deletions | additions       diff --git a/Gassmann's model.tex b/Gassmann's model.tex  index ef14aac..3803b2f 100644  --- a/Gassmann's model.tex  +++ b/Gassmann's model.tex 
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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 relation is largely used to predict the bulk modulus of a fully saturated rock ($K_{sat}$), from the bulk modulus of the dry rock ($K_{dry}$), the fluid ($K_f$), the mineral assemblage ($K_s$), and the rock porosity ($\phi$) \cite{Gassmann}:  \begin{equation}   \label{eq:eq1} \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 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, the effective mineralbulk modulus can be assumed as monomineralic.\\ 
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\begin{equation}   \xi(K_{\pm},G_{\pm}) = \frac{G_{\pm}}{6} \bigg(\frac{9K_{\pm}+8g_{\pm}}{K_{\pm}+2G_{\pm}}\bigg)   \end{equation}  the subscript $\pm$ denote the maximum and the minimum of the grain constituents and $K_i$ and $G_i$ are bulk and shear moduli of the $i^{th}$ grain constituent. constituent obtained from \cite{Mavko2009}.  The brackets $\langle \cdot \rangle$ indicate an average over the grain constituents weighted by their volume fractions. The pressure and temperature dependent bulk modulus ($K_f$) and density ($\rho_f$) of the CO$_2$ were determined from the thermodynamic properties obtained from NIST’s online chemistry webBook \citep{Lemmon2014}.  Porosity ($\phi$) and grain density ($\rho_s$) of the samples are determined by means of Hg porosimetry.\\  The P- and S- wave velocity are then calculated using:  \begin{equation}   V_p = sqrt\bigg(\frac{K_{sat}+\frac{4}{3}G_{sat}}{\phi}\bigg)  \end{equation}  Since shear modulus of the saturated rock ($G_{sat}$) is also the shear modulus of the frame ($G_{dry}$), the saturated bulk modulus obtained with equation \ref{eq:Kdry} and the shear