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 full-wave seismic profile (VSP) for near, mid and large offset (i.e.\ VSP$_{observed}$) is modeled using the code described in \citet{Giroux2012}. As the ultimate aim of the inversion workflow is to obtain an elastic model to predict the CO_2 distribution over time, modeling near, mid and far offset allow us to take into account every amplitude variation versus offset due to the fluid substitutions.

Starting from three hypothetical well-log of the reference model, we co-simulate five sets of 100 realizations, each using the SGS algorithm. These realizations are then linearly combined using the gradual deformation parametrization. At each parametrization step, a full-wave VSP for near, mid and large offset (i.e, VSP$_{synth}$) is computed using an elastic finite-difference time-domain approach \citep{Bohlen_2002} and compared to VSP$_{observed}$. At the end of this iterative process, we obtain the best seismic matched realization for $V_p$, $V_s$, $\rho$ and $\phi$.

The five best seismic matched realizations are then combined again in a gradual deformation parametrization where at each step we run a CO_2 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 = \dfrac{1}{72}\dfrac{\phi^3}{(1-\phi)^2\tau^2}d^2,
\end{equation}
where $k$ is the permeability, $\phi$ the porosity and $\tau$ the tortuosity.
The CO_2 is injected during 200 days in the Covey Hill formation that is a low porosity sandstone with a mean spheres diameter of 5.5x10^{-5}m. 
At each step, the new CO_2 saturated elastic properties of the model are calculated using equations (\ref{eq:rho}),(\ref{eq:Vp}) and (\ref{eq:Vs}), the VSP$_{synth_{200d}}$) is computed and the mismatch against VSP$_{observed_{200d}}$ is evaluated.

The final inversion workflow results for $V_p$, $V_s$, $\rho$ and $\phi$ that best honor the static data (i.e.\ well-log) and the CO_2 flow within the reservoir, are shown in Figure \ref{fig:final-model}. These inversion results are then used to predict CO_2 distributions over time that are validated by the monitoring data.
    
  