Principles of Single-Phase Flow Through Porous Media - Advances in

5 Principles of Single-Phase Flow Through Porous Media Shijie Liu and Jacob H. M a s l i y a h *

Downloaded by CORNELL UNIV on August 25, 2016 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/ba-1996-0251.ch005

D e p a r t m e n t of C h e m i c a l E n g i n e e r i n g , U n i v e r s i t y of A l b e r t a , E d m o n t o n T6G 2 G 6 , C a n a d a

Porous media are both permeable and dispersive to a traversing fluid. Flow of a single-phase fluid in porous media is not only of practical interest but also of fundamental significance in charac terizing the porous media. In this chapter, the characteristics of porous media are introduced from both fundamental and appli cation points of view. A continuum approach is used. The volume -averaged equations are used to describe the flow, where the momentum dispersion has been neglected. The relations between Darcy's law-Brinkman's equation and the volume-averaged Navier-Stokes equation are described. The Forchheimer hypothesis, Ergun equation, and Liu-Afacan-Masliyah equation are briefly described in terms of coupling of the viscous and inertial effects on the single-phase flow in porous media. Discussions are provided on the concept and modeling of areal porosity, tortuosity, perme ability, and shear factor. A curved passage model is discussed in terms of the shear factor and pressure-drop modeling for flow through porous media. Bounding wall effects are discussed through a simple approach. Examples of flow simulations in porous media (i.e., slightly compressible flow in oil reservoirs and incompressible flow in fixed beds) are provided.

Definitions Porous medium is a material consisting of a solid matrix with intercon nected pores. T h e interconnected pores are responsible for allowing a fluid to traverse through the material. F o r the simplest situation, the medium is saturated with a single fluid ("single fluid flow"). In " m u l tiphase fluid flow," several fluids (liquids and/or gas) share the open pores. Porous media are classified as unconsolidated and consolidated. * Corresponding author.

0065-2393/96/0251-0227$22.00/0 © 1996 A m e r i c a n C h e m i c a l Society

Schramm; Suspensions: Fundamentals and Applications in the Petroleum Industry Advances in Chemistry; American Chemical Society: Washington, DC, 1996.


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SUSPENSIONS: F U N D A M E N T A L S & APPLICATIONS IN P E T R O L E U M INDUSTRY

Beach sand, packed or fixed beds, soil, and gravel are unconsolidated porous media. C l o t h , most naturally occurring rocks such as sandstone and limestone, concrete, bricks, paper, and wood are consolidated porous media. The most common way of deriving the laws governing the average or macroscopic variables is to begin with the standard continuity and momentum equations and to average them over volumes or areas con taining many pores. A macroscopic variable is defined by an appropriate average over a sufficiently large representative elementary volume (REV). R E V is a conceptual space unit that bears the same meaning as the physical " p o i n t " for fluid continuum. A n R E V is the m i n i m u m space unit at which the porous medium of concern can be treated as a contin uous medium. At continuum, the porous medium grain or pore structure is invisible. T h e length scale of the R E V is larger than the grain or pore scale but smaller than the length scale of the entire flow domain. The porosity, e, of a porous medium is the fraction of the total volume of the medium that is occupied by the open pores. The porosity is an average quantity, or a bulk property. In practice, only the effective porosity, that is, only the interconnected pores, are useful for traversing fluids. D e a d ends and isolated pores have a negligible effect on a single fluid flow. M e d i a that have the same average value of porosity may be very different in their pore structure and flow capacity. A common method for measuring the porosity of a porous medium is the liquid displacement test. T h e basic assumption for the success of the liquid displacement test is that the open pores can be filled or replaced by a working liquid. By measuring the weight difference between the porous medium itself and with a liquid saturated in the pores, the void space can be calculated. F o r natural porous media, the porosity does not normally exceed 0.6. F o r petroleum reservoirs, the porosity is typically between 0.1 and 0.4. F o r beds of solid spheres of uniform diameter, the porosity can vary between the rhombohedral packing of 0.2595 and the cubic packing of 0.4764. Nonuniform grain size tends to y i e l d smaller porosities than uniform grains. Industrial random packs and porous foams, on the other hand, can offer a porosity of as high as 0.99. Table I shows the porosities of common porous materials. The specific surface or surface per unit volume, a , of a porous me dium is defined as the ratio of the total open pore surface area to the volume of the solids. The equivalent spherical diameter, d , is the d i ameter of an equivalent sphere that has the same surface area per unit volume of the solid material forming the porous medium. The flow properties that describe the matrix from the view point of a flowing fluid are sometimes called pseudotransport properties, such as the permeability, dispersion coefficient, and tortuosity. The permeP

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L i u & MASLIYAH

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Single-Phase Flow Through Porous Media

Χ ι-

χ

S. S α , S *\

r# C0 05

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w

χ

I


00 00 CD

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χ CD

in

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f

0

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f„

r P

«D/2>0

(125)

1

H e n c e , we expect that the larger the multidimensional effect is, the larger the deviation of the center line velocity from the average velocity. In the extreme case where F = 0, or the medium is infinitely permeable, the velocity ratio reaches 2, whereas the coefficient of the m u l t i d i m e n sional effect is infinite.

Steady Flow in Packed Beds of Monosized Spherical Particles. Steady incompressible fully developed flow in porous media confined in a circular pipe can be treated w i t h a single differential equa tion as given by equation 111. The inertial effects are only reflected in the shear factor term. T w o purposes are served in this section: to verify the integrity of the models presented earlier, including the passage model on shear factor and wall effects on the flow, and to show the flow behavior itself. T h e flow problem is solved numerically with a central difference method. A n abundance of experimental data are available in the literature. However, we confine ourselves to the laminar flow regime for a packed b e d of spherical particles. W e make use of the latest avail able data presented by F a n d et al. (JJO) for a packed bed with weak wall effects and the experimental data of L i u et al. (32). Figures 14 and 15 show the normalized pressure drop factor for a densely packed b e d of monosized spherical particles. F o r Re < 7,f is fairly independent of R e , and at high Re values, it increases fairly linearly with Re . The data points are the experimental results taken from F a n d et al. (J JO), where the bed diameter is D = 86.6 mm and the particle diameter is d = 3.072 mm. O n e can observe that the 2-dimensional model of L i u et al. (32), referred to as equation 107, agrees with the experimental data fairly w e l l in the whole range of the modified Reynolds number. F r o m F i g u r e 14, one observes a smooth transition from the Darcy's flow to Forchheimer flow regime. The one-dimensional model of L i u et al. (32) (i.e., equation 106) showed only slightly smaller f value. H e n c e , the no-slip effect or two-dimensional effect for this bed is small. As shown in Figures 14 and 15, the E r g u n equation consistently underpredicts the pressure drop. The deviation becomes larger when flow rate is increased. Figures 16 and 17 show the experimental results of L i u et al. (32) as w e l l as the theoretical predictions for the packed bed of large glass beads where D = 4.47 mm, d, = 3.184 mm, and 6 = 0.6007. One observes m

m

m

m

s

v


v


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SUSPENSIONS: FUNDAMENTALS & APPLICATIONS IN P E T R O L E U M INDUSTRY

Figure 14. Variation of pressure drop factor with modified Reynolds num ber for a densely packed bed of monosized spherical particles at low flow rates. The symbols are experimental data taken from reference 110.

d /D= 0.03547, ε = 0.360 s

Liu et al. Liu etal. Modified Modified

100

(32),eq107, 2-D (32), eq 106, 1-D Ergun equation, eq 109, 2-D Ergun equation, eq 108,1-D

200

300

400 Re

500

600

700

800

m

Figure 15. Variation of pressure drop factor with modified Reynolds num ber for a densely packed bed of monosized spherical particles at high flow rates. The symbols are experimental data taken from reference 110.


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Liu & MASLIYAH

273


4500

I 1 I ι ι I I I ι I I i I I I I i ι I ' M

' » '

M

d /D = 0.7123, ε =0.6007 4000 t Modified Ergun equation, eq 109,2-D Modified Ergun equation, eq 108,1-D 3500 Liu et al. (32), eq 107, 2-D Liu etal. (32), eq 106,1-D 3000 s


2500 2000 1500 1000 500 0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Figure 16. Pressure drop factor variation with modified Reynolds number for a packed bed having large glass heads in the high R e range (32). m

r *

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— ο °o ο

Uuetal. (32), eq 106,1-Q 300

10

20

50

40

30 Re

m

Figure 17. Pressure drop factor variation with modified Reynolds for a packed bed of large glass beads in the low R e range (32).

number

m


274

SUSPENSIONS: FUNDAMENTALS & APPLICATIONS I N PETROLEUM INDUSTRY

that the two-dimensional model of L i u et al. (32) predicts the experi mental results fairly w e l l for the entire range studied, Re < 6000. T h e deviation between the two-dimensional model predictions and the ex perimental data is within 10%. T h e wall effect modified E r g u n equation overpredicts the experimental data in the entire range studied. Figures 18 and 19 show the experimental data of L i u et al. (32) for the packed bed of small glass beads as compared with the theoretical predictions. H e r e , D = ΑΛΊ m m , d = 1.917 mm, and e = 0.4529. W e observe that the 2-dimensional model of L i u et al. (32) agrees fairly w e l l with the experimental results. T h e modified E r g u n equation predicts the experimental data w e l l in the low Re range as is shown in F i g u r e 16 and underpredicts the experimental data as Re is increased (Fig ure 18). Table I V gives a summary of the packed beds that we made use of in this section. T h e term C i reflects the two-dimensional effects for Re = 0. A value of zero, for 100(C / — 1), w o u l d indicate no twodimensional effects. W e can observe that the wall effects on the viscous term, C , range from about 6% for the experimental data of F a n d et al. (110) to 2 7 4 % for the packed bed of L i u et al. (32) with a d /D = 0.7123 as used here. The wall effects on the inertial term, C range from around 1 to 19%. T h e two-dimensional effects are also significant for the packed beds of large particle to tube diameter ratios. T h e 2dimensional model of L i u et al. (32) predicts quite w e l l over a wide range of wall effects. In contrast, the wall modified E r g u n equation sig nificantly underpredicts at low porosity (Figures 15 and 19) and overpredicts at high porosity (Figures 16 and 17) the experimental data. m

s


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m

mc

m

m(

2

w

s

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Steady Flow Through a Cylindrical Bed of Fibrous Mat. The model of L i u et al. (32) was also tested against high porosity cases for the steady incompressible flow through a fibrous mat of D = 25.5 mm, d = 0.167 m m , and e = 0.9301. F i g u r e 20 shows the calculated pressure drop factor and the ex p e r i m e n t a l values. W e observe that the model of L i u et al. (32) p r e dicts the experimental pressure drop b o t h i n the D a r c y ' s flow regime, the transition, and the F o r c h h e i m e r regimes. T h e t w o - d i m e n s i o n a l model gives a m u c h better p r e d i c t i o n than that using the o n e - d i m e n sional m o d e l . T h e E r g u n equation significantly overpredicts the ex p e r i m e n t a l data. The coefficient for multidimensional effect, equation 121 or 123, is strictly speaking valid for Darcy's flow only. H o w e v e r , it is possible to estimate the multidimensional effect. Because the multidimensional ef fect is only significant near the wall, the shear factor used for evaluating the multidimensional coefficient may take an average value in the boundary layer. L e t s



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275


Figure 18. Pressure drop factor variation with modified Reynolds for a packed bed of small glass beads in the low R e range (32).

number

m

Figure 19. Pressure drop factor variation with modified Reynolds number for a packed bed of small glass beads in the high R e range (32). m


276



Table I V .

S u m m a r y of the Characteristics of the Packed Beds Used i n T h i s Study

D, mm

d , mm

d /D

ε

Ref

86.6 4.47 4.47

3.072 3.184 1.917

0.03547 0.7123 0.4289

0.360 0.6007 0.4529

110 32 32

s

s

100(C

Cj

md

1.0589 3.7405 1.9894

0.9857 0.8114 0.8614

15

20

- 1)

0.38 15.08 5.56

200 190 180 170 160 150 140 130 120 110 100 90

0

5

10

25

Re Figure 20. Normalized pressure drop factor variation with modified Rey nolds number for the fine foam (32). m


5.

L i u & MASLIYAH

277


F

d

~

4X85.2Jfc

4

(

1

2

6

)

where f is the normalized pressure drop factor, f , evaluated by a one-dimensional model, for example, equation 126 for a granular bed, and/ is the value o f f at Re = 0. T h e coefficient for m u l t i d i m e n sional effect can then be given by vlL>

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=/

e I D

[ l + 0 . 0 0 0 4 3 ( ^ ° ' - 1)]

(128)

Figures 21 and 22 show the normalized pressured drop estimated by equation 128 for a packed D = 5.588 m m , d = 3.040 m m , and e = 0.5916. T h e experimental data are taken from F a n d and Thinakaran (92). W e can observe that the approximate solution, equation 128, pre dicts fairly well the experimental results and is very good in representing the exact numerical solution of the governing equations. F o r clarity, Figure 21 is an expanded region for the small Re values. E v e n with this scale, we observe that the approximate solution is very close to the exact numerical solution. It was found that equation 128 is a good prediction for the normal ized pressure drop factor when used for data in Figures 1 4 - 2 2 . T h e use s

m

360 d / D = 0.5441, s

340

ε =0.5916

N u m e r i c a l solution, with e q 107 ---•Eq106 A p p r o x i m a t e solution, e q 128

320 ^300 280 260 240

10

30

20

40 Re

50

60

70

m

Figure 21. f variation with R e for a packed bed of monosized spherical particles at low flow rate. The symbols represent the experimental data taken from reference 92. v

m


278

SUSPENSIONS: F U N D A M E N T A L S & APPLICATIONS IN P E T R O L E U M INDUSTRY 1800 • d /D = 0.5441, ε =0.5916 s

1600

Approximate solution, eq 128 Numerical solution, with eq 107 Eq106

1400 : 1200


^ 1000 800 600 400 200,

•

0

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200

.

400

600

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•

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ι

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1000 1200 1400 1600 1800 2000

Figure 22. Normalized pressure drop factor variation with Re for a packed bed of monosized spherical particles at highflowrate. The symbols represent the experimental data taken from reference 92. m

of equation 128 eliminates the need to solve for the two-dimensional flow i n a cylindrical porous medium bed, and its use is of the same complexity as that for a one-dimensional problem. H o w e v e r , the pre dictive accuracy has been vastly improved.

Summary In this chapter, we introduced the characteristics of porous media from both fundamental and application points of view. W e listed some relevant definitions and properties and useful models dealing with flow though porous media and reviewed the description of porous media to introduce to the reader the techniques used i n modeling the porous media and the properties of the porous media and their relationship to the models. F i e l d level characterization is actually a direct utilization of the models for transport in porous media, and our section on this may be considered as a study on the transient compressible flows. Because of the importance of the tortuosity and areal porosity, we devoted a section to discuss them. These two properties are not newly defined, but they do generate some confusion i n the literature. F o r instance, the areal porosity was



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279

thought to be identical to the porosity for all the porous media, although experimental observations indicate otherwise. This is by and large due to the view point that the pores in a porous medium may be treated as bundles of straight nonintersecting channels. The tortuosity is a classical concept; however, when modeling a transport phenomenon in porous media, different groups of investigators used different values. E v e n for the same porous medium when dealing with different transport phe nomena, different values were assigned. W e then discussed the modeling for single-fluid phase flow in porous media. In particular, the shear factor and permeability model of L i u et al. (32) is discussed in detail. T h e bounding wall effects are presented. This section completed the modeling requirements for single-phase i n compressible flow in porous media. W e showed how to solve the gov erning equations for flow in porous media and an approximate solution of the pressure drop for an incompressible flow through a cylindrically bounded porous bed was constructed. The usage of the flow equations can be summarized as follows. F o r the case of a one-dimensional single fluid flow, either equation 106 or 108 can be used to predict the normalized pressure drop factor in a porous medium. T h e determined normalized pressure drop factor is related to the pressure drop by equation 11. F o r the simple case of packed spherical beads, d and e are known a priori. T h e Reynolds n u m ber is evaluated using equation 93. F o r random packs of nonspherical particles, the particle's sphericity needs to be known. Equation 73 can be used to estimate d . F o r the case of consolidated porous m e d i u m , one can estimate d from the knowledge of the intrinsic permeability using equation 14. F o r the case of multidimensional or composite flows, the complete averaged N a v i e r - S t o k e s equation needs to be solved. T h e y are given by equation 19 and 20. The shear factor F is given by either equation 107 or 109. The evaluation of d follows that for the case of one-di mensional flow above. It should be noted that the porosity e of the porous b e d used in equation 20 is not assumed to be constant except for short range (particle diameter scale) variations. Therefore, when solving a problem having an interface of free space and porous medium, there is no need to specify a boundary condition for the interface. T h e variation of e w o u l d suffice to distinguish between the porous medium and the free flow. Moreover, in the free flow region, F becomes zero. W h e n the porous medium is contained in a cylindrical bed, the m u l tidimensional effect is given by equation 127. W h e n the pressure drop is needed, there is no need to solve the governing equation numerically. Equations 106 and 128 can be used for the estimation of pressure drop as well as scaling-up of a packed bed. s

s

s

s


280


Comparison between the models given by the modified Ergun equa tion 108 or 109 and by L A M equation 106 or 107 with available ex perimental data would indicate a preference to using L A M - t y p e equations.

List of Symbols first

A A A

p


t

A

w

a

a a

c

E

a Β

v

B Β B

Dn

υ

w

b b C

E

C C

D

Dn

C C

md

w

C

wi

c c D ¥

(viscous) constant of L A M equation, dimensionless, A = 85.2 areal porosity, dimensionless total cross-sectional area of the porous medium perpendicular to the main flow direction, dimensional wall effect corrected first L A M equation constant, d i m e n sionless radius of a circle, long axis of an ellipse, long side w i d t h of a rectangle, radius of circular arcs, or side width of an equilateral triangle, all are dimensional constant, Archie's law, dimensionless passage shape factor: the first (viscous) constant of the E r g u n equation, dimensionless. specific surface or surface per unit volume, dimensional second (inertial) constant of L A M equation, dimensionless, Β = 0.69 inertial term constant for a curved duct flow, dimensionless formation volume factor, dimensionless wall effect corrected second L A M equation constant, d i m e n sionless short axis of an ellipse, w i d t h of a parallel plate or short side width of a rectangle, all are dimensional second (inertial) constant of E r g u n equation, dimensionless third (viscous dominant to inertial dominant flow transition) constant of L A M equation, dimensionless wellbore storage constant, dimensionless viscous to inertial flow transitional parameter for curved duct flow, dimensionless multidimensional effect coefficient, dimensionless wall effect coefficient (viscous) constant for L A M or Ergun equation, dimensionless wall effect coefficient (inertial) constant for L A M or Ergun equation, dimensionless compressibility, dimensional form-drag coefficient, Ward's model, dimensionless diameter of containing cylinder for a porous medium bed, dimensional


5.

Liu & MASLIYAH

Dn d e

d d Ε Eif(.) £i(.) F F F F(.) /(.) / f f f f h H / /()(*) I\(x) / k k ^a p k k k L p

s

D


t

m

v

vlD

c l D

P

c

0

Y

/ LAM LDV m M η η ρ p* .

281

dean number, dimensionless hydraulic diameter of pores or equivalent passage diameter, dimensional modified particle diameter, dimensional equivalent particle diameter, dimensional electrical potential, dimensional modified exponential integral function exponential integral function shear factor, dimensional average shear factor, dimensional formation factor, dimensionless probability distribution function distribution function fanning friction factor, dimensionless modified Fanning friction factor, dimensionless pressure drop factor, dimensionless value off evaluated by one-dimensional model value o f / at Re = 0 formation height of a reservoir, dimensional constant in equation 115 electrical current, dimensional modified Bessel function of zeroth order modified Bessel function of first order electrical current density, dimensional intrinsic permeability, dimensional second-order tensorial k apparent intrinsic permeability, dimensional calculated value of k, dimensional K o z e n y - C a r m a n constant, dimensionless flow passage shape factor, dimensionless thickness of the porous medium parallel to the mainflowd i rection, dimensional radius of the reservoir, dimensional Liu-Afacan-Masliyah laser-Doppler velocimetry cementation factor wall effect factor, dimensionless index out normal of surface S, pressure, dimensional microscopic pressure, dimensional .) dimensionless pressure solution of equation 36 reference pressure, dimensional initial pressure (t = 0), dimensional v

vlD0

Po Pi


m


282

Y

capillary pressure, dimensional bottom-hole pressure, dimensional extrapolated bottom-hole pressure, dimensional flow rate, dimensional superficial unidirectional (1-dimensional) or average axial flow velocity, dimensional modified Reynolds number, dimensionless local modified Reynolds number, dimensionless truncation error of a function at nth term pore Reynolds number, dimensionless representative elementary volume radial coordinate, dimensional mean radius of the capillary pore, dimensional wellbore radius, dimensional effective wellbore radius, dimensional meniscus radii of the immicible fluids in the porous medium pore, dimensional saturation ratio, dimensionless solid-free space interface skin effect factor, dimensionless saturation ratio of the nonwetting phase, dimensionless irreducible saturation ratio of the nonwetting phase, d i m e n sionless surface area of particle, dimensional saturation ratio of the wetting phase, dimensionless irreducible saturation ratio of the wetting phase, dimensionless time, dimensional dimensionless time, equation 41 dimensionless time based on the image w e l l , equation 5 2 dimensionless time based on the real w e l l , equation 5 2 superficial axial velocity, dimensional characteristic pore velocity, dimensional volume, dimensional particle volume, dimensional volume of the free space in an R E V superficial radial velocity, dimensional superficial velocity vector, dimensional microscopic velocity vector, dimensional variable or axial coordinate variable defined by equation 113

Greek a a

first Forchheimer constant, dimensional second-order tensorial a

Pc Pw PJ

Q q Re Re.

m

RN(.)

Re REV r r r p



m

w

r ' we

S s, Skin

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S

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D

image ^Dreal U U

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ν*


5.

L i u & MASLIYAH

283

a β β

hydraulic diffusivity, dimensional second Forchheimer constant, dimensional second-order tensorial β

Δ δ δ/, b e θ \ μ μ ρ Σ a σ

difference pore size, dimensional pore body diameter, dimensional entry pore diameter, dimensional porosity, dimensionless contact angle characteristic curvature ratio, dimensionless viscosity, dimensional effective viscosity, dimensional density, dimensional summation electric conductivity of a fluid electric conductivity of a porous medium saturated with a con ducting fluid tortuosity, dimensionless calculated value of tortuosity, dimensionless particle sphericity, dimensionless

k

e

m



w

0

τ T Φ. C

Α

Acknowledgments W e thank the Natural Sciences and Engineering Research C o u n c i l of Canada and N C E for financial support.

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for review July 28, 1994. ACCEPTED revised manuscript June 6, 1995.


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