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\subsubsection{CO$_2$ \subsection{CO$_2$  injection modelisation} Seismic monitoring of CO$_2$ injection requires careful computational models to predict the system behaviour. The classic approcach employs fully three-dimensional numerical methods to solve the system with a high degree of accuracy. However, this involves high computational efforts that are not always feasable. In recents years, approach that employ semi-analytical methods have been increasingly developed \cite{Nordbotten2005a, Nordbotten2009}. One promising simulation tool for fast and accurate modeling of CO$_2$ sequestration is based on the vertical equilibrium (VE) assumption. VE models have a long tradition for describing flows in porous media; in hydrology it is known as the Dupuit approximation, whereas in the oil industry is used to simulate two-phase and three phase segregated flow \cite{Martin1958,Coats1967,Martin1968}. In recent years, VE methods have been employed to simulate large scale CO$_2$ injection and migration, for which a sharp interface assumption with vertical equilibrium may be reasonable due to the large density difference between supercritical CO$_2$ and brine \cite{Nordbotten2005a,Celia2006, Nordbotten2006}. \\  For the purpose of this study, we used the VE solvers included in the Matlab Reservoir Simulation Toolbox (MRST) \cite{Lie_2011}, to model the CO$_2$ injection. Two different scenario are proposed: a Bécancour-like scenario (Fig \ref{fig:modelCO2}-A), where the stochastic geological model computeed in the previous section is used as input and an optimistic scenario (Fig \ref{fig:modelCO2}-B), where porosity and permeability reflect those of the Ketzin pilot project \cite{Michael_2010}. The model simulate the CO$_2$ injection in the Potsdam's formations during 15 years at an injection rate of 45 tonnes per day that is comparable to the average rate injection at Ketzin \cite{Martens_2012} as well as a migration time of 35 years. The total storage of CO$_2$ is 245 kt. The properties of the model are summarized on Table 2. The CO$_2$ plume for the Bécancour-like scenario is limited to a few hundreds of meters around the well due to its low permeabilities and porosity. The results are consistent with those obtained by \citet{TranNgoc2013} using TOUGH2. For the optimistic scenario, the plume extends for more than a kilometer. During the migration time the CO$_2$ is partially dissolved, however this is not the case in the Bécancour-like scenario.