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\subsubsection{CO$_2$ injection modelisation}  The classic approcach to modeling CO$_2$ injection 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, thera has been a renewed interset in VE methods as a means to simulate large-scale CO$_2$ 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 Becancour like scenario (Fig \ref{fig:modelCO2}-A), where the geological model compute 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 properties of the model are summarized on Table \ref{fig:tab2}. 2.  The CO$_2$ plume for the Becancour like scenario is limited to few hundreds of meters around the well due to its low permeabilities and porosity. For the optimistic scenario, the plume extends for more than a kilometer. The Becancour-like simulation is consistent with the results obtained by \cite{TranNgoc2013} using TOUGH2.