Abstract
In this second paper, the averaging rules presented in Part 1 are employed in order to develop a general macroscopic balance equation and particular equations for mass, mass of a component, momentum and energy, all of a phase in a porous medium domain. These balance equations involve averaged fluxes. Then macroscopic equations are developed for advective, dispersive and diffusive fluxes, all in terms of averaged state variables of the system. These are combined with the macroscopic balance equations to yield field equations that serve as the core of the mathematical models that describe the transport of extensive quantities in a porous medium domain.
It is shown that the methodology of averaging leads to a better understanding of the effective stress concept employed in dealing with transport phenomena in deformable porous media.
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Abbreviations
- a (α) ijkm :
-
components of porous medium dispersivity of the α-phase
- a L , a T :
-
longitudinal and transversal dispersivities of an isotropic porous medium
- c :
-
concentration of γ-component in a-phase (= ϱαγ)
- C f :
-
shape factor of (U of )
- C s :
-
coefficient of heat capacity of solid
- C ∀ :
-
fluid's heat capacity at constant volume
- D:
-
coefficient of dispersion
- D d fγ :
-
coefficient of molecular diffusion of binary system in fluid phase
- D d* fγ :
-
coefficient of molecular diffusion of fluid phase in a porous medium (D d fγ T * f )
- e α :
-
density of E of α-phase
- E α :
-
extensive quantity of α-phase
- ∫:
-
subscript indicating fluid phase; source term in (93)
- F α :
-
body force acting on α-phase
- g :
-
gravity acceleration
- G α :
-
tensorial property of α-phase
- I :
-
unit tensor (components δ ij )
- J dγ α :
-
J m αγ U α = diffusive flux of the γ-component in the α phase
- J E α U α :
-
diffusive flux of E α relative to α-phase, (=e α (V E α -V α ))
- J h α :
-
conductive heat flux of α-phase
- k α :
-
permeability of α-phase (component (k α)ij)
- l :
-
characteristic size of REV
- m α :
-
mass of α-phase
- m αγ :
-
mass of γ-component in α-phase
- M αγ :
-
linear momentum of α-phase
- n :
-
porosity
- nw :
-
subscript for nonwetting fluid phase
- N Re :
-
Reynolds number
- p αγ :
-
pressure in α-phase
- p c :
-
capillary pressure (= P nw - P w )
- q α :
-
specific discharge of α-phase (\(( = n\overline {V_\alpha } ^\alpha )\))
- q r :
-
specific discharge of α-phase relative to solid (\(( = n\overline {V_\alpha } ^\alpha - \overline {V_s } ^s ))\))
- q m f :
-
mass weighted specific discharge of fluid
- q supm infr f :
-
q m f relative to solid
- r α :
-
correlation coefficient
- R :
-
radius of spherical REV
- s v :
-
length measured along v
- s :
-
subscript indicating solid phase
- (S αβ ):
-
interface surface between α-phase and all other phases in REV (e.g., r αs )
- S αβ :
-
area of (S αβ )
- S 0 :
-
surface area of sphere surrounding U 0
- S αα :
-
surface area of α-phase on S 0
- S 0α :
-
total surface area of (U 0α )
- S w , S wn :
-
saturation of wetting (= θ w /n) and nonwetting (= θ nw /n) phases
- t :
-
time
- t′, t″:
-
unit vectors in plane tangent to α-s surface
- (T * f ) ij :
-
components of a coefficient related to the configuration of (U 0f )
- T iαj :
-
(dx i /ds)(dx j /ds) in α-phase
- T α :
-
temperature of α-phase (T f and T s )
- u :
-
velocity of (S αβ )
- U 0 :
-
volume of REV
- U 0α :
-
volume of α-phase in U 0 (also U 0β , U 0s )
- υ α :
-
specific volume of α-phase
- v :
-
subscript denoting void space
- V α :
-
volume weighted velocity of α-phase (also V s )
- V E σ :
-
velocity of E of α-phase
- V m α :
-
mass weighted velocity of α-phase
- V E αγ :
-
velocity of γ-component of α-phase
- w :
-
subscript for wetting fluid
- x :
-
position vector of a point
- x 0 :
-
position vector of centroid of REV
- \(\mathop x\limits^ \circ\) :
-
x - x 0; \(\mathop x\limits^ \circ\) α =x-x 0α
- z :
-
vertical coordinate (positive upward)
- α :
-
a phase; also as subscript indicating a phase
- β :
-
subscript for all other phases (except α) in REV
- β p :
-
coefficient of fluid compressibility at constant temperature (=∂ ρf /∂ pf )| T f
- β T :
-
coefficient of fluid compressibility at constant pressure (=-∂ ρf /∂T f )| ρf
- β b :
-
bulk compressibility of porous medium
- β s :
-
solid's compressibility
- β * :
-
scalar coefficient
- γ :
-
a component of α-phase; β s /β b
- ΓE α :
-
net rate of production of E from sources within α-phase
- δ ij :
-
components of Kronecker's delta
- Δ:
-
characteristic distance from solid wall to interior of the phase; hydraulic radius
- ε s :
-
solid's dilatation
- η :
-
thermoelastic coefficient
- θ α :
-
volumetric fraction of α-phase
- λ α :
-
thermal conductivity of α-phase (also λ s )
- λ * α :
-
thermal conductivity of α-phase in a porous medium
- Λ:
-
thermal conductivity of a porous medium
- Μ :
-
dynamic viscosity of fluid
- ν α :
-
unit normal vector on (S αβ ) or (S αα ) pointing outward of (U 0α )
- v :
-
unit vector along axis of symmetry
- ρ α :
-
mass density of α-phase
- \(\underline {\sigma _\alpha }\) :
-
stress in α-phase
- \(\underline {\sigma _s^{**} }\) :
-
Verruijt's effective stress
- σ ** s :
-
intergranular stress (usually called effective stress)
- ∑ ′ αβ :
-
specific area of α-phase (=S αβ /U 0)
- τ f :
-
viscous stress in fluid phase
- (°):
-
deviation of ( ) from intrinsic phase average of ( )
- \(\overline {( )}\) :
-
phase average of ( )
- \(\overline {( )} ^\alpha\) :
-
intrinsic phase average of ( )
- \((\widetilde{}\widetilde{}\widetilde{})^{\alpha \beta }\) :
-
average of ( ) over (S αβ )
- [( )]1,2 :
-
( )¦1 - ( )¦2 = jump in ( ) from side 1 to side 2 of a surface
- D f ( )/Dt :
-
material derivative of ( ) (= ∂( )/∂t + V m f · ▽)
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Bear, J., Bachmat, Y. Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass, momentum and energy transport. Transp Porous Med 1, 241–269 (1986). https://doi.org/10.1007/BF00238182
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DOI: https://doi.org/10.1007/BF00238182