While fluid percolates through porous media in infinitesimal volume $\begin{array}{l}\delta V\end{array}$ generates the noise of the power $\begin{array}{l}\delta N\end{array}$ in a wide frequency range:

(1)	$\begin{array}{l}\displaystyle \delta N = A(f) \cdot {\bf u} \cdot \nabla p \cdot \delta V\end{array}$

where

$\begin{array}{l}{\bf u}\end{array}$	flow velocity
$\begin{array}{l}p\end{array}$	fluid pressure
$\begin{array}{l}A(f)\end{array}$	normalised noise spectrum, $\begin{array}{l}\displaystyle \int_0^\infty A(f) \, df = 1\end{array}$
$\begin{array}{l}f\end{array}$	noise frequency

While propagating through the rocks the different frequencies will decay at different rate $\begin{array}{l}\alpha(f)\end{array}$ and if noise sensor is located at $\begin{array}{l}{\bf r}_0 = \{0, \, 0, \, 0\}\end{array}$ then the it will capture:

(2)	$\begin{array}{l}\displaystyle \delta N_S = \int_V A(f) \cdot {\bf u} \cdot \nabla p \cdot \exp[-\alpha(f)r] \cdot \delta V\end{array}$

The decay decrement $\begin{array}{l}\alpha(f)\end{array}$ is growing with frequency: $\begin{array}{l}\displaystyle \frac{d \alpha}{df} > 0\end{array}$ .

There is no universal model but it can be approximated by a linear dependance:

(3)	$\begin{array}{l}\displaystyle \alpha = \alpha_1 \cdot f\end{array}$

with $\begin{array}{l}\alpha_1(f)\end{array}$ having much slower dependance on frequency than $\begin{array}{l}\alpha(f)\end{array}$ .

Reference

McKinley R.M., Bower F.M., Rumble R.C. 1973. The Structure and Interpretation of Noise From Flow Behind Cemented Casing, Journal of Petroleum Technology, 3999-PA

McKinley, R.M. 1994. Temperature, Radioactive Tracer, and Noise Logging for Well Integrity: 112-156

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