Abstract
Experimental observations have established that the proportionality between pressure head gradient and fluid velocity does not hold for high rates of fluid flow in porous media. Empirical relations such as Forchheimer equation have been proposed to account for nonlinear effects. The purpose of this work is to derive such nonlinear relationships based on fundamental laws of continuum mechanics and to identify the source of nonlinearity in equations.
Adopting the continuum approach to the description of thermodynamic processes in porous media, a general equation of motion of fluid at the macroscopic level is proposed. Using a standard order-of-magnitude argument, it is shown that at the onset of nonlinearities (which happens at Reynolds numbers around 10), macroscopic viscous and inertial forces are negligible compared to microscopic viscous forces. Therefore, it is concluded that growth of microscopic viscous forces (drag forces) at high flow velocities give rise to nonlinear effects. Then, employing the constitutive theory, a nonlinear relationship is developed for drag forces and finally a generalized form of Forchheimer equation is derived.
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Abbreviations
- a :
-
coefficient in Equations (1) to (3) and (22); also a 1 and a 2 in (23) and (24)
- b :
-
coefficient in Equations (1) to (3) and (22); also b 1 and b 2 in (23) and (24)
- c :
-
coefficient in (2); also c 1 and c 2 in (23)
- d :
-
coefficient in (3)
- da :
-
microscopic infinitesimal element of area
- d kl :
-
deformation rate tensor, 1/2(ν k,l+ ν nl,k)
- e 1 :
-
coefficient in (24)
- E KL :
-
solid-phase deformation tensor
- g k :
-
gravity vector
- l :
-
microscopic (pore) characteristic length of the porous medium
- L :
-
macroscopic characteristic length of the porous medium
- M :
-
specific surface of the solid phase
- n supfsinfl :
-
unit vector normal to the fluid-solid interfaces
- p :
-
(macroscopic) thermodynamic pressure
- p′ :
-
microscopic (pore) pressure
- q :
-
order of magnitude of flow velocity
- R :
-
coefficient in Equation (20), also R kl, R klm, and R klmnin (14)
- Re:
-
Reynolds number defined as pql/μ
- t′ kl :
-
microscopic fluid stress tensor
- T :
-
characteristic time, assumed to be equal to L/q
- \(\hat T_k\) :
-
solid-fluid interfacial drag force
- υ :
-
magnitude of velocity in Equations (1) to (3)
- υ k :
-
macroscopic fluid velocity vector
- ~υ k :
-
deviation of pore velocity from average velocity, \~υ k = υ′k − υk
- υ supdinfk :
-
macroscopic velocity of fluid relative to the solid
- z k :
-
an arbitrary objective vector in Equations (15) to (20)
- Z :
-
a scalar product defined in (15)
- δ kl :
-
Kronecker delta
- δV :
-
volume of the representative element of volume (REV)
- δA fs :
-
solid-fluid interfacial surfaces within an REV
- ε :
-
porosity
- θ :
-
temperature
- μ :
-
(microscopic) fluid viscosity
- ξ a :
-
a set of invariants defined in (18)
- ϱ :
-
macroscopic fluid density
- ϱ′ :
-
microscopic fluid density
- \(\hat \tau _k\) :
-
dissipative part of \(\hat T_k\)
- τ kl :
-
dissipative part of macroscopic fluid stress tensor
- 〈〉:
-
averaging sign
- ∘ :
-
order of magnitude
References
Ahmed, N. and Sunada, D. K., 1969, Nonlinear flow in porous media, J. Hyd. Div. Proc. ASCE 95, 1847–1857.
Barak, A. Z. and Bear, J., 1981, Flow at high Reynolds numbers through anisotropic porous media, Adv. Water Resour. 4, 54–56.
Bear, J., 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York.
Blick, E. F., 1966, Capillary orifice model for high speed flow through porous media, I&EC, Process Design and Development, No. 1, 5, 90–94.
Coulaud, O., Morel, P., and Caltagirone, J., 1986, Effets non-linéaires les écoulments en milieu poreux, C. R. Acad. Sci. Paris, Ser. II, 6, 263–266.
Cvetković, V. D., 1986, A continuum approach to high velocity flow in a porous medium, Transport in Porous Media 1, 63–97.
deVries, J., 1979, Prediction of non-Darcy flow in porous media, J. Irrig. Drain. Div. ASCE IR2, 147–162.
Dullien, F. A. L. and Azzam, M. I. S., 1973, Flow rate-pressure gradient measurements in periodically nonuniform capillary tubes, AIChE J. 19, 222–229.
Dybbs, A. and Edwards, R. V., 1984, A new look at porous media fluid mechanics - Darcy to turbulent, in J. Bear and M. Y. Corapcioglu (eds.), Fundamentals of Transport Phenomena in Porous Media, Martinus Nijhoff, Dordrecht, pp. 199–256.
Firoozabadi, A. and Katz, D. L., 1979, An analysis of high-velocity gas flow through porous media, J. Petrol. Technol. (February 1979), 211–216.
Geertsma, J., 1974, Estimating the coefficient of inertial resistance in fluid flow through porous media, Soc. Petrol. Eng. J. (October, 1974), 445–450.
Hannoura, A. and Barends, F. B. J., 1981, Non-Darcy flow, a state of art, in A. Verruijt and F. B. J. Barends (eds.), Proc. Euromech 143, Delft, pp.37–51.
Happel, J. and Brenner, B., 1965, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ.
Hassanizadeh, M. and Gray, W. G., 1979a, General conservation equations for multi-phase systems, I. averaging procedure, Adv. Water Resour. 2, 3, 131–144.
Hassanizadeh, M. and Gray, W. G., 1979b, General conservation equations for multi-phase systems, II. Mass, momenta, energy, and entropy equations, Adv. Water Resour. 2, 4, 191–203.
Hassanizadeh, M. and Gray, W. G., 1980, General conservation equations for multi-phase systems, III. Constitutive theory for porous media flow, Adv. Water Resour. 3, 1, 25–40.
Hubbert, M. K., 1956, Darcy's law and the field equations of the flow of underground fluids, Trans. AIME 20, 222–239.
Irmay, S., 1958, On the theoretical derivation of Darcy and Forchheimer formulas, J. Geophys. Res. 39, 702–707.
MacDonald, I. F., El-Sayed, M. S., Mow, K., and Dullien, F. A. L., 1979, Flow through porous media - the Ergun equation revisited, Ind. Eng. Chem. Fundam. 18, 3, 199–208.
Polubarinova-Kochina, P. Ya., 1952, Theory of Groundwater Movement (in Russian), English translation by R. J. M. DeWiest, Princeton University Press, Princeton, NJ.
Scheidegger, A. E., 1974, The Physics of Flow Through Porous Media, 3rd edn., University of Toronto Press, Toronto.
Schneebeli, G., 1955, Expériences sur la limite de validité de la loi de Darcy et l'apparition de la turbulence dans un écoulement de filtration, La Huille Blanche, No. 2, 10, 141–149.
Slattery, J. C., 1972, Momentum, Energy, and Mass Transfer in Continua, McGraw-Hill, New York.
Spencer, A. J. W., Theory of invariants, in A. C. Eringen (ed.), Continuum Physics, Vol. 1, Academic Press, New York, Part III.
Stark, K. P., 1972, A numerical study of the nonlinear laminar regime of flow in an idealized porous medium, in IAHR, Fundamentals of Transport Phenomena in Porous Media, Elsevier, Amsterdam, pp. 86–102.
Sunada, D., 1965, Turbulent flow through porous media, Water Resources Center, Contribution No. 103, University of California, Berkeley, California.
Swartzendruber, D., 1962, Non-Darcy flow behavior in liquid-saturated porous media, J. Geophys. Res. 67, 5205–5213.
Tek, M. R., 1957, Development of a generalized Darcy equation, Trans. AIME, 210, 376–377.
Whitaker, S., 1969, Advances in theory of fluid motion in porous media, Ind. Eng. Chem. 61, 14–28.
Wright, D. E., 1968, Non-linear flow through grannular media, J. Hyd. Div. Trans. ASCE, 94, 851.
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Hassanizadeh, S.M., Gray, W.G. High velocity flow in porous media. Transp Porous Med 2, 521–531 (1987). https://doi.org/10.1007/BF00192152
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DOI: https://doi.org/10.1007/BF00192152