deletions | additions       diff --git a/The_procedure_followed_in_this__.tex b/The_procedure_followed_in_this__.tex  index bd47950..6946927 100644  --- a/The_procedure_followed_in_this__.tex  +++ b/The_procedure_followed_in_this__.tex 
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The procedure followed in this study to perform seismic modeling on the reference model is directly inspired by the work of \citet{Carcione2006} who present an application of poro-viscoelastic modeling for monitoring underground CO$_2$ storage. The poro-viscoelastic formulation represents perhaps the most effective tool to study the effect of the saturating fluid on the seismic response because fluid properties are directly taken into account in the equations. A vertical seismic profile (VSP) for near, mid and large offset (i.e VSP$_{observed}$) is modeled using the code described in \citet{Giroux2012}.\\  Starting from the reference model, we co-simulate \num{5} set of \num{100} realizations each using the SGS algorithm that are then linearly combined using the gradual deformation parametrization. At each parametrization step, a VSP for near, mid and large offset (i.e VSP$_{synth}$) is computed using a viscoelastic finite-difference time-domain approach \citep{Bohlen_2002} and compared to the VSP$_{observed}$. At the end of this iterative process we obtain the best seismic matched realization for $V_p$, $V_s$, $\rho$ and $\phi$. $\phi$.\\  The five best seismic matched realization are then combined again in a gradual deformation parametrization where at each step we run a \ce{CO2} flow simulator. The permeability is derived using an extension of the Kozeny-Carman equation \citep{Kozeny1927,Carman1938} for a packing of identical spheres of diameter $d$ \citep{Mavko2009}:  \begin{equation}  k = \frac{1}{72}\frac{\phi^3}{(1-\phi)^2\tau^2}d^2,  \end{equation}  where k is the permeability, $\phi$ the porosity and $\tau$ the tortuosity.  The \ce{CO2} is injected during \num{200} days in the Covey Hill formation that is a low porosity sandstone with a mean spheres diameter of \SI{5.5e-5}{\metre}.\\  The new \ce{CO2} saturated rock properties are calculated using equations (\ref{eq:rho}),(\ref{eq:Vp}) and (\ref{eq:Vs}). At each step, the seismic VSP is computed and the mismatch against the equivalent time lapse VSP$_{observed}$ is evaluated.\\  The inversion results for $V_p$, $V_s$, $\rho$ and $\phi$ that best honor the static data (i.e., well-log) and the \ce{CO2} propagation within the reservoir, are shown in Figure \ref{fig:final-model}.