are strongly analogous to e ach othe r. B oth F and m are affe cte d by the shape of grains and pore s; the y incre ase with incre asing angularity of the grains and de cre ase with incre asing sphe ricity of the grains. Conseque ntly, the progre ssive incre ase betwe e n K cc and F sugge sts that K cc is affe cte d (similar to F ) by the de gre e of angularity-sphe ricity of the grains. Also , the positive corre lation be twe e n K cc and F c an be conside re d a good indic ator of the variations of porosity and e le ctrolyte salinity (i.e ., highe r K cc ; lowe r f ; highe r F ; lowe r R w ; highe r wate r conductivity; highe r salinity ). The re lationships be twe e n K cc 1 ( s t S hf ) , s s , and k for the Hibe rnia B-27 we ll wFigure s 3 and 4; E qs. (18 ) and (19 ) x show value s of R c of 0.92 and 0.90 , re spe ctive ly:

K cc1 s y 10 .217 q 4 .6406 log s s

(18 )

K cc1 s 7 .3431 y 1 .2652 log k

(19 )

570

H. S. Salem


The high value s of R c for both re lationships indic ate a strong inte rconne ction am ong the thre e param e ters ( K cc , s s , k ) and a strong de pe nde nce on e ach othe r. The value s of K cc 1 , ranging from f 3.5 to 10 , corre spond to a range of s s from y f 1.1 = 10 3 to 1.9 = 10 4 cm 1 (Figure 3 ) and to a range of k from f 0.1 to 365 mD (Figure 4 ). The pre sence of high content of clays (clastic and authe ge nitic ) , c alcite , and quartz , growing in the pore s and pore channe ls , te nds to incre ase s s and K cc and to de cre ase k . The spe cific surface are a is a minimum for any spe cific grain size whe n the grains are pe rfe ct sphe re s , but strongly incre ase s whe n irre gularity of the grains incre ase s. A gre ate r de gre e of compaction , re sulting in tighte r packing of the grains , incre ase s s s and K cc and de cre ase s k .

Conclusions The Koze ny-C arm an coe fficie nt ( K cc ) of the Koze ny-C arm an e quation is obtaine d as K cc1 s t S hf and K cc2 s t m . The value s of K cc1 and K cc2 are close to e ach othe r , with ne gligible diffe re nces. Thus , the shape factor ( S hf ) of the Koze ny-C arm an e quation is strongly an alogous , physic ally and ge ome trically, to the Archie ce me ntation factor ( m ) of the Archie -W insaue r e quation , and thus , both param e ters ( S hf and m ) can replace e ach othe r in both e quations. For the Hibe rnia and Te rra Nova rese rvoirs , consisting of consolidate d shaly sandstone s and saturate d with multiphase fluids (oil , gas , and brine ) , an ave rage value of 7.5 was obtaine d for K cc and an ave rage value of 3.3 was obtaine d for t , resulting in an ave rage value of 2.28 for S hf (an alogous to m ). The Koze ny-C arm an coe fficie nt is affe cte d by various physical param e te rs and lithological attribute s. The re lationships betwe e n K cc , form ation resistivity factor ( F ) , spe cific surface are a ( s s ) , and pe rm e ability ( k ) , with coe fficie nts of corre lation ranging from 0.90 to 0.95 , show that K cc incre ase s with incre asing F and s s and with de cre asing k . The e quations obtaine d in this study e nable one to de termine K cc if any of the param e te rs: s s , k , or F is known , or vice ve rsa. The re lationships obtaine d sugge st that the Hibe rnia and Te rra Nova re servoirs e xhibit gre ate r value s of K cc (highe r t ) whe n e le ctric curre nt passe s through form ations of lowe r e le ctric conductivity (highe r F ) , and whe n fluid passe s through form ations of lowe r hydraulic conductivity (lowe r k ; highe r s s ). Also , K cc is strongly affe cte d by the shape of pore channe ls , pore s , and grains; type , size , and packing of grains; clay content; wate r saturation; anisotropy; e tc. Furthe rm ore , highe r value s of K cc sugge st lowe r porosity and highe r salinity (lowe r resistivity ) of the pore wate r , and highe r de gre e s of compaction , consolidation , and ce me ntation.

Nomenclature a A ff F g k K cc

pore ge ome try factor , dime nsionle ss cross-se ctional are a , cm 2 friction factor , dime nsionle ss form ation re sistivity factor , dime nsionle ss gravity acce le ration , cm r s 2 y pe rme ability, D arcy, m D , or cm 2 (D arcy s 9.87 = 10 9 cm 2 s 0.987 m m 2 ) Koze ny-C arm an coe fficie nt , dime nsionle ss


Fluid-Flow Characterization (Jeanne d’A rc Basin Reserv oirs) K cc 1 K cc 2 L Lc L ci Le m n N q rc rci rh Rb Rc Rn Rw ss

S hf Sw v ac v ap

Z D

P D Z m

f r t

c

¥

571

Koze ny-C arm an coe fficie nt ( s t S hf ) , dime nsionle ss Koze ny-C arm an coe fficie nt ( s t m ) , dim e nsionle ss bed thickne ss , or sam ple le ngth , cm le ngth of channe ls , cm le ngth of channe l i , cm e ffe ctive le ngth of fluid flow , cm Archie ce me ntation factor (an alogous to shape factor; S hf ) , dime nsionle ss num ber of channe ls in a group , dime nsionle ss num ber of tubes (groups of channe ls ) in a capillary syste m , dime nsionle ss volume rate of fluid flow , cm 3 r s radius of a group of channe ls , cm radius of channe l i , cm me an hydraulic radius , cm bulk re sistivity, V m corre lation coe fficie nt , dim e nsionle ss Re ynolds numbe r , dim e nsionle ss pore -wate r re sistivity, V m y spe cific surface are a (in ge ne ral ); pe r unit of bulk volume , sb , cm 1 ; pe r unit y1 y1 of grain volume , s g , cm ; pe r unit of pore volume , s p , cm ; or pe r unit of we ight , s wt , cm 2 r g shape factor (an alogous to Archie ce me ntation factor; m ) , dime nsionle ss wate r saturation , fraction or % actual (microscopic ) ve locity of fluid flow , cm r s appare nt (m acroscopic ) ve locity of fluid flow , cm r s de pth , m pressure drop in the dire ction of fluid flow , atm , dyne s r cm 2 , or g r cm 2 sam pling-de pth inte rval , m fluid viscosity, cP , or Poise s g r cm s porosity, fraction or % fluid de nsity, g r cm 3 tortuosity, dime nsionle ss characte ristic line ar dime nsion indicating ge ome try; in the case of channe ls ¥ s rc s channe l radius , in cm coe fficie nt de pe nding on the type of channe l ge ome try, dim e nsionle ss

References Archie , G. E. 1942. The ele ctrical re sistivity log as an aid in dete rmining some re se rvoir characte ristics. T rans. A IME 146:54 ] 62. Atkins, E. R., Jr., and G. H. Smith. 1961. The significance of particle shape in form ation factor-porosity re lationships. J. Pet. Technol. 13:285 ] 291. C arman, P. C. 1937. Fluid flow through granular be ds. T rans. Inst . Chem . E ng. 15:150 ] 156. C arman, P. C. 1938. The dete rmination of the spe cific surface of powde rs I . J. Soc. Chem . Ind. 57:225 ] 234. Chilingar , G. V ., R. Main , and A. Sinnokrot. 1963. Relationship be twe en porosity, pe rmeability, and surface are as of se dime nts. J. Sed. Petrol. 33:759 ] 765. Corne ll, D., and D. L. Katz. 1953. Flow of gases through consolidate d porous me dia. J. Ind . Eng. Chem . 45:2145 ] 2152. D arcy, H. P. G. 1856. Le s fountains publiques de la V ille de Dijon , 590 ] 594. P aris: V ictor Dalmont.


572

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Kozeny, J. 1927. Uebe r kapillare Le itung de s W assers im Boden. Sitzungsber. A kad. Wiss. (Wien ) 136a:271 ] 306. Muskat, M., and H. G. Botset. 1931. Air flow under high pressure through a be d of glass particle s. Physics 1:27 ] 47. Rose , W. D., and W. A. Bruce . 1949. Evaluation of capillary character in pe troleum re se rvoir rocks. Trans. A IME 186:127 ] 141. Sale m , H. S. 1992. A the oretical and practical study of pe trophysical, e le ctric, and elastic paramete rs of se diments. Ph.D. dissertation , Kie l University, F.R. Germ any, publishe d by University Microfilm International (UMI ) , Ann Arbor , MI. Sale m , H. S. 1994. The ele ctric and hydraulic anisotropic be havior of the Je anne d’Arc Basin re se rvoirs. J. Pet. Sci. Eng. 12:49 ] 66. Sale m , H. S. 2000a. Interre lationships among water and hydrocarbon saturations, perme ability, and tortuosity for shaly sandstone re se rvoirs in the Atlantic O ce an. Energy Sources 22 (3 ) :333 ] 345. Sale m , H. S. 2000b. Compute r mode lling of porosity and lithology for complex reservoirs using we ll-log me asure me nts. Energy Sources , in pre ss. Sale m , H. S., and G. V . Chilingarian. 1998a. Physical and m athem atical aspe cts of tortuosity in regard to the fluid flow and e lectric-curre nt conduction in porous me dia (on the example of the Hibe rnia and Te rra Nova reservoirs, off the e aste rn coast of C anada). Energy Sources 22 (2 ):137 ] 145. Sale m , H. S., and G. V . Chilingarian. 2000. Influence of porosity and dire ction of flow on tortuosity in unconsolidated porous me dia. Energy Sources, in press. Sale m , H. S., and G. V . Chilingarian. 1999. De termination of spe cific surface are a and me an grain size from we ll-log data and their influence on the physical be havior of offshore re se rvoirs. J. Pet. Sci. Eng. 22 (4 ):241 ] 252. Sim andoux, P. 1963. Me sure s die lectriques e n milleu poreux application a la mesure des saturations en e au etude du comporteme nt de s m assifs argileux. Rev . Inst . Fr. Pet . 18:193 ] 215. Sullivan , R. R., and K. L. Hertel. 1942. Pe rme ability me thod for de te rmining specific surface of fibers and powders. A dv . Colloids Sci. 1:37 ] 80. Tiab, D., and E. C. Donaldson. 1996. Petrophysics . Theory and practice of m easuring reserv oir rock and fluid transport properties. Houston , TX : Gulf. Timur, A. 1968. An inve stigation of perme ability, porosity and re sidual wate r saturation re lationships for sandstone re servoirs. Trans. SPWL A , 9th Annu. Logging Symp., 23 ] 26 June , pp. JJ1 ] 18. Winsaue r, W. O ., H. M. She arin, P. H. Masson, and M. Williams. 1952. Resistivity of brine-saturated sands in re lation to pore ge ome try. Bull. A m . A ssoc . Pet. G eol. 36:253 ] 277. Wyllie , M. R. J., and A. R. Gre gory. 1955. Fluid flow through unconsolidate d porous aggre gates: Effects of porosity and particle shape on Kozeny-Carm an constant. J. Ind . Eng. Chem . 47:1379 ] 1388. Wyllie M. R. J., and W. D. Rose. 1950. Application of the Kozeny equ ation to consolidated porous me dia. N ature 165 (June ):972. Wyllie , M. R. J., and M. B. Sp angle r. 1952. Application of electrical re sistivity me asurements to problem of fluid flow in porous media. Bull. A m . Assoc. Pet . G eol. 36:359 ] 403.