# Porous Media Transport Phenomena (Civan/Transport Phenomena) || Mass, Momentum, and Energy Transport in Porous Media

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CHAPTER 9 MASS, MOMENTUM, AND ENERGY TRANSPORT IN POROUS MEDIA

9.1 INTRODUCTION

This chapter demonstrates the applications of coupled mass, momentum, and energy conservation equations for various problems. * The transport of species through porous media by different mechanisms is described. Dispersivity and dispersion in heterogeneous and anisotropic porous media issues are reviewed. Formulation of compositional multiphase fl ow through porous media is presented in the following categories: general multiphase fully compositional nonisothermal mixture model, isothermal black oil model of nonvolatile oil systems, isothermal limited composi-tional model of volatile oil systems, and shape - averaged models. Formulation of source/sink terms in conservation equations is discussed. Analyses and formulations of problems involving phase change and transport in porous media, such as gas condensation, freezing/thawing of moist soil, and production of natural gas from hydrate - bearing formations, are presented. Typical applications and their results are provided for illustration of the theoretical treatise.

Porous Media Transport Phenomena, First Edition. Faruk Civan. 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

281

* Parts of this chapter have been reproduced with modifi cations from the following: Civan, F. 2000d. Unfrozen water content in freezing and thawing soils Kinetics and correlation. Journal of Cold Regions Engineering, 14(3), pp. 146 156, with permission from the American Society of Civil Engineers; Civan, F. and Sliepcevich, C.M. 1984. Effi cient numerical solution for enthalpy formulation of conduction heat transfer with phase change. International Journal of Heat Mass Transfer, 27(8), pp. 1428 1430, with permission from Elsevier; Civan, F. and Sliepcevich, C.M. 1985b. Comparison of the thermal regimes for freezing and thawing of moist soils. Water Resources Research, 21(3), pp. 407 410, with permission from the American Geophysical Union; and Civan, F. and Sliepcevich, C.M. 1987. Limitation in the apparent heat capacity formulation for heat transfer with phase change. Proceedings of the Oklahoma Academy of Science, 67, pp. 83 88, with permission from the Oklahoma Academy of Science.

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282 CHAPTER 9 MASS, MOMENTUM, AND ENERGY TRANSPORT IN POROUS MEDIA

9.2 DISPERSIVE TRANSPORT OF SPECIES IN HETEROGENEOUS AND ANISOTROPIC POROUS MEDIA

Spontaneous irreversible dispersive transport of species in fl uid media may occur under the effect of various driving forces (Bird et al., 1960 ), including the presence of the gradients (difference over distance) of species concentration, total pressure, potential energy or body force, and temperature (the Soret effect), and by hydrody-namic spreading and mixing (Bear, 1972 ; Civan, 2002e, 2010b ). However, the dominant mechanism for spontaneous species transport in fl uids present in porous media is the hydrodynamic mixing or dispersion phenomenon caused during fl ow through irregular hydraulic fl ow paths formed inside the porous structure. Under special circumstances, other factors may also cause spontaneous species transport. For example, live organisms, such as bacteria, have the natural tendency to move from scarce to abundant nutrient - containing locations. This phenomenon is referred to as chemotaxis (Chang et al., 1992 ).

The total fl ux of spontaneous transport of species i in phase j is expressed as the sum of the fl uxes of spontaneous transport of species by various factors:

J J J J J J Jij ijM ijD ijP ijB ijT ijOO

= + + + + + , (9.1) where the superscripts M , D , P , B , T , and O refer to spontaneous species transport owing to molecular diffusion, convective or mechanical dispersion, pressure, body force, temperature, and other factors, respectively. Table 9.1 presents the fundamen-tal expressions of fl ux. Bear (1972) defi nes the sum of the molecular diffusion and convective mixing effects as hydrodynamic dispersion. Therefore, the spontaneous hydrodynamic dispersion fl ux can be defi ned as

J J JijH ijM ijD= + , (9.2) where the superscript H refers to hydrodynamic dispersion.

Frequently, the fl ux of species is expressed by the gradient model, expressing the fl ux as being proportional to the gradient of the variable causing the spontaneous transport, associated with a usually empirically determined proportionality coeffi -cient, referred to as the diffusion coeffi cient. In the following, commonly used gradi-ent models for expressing species fl ux by various driving mechanisms are reviewed.

TABLE 9.1 Fundamental Expressions of Flux

Driving Force Gradient Law Species Flux

Flow potential

Hydraulic dispersion or convective mixing effect

Molecular diffusion Soret effect (temperature gradient)

u Kjrj

jj

k=

J u mijB ij j j ij jw w= =

m DjH jb j j= ( ) J mijH ij jHw=J DjM ij ijw= J JijM ij jMw=J DjT jT jT= J JijT ij jTw=

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9.2 DISPERSIVE TRANSPORT OF SPECIES 283

9.2.1 Molecular Diffusion

The diffusive species fl ux is given by

J DijM j j ijM ijw= , (9.3) in which the molecular diffusion coeffi cient tensor, considering the effect of tortuous fl ow paths in porous media, is given by

D T IijM ijD= 1 , (9.4) where Dij is the molecular diffusion coeffi cient of species i present in phase j , T is the tortuosity tensor of porous media, and I is the unit diagonal tensor. DijT1 rep-resents the extended molecular diffusion coeffi cient of species i traveling through tortuous fl ow paths, generally longer than the porous media.

The molecular diffusion coeffi cient is usually estimated by empirical correla-tions, such as the Wilke and Chang correlation given by (Wilke, 1950 ; Perkins and Geankopolis, 1969 )

D TMv

ii

=

7 4 10 8 0 50 6

.

,

.

. (9.5)

where D i denotes the molecular diffusivity of species i in the square centimeter per second units, and M and represent the molecular weight (gram per mole) and viscosity (centipoises) of the fl uid phase in which species i diffuses; hence, M and do not include species i . v i denotes the partial molar volume of species i (cubic centimeter per mole), and T is the absolute temperature (Kelvin).

9.2.2 Hydrodynamic Dispersion

The convective or mechanical mixing and dispersion of fl uids is a dominant factor causing spontaneous spreading of species in porous media. Convective dispersion and mechanical mixing occur owing to the inhomogeneity of intricately complicated interconnected pore structure in porous media (Bear, 1972 ). Simultaneously, the molecular diffusion phenomenon takes place owing to species concentration difference.

Hydrodynamic dispersion can be primarily explained by means of the stream - splitting and mixing, and mixing - cell theories, although other mechanisms may also promote additional mixing affects (Skjaeveland and Kleppe, 1992 ). The stream - splitting and merging theory is based on the phenomenon of consecutive splitting and merging of fl ow as the fl uid encountering the porous media grains is forced to fl ow around them. As a result, a lateral spreading of fl uid and transverse dispersion of species contained in the spreading fl uid occurs. The mixing - cell theory considers the mechanical mixing as a result of consecutive expansion and compression of fl uid as it fl ows through pore bodies and pore throats, causing repetitive fl uid accel-erations and decelerations, respectively, and therefore a longitudinal dispersion of species. Insuffi cient pore connectivity, local fl uid recirculation, fl ow constriction, dead - end pores, and retardation by sorption processes also affect convective mixing (Skjaeveland and Kleppe, 1992).

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284 CHAPTER 9 MASS, MOMENTUM, AND ENERGY TRANSPORT IN POROUS MEDIA

The dispersive species fl ux due to convective mixing is usually expressed by (Lake at al., 1984, Bear and Bachmat, 1990 ) J DijD j j ijbD ijw= , (9.6) where DijD is the hydrodynamic dispersion coeffi cient tensor in the length squared over time units ( L 2 / T ). Although used frequently, this particular form or its variations may have some limitations. For example, when the species mass fraction is uniform, that is, w ij = ct., Eq. (9.6) implies a vanishing dispersive fl ux of species. However, dispersive fl ux of species occurs as long as convective mixing prevails. Therefore, extending the formulation of Civan (2002e, 2010b) , and by intuition, a more appro-priate gradient model can be proposed as (see Chapter 3)

J D D D

D

ijD

ijbD

ij b ijbD

j ij j ijbD

j j ij j

ij

w= = ( ) = ( )=

bbD

j j j ij j j ij jw w + ( ) . (9.7) If the average of the product of deviations is neglected and the convention for

averaging is dropped for convenience, Eq. (9.7) can be simplifi ed as J DijD ijbD j j ijw= ( ) . (9.8)

Eq. (9.8) simplifi es to Eq. (9.6) only when the volume fraction and density of the j - phase are uniform in porous media. Eq. (9.8) implies that dispersive species fl ux can occur so long as the volume fraction, density, and species concentration in the j - phase are not uniform in porous media.

The predictability of the hydrodynamic dispersion coeffi cient has occupied many researchers, including Bear and Bachmat (1990) and Liu and Masliyah (1996) . However, further work is necessary for improved means of estimating the hydrody-namic dispersion coeffi cient in terms of the relevant parameters of porous media and pore fl uids. Under certain simplifying assumptions, such as isochoric fl ow, uniform density, conservative extensive quantity, and impervious interface for dif-fusion, Bear and Bachmat (1990) derived an expression for the hydrodynamic dis-persion coeffi cient. This equation will be adapted in the following form for the components of the hydrodynamic dispersion coeffi cient tensor:

D a v v f Pe rnm nklm k l i= ( )v

, , (9.9)

where n , m = x - , y - , and z - Cartesian directions, and a nklm denote the components of the isotropic dispersivity, given by

a aa a

nklm T nm klL T

nk lm nl mk= +

+( ) 2

, (9.10)

in which a L and a T are referred to as the longitudinal and transverse dispersivity coeffi cients, respectively, in the length dimensions ( L ) and relative to the primary fl ow direction. pq is the Kronecker delta, whose value is zero when p q and unity when p = q . The function f ( Pe i , r ) is given by

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9.2 DISPERSIVE TRANSPORT OF SPECIES 285

f Pe r PePe ri

i

i, ,( )

+ + =

1 (9.11)

where Pe i is the species i Peclet number and r is the ratio of the characteristic length along the fl ow direction to the characteristic length normal to the fl ow direction inside a pore space. Hence, Eq. (9.11) indicates that

Limit 1f Pe rPe

i

i

,

.

( ) =

(9.12)

Bear and Bachmat (1990) justify that r = O (1) and therefore, f ( Pe i , r ) 1 when Pe i >> 1, and f ( Pe i , r ) = O ( Pe i ) when Pe i

286 CHAPTER 9 MASS, MOMENTUM, AND ENERGY TRANSPORT IN POROUS MEDIA

in which the fl ow potential gradient is defi ned by

= + j j jp g z. (9.19) Darcy s equation given above considers laminar fl ow due to viscous forces.

9.2.4 Correlation of Dispersivity and Dispersion

Auset and Keller (2004) observed, by means of experiments conducted using micro-models, that the magnitude of particle dispersion depends primarily on their prefer-ential paths and velocities. Smaller particles tend to move along longer complicated tortuous paths and therefore are slower, and larger particles tend to move along shorter, straighter paths and therefore are faster. Hence, the dispersivity and disper-sion coeffi cient of particles in porous media increase with decreasing particle size and vice versa, depending on the pore structure and the pore channel - to - particle size ratio.

Let u denote the volumetric fl ux, k B = 1.38E - 23 m 2 kg/s 2 /K is the Boltzmann constant, is viscosity, T is the absolute temperature, D g is the porous media mean grain diameter, D p is the mean diameter of diffusing particle (or molecule), and D denotes the bulk molecular diffusivity.

The Peclet number N Pe based on grain diameter D g is given by

NuDDPe

g=

. (9.20)

The Stokes Einstein equation of bulk molecular diffusivity D of particles (or molecules) is given by

D k TD

B

p

=

3. (9.21)

Many attempts have been made at developing empirical correlations for the estimation of the dispersion coeffi cients. In the following, two of the frequently used correlations are presented.

The correlation of Hiby (1962) for the dispersion coeffi cient is given by

DD

NN

NPePe

Pe

= ++

0 67 0 651 6 7

0 01 1001 2..

.

, . ./ (9.22)

The correlation of Blackwell et al. (1959) for the dispersion coeffi cient is given by

DD

N NPe Pe

= >8 8 0 51 17. , . .. (9.23)

Let v denote the interstitial pore fl uid velocity, given by (Dupuit, 1863 )

vu

=

. (9.24)

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9.2 DISPERSIVE TRANSPORT OF SPECIES 287

Thus, substituting Eq. (9.20) for the Peclet number into Eq. (9.23) and then applying Eq. (9.24) yields (Civan, 2010c )

DD

uDD

v DD

Ng g Pe

= = >8 8 8 8 0 5

1 17 1 17

. . , . .

. .

(9.25)

We can obtain the following approximation in terms of the superfi cial velocity by rounding the exponent to 1.0:

D u N DPe g > = 1 10 5 8 8, . , . . (9.26) Alternatively, we can obtain the following approximation in terms of the

interstitial velocity:

D v ND

Peg

> =

, . ,

.

.0 5 8 8 (9.27)

Figure 9.1 shows a comparison of the correlation developed by the author and the Unice and Logan (2000) correlation of the dispersivity (meter) data of Gelhar et al. (1992) . Unice and Logan (2000) correlated the data of Gelhar et al. (1992) , where x (meter) denotes the scale, by = ( )[ ]10 1 120exp / .x (9.28)

Unice and Logan (2000) also provided a more consevative correlation of the data of Gelhar et al. (1992) by = ( )1 36exp / .x (9.29)

Figure 9.1 Comparison of the present and Unice and Logan (2000) correlation of the dispersivity data of Gelhar et al. (1992) (prepared by the author).

1000

100

10

1

0.1

0.01

0.0011 10

Scale, x (m)100

Gelhar et al. (1992)Civan (present)Unice and logan (2000)

1000

Dis

pers

ivity

, a (m

)

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