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\title{SIMULATION OF CARBON DIOXIDE STORAGE APPLYING ACCURATE PETROPHYSICS, 
FLUID-FLOW AND SEISMICS MODELS}

\author[a]{Gabriela B. Savioli}
\author[b]{Juan E. Santos}
\author[c]{Jose  M. Carcione}
\author[c]{Davide Gei}

%\author[a]{Fourth D. Author}
%
\affil[a]{Laboratorio de Ingenier\'\i  a de Reservorios, Instituto del Gas y del Petr\'oleo and Dto. de 
Ing. Qu\'\i  mica, Facultad de Ingenier\'\i  a, Universidad de Buenos Aires\\
Av. Las Heras 2214 Piso 3  C1127AAR Buenos Aires, Argentina\\
gsavioli@di.fcen.uba.ar}
%
\affil[b]{CONICET, Instituto del Gas y del Petr\'oleo, 
Facultad de Ingenier\'\i  a, Universidad de Buenos Aires and\\
Universidad de La Plata and\\  
Department of Mathematics, 
Purdue University, 150 N. University  Street, 
West Lafayette, Indiana, 47907-2067, USA, 
santos@math.purdue.edu} 
%\url{http://www.math.purdue.edu/~santos}


\affil[c]{Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), 
Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy jcarcione@inogs.it}
%\url{http://www.math.purdue.edu/~santos}



%% NOTE: IF ALL AUTHORS BELONG TO THE SAME AFFILIATION
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%% \author[\voidaffil]{Second B. Author}
%% \author[\voidaffil]{Third C. Author}
%% \author[\voidaffil]{Fourth D. Author}
%% %
%% \affil[\voidaffil]{Grupo de Mec\'anica Computacional,
%% Universidad Nacional de Villa Carolina,
%% Los Alerces 3492, 4200 Villa Carolina, Argentina,
%% gmc@uncarolina.edu.ar, http://www.uncarolina.edu.ar/gmc}

\begin{document}
\vspace{3cm}

\maketitle

\begin{keywords}
CO$_2$  Sequestration, Finite Elements, 
Wave Propagation
\end{keywords}

\begin{abstract}
Capture and storage of Carbon dioxide in aquifers and reservoirs is one of the solutions 
to mitigate the greenhouse effect.
Geophysical methods can be used to monitor the location and migration of the gas
in the underground. To perform this task properly, a suitable
geological model is important, which simulates the geometry and petro-elastical properties
of the different formations. 
In this work we integrate numerical simulators of CO$_2$-brine flow and seismic wave
propagation to model and monitor CO$_2$ storage in saline aquifers. We also build a 
petrophysical model of a shaly sandstone based on porosity and clay content 
and considering the variation of properties with pore pressure and fluid saturation.
 The pressure map before the injection of CO$_2$ is assumed to be hydrostatic for which a reference porosity map 
is defined. The permeability is assumed to be anisotropic and is obtained from first principles as a function
 of porosity and grain sizes. The density is the usual arithmetic average of the sandy and shaly components.
The numerical simulator of the CO$_2$-brine flow is based on the Black-Oil formulation for two phase flow in
 porous media, which uses the Pressure-Volume-Temperature (PVT) behavior as a simplified thermodynamic model. 
The propagation of waves in porous media is described using a viscoelastic
model that takes into account the dispersion and attenuation effects due to the presence of
heterogeneities in the fluid and solid phase properties. We introduce the P-wave attenuation 
following White's model of porous layers alternately saturated with brine and  CO$_2$. 
 S-wave attenuation is considered with a mechanism related to the P-wave one.
Numerical examples of CO$_2$ injection and time-lapse seismics in the Utsira formation at the Sleipner field are analyzed. 
The Utsira formation is represented using the new petrophysical model that allows a realistic inclusion of shale seals 
and fractures. The  results of the simulations show the capability of the proposed methodology  to monitor 
the migration and dispersal of the CO$_2$ plume and to make long term predictions.
\end{abstract}

\section{INTRODUCTION}
Geologic storage of CO$_2$ consists in injecting the gas into a geologic formation 
at depths typically greater than 1000 m, where it is present at supercritical conditions  \citep{Arts} .
Saline aquifers are suitable as storage sites due to their large volume and their common occurrence in nature.

In the Sleipner gas field (North Sea), the first industrial scale CO$_2$  injection project is being developed 
\citep{Chadwick}.
CO$_2$ separated from natural gas produced at Sleipner is currently being injected 
into the Utsira Sand, a saline aquifer
some 26000 km$^2$ in area. 

 Numerical modeling of CO$_2$ injection and seismic monitoring are important tools to understand the behavior
 of CO$_2$ after its injection and to make long term predictions. To perform this task properly,
a reliable geological model is important, which simulates the geometry and petroelastical
properties of the different formations.  This model assumes a
shaly sand, which represents the Utsira Sand or shale formation, depending on the
clay content. The permeability is assumed to be anisotropic and is obtained from
first principles as a function of porosity and grain sizes.We generate the geological model 
from the initial porosity
(at hydrostatic pore pressure) and clay content.
The simultaneous flow of brine and CO$_2$ is modeled with the Black-Oil 
formulation for two-phase flow in porous media \citep{Aziz}.
After  injection, the CO$_2$ migrates upwards, accumulating below thin shale (mudstone)
layers. These layers allow the vertical CO$_2$ migration to the top 
because they are not fully sealing.
The fluid pressures and saturations computed by the fluid simulator are used to  obtain the properties of the CO$_2$  at in-situ
conditions and the petro-physical properties of the Utsira Sand. These are obtained
with White’s mesoscopic model \citep{Carcione, 2007}.

Wave propagation is based on a viscoelastic model that considers dispersion and attenuation effects. In regions 
with partial CO$_2$ saturation, following White's theory \citep{White}, we consider P-wave attenuation due to 
wave-induced fluid flow at mesoscopic scales using a model of porous layers alternately saturated with brine
 and CO$_2$. The results of the flow simulator allow us  
to determine  the phase velocities and attenuation coefficients of P and S waves  at each computational cell using White's model. 

The  numerical examples present simulations of CO$_2$ injection and time-lapse seismograms in 
the Utsira Sand aquifer at Sleipner field. We build a petrophysical model of the Utsira formation based on 
porosity and clay content and taking into account the variation of properties with pore pressure and CO$_2$ 
saturation. This model is able to simulate  embedded mudstone layers of very 
low permeability that accumulate CO$_2$ but also allow its migration. 
The proposed methodology is able to identify the spatio-temporal distribution of CO$_2$ after its injection. 
Attenuation and dispersion effects are clearly observed in the snapshots 
and the recorded traces. 
This methodology  allowed us   to identify the horizontal and vertical saturation distribution of CO$_2$ over long periods of time.
\vskip1cm




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{PETROPHYSICAL MODEL OF THE UTSIRA FORMATION}

The petrophysical model assumes a shaly sand and is generated from the initial porosity $\phi_0$
(at hydrostatic pore pressure $p_H$) and clay content $C$.

The pressure dependence of properties is based on the following relationship between porosity and
 pore pressure $p$, 

\begin{equation}\label{exact}
\frac{(1-\phi_c)}{K_s}(p-p_H)=\phi_0-\phi+\phi_c\ln \frac{\phi}{\phi_0},
\end{equation}
where $\phi_c$  is a critical porosity. 


The bulk modulus of the dry matrix is estimated from the pressure dependent porosity using the Krief equation  [9], 

\begin{equation}\label{Krief}
 K_m = K_s(1-\phi)^{A/(1-\phi)},
\end{equation}
where the bulk modulus of the solid grains  $K_s$ is the arithmetic average of the Hashin Shtrikman  
upper and lower bounds. The rock is formed with quartz (bulk modulus of 40 GPa) and clay (bulk modulus 
of 15 GPa).

We assume the grain to be a Poisson medium, then the shear modulus of the solid grains 
is $\mu_s = 3 K_s/5$ and we set

\begin{equation}\label{Poisson}
\mu_m =\mu_s(1-\phi)^{A/(1-\phi)}.
\end{equation}

The black-oil simulator needs the compressibility $C_{pp}$ defined by Zimmerman et al. [20],
which is given by

\begin{equation}\label{compres}
C_{pp}=\frac{1}{\phi}\bigg(\frac{1}{K_m}-\frac{1+\phi}{K_s}\bigg)
\end{equation}

As permeability is anisotropic, we consider horizontal permeability $\kappa_x$ and 
vertical permeability $\kappa_z$. Carcione et al. [7] derived a model depending on 
porosity and clay content, assuming that a shaly sandstone is composed of
 a sandy matrix and a shaly matrix with partial permeabilities

\begin{equation}\label{permeab}
\kappa_q=\frac{R^2_q \phi^3}{45(1-\phi)^2 (1-C)} \; \; \;\textrm{and}\; \;\; \kappa_c=\frac{R^2_c \phi^3}{45(1-\phi)^2C}
\end{equation}
where $R_q$ and $R_c$ denote the average radii of sand and clay particles, respectively.

Therefore, the average permeability of the shaly sandstone along the horizontal
direction is given by

\begin{equation}\label{average}
\frac{1}{\kappa_z}=\frac{1-C}{\kappa_q}+\frac{C}{\kappa_c}
\end{equation}

Following Carcione et al. [8] , we assume

\begin{equation}\label{anisotropy}
\frac{\kappa_x}{\kappa_z}=\frac{1-(1-0.3a)\sin \pi S_w}{a(1-0.5\sin \pi S_w)},
\end{equation}
where $a$ is a permeability-anisotropy parameter and $S_w$ is the brine saturation.

Note that for $S_w=0$ or $1$, $\kappa_z = a \kappa_x$. It is $\kappa_x > \kappa_z$ at full water saturation, due to pore cross sections which are larger in the $x$-direction.

As water saturation is reduced, and the larger pores drained  first, a saturation is
reached at which $\kappa_x = \kappa_z$. Then, as saturation is further reduced, $\kappa_x < \kappa_z$. At the
other end (full gas saturation), we have again $\kappa_x > \kappa_z$.


\section{THE BLACK-OIL FORMULATION OF TWO-PHASE FLOW IN POROUS MEDIA}

The simultaneous flow of brine and CO$_2$ in  porous media 
is described by the well-known Black-Oil 
formulation applied to two-phase, two component fluid flow [2]. In this way, CO$_2$ component may dissolve in the brine 
phase but the brine is not allowed to vaporize into the CO$_2$ phase.
The differential equations,obtained by combining the mass conservation equations with Darcy's empirical law, are

\begin{eqnarray}\label{bo1}
&&\nabla \cdot (\underline {\kappa} ( \dfrac{k_{rCO2}}{B_{CO2}\eta_{CO2}}(\nabla p_{CO2} - \rho_{CO2} g \nabla D) 
+ \dfrac{R_s k_{rb} }{B_b \eta_b } (\nabla p_b - \rho_b g \nabla D)))+ \dfrac{q_{CO2}}{\rho^{SC}_{CO2}} \\
&&\hskip5.8cm = \ds {\Big [\phi \left (\dfrac{S_{CO2}}{B_{CO2}} +\dfrac{R_s S_b}{B_b} \right )\Big ]}t,\nonumber
\end{eqnarray}


\begin{eqnarray}\label{bo2}
&&\nabla \cdot (\underline {\kappa} \dfrac{k_{rb}}{B_b \eta_b }(\nabla p_b - \rho_b g \nabla D)+\dfrac{q_b}{\rho^{SC}_{b}} 
=
\ds {\Big [\phi \dfrac{ S_b}{B_b }\Big ]}t.
\end{eqnarray}
Subscripts $\beta = b, CO2$ denote brine and CO$_2$ phases, respectively. 
The unknowns for the Black-Oil model are the fluid pressures
$p_{\beta}$ and saturations $S_{\beta}$.
Also $\rho_{\beta}$ is density, $q_{\beta}$ mass rate of injection per unit volume,
and the  funcions $k_{r \beta}$ and $\eta_{\beta}$ are the relative permeability and
viscosities, respectively. Finally $\underline {\kappa}$ is the absolute permeability 
tensor and $\phi$ is porosity. 

The Black-Oil formulation uses as a simplified thermodynamic model, the PVT data
 $R_s$ and $B_{\beta}$  defined as

\begin{itemize}

\item  
 $R_s=\dfrac{V_{dCO2}^{SC}} {V_b^{SC}}$, CO$_2$ solubility in brine; 
\item
$B_{CO2}=\dfrac{V_{CO2}^{res}} {V_{CO2}^{SC}}$, CO$_2$ formation volume factor;  
\item
 $B_b=\dfrac{(V_{dCO2}^{res}+V_b^{res})} {V_b^{SC}}$, brine formation volume factor; 
\end{itemize}
with $V_{CO2}^{res}$, $V_b^{res}$ and  $V_{dCO2}^{res}$ the volume of CO$_2$, brine and dissolved CO$_2$ in brine at 
reservoir conditions; and $V_{CO2}^{SC}$,  $V_b^{SC}$ and $V_{dCO2}^{SC}$ are the volume of CO$_2$, brine and dissolved CO$_2$ in brine at standard conditions, respectively. Also, $\rho^{SC}_{b}$ and $\rho^{SC}_{CO2}$ are the CO$_2$ and brine densities at standard conditions.  

$R_s$ and $B_b$ can be expressed in terms of the equilibrium properties obtained from an equation of state [14]: 


\begin{itemize}

\item  
$R_s=\dfrac{\tilde \rho_{b}^{SC} \chi_{CO2}} {\tilde \rho_{CO2}^{SC} (1-\chi_{CO2})}$

\item  
$B_b=\dfrac{ \rho_{b}^{SC}} {\rho_{b} (1-\omega_{CO2})}$
\end{itemize}
The conversion from compositional data from equations of state into the Black-Oil PVT data
is performed applying  an algorithm 
developed by Hassanzadeh et al  [\citep{Hassan}] which uses the brine and CO$_2$ molar density at standard conditions; and 
the  CO$_2$ mole and mass fraction in the brine phase. 

Two algebraic equations relating the saturations and pressures complete the system: 
\begin{eqnarray}\label{bo}
&S_b+S_{CO2}=1,\qquad 
&p_{CO2}-p_b=P_C(S_b),
\end{eqnarray}
where $P_C$ is the capillary pressure. 


\vskip0.5cm

\vskip0.5cm
 
The numerical solution  was obtained  employing the public 
domain software BOAST \citep{Fanchi}. BOAST solves the differential equations 
using IMPES (IMplicit Pressure Explicit Saturation), 
a semi-implicit finite difference technique  \citep{Aziz}. 














%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A VISCOELASTIC MODEL FOR WAVE PROPAGATION}
The propagation of waves in  fluid-saturated porous media is 
described using an equivalent viscoelastic model that takes  into account 
the dispersion and attenuation effects due to the presence of 
heterogeneities in the fluid and solid phase properties.
    
The equation of motion in a  2D  isotropic viscoelastic  domain  $\O$ 
with boundary $\p\O$ can be stated in the space-frequency domain as  
\begin{eqnarray}\label{mod3}
&&\hskip-.5cm-\o^2 \rho u  -  \nabla\cdot\sigma(u) = f(x,\o),\quad \O\\
&&-\sigma(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\label{mod4}
\end{eqnarray}
where $u =(u_x, u_y)$ is the displacement vector.
Here  $\rho$ is the  bulk density and  
\eqref{mod4} is a first-order 
absorbing boundary condition 
 using the positive definite matrix ${\mathcal D}$, which definition 
 is given in  \citep{hass02}. 

The stress tensor $\sigma(u)$  
 is  defined in the space-frequency domain by 
\begin{eqnarray}\label{mod5}
&&\sigma_{jk}(u) = \lambda_G(\o) \nabla\cdot u \delta_{jk}  
+ 2 N(\o) \varepsilon_{jk}(u) ,\quad \O,
\end{eqnarray}   
where $\varepsilon_{jk}(u)$ denotes the strain tensor and $\delta_{jk}$ is the
Kroenecker delta.  

The coefficient $N$  in \eqref{mod5} 
is the shear modulus of the dry matrix, while the Lam\'e coefficient is $\lambda_G = K_G - \frac23 \mu $ in 3D and 
$\lambda_G = K_G - \mu$  in 2D. $K_G$ is  the Gassmann's undrained   
bulk modulus, computed as follows:
\begin{eqnarray}\label{mod.3}
&& K_G = K_m + \alpha^2 K_{av}\nn\\
&&\alpha = 1 - \dfrac{K_m}{K_s}, 
\nn\\
&& K_{av} = \left[\dfrac{\alpha- \phi}{K_s} 
+ \dfrac{\phi}{K_f}\right]^{-1}.\nonumber\nn
\end{eqnarray} 
where 
\begin{itemize}

\item
$  K_m$: bulk modulus of the dry matrix
\item
$K_s$: bulk modulus of the solid grains
\item
$  K_f$: bulk modulus of the saturant fluid
\end{itemize}

Following \citep{White}, 
we consider P-wave attenuation due to wave induced fluid flow at mesoscopic scale 
using a model of porous layers 
alternately saturated with brine and CO$_2$ respectively. This approach yields  a complex and frequency dependent P-wave
 modulus for the formation.

S-wave attenuation is also taken into account making the shear modulus complex an 
freuqency dependent  
using another relaxation mechanism 
related to the P-wave mechanism. 

The approximate solution of \eqref{mod3} with the boundary conditions 
\eqref{mod4} is obtained using an iterative  finite element domain decomposition procedure.  
To approximate each component of the solid displacement vector we employed the
nonconforming finite element space defined in \citep{dssy_99}, since it 
generates less numerical dispersion than the standard bilinear elements
\citep{zgs03}.
 It can be  demonstrated that the  error associated with  this numerical  
procedure measured in the energy norm is of order $h$ \citep{hass02}, where $h$ is the size of the computational mesh.  



\section{NUMERICAL EXPERIMENTS}

\section{Model of the Utsira Formation and CO$_2$ Injection.}
We consider a 2D model of the Sleipner field 
constructed using  
an initial porosity at hydrostatic pressure and the clay content of the formation.
%Both porosity and clay content were modeled  as stochastic fractals.   
The model  has $400$ m thickness (top at $700$ m and
bottom $1100$ m b.s.l.).  
Within the formation,
there are several  mudstone layers which act as barriers to the vertical 
motion of the CO$_2$ and can be observed in Figure \ref{label1} .
The viscosity, density and bulk modulus of CO$_2$  were 
obtained from the Peng-Robinson equations as a function of temperature 
 and pore pressure \cite{Hassanzadeh08}.

%$p_H = \rho_w gz$ is the 
%hydrostatic pressure, with $\rho_w = 1040\,kg/m^3$ 
%the density of brine and $g$ the gravity constant . In the Utsira sandstone,
% $\rho= (1-\phi )\rho_s +\phi \rho_f$, where $\phi(z)$ is the porosity, 
% $\rho_s$ is the grain density and $\rho_f$ is
%the fluid density. The Utsira grain density is $\rho_s = 2550\,kg/m^3$ (both quartz and clay), while $\rho_f = (1 - S_w)\rho_g + S_w  \rho_w$, where $S_w$ is the water saturation and $\rho_g$ is
%the CO$_2$ density, which is obtained from the Peng-Robinson equations as a function of $T$ and $p$, 
%where $p$ is the pore pressure [9].
CO$_2$ is injected at a constant flow rate of one millon tons per year. The injection point is located at the bottom 
of the Utsira formation: $x=600$ m, $z=1060$ m. 
Figure \ref{label2} shows the saturation field after 2 years (left) and 4 years (right) of CO$_2$ injection,
 computed using the BOAST flow simulator. 


\begin{figure}[h]
\begin{center}
%\vspace*{-.4cm}
\includegraphics[width=11cm,height=8cm]{por_sin_tope_ni_sello.eps}\\ 

%\vspace*{0.9cm}
%\begin{minipage}{2cm}
\caption{\label{label1}  Porosity map }
%{ {\bf Figure 1.} Idealized model of the Utsira formation. The injection point is located at x= 400 m, z= 1060 m}
%\end{minipage}
\end{center}
\end{figure}


\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5 cm,height=5 cm]{sat_CO2_2a_600x400.eps}\quad
\includegraphics[width=7.5 cm,height=5 cm]{sat_CO2_4a_600x400.gnu.eps}\\
\caption{\label{label2}  CO$_2$ saturation distribution after 2 years (left) and 
4 years of injection. The injection point is located at x= 600 m, z= 1060 m }
%{ {\bf Figure 1.} Idealized model of the Utsira formation. The injection point is located at x= 400 m, z= 1060 m}
%\end{minipage}
\end{center}
\end{figure}


Figure \ref{label3} displays the distribution of P-wave phase velocity after 2 years of CO$_2$ injection.

\begin{figure} [!h]
\begin{center}
%\vspace*{-.5cm}
\includegraphics[width=7.5cm,height=5cm]{vp_map_gnu.eps}\quad 
\includegraphics[width=7.5cm,height=5cm]{inv_qp_map.eps}\\
\caption {\label{label3} P-wave phase velocity (left) and attenuation coefficient (right)  after 2 years of CO$_2$ injection}
\end{center}
\end{figure}



\section{Time-Lapse Seismic Monitoring}


To analyze the capability of 
seismic monitoring to identify zones of CO$_2$ 
accumulation, the media is excited with a compressional  point source 
located at $x= 600$ m, $z= 710$ m before and after 2 and 4 years of CO$_2$ injection. 
Time histories  measured near the surface  are shown in Figures \ref{label6}  
before  CO$_2$ injection, in Figure \ref{label7} and after 2 years and in Figure \ref{label8} after 4 years of 
CO$_2$ injection. 
The  upper reflection in the three figures is due to the direct wave coming from the point source. The lower 
reflection in Figures \ref{label7} and \ref{label8}
is due  to the  CO$_2$ accumulations below the deepest  mudstone
 layer. Figure \ref{label8} shows another reflection due to the upper accumulation, which appears in the right side.
Finally, Figure \ref{label9} displays the traces measured at x= 750 m, 
z = 710 m shown in Figure \ref{label7} .
 
\begin{figure}[h]
\begin{center}
%\vspace*{-.4cm}
\includegraphics[width=7.5 cm,height=5 cm]{gather_before.eps}\quad
\includegraphics[width=7.5 cm,height=5 cm]{gather_after.eps}\\
%\vspace*{0.5cm}
%\vspace*{0.5cm}
%\begin{minipage}{13cm}
\caption{\label{label6} Traces of the  z-component of the displacement before (left) and after 
2 years (right) CO$_2$ injection. The  upper reflection is due to the 
direct wave coming from the point source. The  second reflection is due to the 
deepest  CO$_2$ accumulation }
%{ {\bf Figure 1.} Idealized model of the Utsira formation. The injection point is located at x= 400 m, z= 1060 m}
%\end{minipage}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
%\vspace*{-.4cm}
\includegraphics[width=7 cm,height=5 cm]{gather_after.eps}\quad
\includegraphics[width=5 cm,height=7 cm,angle=-90]{Gather_uz_4a_point_600x200.ps}\\
%\vspace*{0.5cm}
%\begin{minipage}{13cm}
\caption{\label{label7} Traces of the  z-component of the displacement after 
2 years (left) and 4 years (right) of CO$_2$ injection. 
The  lower reflection is due to the 
deepest  CO$_2$ accumulation. The intermediate reflection in the right is due to the 
upper  CO$_2$ accumulation}
%{ {\bf Figure 1.} Idealized model of the Utsira formation. The injection point is located at x= 400 m, z= 1060 m}
%\end{minipage}
\end{center}
\end{figure}


\begin{figure}[h]
\begin{center}
\vspace*{.4cm}
\includegraphics[width=8 cm,height=5 cm]{trace_50_point_2a_600x200_and_solo_brine.eps}\\
%\begin{minipage}{13cm}
\caption{\label{label9} Traces of particle velocity of the solid phase  before  and after 2 years of CO$_2$ injection. }
%{{\bf Figure 3.} Traces of particle velocity of the solid phase  before  and after 5 years of CO$_2$ injection }
%\end{minipage}
\end{center}
\end{figure}

\section{Conclusions}

In this work we  integrate numerical simulators of CO$_2$-brine flow and  
seismic wave propagation to model and monitor CO$_2$ storage in saline aquifers. 
We also build a petrophysical model of a 
shaly sandstone based on porosity and clay content and considering the variation of properties with 
pore pressure and fluid saturation.
Numerical examples show the effectiveness of this metodology to detect the spatio-temporal distribution of CO$_2$.
 Therefore, it constitutes an important tool to monitor the migration and dispersal of the CO$_2$ plume and to 
make long term predictions.


\section{AKNOWLEDGEMENTS} 
This work was partially funded by CONICET, Argentina (PIP 0952)    

%\nonumsection{References}

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