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High velocity flow in porous media

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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

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Authors and Affiliations

  1. National Institute of Public Health and Environmental Hygiene (RIVM), P.O. Box 1, 3720 BA, Bilthoven, The Netherlands

    S. Majid Hassanizadeh

  2. Department of Civil Engineering, University of Notre Dame, 46556, Notre Dame, Indiana, USA

    William G. Gray

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  1. S. Majid Hassanizadeh
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  2. William G. Gray
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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

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