<p>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<br>
V: pore volume<br>
S: specific surface area<br>
$c$: constant</p>

<p>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<br>
T$_2$: Caracteristic decay time<br>
$a$, $b$, $c_\mathrm{SDR}$: constants</p>

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

<p>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</p>

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

<p>Comparing this equation to the right side of the Kozeny-Carman equation I obtain:</p>

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

<p>So I get a=1 and b=2 what is not in accordance with the exponent of 4 mentioned in this site <a href="http://petrowiki.org/Permeability_estimation_with_NMR_logging" rel="nofollow">petrowiki.org</a>:</p>

<blockquote>
  <p>Both models treat permeability as an exponential function of porosity, $\phi^4$.</p>
</blockquote>

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

CC BY-SA 3.0
2016-05-06T16:42:33.980Z
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.
Matt Hall
CC BY-SA 3.0
2016-05-09T18:01:19.110Z
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
<rocks><fluid-dynamics><hydrogeology>
Link between different ways of modeling permeability
Zoran
# Link between different ways of modeling permeability

(Asked by: username_0 on 2016-05-06)

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](http://petrowiki.org/Permeability_estimation_with_NMR_logging):

> 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 $\phi$ ?

**Tags:** rocks, fluid-dynamics, hydrogeology

> **Comments:**
> **username_1**: 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.  
> **username_0**: I suppose it is possible as I read in a hydrogeophysics course that the SDR equation is directly based on the Kozeny-Carman equation.  

---

## 0 answers

