# Link between different ways of modeling permeability

The most famous equation describing the permeability $k$ of a soil is the Kozeny-Carman equation: $$k = \frac{\phi}{8}r^2 \approx \frac{\phi}{c}\left(\frac{V}{S}\right)^2$$ $\phi$: porosity
V: pore volume
S: specific surface area
$c$: constant

Another equation that describe the permeability $k$ is the Schlumberger-Doll Research (SDR) equation: $$k = c_\mathrm{SDR}\phi^a T_2^b$$ $\phi$: porosity
T$_2$: Caracteristic decay time
$a$, $b$, $c_\mathrm{SDR}$: constants

The latter equation has been derived from the former and I tried do redo the derivation:

Knowing that: $$T_2^{-1} \approx \rho_2 \frac{S}{V}$$ $$\Leftrightarrow T_2 = \frac{1}{\rho_s}\frac{V}{S}$$ $\rho_s$: surface relaxivity

Inserting in the SDR equation: $$k = c_\mathrm{SDR}\phi^a\left(\frac{1}{\rho_s}\frac{V}{S}\right)^b = c_\mathrm{SDR}\phi^a\frac{1}{\rho_s^b}\left(\frac{V}{S}\right)^b$$

Comparing this equation to the right side of the Kozeny-Carman equation I obtain:

$$\phi = \phi^a \Leftrightarrow a = 1$$ $$\left(\frac{V}{S}\right)^2 = \left(\frac{V}{S}\right)^b \Leftrightarrow b=2$$ $$\frac{1}{c} = \frac{c_\mathrm{SDR}}{\rho_s^b} \Leftrightarrow c = \frac{\rho_s^2}{c_\mathrm{SDR}}$$

So I get a=1 and b=2 what is not in accordance with the exponent of 4 mentioned in this site petrowiki.org:

Both models treat permeability as an exponential function of porosity, $\phi^4$.

Can somebody explain me how they get an exponent 4 and not 1 for $\phi$ ?

• I don't think you can do what you are trying to do. The equations are trying to model permeability in different ways, they aren't rearrangements of one another. – kwinkunks May 6 '16 at 16:42
• I suppose it is possible as I read in a hydrogeophysics course that the SDR equation is directly based on the Kozeny-Carman equation. – Zoran May 9 '16 at 18:01