Abstract
A methodology for eliminating nondominant effects in models that describe transport phenomena in porous media is presented. The methodology is based on the introduction of dimensionless numbers and on a proper evaluation of the order of magnitude of terms. These dimensionless numbers are redefined as characteristics of transport and transformation phenomena in porous media. It is shown that different time scales and different length scales may have to be employed for different variables. A method for evaluating the order of magnitude of the error of prediction when terms are deleted, is presented.
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Abbreviations
- a ijkl :
-
Component of porous medium dispersivity
- a L :
-
Longitudinal dispersivity of an isotropic porous medium
- c :
-
Concentration. As subscript, a symbol denoting a characteristic value
- C s :
-
Specific heat of a solid at constant strain
- C v :
-
Specific heat of a fluid at constant volume
- D E α :
-
Coefficient of dispersion ofE in anα-phase
- D γ α :
-
Coefficient of molecular diffusion of aγ-phase component in anα-phase
- D *γ α :
-
Coefficient of molecular diffusion of aγ-component in anα-phase within a porous medium
- e :
-
Density ofE (=amount ofE per unit volume of a phase)
- f :
-
As subscript, a symbol denoting a fluid
- g :
-
Gravity acceleration
- k ij :
-
Permeability component
- L :
-
Length of a domain
- L (u) :
-
Length assigned to a variableu
- n :
-
Porosity
- p :
-
Pressure
- q :
-
Specific discharge of a phase (=n V)
- q r :
-
Specific discharge of a phase relative to the solid
- t :
-
Time
- T :
-
Temperature
- T * ij :
-
Tortuosity component
- V :
-
Mass weighted velocity
- x :
-
Horizontal coordinate
- y :
-
Horizontal coordinate
- z :
-
Vertical coordinate (positive upward)
- α :
-
As subscript, a symbol for anα-phase
- Β p :
-
Coefficient of fluid compressibility at constant temperature and concentration
- Β T :
-
Coefficient of thermal expansion at constant pressure and concentration
- γ :
-
As subscript, a symbol denoting aγ-component
- δ f :
-
Hydraulic radius of the fluid filled part of void space
- λ α :
-
Thermal conductivity of anα-phase
- λH :
-
Combined thermal conductivity in the fluid and the solid in a saturated porous medium
- Μ :
-
Dynamic viscosity
- v :
-
Kinematic viscosity
- ρ :
-
Mass density of fluid
- Da:
-
Darcy number (≡Da(V)
- Eu:
-
Euler number
- Fr:
-
Froude number
- Pe:
-
Péclet number
- Pr:
-
Prandtl number
- Re:
-
Reynolds number
- Ri:
-
Richardson number
- St:
-
Strouhal number (e.g., St(K)
- (..)f :
-
Intrinsic phase average of (..)
- (..)*:
-
Dimensionless variable
- ∇:
-
Differential operator (nabla)
- ∇*(u):
-
Dimensionless differential operator based on the length scale of the variableu
References
Bear J. and Bachmat, Y., 1986, Macroscopic modelling of transport phenomena in porous media, 2. Applications to mass, momentum and energy,Transport in Porous Media,1, 241–270.
Bear J. and Bachmat, Y., 1990,Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht.
Guchman, A. A., 1963,Introduction to the Theory of Similarity (in Russian), Gosizdat Vysshaya Shkola, Moscow.
Lin, C. C. and Segel, L. A. 1974,Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan, New York.
Perry, J. H. (Ed). 1950,Chemical Engineers' Handbook, McGraw Hill, New York.
Prager, W., 1961,Introduction to Mechanics of Continua, Ginn and Co., Boston.
Sedov, L. I., 1965,Methods of Similarity and Dimensions in Mechanics (in Russian), Izdat. Nauka., Moscow.
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Bear, J., Bachmat, Y. Deletion of nondominant effects in modeling transport in porous media. Transp Porous Med 7, 15–38 (1992). https://doi.org/10.1007/BF00617315
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DOI: https://doi.org/10.1007/BF00617315