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