Abstract
A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.
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Abbreviations
- A p :
-
pore cross-sectional area
- d :
-
microscopic characteristic length
- d e :
-
flow length within RUC
- d p :
-
pore width
- f :
-
friction factor,\(\left( {\frac{{\partial p}}{{\partial x}} \cdot 2d_p } \right)/(p\upsilon _p^2 ),\)
- f app :
-
friction factor, (2d pΔp)/(γν p2 Δx)
- g :
-
gravity constant
- I :
-
integral expression
- K :
-
hydrodynamic permeability
- l f :
-
frictional flow length within RUC
- n :
-
porosity
- p :
-
pressure
- q :
-
specific discharge (= 〈v〉)
- Re:
-
Reynolds number,γυ pdp/μ
- Re qd :
-
Reynolds number,γqd/μ
- S :
-
surface
- S fs :
-
fluid-solid interface within RUC or REV
- T :
-
tortuosity
- V :
-
volume
- v :
-
fluid velocity withinV n
- v n :
-
average fluid velocity withinv n
- v p :
-
mean fluid velocity within pore section
- x :
-
axial distance
- x + :
-
dimensionless axial distance
- β :
-
inertia parameter
- μ :
-
fluid dynamic viscosity
- μ′ :
-
Brinkman effective viscosity
- ν :
-
normal vector onS fs
- ϱ :
-
fluid density
- φ :
-
extensive tensorial property
- c :
-
as subscript, denotes central or critical value
- i :
-
as subscript, denotes inflection point
- n :
-
as subscript, pertaining to void volume
- o :
-
as subscript, denotes total volume of RUC or REV
- p :
-
as subscript, pertaining to pore section
- μ :
-
a subscript, denotes streamwise shear term
- ▽:
-
vector operator ‘del’
- (·):
-
deviation of ( ) from 〈( )〉 n , tensor inner product
- 〈( )〉:
-
volumetric phase average of ( )
- 〈( )〉 n :
-
volumetric intrinsic phase average of ( )
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Du Plessis, J.P., Masliyah, J.H. Mathematical modelling of flow through consolidated isotropic porous media. Transp Porous Med 3, 145–161 (1988). https://doi.org/10.1007/BF00820342
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DOI: https://doi.org/10.1007/BF00820342