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The microscopic analysis of high forchheimer number flow in porous media

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Abstract

High Forchheimer number flow through a rigid porous medium is numerically analysed by means of the volumetric averaging concept. The microscopic flow mechanisms, which must be known in order to understand the macroscopic flow phenomena, are studied by utilising a periodic diverging-converging representative unit cell (RUC). The detailed information for the microscopic flow field, in association with the locally averaged momentum balance, makes it possible to quantitatively demonstrate that the microscopic inertial phenomenon, which leads to distorted velocity and pressure fields, is the fundamental reason for the onset of nonlinear (non-Darcy) effects as velocity increases. The hydrodynamic definitions for Darcy's law permeabilityk, the inertial coefficientΒ and Forchheimer number Fo are obtained by applying the averaging theorem to the pore level Navier-Stokes equations. Finally, these macroscopic parameters are numerically calculated at various combinations of micro-geometry and flow rate, and graphically correlated with the relevant microscopic parameters.

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Abbreviations

a i :

body force acceleration (m/s2)

A :

viscous integral term defined in (4.6)

A f :

area of entrance and exist of RUC (m2)

A fs :

interfacial area between the fluid and solid phases (m2)

B :

pressure integral term defined in (4.4)

d :

throat diameter of RUC (m)

D :

pore diameter of RUC (m)

Fo:

Forchheimer number defined in (4.1) and (4.10)

g :

gravitational acceleration (m/s2)

i, j :

microscopic unit vector for RUC

k :

Darcy's law permeability (m2)

k v :

velocity dependent permeability defined in (4.1) (m2)

L :

length of a unit cell (m)

L p :

pore length of RUC (m)

L t :

throat length of RUC (m)

n :

unit outwardly directed vector for the fluid phase

p :

microscopic fluid pressure (N/m2)

P :

macroscopic fluid pressure (N/m2)

\(\bar P\) en :

mean pressure at entrance of RUC (N/m2)

\(\bar P\) ex :

mean pressure at exit of RUC (N/m2)

r i,r :

coordinate on the macroscopic scale (m)

Re d :

Reynolds number defined in (4.5)

u i,u :

microscopic velocity (m/s)

\(\bar U\) :

specific discharge (m/s)

\(\bar U\) d :

mean velocity at the throat of RUC (m/s)

v :

microscopic velocity (m/s)

V b :

representative elementary volume (REV) (m3)

V f :

volume occupied by the fluid within REV (m3)

V s :

volume occupied by the solid within REV (m3)

x i,x :

coordinate on the microscopic scale (m)

X i,X :

coordinate on the macroscopic scale (m)

Β :

the inertia coefficient (1/m)

Μ :

viscosity coefficient (Ns/m2)

Ν i :

microscopic unit vector

ξ :

areosity at the entrance and the exit cross-section of RUC

ρ :

fluid density (kg/m3)

Φ :

porosity

ψ f :

a general property of the fluid phase

〈 〉f :

intrinsic phase average

\(\tilde \psi \) :

the fluctuating part ofψ f

\(\bar \psi \) :

the mean value ofψ f

ψ *f :

the dimensionless value ofψ f

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Manitoba, R3T2N2, Winnipeg, Canada

    H. Ma & D. W. Ruth

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  1. H. Ma
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  2. D. W. Ruth
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Ma, H., Ruth, D.W. The microscopic analysis of high forchheimer number flow in porous media. Transp Porous Med 13, 139–160 (1993). https://doi.org/10.1007/BF00654407

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  • DOI: https://doi.org/10.1007/BF00654407

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