Permeability Derivation based on Capillary Tube Model

Hagen-Poisseuille Law
Hagen-Poisseuille Law gives,
q = (nπr ΔP)/(8μL),
where,
q = flow rate, m /sec,
n = number of tubes, dimensionless,
r = radius of capillary tube, m,
μ = viscosity of fluid, Pa*s,
ΔP = pressure drop along the rock sample, Pa,
L = length of sample, m.
In capillary Tube Model of Permeability, every tube is assumed straight, so tube length, l, may substitute sample length, L.
Thus, we have,
q = (nπr ΔP)/(8μl). Eq.1
Combination with Kozeny Equation
n = φAL/(πr l), Eq.2
where,
φ = porosity, dimensionless
Combine Eq. 1 & Eq. 2,
q = φΑLr ΔP/(8l μ), Eq.3
Combination with Darcy's Law
From ,
q = kAΔP/(μL), Eq. 4
where,
q = flow rate, m /sec,
k = permeability of the rock sample, m
A = cross-section area of the sample, m ,
μ = viscosity of fluid, Pa*s,
ΔP = pressure drop along the rock sample, Pa,
L = length of sample, m.
Combine Eq. 3 and Eq. 4,
k φr L /(8l ) (φr )/(8τ ),
where,
τ tortuosity l/L, dimensionless.

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