deletions | additions       diff --git a/Gassmann's model.tex b/Gassmann's model.tex  index 0b68c87..e3c1d14 100644  --- a/Gassmann's model.tex  +++ b/Gassmann's model.tex 
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\subsection{Gassmann's model} \subsection{Gassmann modeling}  Applying the laboratory results to seismic monitoring will require the high frequency data to be scaled down to low frequencies by adequate rock physics models.  Gassmann’s Gassmann's  relation is largely used to predict the bulk modulus of a fully saturated rock ($K_{sat}$). It is a function of the bulk modulus of the dry rock ($K_{dry}$), the modulus of the fluid ($K_f$), the modulus of the mineral assemblage ($K_s$), and the rock porosity ($\phi$) \cite{Gassmann}: \begin{equation}   \label{eq:Kdry}  K_{sat} = K_{dry} + \frac{1 - (K_{dry}/K_s)^2}{(\phi/K_f) + ((1 - \phi)/K_s)-(K_{dry}/K_s^2)}.  \end{equation}  Application of Gassmann's equation is based on the assumption that the pore space is completely connected and the porous frame consists of a single solid material. The bulk modulus of the minerals composing the studied sandstones samples are known and not highly variable. As such, the effective mineral bulk modulus can be assumed to be monomineralic.\\  The bulk modulus of the solid ($K_s$) is estimated by applying the aritmetic arithmetic  average of the upper and lower Hashin-Shtrikman bounds \citep{Hashin1963}. For a rock made up of different minerals, these bounds are formulated as \citep{Mavko2009}: \begin{equation}   \label{eq:Ks}  K^{HS\pm} = \Lambda(G_{\pm}), 
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\rho_{sat} = (1-\phi)\rho_s + \phi \rho_f.  \end{equation}  Porosity ($\phi$) and grain density ($\rho_s$) of the samples are determined by means of Hg porosimetry.\\  Figure \ref{fig:fig4} \ref{fig:results_lab}  shows the modeled Gassmann's velocities for $P$- and $S$- wave as dashed lines. There is a general agreement between modeled and measured velocities, though Gassmann's model overpredicts over-predicts  the measured velocities when the pore fluid is gaseous CO$_2$, except for the CS sample at 35 $^{\circ}$C and underpredicts under-predicts  the observed velocities when pore fluid is liquid or supercritical CO$_2$.\\ The model assumes that the pores are in connection. The rock samples used for the ultrasonic measurements, have a very small pore size (see Fig \ref{fig:fig5}), \ref{fig:poresize}),  that would prevent the formation of a connective network. Gassmann's model also assumes that the pressure at the level of the pores is in equilibrium. It is possible that during the measurements, the samples did not have sufficient time to stabilize during the changing of pore and confining pressure. These factors may have limited the accuracy of the Gassmann’s Gassmann's  modeling.