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Lorenzo Perozzi added subsubsection_Elastic_properties_ELastic_properties__.tex
about 9 years ago
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\subsubsection{Elastic properties}
ELastic properties are usually computed through a rock-physics model. This model is a set of equation that transform petrophysical variables, typically porosity, mineralogy and fluid saturations into elastic properties such as $P$- and $S$-wave velocity and density. The new CO$_2$ saturated rock properties $K_{sat}$ and $G_{sat}$ are calculated using the Gassmann's relation \cite{Gassmann} and the saturated density $\rho_{sat}$ is computed as a linear combination of the solid density $\rho_s$ and fluid density $\rho_{fl}$ weighted by their respective volume fractions
\begin{equation}
\rho_{sat} = \phi\ \rho_{fl} + (1 - \phi)\rho_s
\end{equation}
and $P$- and $S$-wave velocity are calculated as function of saturated elastic properties $K_{sat}$ and $G_{sat}$ and density $\rho$:
\begin{equation}
\label{eq:Vp}
V_p = \sqrt{\bigg(\frac{K_{sat}+\frac{4}{3}G_{sat}}{\rho_{sat}}\bigg)},
\end{equation}
and
\begin{equation}
\label{eq:Vs}
V_s = \sqrt{\frac{G_{sat}}{\rho_{sat}}}.
\end{equation}
Then, the synthetic seismic response $d_{synth}$ is computed using a viscoelastic finite-difference time-domain approach \cite{Bohlen_2002}.
The mismatch between $d_{synth}$ and $d_{obs}$ is evaluated using the same objective function of the previous step. \\
At the end of the history-matching process we obtain the field of $V_p$, $V_s$, density ($\rho$) and porosity ($\phi$) that best honor static and dynamic data.
