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\title{Modeling mesoscopic attenuation in a highly 
heterogeneous Biot's medium employing an equivalent viscoelastic model}
%in heterogeneous porous media}
%\subtitle{in Fluid Saturated  Poroviscoelastic Media}
\author{Juan E. Santos,$^\dag$, J. G. Rubino and C. L Ravazzoli }
\institution{%
$^\dag$ CONICET, Departamento de Geof\'\i sica Aplicada,\\
Facultad de Ciencias Astron\'omicas y Geof\'\i sicas,
 Universidad Nacional de La Plata\\
and Department of Mathematics, Purdue University. \\ $ $ \\
{\magenta  SEG Annual Meeting, Las Vegas, 9-14 November 2008}}
%\subtitle{{\magenta 2007 SEG Annual Meeting, San Antonio, Texas, September 
%23-28, 2007} }

\begin{document}
\maketitle




%----------------------------------------------- SLIDE -
\begin{slide}{\small Mesoscopic attenuation and dispersion effects. I}
{\small
\begin{itemize}
\item One important mechanisms of P-wave attenuation and dispersion 
at seismic frequencies is known as {\magenta {}mesoscopic loss}, and is caused by
the presence of heterogeneities larger than the pore size but much 
smaller than the predominant wavelengths 
{\magenta { }(mesoscopic-scale heterogeneities)}. 


\item
These effects are caused by 
equilibration of wave-induced fluid pressure gradients via a
{\magenta { } slow-wave diffusion} process.


\item  It is also the  case that the exact spatial distribution of these heterogeneities is  in general unknown. Here they will be assumed to be {\magenta { }stochastic 
functions} characterized by their statistical properties.

\end{itemize} 
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small Mesoscopic attenuation and dispersion effects. II}
{\small
\begin{itemize}
\vskip-.3cm
\item  Due to the {\magenta { }extremely fine meshes} 
needed to properly represent these type of media, numerical simulations 
 using Biot's equations of motion at the macroscale 
is very expensive or even not feasible.
\item
{\magenta { }Suggested  approach:} 
First, determine {\magenta { }  equivalent  complex P-wave and shear   moduli}  solving   
  at the mesoscale and for a finite number of frequencies  boundary value problems   representing oscillatory compressibility  and shear tests   on   a  representative  volume  of bulk material containing 
 stochastic heterogeneities.

\item  The 
oscillations of these  
mesoscopic scale heterogeneous samples are assumed to obey {\magenta { }  Biot's} equations of motion.
\end{itemize} 
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small Mesoscopic attenuation and dispersion effects.  III}
{\small
\begin{itemize}
\item For a given  realization of the stochastic parameters, the  computed displacements allow to determine {\magenta { }equivalent  complex P-wave and shear moduli} as functions of frequency. 

\item 
These complex moduli represent the behavior of the heterogeneous material at the {\magenta { }macroscale}.

\item
Finally, to insure causality   we  define an   {\magenta { }equivalent  viscoelastic model} that  fits  the 
behaviour of the  computed complex  moduli in the range  of frequencies being considered. 

\end{itemize} 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small Mesoscopic attenuation and dispersion effects. VI}
{\small
\begin{itemize}


\item  The {\magenta { }equivalent viscoelastic model} is  used at the {\red { } macroscale}  to perform numerical simulations of wave propagation phenomena in these type of highly heterogeneous media.

\item The numerical solution of the oscillatory compressibility tests and the equivalent viscoelastic  model was obtained using  {\magenta { } finite element procedures}. 

\item Numerical experiments show the implementation of the  methodology  
 to simulate and analyze the 
effect of {\magenta { }underground carbon dioxide 
 (CO$_2$)  accumulations }
on the amplitude and attenuation of seismic waves. 
\end{itemize} 
}
\end{slide}

\begin{slide}{{Methodology}}
{\bf {\small
$\O$: a 2D   isotropic fluid-saturated   porous  material  
with boundary $\p\O$ containing  stochastic multiscale mesoscopic  heterogeneities in the fluid and/or solid matrix properties.  
\vskip.5cm

$\O$ is  composed of several subdomains $\O_m, m=1,\cdots M$ and each $\O_m$ is statistically homogeneous  (described by its statistical properties).

\vskip.5cm   
$\O_{R,m}$ is  a part of bulk fluid-saturated material in $\O_m$  containing a 
representative number of   such heterogeneities. 

\vskip.5cm
Assume that $\O_{R,m} = (0,L)^2$.   


}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{Biot's equation of motion in $\O_{R,m}$}}
{\bf 
{\tiny
\begin{eqnarray*}\label{mod1}
&&-\o^2 \rho u^{s}(\tilde x,\o) -\o^2 \rho_f u^{f}(\tilde x,\o)- 
\nabla\cdot\tau(u^{s},u^{f}) = 0,\quad \tilde x\in \O_{R,m},\\
&&-\o^2 \rho_f u^{s}(\tilde x,\o)
 -\o^2 \dfrac{S \rho_f}{\phi}  u^{f}(\tilde x,\o) 
+ i \o \dfrac{\eta}{\kappa} u^{f}(\tilde x,\o)+\nabla p_f(u^{s},u^{f})= 0,\quad\tilde x \in  \O_{R,m}.\label{mod2}
\end{eqnarray*}
\noi
$\tilde x$:  the local coordinate system at the mesoscale in $\O_{R,m}$. 
\vskip.2cm
\noi
$u^{s}$ and $u^{f}$:  Fourier transforms of the solid  
and relative fluid   displacement vectors, respectively.

\vskip.2cm
\noi  
$\rho_s$ and $\rho_f$:  
mass densities of the solid grains 
and the saturating fluid
\vskip.2cm
$\rho = (1 - \phi) \rho_s  + \phi \rho_f$: bulk density of the material

\vskip.2cm
\noi $\phi$:   effective porosity. 

\vskip.2cm
\noi
$\eta$: fluid viscosity, \hskip.5cm   $\kappa$: absolute permeability,\hskip.5cm $S$: tortuosity factor. 

\vskip.2cm 
\noi
$\tau(u^{s},u^{f})$:   total stress tensor \hskip.5cm $p_f(u^{s},u^{f})$: fluid pressure

}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\small{Schematic representation of 
an oscillatory compressibility test in $\O_{R,m}$ }}
\begin{center}
\vspace*{-0.7cm}
\includegraphics[width=4.cm]{fig1_test_compres.eps}\\
\vspace*{-0.2cm}
%{\tiny Fig. 1. Compressional test}
\end{center}
\vskip.5cm 
\tiny{\bf
The (complex) oscillatory volume change suffered by the sample, 
$\Delta V(\omega)$, computed using Biot's equations,  yields the 
{\red { } equivalent undrained complex P-wave modulus} $\overline {M^m_c}(\omega)$:
\vskip.1cm
\[
\frac{\Delta V(\omega)}{V}=-\frac{\Delta P}{\overline {M^m_c}\omega)}.
\]
}
\end{slide}


\begin{slide}{\small{Schematic representation of 
an oscillatory shear test in $\O_{R,m}$}}
\begin{center}
\vspace*{-0.7cm}
\includegraphics[width=4.cm]{fig2_test_shear.eps}\\
\vspace*{-0.2cm}
%{\tiny Fig. 1. Compressional test}
\end{center}
\vskip.2cm
\tiny
{\bf{
The change in shape of the sample allows to recover its {\red { } equivalent   
undrained complex shear modulus } $\overline \mu^m_c(\omega)$:
\vskip.3cm
$
tg(\theta^m(\omega))=\frac{\Delta T}{\overline \mu^m_c(\omega)}\hskip.5cm 
$
$\theta(\omega)$:  departure angle between the original positions of the 
lateral boundaries
and those after the shear test. 

$tg(\theta^m)\approx u^{s,T}_x(\omega)/L$, \hskip.2cm 
$u^{s,T}_x(\omega)$  average of the horizontal displacement suffered by the top boundary, computed using Biot's equations.
}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{\small{Local degrees of freedom for the FEM solution of the oscillatory tests}}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=7.cm]{apreton_dofs.eps}\\
\end{center}
\small{\bf  The solution of the oscillatory compressibility and shear tests was computed using a finite element procedure. The figure displays the local degrees of freedom (dofs) associated with each component of the solid displacement and the fluid displacement vectors. 
}
\end{slide}




%--------------------------------------------------------Slide
\begin{slide}{\small{The equivalent viscoelastic model. I}}
{\bf 
{
\vskip1cm 
\noi
After for each subdomain $\O_m$  the {\magenta { } equivalent} complex 
frequency dependent  plane wave modulus $\overline M^m_c(x,\o)=\overline  \lambda^m_c(x,\o) + 2 \overline \mu_c^m(x,\o)$  and  shear modulus $\overline  \mu_c^m(x,\o)$  are determined at a finite number of frequencies, they are used to define an {\magenta { } equivalent viscoelastic model} that fits the behaviour of $\overline M^m_c(x,\o)$ and 
$\overline \mu_c^m(x,\o)$  in the range  of frequencies being considered. 
}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{\small{The equivalent viscoelastic model.II}}
{\bf 
{\small
At this stage, 
we solve the following boundary value problem at the macroscale (in $\O$):  
\begin{eqnarray*}\label{mod3}
&&\hskip-.5cm-\o^2 \overline \rho_b u  -  \nabla\cdot\sigma(u) = F,\quad \O\\
&&-\sigma(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\label{mod4}
\end{eqnarray*}
$u =(u_x, u_z)$: displacement vector, 

\vskip.2cm
$\overline \rho_b$: average bulk density.
\vskip.3cm
\noi
$\sigma(u)$:  stress-tensor of our {\magenta { } equivalent viscoelastic material}, defined in terms of $\overline M^m_c(x,\o)$ and 
$\overline  \mu_c^m(x,\o)$.
\vskip.2cm   
We are employing   an  
absorbing boundary condition with a  positive definite matrix ${\mathcal D}$
  
}
}
\end{slide}


% --------------------------------------------Slide

\begin{slide}{\small{Application to seismic modelling in a CO$_2$ injection site. I  }}
\vskip-.3cm
{\bf 
{\small
Since  1996,  CO$_2$ is  separated from natural gas and 
re-injected into a shallow saline aquifer, the Utsira formation at Sleipner field.

\vskip.1cm
\noi
This formation is a high porosity unconsolidated sandstone with several 
{\magenta { } thin intra-reservoir shale layers}, which  act as {\magenta { } temporary seals} causing
 accumulations of   CO$_2$  beneath them.

\vskip.1cm
\noi
The CO$_2$ is in {\magenta { }supercritical state (at about  10 MPa and 36 C)},  with density $\rho_g = 0.485  gr/cm^3$,
 bulk modulus $K_g =  0.02$ GPa and  viscosity $\eta_g = 0.0015$ Poise.  


\vskip.1cm
\noi 
According to seismic interpretations and reservoir flow simulations 
these shales allow for the vertical  migration of CO$_2$, giving rise to 
the formation of {\magenta { }chimneys } at depth levels above the injection point.
}
}
\end{slide}


% --------------------------------------------Slide

\begin{slide}{\small{Application to seismic modelling in a CO$_2$ injection site. II  }}
{\bf 
{\small
\noi
The marked contrast between the physical and acoustical properties 
of natural reservoir fluids and those of CO$_2$ allows for the 
utilization of seismic methods as a tool for monitoring the spatio-temporal
distribution of CO$_2$ near the injection point.

\vskip.2cm
\noi
Seismic data recorded in this 
injection site show a {\magenta { }strong amplitude decay} and delay of the seismic signals through the
{\magenta { }chimneys}. 
\vskip.2cm
\noi
With these motivations, our {\magenta { } equivalent viscoelastic model} is employed  to simulate 
the seismic response of wave propagation using the parameters of the Utsira formation.
\vskip.2cm
\noi
We designed an idealized  domain $\O=(0,L)^2$  with $L=$ 1 Km  
consisting of  different regions  $\O_i, i=1,2,3,4$ 
}
}
\end{slide}
%---------------------------------------------------------------%

% -----------------------------------------Slide -------------

\begin{slide}{\small{The idealized model of CO$_2$ accumulation}}
\begin{center}
\vspace*{-0.5cm}
\includegraphics[width=6.cm]{fig7_modelo.ps}\\
\vspace*{0.2cm}
\end{center}
\tiny{\bf 
$\O_1$: brine-filled Utsira sand, $vp_1 \approx 2.1$ km/s \hskip0.5cm $\O_2$: brine filled  thin shale layer 
$vp_2 \approx 2.2 $ km/s.
 
$\O_3$ and $\O_4$: regions within the brine-filled Utsira sand 
containing patches of  CO$_2$ with fractal-type spatial distribution.
 \vskip.1cm
\hskip3cm $\O_3$: 50\% CO$_2$ patchy  saturation, $vp_3 \approx 1.18 $ km/s, $Qp_3\approx 20 $

\hskip3cm  $\O_4$: 10\% CO$_2$ patchy saturation, $vp_4 \approx 1.3 $ km/s, $Qp_4\approx 8 $
}
\end{slide}




% --------------------------------------------Slide

\begin{slide}{\small{Application to seismic modelling in a CO$_2$ injection site. III  }}
{\bf 
{\small
\noi
The thin layers of shale, referred  to as $\O_2$,  
are used  to model the {CO$_2$ accumulations } at
different depths. 
\vskip.2cm
\noi
Region $\O_4$ contains a 90\% of brine and a 10 \% of CO$_2$ 
 in the  form of {\magenta { }patches } with a fractal-type spatial distribution.
\vskip.2cm
\noi
Region $\O_3$ contains {\magenta { } patches} of CO$_2$, but with 50\%  CO$_2$
saturation and 50\% brine saturation. 

\vskip.2cm
\noi
Regions $\O_1$ (top of the Utsira)  
 and $\O_2$ (thin shale layers)   are considered to be  brine-filled and are  represented using a Gassmann's  model.

}
}
\end{slide}
% --------------------------------------------Slide

\begin{slide}{\small{Application to seismic modelling in a CO$_2$ injection site. IV  }}
{\bf 
{\small
\noi
The  viscoelastic moduli associated with the  
patchy saturated regions  $\O_3$ and $\O_4$ were determined   
using  {\magenta { }stochastic fractal fields}  based 
on the  von Karman self-similar correlation  functions. 
\noi
The {\magenta { }von Karman spectral density}  is:
\begin{equation}\nonumber
S_d(k_x,k_y)= S_0 (1 + k^2 a^2)^{-(H + E/2)}
\end{equation}
\vskip.3cm
\noindent 
$k=\sqrt{k_x^2+k_y^2}$: radial wavenumber \hskip1cm  $a$:  correlation
 length
\vskip.3cm
\noindent
 $H$ : a self-similarity coefficient ($0 < H < 1$) 
\vskip.3cm
\noindent
$S_0$:  normalization constant.

\vskip.3cm
\noindent
This  corresponds to a fractal process of {\magenta { } fractal dimension}
$D=E+1-H$ at scales smaller than $a$.  

}
}
\end{slide}
% -----------------------------------------Slide -------------

\begin{slide}{\small{\bf The reference sample $\O_{r,4}$  for  10\% CO$_2$ patchy saturation}}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=6.cm]{fig4_patch_10CO2_a_10cm.eps}\\
\vspace*{0.2cm}
\end{center}
\small{\bf  Fractal dimension  D=2.2  (H=0.8), correlation length a = 10cm 
}
\end{slide}
% --------------------------------------------Slide

\begin{slide}{\small{Application of the oscillatory tests for the case of  CO$_2$ patchy saturation}}
{\bf 
{\small
\noi 
In the regions $\O_3$ and $\O_4$ the corresponding {\magenta { } equivalent  P-wave   modulus} $\overline M_c(\o)$  was determined performing  the numerical compressibility tests 
over the reference samples $\O_{R,3}$ and  $\O_{R,4}$. 

\vskip.5cm
The numerical shear tests applied over these samples  gave  as expected 
 negligible  values of attenuation and dispersion and consequently the corresponding {\magenta { } equivalent  shear  modulus } $\overline \mu_c(\o)$ was taken to be real.


}
}
\end{slide}

% -----------------------------------------Slide -------------

\begin{slide}{\tiny{\bf Computed and fitted  P-wave phase velocity for  $\O_{r,4}$ with 
10\% CO$_2$ patchy saturation}}
\begin{center}
\vspace*{.5cm}
\includegraphics[width=9.cm]{fig5_vp_10CO2_a_10cm.eps}
%\vspace*{-0.6cm}
%\includegraphics[width=4.cm,height=7.2cm,angle=-90]{fig6_inv_qp_10CO2.eps}
%\vspace*{0.2cm}
\end{center}
%\vskip.5cm 
\tiny{\bf
}
\end{slide}

% -----------------------------------------Slide -------------

\begin{slide}{\tiny{\bf Computed and fitted  
inverse quality factor for  $\O_{r,4}$ with 
10\% CO$_2$ patchy saturation}}
\vspace*{.5cm}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=9.cm]{fig6_inv_qp_10CO2_a_10cm.eps}\\
\vspace*{0.2cm}
\end{center}
\vskip.5cm 
\tiny{\bf
}
\end{slide}

% -----------------------------------------Slide -------------

\begin{slide}{\tiny{\bf Phase velocity for 
10\% CO$_2$ patchy saturation and that corresponding to White's layered model }}
\begin{center}
\vspace*{.5cm}
\includegraphics[width=8.cm]{vp_white_and_compress.eps}\\
\vspace*{0.2cm}
\end{center}
\vskip.5cm 
\tiny{\bf
}
\end{slide}

% -----------------------------------------Slide -------------

\begin{slide}{\tiny{\bf Inverse quality factor  for  $\O_{r,4}$ with 
10\% CO$_2$ patchy saturation and that corresponding to White's layered model }}
\begin{center}
\vspace*{.5cm}
\includegraphics[width=8.cm]{invqp_white_and_compress.eps}\\
\vspace*{0.2cm}
\end{center}
\vskip.5cm 
\tiny{\bf
}
\end{slide}
%\end{document}



% -----------------------------------------Slide -------------

\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=175 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig8_vsz_t175_3d_color.eps}\\
\end{center}
\tiny{\bf  The figure shows  reflected and transmitted P-waves at the top interfaces  
between regions $\O_1$, $\O_3$ and $\O_4$. 
% Also, the wavefront travelling inside $\O_4$ suffers a delay due to the lower phase %velocity in that region.
}
\end{slide}

% -----------------------------------------Slide -------------

\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=225 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig9_vsz_t225_3d_color.eps}\\
\end{center}
\tiny{\bf  The figure shows the  reflected and transmitted P-waves at the top interface 
between regions $\O_1$, $\O_3$  and $\O_4$. Also, the wavefront travelling inside $\O_4$ suffers a delay due to the lower phase velocity in that region.
}
\end{slide}
% -----------------------------------------Slide -------------

\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=350 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig10_vsz_t350_3d_color.eps}\\
\end{center}
\tiny{\bf  The  P-waves have been  reflected and transmitted at the interface between $\O_1$ and the lower CO$_2$ accumulations in $\O_3$.  The  wavefront 
of smaller amplitude  is the  downwards transmitted wave generated at the upper   CO$_2$ bearing layer and travelling inside  the  chimney region $\O_4$. 
}
\end{slide}





\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=400 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig12_vsz_t400_3d_color.eps}\\
\end{center}
\tiny{\bf  The transmitted P-waves have been  reflected and transmitted at the interface between $\O_1$ and the lower CO$_2$ accumulations in $\O_3$.  The  wavefront 
of smaller amplitude  is the  downwards transmitted wave generated at the upper   CO$_2$ bearing layer and arriving at the lower CO$_2$ accumulation layer  in  $\O_3$.
}
\end{slide}
\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=500 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig13_vsz_t500_3d_color.eps}\\
\end{center}
\vskip-.5cm
\tiny{\bf  The  P-waves  reflected  at the interface between $\O_1$ and the lower CO$_2$ accumulations in $\O_3$ have been transmitted (travelling upwards)  and reflected  (travelling downwards, first  multiple) at the upper accumulation of CO$_2$  layer in $\O_3$.  The  center of the wavefront travelling upwards  has been totally attenuated after the two-way travel across the   the chimney region $\O_4$.  There is a small amplitude transmitted wave generated at the interface between $\O_4$ and the lower layer of $\O_3$.
}
\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\tiny{Time histories  measured near the
surface.  CO$_2$  accumulations  in  the lower layer  of $\O_3$}}

\vskip -.2cm
\begin{center}
\includegraphics[width=6.cm,height=9.cm,angle=-90]{Gather_modelo5_test3.eps}\\
\end{center}
{\tiny {\bf The  reflection  at about  600 ms  is  
  generated by the presence of  CO$_2$-accumulation in the lower  layer  of  
 $\O_3$.}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\tiny{Time histories  measured near the 
surface.  CO$_2$ accumulations  in  the lower layer  of $\O_3$ and  in $\O_4$}}

\vskip -.2cm
\begin{center}
\includegraphics[width=6.cm,height=9.cm,angle=-90]{Gather_modelo5_test4.eps}\\
{\tiny The  first reflection at about 250 ms 
is due to the presence of the {\magenta{} chimney} $\O_4$.  The second reflection at about  600 ms 
is  generated by CO$_2$ accumulations in the lower  layer  of  
 $\O_3$. The center of the second reflection  is strongly attenuated due to
the two-way travel across the {\magenta{}  chimney $\O_4$}.
}
\end{center}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\tiny{Time histories  measured near the
surface. CO$_2$ accumulations  in  $\O_3$ and  the chimney  $\O_4$}}

\vskip -.2cm
\begin{center}
\includegraphics[width=6.cm,height=9.cm,angle=-90]
{Gather_modelo5_test6_10CO2_a_10cm.eps}\\
{\tiny The 
first and second reflections at about 250 ms and 600 ms, respectively, 
are  generated by the presence of CO$_2$ accumulations in the upper and lower layers 
of   $\O_3$. The center of the second reflection  is strongly attenuated due to
the two-way travel across the {\magenta{}  chimney $\O_4$}.
}
\end{center}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\tiny{Time histories  near the
surface. 90\%  CO$_2$ accumulations   in  $\O_3$ and   $\O_4$  using a  Gassmann model. }}

\vskip -.2cm
\begin{center}
\includegraphics[width=6.cm,height=9.cm,angle=-90]
{Gather_modelo10_test1_90CO2_gassmann.eps}\\
{\tiny The  first reflection at about  250 ms corresponds to the top CO$_2$ layer.
The reflection from the  bottom CO$_2$ layer  is located at about 600 ms  except 
at the center, showing  a delay due to the low velocity in region $\O_4${\magenta{}  (pushdwon effect)}.
}
\end{center}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{Conclusions. }
{\small
\begin{itemize}
\item We presented an approach to model  
 wave propagation and attenuation and dispersion effects in highly heterogeneous fluid-saturated porous formations employing an equivalent viscoelastic model.

\item The  methodology allows 
to  represent at the macroscale the attenuation effects caused by mesoscopic-scale heterogeneities, thus reducing  the number of degrees of freedom  in several orders of magnitude.


\item  Numerical simulations  of  the seismic response  in 
 a   CO$_2$ injection site at Sleipner field   give a possible explanation for  
the high levels of attenuation observed in  seismic sections.


%\item The procedure can be 
%a useful tool in seismic monitoring, for the interpretation 
%of time lapse seismic, delineation of CO$_2$  plumes  and saturation  
%characterization.  

\end{itemize} 
}
\end{slide}
\end{document}

\begin{slide}{\small{Snapshot of the vertical component of the velocity field at t=375 ms}}
\vspace*{-1.cm}
\begin{center}
\includegraphics[width=6.4cm]{fig11_vsz_t375_3d_color.eps}\\
\end{center}
\tiny{\bf  The  P-waves have been  reflected and transmitted at the interface between $\O_1$ and the lower CO$_2$ accumulations in $\O_3$.  The  wavefront 
of smaller amplitude  is the  downwards transmitted wave generated at the upper   CO$_2$ bearing layer and travelling inside  the  chimney region $\O_4$.
}
\end{slide}
















