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\title{ {SEISMIC MONITORING OF CARBON DIOXIDE FLUID FLOW}}
%in heterogeneous porous media}
%\subtitle{IC-MSQUARE 2012, September 3-7, 2012}
\author{J. E. Santos$^1$, G. B. Savioli$^2$, J. M. Carcione$^3$, D. Gei$^3$}
\institution{
$^1$ CONICET, IGPUBA,  Fac. Ing., UBA 
\\ and  Univ. Nac. de La Plata, ARGENTINA \\
and Department of Mathematics, Purdue University, USA \\$ $ 
$^2$ IGPUBA - Fac. Ing., UBA, ARGENTINA$ $ 

%$^2$ CONICET, Inst. del Gas y del Petr\'oleo - FI, Univ. de Buenos Aires and \\ Univ. Nacional de La Plata, ARGENTINA \\
%and Department of Mathematics, Purdue University, USA \\$ $ 
$^3$ Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Trieste, ITALY  \\$ $
 {\magenta VII ITLA 2012, 17-21 diciembre 2012 Rosario, Santa F\'e, Argentina }
}
%$^*$ Instituto del Gas y Petr\'oleo, Facultad de Ingenier\'\i a, Universidad Nacional %de Buenos Aires \\$ $ \\
%{\magenta  WCCM 2012,  July 8-13, 2012}}
%\subtitle{{\magenta 2007 SEG Annual Meeting, San Antonio, Texas, September 
%23-28, 2007} }

\begin{document}
\maketitle
%----------------------------SLIDE -------------------------------
\begin{slide}{\small {Porosity  map of the formation.}}
\begin{center}
{\small
%\vspace*{-.4cm}
\includegraphics[width=12cm,height=6.5cm]{gather_vz_line_600x200_2a_con_base_solo_brine.ps}
%\vspace*{-0.2cm}
}
\end{center}
\end{slide}
\end{document}
%----------------------------------------------- SLIDE -
\begin{slide}{Introduction. I}
{\small
\begin{itemize}
\item Storage of CO$_2$ in geological formations is a procedure employed to 
reduce the amount of greenhouse gases in the atmosphere  to slow down  
global warming.


\item Geologic sequestration
involves {\magenta { } injecting CO$_2$  into a target
geologic formation}  at depths typically >1000 m where
pressure and temperature are above the critical point for
CO$_2$ (31.6C, 7.38 MPa).

\item First industrial scale CO$_2$  injection
project:    Sleipner gas field (North Sea). 
\end{itemize} 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{Introduction. II}
{\small
\begin{itemize}



\item CO$_2$ is separated from natural gas produced and is
currently being injected into the Utsira Sand, 
a saline aquifer at the Sleipner field,  
some 26000 km$^2$ in area. 

\item Injection started in 1996 and is
planned to continue for about 20 years, at a rate of about one
million tonnes per year.

\item {\magenta { }Time-lapse seismic surveys}  aim to  
 monitor the migration and dispersal of the
CO$_2$ plume after injection. 

\item Very little is known about the effectiveness of  CO$_2$ sequestration 
over very long periods of time.
\end{itemize} 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{Introduction. III }
{\small
\begin{itemize}



\item The analysis  of    CO$_2$   
underground storage safety  in the long
term is a current area of research.

\item We present a methodology integrating  {\magenta { } numerical simulation of 
 CO$_2$-brine flow
 and seismic wave propagation } to model and monitor  CO$_2$ injection. 

 \item The model of the formation  is based on the  porosity and clay content 
distribution  considering  the variation of properties at the 
site with pressure and saturation.
\end{itemize} 
}
\end{slide}

%


% -------------------------------SLIDE---------------------------

\begin{slide}{Presentation Outline}
{\small
\begin{itemize} 
\item
Present  the two-phase fluid flow equations 
used to simulate {\magenta { } CO$_2$ injection}. 

\item Describe   a {\magenta { } viscoelastic model for wave propagation} 

\item Present  a petrophysical model of a shaly sandstone based 
on fractal porosity and clay content, considering
the variation of properties with pressure and saturation. 

%\item
%Outline the numerical procedures employed  to 
%compute approximate solutions to the model.


\item Show {\magenta { } numerical simulations of CO$_2$ injection and time-lapse 
seismics}   to monitor the migration and dispersal  of CO$_2$ after 
injection in the Utsira formation at the Sleipner field.

\end{itemize} 
}
\end{slide}

% -------------------------------SLIDE---------------------------

\begin{slide}{ The  Black-Oil formulation }
{\bf 
{\small

\begin{itemize}

\item The simultaneous flow of brine and CO$_2$  is described by the well-known 
{\magenta { }Black-Oil 
formulation } applied to two-phase, two component fluid flow. 

\item In the {\magenta { }Black-Oil } model employed, brine is NOT present, 
OIL is identified with brine and  CO$_2$ is identified with  GAS.


\item Also, CO$_2$ may dissolve in  brine (OIL)  
but brine (OIL)  is not allowed to vaporize into the CO$_2$ phase.
 
\item
This formulation uses, as a simplified thermodynamic model, the quantities 
 $R_s, B_b $ and $  B_{CO2}$  as  PVT data:
\end{itemize}
}
}

\end{slide}
% -------------------------------SLIDE---------------------------

\begin{slide}{ The  Black-Oil formulation }
{\bf 
{\small

\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}
An algorithm 
developed by Hassanzadeh (2008) can be used to  estimate  the above PVT data.
\vskip.3cm  
The Black-Oil equations for two-phase flow in porous media  are obtained 
combining conservation of mass of each component  with two-phase Darcy's law.
}
}

\end{slide}






% -------------------------------SLIDE---------------------------

\begin{slide}{{\small The  Black-Oil formulation of two-phase flow 
in porous media}} 
{\bf 
{\tiny
\vskip -0.6cm 
\begin{eqnarray}\label{bo1}
&&\nabla \cdot \Big [\dfrac{\underline {\kappa} k_{rCO2}}{B_{CO2}\eta_{CO2}}(\nabla p_{CO2} - \rho_{CO2} g \nabla D) 
+ \dfrac{\underline {\kappa} R_s k_{rb} }{B_b \eta_b } (\nabla p_b - \rho_b g \nabla D)\Big ]\nonumber\\
&&\hskip5 cm + q_{CO2} =\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 \Big [ \dfrac{\underline {\kappa} k_{rb}}{B_b \eta_b }(\nabla p_b - 
\rho_b g \nabla D\Big ]+q_b 
=
\ds {\Big [\phi \dfrac{ S_b}{B_b }\Big ]}t\nonumber
\end{eqnarray}

Two algebraic equations complete the system: 
\begin{eqnarray}
&S_b+S_{CO2}=1,\qquad 
&p_{CO2}-p_b=P_C(S_b)\nonumber
\end{eqnarray}

}
}
\end{slide}

% -------------------------------SLIDE---------------------------

\begin{slide}{\small { Numerical solution of the  Black-Oil 
formulation of two-phase flow in porous media.}} 
{\bf 
{\small
The unknowns for the Black-Oil fluid-flow model are the {\magenta { } 
fluid pressures}
$p_{CO2}, p_b$ and the {\magenta { } 
saturations $S_{CO2}, S_b$ }for the CO$_2$ and brine phases.

\vskip0.5cm

They were computed using the public domain software {\magenta { }BOAST},
which solves the differential equations applying {\magenta { }IMPES}, a 
finite difference technique. 

\vskip0.5cm


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


\item
These effects are  due to the 
equilibration of wave-induced fluid pressure gradients via a
{\magenta { } slow-wave diffusion} process (Type II waves).

\item White et al. (1975) were the first to introduce the 
mesoscopic-loss mechanism in the framework of Biot's theory, which is
illustrated in the next figures. 
 


\end{itemize} 
}
\end{slide}

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

\begin{slide}{\tiny {Traces of waves travelling through a porous media saturated by 
gas (top left), water (top right) and periodic gas-water (bottom)} }
\begin{center}
\vspace*{0.3cm}
\hskip-.2cm
\includegraphics[width=4.cm,height=3.cm]{figure2_gas.eps}\hskip.8cm 
\includegraphics[width=4.cm,height=3.cm]{figure2_water.eps}\\ 
\vskip.5cm
\includegraphics[width=4cm,height=3.cm]{figure2_gas_water.eps}
%\\ 
%{\tiny Figure  2. Top left: gas, Top right: water, Bottom: periodic gas-water }
\end{center}
\end{slide}


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

\begin{slide}{\tiny {Traces of waves for the gas, water and periodic gas-water saturation
cases } }
\begin{center}
\vspace*{0.5cm}
%\vskip.2cm
\includegraphics[width=8cm,height=4.5cm]{figure3.eps}\\ 
\vskip.1cm
{\tiny The delay in in the arrival 
time in the periodic case is due to the velocity dispersion 
caused by the mesoscopic scale
heterogeneities. Attenuation  adn dispersion in the periodic 
gas-water case is in perfect 
agreement with that predicted by White's theory.}
\end{center}
\end{slide}
%------------------------------Slide
% -----------------------------------------Slide -------------

\begin{slide}{\tiny{\bf Representative sample for  10\%  fluid 
mixture CO$_2$ patchy saturation.}}
\begin{center}
\vspace*{.5cm}
\includegraphics[width=5.cm]{leakage1_patch_10_fluid_mix_a_5cm.eps}\\
\vspace*{0.2cm}
\end{center}
\tiny{\bf  Side length 50 cm, fractal 
dimension  D=2.5, correlation length a = 5cm.  
Black zones: pure CO$_2$ saturation, white zones: pure brine saturation. 
}
\end{slide}

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

\begin{slide}{\tiny{\bf Computed  Vp (P-wave phase velocity)   
as function of CO$_2$  patchy  saturation}}
\begin{center}
\vspace*{.7cm}
\includegraphics[width=7.5cm]{vp_leakage1.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  CO$_2$ is in gaseous state, fractal dimension 
 D=2.5, correlation length  a = 5 cm 
}
\end{slide}


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

\begin{slide}{\tiny{\bf Computed  
inverse quality factor 
as function of    fluid mixture  CO$_2$ patchy saturation}}
\vspace*{.7cm}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=7.5cm]{inv_qp_leakage1.eps}\\
\vspace*{0.2cm}
\end{center}
%\vskip.5cm 
\tiny{\bf CO$_2$ is in gaseous state, fractal dimension  
D=2.5, correlation length  a = 5 cm
}
\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small Seismic modeling. Mesoscopic attenuation  effects. II}
%\vskip-0.2cm
{\small
\begin{itemize}
\item  Due to the {\magenta { }extremely fine meshes} 
needed to properly represent these type of media, numerical simulations 
 at the macroscale is very expensive or even not feasible.

\item outside the Utsira formation, 
 the  {\magenta { } complex bulk and shear moduli } as function of frequency 
were determined  
using a Zener model. 

  
\end{itemize}
The dimensionless complex modulus of a Zener element is 
\[
Z(\omega) = \frac{\tau_{\sigma}}{\tau_{\epsilon}} 
\left[\frac{1 + i \omega \tau_{\epsilon}}{1 +  i \omega \tau_{\sigma}} \right] \qquad \tau_{\epsilon}, \tau_{\sigma}: \text{relaxation times}.
\] 
}
\end{slide}
% -----------------------------------------Slide -------------
\begin{slide}{{\bf  The Zener  model}}
{\bf 
{\small
\vskip-.2cm
The associated quality factor is 
\[
Q(\omega) = \frac{\text{Re}(Z(\omega))}{\text{Im}(Z(\omega))}
\] 
\vskip-.2cm
Its  minimum value is equal to 
\vskip-.4cm
\begin{eqnarray}\label{eq1}
Q_0 = \frac{2x}{x^2 -1}, \qquad 
x=\left(\frac{\tau_{\epsilon}}{\tau_{\sigma}}\right)^{1/2}
\end{eqnarray}
and is located at
$
\o_0 = \frac{1}{ (\tau_{\sigma} \tau_{\epsilon})^{1/2}}. 
$ 

If $\o_0$ and $Q_0$ are chosen as data, then
\[
\tau_{\epsilon} = \frac{x}{\o_0},\qquad \tau_{\sigma} = \frac{x^{-1}}{\o_0},\quad \text{where}\, \, x \,\, \text{solves} (1)
\]
where $x$ solves  (1)
}
}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small Seismic modeling. Mesoscopic attenuation  effects. II}
%\vskip-0.2cm
{\small
\begin{itemize}

\item 
Within the Utsira formation, determine {\magenta { }  
complex and frequency dependent P-wave}  modulus  
\[
E(\o)= \lambda(\o) + 2 \mu(\o)
\]
at the mesoscale using {\magenta { } White's  theory for patchy saturation}.

\item $\lambda(\o), \mu(\o)$ : Lam\'e coefficients 

\item$\o$ : angular frequency

\end{itemize} 
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small Seismic modeling. Mesoscopic attenuation  effects. II}
{\small
\begin{itemize}
\item 
Shear wave attenuation is taken into account using another relaxation mechanism 
related to the P-wave White mechanism to make the shear modulus 
\[
\mu(\o)
\] 
complex and frequency dependent. 
\item 
These complex moduli define an {\magenta { }equivalent  viscoelastic model} 
at the {\magenta { }macroscale} 
that take into account  dispersion and attenuation effects occuring at the mesoscale. 

\end{itemize} 
}
\end{slide}





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

\begin{slide}{\small {Seismic modeling. Constitutive Relations}}
{\bf 
{\small

$u = u(\o) =(u_x(\o), u_z(\o))$: Time-Fourier transform of the displacement vector

Stress-strain relations  in the space-frequency domain:
\begin{eqnarray}\label{mod5}
&&\sigma_{jk}(u) = \lambda(\o) \nabla\cdot u \delta_{jk}  
+ 2 \mu(\o) \varepsilon_{jk}(u) ,\nonumber
\end{eqnarray}   

\vskip0.2cm
$\sigma_{jk}(u)$: stress tensor\qquad 
$\varepsilon_{jk}(u)$: strain tensor\\
$\delta_{jk}$: Kroenecker delta\\

\vskip0.2cm  
$\lambda(\o), \mu(\o)$: complex Lam\'e coefficients determined using 
{\magenta { } White's theory}.

}
}
\end{slide}


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

\begin{slide}{\small Seismic modeling. Phase velocities and attenuation 
coefficient}
{\bf 
{\small
For viscoelastic solids,  the frequency dependent {\magenta { }phase velocities $v_p(\o)$ 
and quality factors $Q(\o)$} 
are  defined by the relations
\begin{eqnarray}\label{def_vpQ}
v_p(\o) = \left[{\rm Re} \left( \frac { 1 }{ vc_p(\o) } \right) \right]^{-1},\qquad 
Q(\o) = \dfrac {\text{Re}(vc_p(\o)^2)}{\text{Im}(vc_p(\o)^2)},\nonumber
\end{eqnarray}
$vc_p(\o)$: complex and frequency dependent compressional  velocity defined as  
\begin{equation}\label{def_vcp}
vc_p(\o) = \sqrt{\dfrac{E(\o)}{\rho} }.\nonumber
\end{equation}
$\rho$: bulk density,
}
}
\end{slide}

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

\begin{slide}{\small Seismic modeling. A viscoelastic model 
for wave propagation. I }
{\bf 
{\small

Equation of motion in a  2D  isotropic viscoelastic  domain  $\O$ 
with boundary $\p\O$: 
\begin{eqnarray}\label{mod3}
&&\hskip-.5cm\o^2 \rho u + \nabla\cdot\sigma(u) = f(x,\o),\quad \O\nonumber\\
\nonumber
\end{eqnarray}
First-order absorbing boundary condition: 
\begin{eqnarray}\label{mod3a}
&&-\sigma(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\label{mod4}\nonumber
\end{eqnarray}
$f(x,\o)$ external source 
}
}

\end{slide}




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


\begin{slide}{\small  Numerical Solution - Finite Element Method. }
{\bf 
{\small
\begin{itemize}
\item 
The FE  was  computed  
at a finite number of frequencies 
in the range of interest  using an {\magenta { }iterative 
 finite element domain decomposition procedure}.  

\item  The time domain solution was obtained using a 
discrete inverse Fourier transform.

\item
To approximate each component of the solid displacement vector we employed a 
{\magenta { } nonconforming finite element space} which 
generate {\magenta { }less numerical dispersion than the standard bilinear elements}.

\item 
The  error measured  in the energy norm is of order $h^{1/2}$,   
where $h$ is the size of the computational mesh.  

\end{itemize}
}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{{A petrophysical model of the Utsira formation.}}

{\bf 
{\small
We consider a 2D model of the Utsira formation  
constructed using  
an {\magenta { }  initial porosity $\phi_0$  (at hydrostatic pore pressure)  
and the clay  content} $C$ of the formation.

\vskip.2cm
The model was constructed  using iteratively a wave propagation simulator
to fit the avalaible seismic data. It has $400$ m thickness (top at $800$ m and
bottom $1200$ m b.s.l.). 

\vskip.2cm
Within the formation,
there are several  mudstone layers which act as barriers to the vertical 
motion of the CO$_2$.
}
}
\end{slide}



%----------------------------SLIDE -------------------------------
\begin{slide}{{\small A petrophysical model considering pressure 
and saturation variations. I}}
%\vskip-.3cm
{\bf
{\small
Our  shaly sand model was generated  using fractal realizations of the initial 
 porosity 
$\phi_0$ and clay content $C$. 
\begin{itemize}
\item
Relationship between porosity $\phi$ and pore pressure $p$: 
\begin{eqnarray}
\frac{(1-\phi_c)}{K_s}(p-p_H)=\phi_0-\phi+\phi_c\ln 
\frac{\phi}{\phi_0},\; \;\phi_c \; \textrm{critical porosity}
\end{eqnarray}

 
\item
The bulk modulus of the dry matrix was determined using a Krief model ($A= 4.5$):
\begin{eqnarray*}
 K_m = K_s(1-\phi)^{A/(1-\phi)}
\end{eqnarray*}

\end{itemize}
Here $K_s$  denote the bulk  modulus of the solid grains.
}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{{\small A petrophysical model considering pressure 
and saturation variations. II}}
%\vskip-.3cm
{\bf
{\small

\begin{itemize}
\item
Assuming a Poisson medium  $\mu_s = 3 K_s/5$ and the 
shear modulus of the dry matrix is   
\begin{eqnarray*}
\mu_m =\mu_s(1-\phi)^{A/(1-\phi)}\; \; \;\textrm{,}\; \;\; \mu_s = 3 K_s/5
\end{eqnarray*}
Here $\mu_s$  denote the shear modulus of the solid grains.
\end{itemize}


}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{{\small A petrophysical model considering pressure 
and saturation variations. III }}
{\bf
{\small

\begin{itemize}


\item
Horizontal $\kappa_x$ and vertical permeability $\kappa_z$,
\begin{eqnarray*}
\frac{1}{\kappa_x}=\frac{45(1-\phi)^2}{\phi^3}\bigg(\frac{(1-C)^2}{R^2_q}+\frac{C^2}{R^2_c} \bigg)
\end{eqnarray*}

 $R_q$, $R_c$: average radii of sand and clay particles

\begin{eqnarray*}
\frac{\kappa_x}{\kappa_z}=\frac{1-(1-0.3a)\sin \pi S_b}{a(1-0.5\sin \pi S_b)}
\end{eqnarray*}
$a$: permeability-anisotropy parameter
\end{itemize}
}

}
\end{slide}




%----------------------------SLIDE -------------------------------
\begin{slide}{\small {Permeability map (Darcy units) of the  formation.}}
\begin{center}
{\small
%\vspace*{-.4cm}
\includegraphics[width=12cm,height=6.5cm]{kx_xz_300x5x400.gnu.eps}
%\vspace*{-0.2cm}
}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{{CO$_2$ Injection model. I}}
{\small
\begin{itemize}
\item
At the Utsira formation CO$_2$ is injected at a constant flow rate 
of one millon tons per year.

\item
The injection point is located at the bottom 
of the Utsira formation: $x=600$ m, $z=1060$ m.

\item
The viscosity, density and bulk modulus of CO$_2$ needed for the flow simulator 
 were 
obtained from {\magenta { } the Peng-Robinson equations} 
as a function of {\magenta { } temperature  and pore pressure}.
 
\end{itemize}

}

\end{slide}



%----------------------------SLIDE -------------------------------

\begin{slide}{\small{Injection Modeling. II}}

%\vskip -.6cm
\begin{center}
\includegraphics[width=5.2cm,height=5.2cm]{sat_CO2_300x400_frac_2a_10km_dt_0.5_test3.gnu.eps}\quad 
\includegraphics[width=5.2cm,height=5.2cm]{sat_CO2_300x400_frac_6a_10km_dt005_test3.gnu.eps}\\

{\tiny CO$_2$ saturation distribution after 2 years (left) and 6 years(right) 
of CO$_2$ injection
}
\end{center}
\end{slide}





%----------------------------SLIDE -------------------------------
\begin{slide}{{Time Lapse . I}}
{\small
\begin{itemize}
\item
The equivalent viscoelastic model, determined using {\magenta { } 
White's model for patchy saturation}, 
assumes an effective  single-phase fluid. 

\item
Effective  fluid  density, viscosity and bulk modulus 
were obtained using the properties of the  CO$_2$ 
and brine weighted by the corresponding saturations 
computed by the BOAST flow simulator. 

\item
The next Figure displays maps of  of P-wave phase velocity $v_p$ (left) 
and attenuation coefficient $1000/Q_p$ (rigth) at 50 Hz  
after 2 years of CO$_2$ injection. 


\end{itemize}

}

\end{slide}

%----------------------------SLIDE -------------------------------

\begin{slide}{\tiny{P-wave phase velocity (m/s) and attenuation coefficient $Q_p$ 
at 75 Hz.}}

%\vskip -.6cm
\begin{center}
\includegraphics[width=5.2cm,height=5.2cm]{vp_map_2a_gnu_ve_con_base.eps}\quad 
\includegraphics[width=5.2cm,height=5.2cm]{qp_map_2a_gnu_ve_con_base.eps}\\

{\tiny Observe a 
decrease in  P- wave velocity $v_p$  in  zones  
of CO$_2$ accumulation (left)  and a corresponding decrease in  attenuation 
coeefficient $Q_p$ (right) . 
}
\end{center}
\end{slide}



%----------------------------SLIDE -------------------------------

\begin{slide}{\tiny{S-wave phase velocity (m/s) and attenuation coefficient $Q_s$ 
at 50 Hz.}}

%\vskip -.6cm
\begin{center}
\includegraphics[width=5.2cm,height=5.2cm]{vs_map_2a_gnu_ve_con_base.eps}\quad 
\includegraphics[width=5.2cm,height=5.2cm]{qs_map_2a_gnu_ve_con_base.eps}\\

{\tiny Observe a 
decrease in  P- wave velocity $v_p$  in  zones  
of CO$_2$ accumulation (left)  and a corresponding decrease in  attenuation 
coeefficient $Q_s$ (right) . 
}
\end{center}
\end{slide}




%------------------------------------ SLIDE -
\begin{slide}{Seismic Monitoring. I}
{\small
\begin{itemize}

\item To analyze the capability of 
seismic monitoring to identify zones of CO$_2$ 
accumulation, the model  is excited with a  {\magenta { } compressional  line source 
located at $x= 600$ m, $z= 710$, } 
with central frequency 50 Hz and highest frequency 180 Hz.

\item 
The viscoelastic wave equation is solved for  200 frequencies, and the time histories  
were obtained using   an approximate inverse Fourier transform.

\item The next Figures display  time histories  measured near the surface 
before  and after 2  an 6 years  of CO$_2$ injection. 


\end{itemize} 
}
\end{slide}



\end{document}



%

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{Time histories before and after 2 years of CO$_2$ injection}}
%\vskip-.7cm
{\bf
{\small

\vspace*{-.4cm}
\includegraphics[width=5.2 cm,height=5.2cm]{gather_vz_line_600x200_2a_con_base_solo_brine.ps}\quad 
\includegraphics[width=5.2 cm,height=5.2cm]{gather_vz_line_600x200_2a_con_base.ps}\\
%\vspace*{-0.2cm}
\tiny{ Traces of the  z-component of the displacement before (left) and after 
2 years (right) CO$_2$ injection. The two reflections on the left are coming from the  
base and top of the Utsira. After 2 year of inyection the additional reflection is due to the 
 CO$_2$ accumulation below the deepest mudstone layer. }
}
}

\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{Time histories before and after 2  and 4 years of CO$_2$ injection}}
%\vskip-.7cm
{\bf
{\small

\vspace*{-.4cm}
\includegraphics[width=5.2 cm,height=5.2cm]{gather_after.eps}\quad 
\includegraphics[width=5.2 cm,height=5.2cm]{gather_4a2.eps}\\
%\vspace*{-0.2cm}
\tiny{ 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  earlier  reflection to the right is due to the 
upper  CO$_2$ accumulation}
}
}

\end{slide}





%------------------------------------ SLIDE -
\begin{slide}{CONCLUSIONS}
{\small
\begin{itemize}

\item The  numerical examples show the effectiveness of   combining 
multiphasic flow simulators in porous media with  
 seismic monitoring  
to  map  the spatio-temporal distribution of CO$_2$ after injection.

\item The wave propagation model includes 
 attenuation and dispersion effects due to mesoscopic scale 
heterogeneities using White's theory.

\item This methodology constitutes a valuable  tool to monitor the
 migration and dispersal of the CO$_2$ plume and to analyze storage integrity,
 providing early warning should any leakage occurs.

%\item


\end{itemize} 
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small AKNOWLEDGEMENTS}

{\small
This work  was partially funded by CONICET, Universidad de Buenos Aires, 
Universidad  Nacional de La Plata  and the 
European Union under project CO2 ReMoVe.  


\vskip.5cm 
{\magenta {} THANKS FOR YOUR ATTENTION !!!!!!}
}
\end{slide}
\end{document}




