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\title{Numerical Simulation in Applied Geophysics. From the Mesoscale to the Macroscale}
\vskip-.5cm
\author{Juan E. Santos
,$^\dag$ Patricia M. Gauzellino $^*$,  Gabriela B. Savioli$^{**}$ and 
Robiel Martinez Corredor$^{\dag,\dag}$
}
\institution{%
$^\dag$ CONICET, Instituto del Gas y del Petr\'oleo,  {\magenta Facultad de Ingenier\'\i a UBA}, UNLP   
and Department of Mathematics, {\magenta Purdue University}. \\ 
$^*$ Facultad de Cs. Astr. y Geof\'\i sicas, UNLP \\
$**$ Instituto del Gas y del Petr\'oleo,  {\magenta Facultad de Ingenier\'\i a UBA
\\
$^{\dag,\dag}$ Facultad de Ingenier\'\i a, UNLP 
}

{\magenta  Jornadas de Cloud Computing, LIDI, Fac. de Inform\'atica, UNLP, 18 Junio 2013}}
\begin{document}
\maketitle
%----------------------------------------------- SLIDE -
\begin{slide}{\small Introduction. I}
{\bf 
{\small
\begin{itemize}
\item {\red {} Seismic wave propagation } is  a common technique   used  
in hydrocarbon exploration geophysics, mining and reservoir  characterization and production.

\item Local variations  in the fluid and solid matrix properties, 
fine layering, fractures and craks  at  the mesoscale 
(on the order of centimeters) 
are common in the earth's crust and induce attenuation, dispersion and anisotropy 
of the seismic waves observed at the macroscale.

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


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

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

\item Due to the {\magenta { }extremely fine meshes} 
needed to properly represent these type of mesoscopic-scale heterogeneities, 
numerical simulations are  very expensive or even not feasible.

\item
{\magenta { }Suggested  approach:} 
Use a {\magenta { } numerical upscaling procedure } to determine 
the complex and frequency dependent stiffness at the macroscale of an equivalent  
viscoelastic medium  including  the 
mesoscopic-scale effects. 

\item To determine the {\blue{} 
 complex stiffness} coefficients of the 
{\magenta{}equivalent  medium}, we 
solve a set  of 
 boundary value problems (BVP's) for the wave equation of motion  
in the frequency-domain using  the finite-element method (FEM).


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

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

\item The BVP's  represent {\blue{} harmonic tests} at a finite number of 
frequencies on a representative sample  of the material.

\item {\blue {}Numerical rock physics} offer an alternative to laboratory measurements.

\item
Numerical experiments are inexpensive and informative since 
the physical process of wave
propagation can be inspected during the experiment. 

\item Moreover, they are repeatable, essentially
free from experimental errors, and may easily be run using 
alternative models of the rock and  fluid properties.


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


%----------------------------------------------- SLIDE -
\begin{slide}{\small The Mesoscale. Anisotropic poroelasticity. I}
{\bf 
{\small
\begin{itemize}
\item To model mesoscopic effects  we  need a suitable differential model.

\item As example, we will describe the numerical experiments on  a 
representative sample of fluid-saturated sample having a dense set 
of horizontal  fractures modeled as very thin and compliant layers.


\item A dense set of horizontal  fractures in a fluid-saturated poroelastic medium    
behaves as a  {\magenta { } viscoelastic  
transversely isotropic (VTI)} medium when the average fracture distance 
is much smaller than the predominant  wavelength of the traveling waves. 

\item This leads to frequency and angular variations of velocity and 
attenuation of seismic waves. 



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




%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Mesoscale. Anisotropic poroelasticity. II}}
\vskip-.5cm
{\small
\begin{itemize}
\item 
For fluid-saturated poroelastic media (Biot's media), 
White et al. (1975) were the first to introduce the 
mesoscopic-loss mechanism in the framework of Biot's theory. 

\item For fine layered poroelastic materials, the theories of  
Gelinsky and  Shapiro (GPY, {\bf 62}, 1997) and  
Krzikalla and  M\"uller (GPY, {\bf 76}, 2011) allow to  
obtain the {\blue { }five} complex and frequency-dependent stiffnesses of 
the {\magenta { }equivalent VTI medium}.  


\item To test the model and provide a more general modeling tool, 
we present a {\magenta { }numerical  upscaling procedure} to obtain the 
complex stiffnesses of the effective VTI medium.
\end{itemize} 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Mesoscale. Anisotropic poroelasticity. III}}

{\small
\begin{itemize}

\item We employ the Finite Element  Method {\blue { } (FEM)}
to solve  {\magenta { } Biot's equation of motion} 
in the space-frequency
domain with boundary conditions representing 
{\blue { } compressibility and shear harmonic} experiments.

\item The methodology is  
applied to saturated isotropic 
poroelastic samples having a dense set 
of horizontal  fractures.

\item The samples contained  mesoscopic-scale heterogeneities due to 
patchy brine-CO$_2$ 
saturation and fractal porosity and consequently, fractal permeability and frame properties.


\end{itemize} 
}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf TIV media and fine layering. I}}
\vskip-.5cm
{\bf 
{\small
Let us consider isotropic fluid-saturated poroelastic layers. 
 
{\blue { }   $ {\mathbf u^s}({\mathbf x}), {\mathbf u^f}({\mathbf x})$ } : time Fourier transform of the 
displacement vector of the solid and fluid relative to the solid   
frame, respectively.
 
\smallskip

{\blue { }${\mathbf u}  = ({\mathbf u^s}, {\mathbf u^f})$}
\vskip.2cm
 
{\magenta { }${\mathbf \sigma_{kl}(u)}, {\mathbf p_f(u)}$}: Fourier transform of the total stress 
and the fluid pressure, respectively 
\vskip.3cm
On each plane layer $n$ in a 
sequence of $N$ layers, the {\magenta { }frequency-domain  stress-strain relations }
 are
\begin{eqnarray*}\label{st_1}
&{\mathbf \sigma}_{kl}(u) =  2 \mu\,\varepsilon_{kl}(u^s)+\delta_{kl}
\left( \lambda_{_G} \,\nabla\cdot u^s  + \alpha M  \nabla\cdot u^f \right),\\
&{\mathbf p_f(u) }=  - \alpha M \nabla\cdot u^s -  M  \nabla\cdot u^f.\label{st_7}
\end{eqnarray*} 

}
}

\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small TIV media and fine layering. II}

{\bf 
{\small
{\magenta { } Biot's equations in the diffusive range:}
\begin{eqnarray*}\label{motion1}
&&\nabla\cdot\sigma(u)  = 0, \\
&& \ii \o  \dfrac{\eta}{\kappa} u^f(x,\o)  + \nabla p_f(u)=0,\label{motion2}
\end{eqnarray*}
\vskip.2cm 
$\o=2\pi f$: angular frequency
\vskip.2cm
$\eta$: fluid viscosity
\hskip1cm
$\kappa$: frame permeability


}
}

\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf TIV media and fine layering. III}}
\vskip-.5cm
{\bf 
{\small

${\mathbf \tau}_{{\mathbf i} {\mathbf j}}$: stress tensor of the equivalent VTI medium

For a {\blue  { }closed system( $\nabla\cdot u^f  = 0) $}, 
the corresponding  {\magenta { }stress-strain relations}, 
stated in the space-frequency domain, are 
\begin{eqnarray*}                                                           
&&\tau_{11}(u) = {p}_{11} ~\epsilon_{11}(u^s) + p_{12} ~\epsilon_{22}(u^s) 
+ p_{13} ~\epsilon_{33}(u^s), \label{st_GL_1}\\
&&\tau_{22}(u) = p_{12} ~\epsilon_{11}(u^s) + p_{11} ~\epsilon_{22}(u^s)    
 + p_{13} ~\epsilon_{33}(u^s), \label{st_GL_2}\\  
&&\tau_{33}(u) =  p_{13} ~\epsilon_{11}(u^s) + p_{13} ~\epsilon_{22}(u^s)      
+ p_{33} ~\epsilon_{33}(u^s), \label{st_GL_3}\\   
&&\tau_{23}(u) =  2~  p_{55} ~\epsilon_{23}(u^s), \label{st_GL_4}\\                    
&&\tau_{13}(u) = 2 ~  p_{55} ~\epsilon_{13}(u^s),\label{st_GL_5} \\                    
&&\tau_{12}(u) = 2 ~  p_{66} ~\epsilon_{12}(u^s). \label{st_GL_6}  
\end{eqnarray*}
{\small   
This approach provides the complex velocities of the fast modes and 
takes into account {\magenta { }interlayer flow effects}.}
}
}

\end{slide}





%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine the 
stiffness coefficients. I}}
\vskip1cm
{\bf 
{\small
To determine the complex stiffness we solve Biot's equation   
 in the 2D case on a reference 
square $\O=(0,L)^2$  with boundary $\G$ in the $(x_1, x_3)$-plane. 
Set {\blue { }$\G = \G^L\cup\G^B\cup\G^R\cup\G^T$}, where  
\begin{eqnarray}
{\blue { }\G^L} = \{ (x_1, x_3)\in \G: {\blue { }x_1=0}\}, 
\quad {\blue { }\G^R} = \{ (x_1,x_3)\in \G: {\blue { }x_1=L}\},
\nonumber\\
\nonumber\\
{\blue { }\G^B} = \{ (x_1, x_3)\in \G: {\blue { } x_3=0}\},
\quad {\blue { }\G^T }= \{ (x_1, x_3)\in \G:{\blue { } x_3=L}\}.\nonumber
\end{eqnarray}



}
}

\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine the 
stiffness coefficients. II}}
\vskip-.5cm
{\bf 
{\small
The sample is subjected to 
{\blue  { } harmonic compressibility and shear tests} described by the following 
sets of {\magenta { } boundary conditions}.
\vskip .1cm
{\blue { } $p_{33(\o)}$}: 
\begin{eqnarray*}\label{bc1a}
&&\sigma(u) \nu \cdot \nu = -\Delta P, \quad (x_1, x_3)\in \G^T,\label{bc1b}\\
&&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G^T\cup\G^L\cup\G^R,\label{bc1c}\\
%&&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G^L\cup\G^R,\label{bc1d}\\
&&u^s\cdot\nu = 0, \quad (x_1, x_3)\in \G^L\cup\G^R,\label{bc1e}\\
&&u^s = 0, \quad (x_1, x_3)\in \G^B,\quad 
u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{bc1g}.
\end{eqnarray*}
{\magenta { } $\nu$}:  the unit outer normal on $\G$
   
{\magenta { } $\chi$}:  a   unit tangent  on $\G$  so that ${\blue { } \{\nu,\chi\}}$ is 
an orthonormal system on $\G$. 
Denote by $V$ the original volume of the sample and by  
$\Delta V(\omega)$
its (complex) oscillatory volume change. 
}
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine 
the stiffness coefficients. III}}
\vskip-.5cm
{\bf 
{\small
In the quasistatic case  
\begin{eqnarray*}\label{c33-dv}
\frac{\Delta V(\omega)}{V}=-\frac{\Delta P}{{\magenta { } p_{33}(\omega)}},
\end{eqnarray*}
Then after computing the  average $u^{s,T}_3(\omega)$ of the  vertical displacements 
on  $\Gamma^T$, we approximate
\[
\Delta V(\omega)\approx L u^{s,T}_3(\omega)
\]
which enable us to compute  ${\magenta { } p_{33}(\omega)}$
\vskip.5cm
To determine {\blue { }$p_{11}(\omega)$}  we solve an identical boundary 
value problem than for {\magenta { }$p_{33}$} 
but for a 90$^{\rm o}$ rotated sample.
}
}
\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine the 
stiffness coefficients. IV}}
\vskip-.5cm
{\bf 
{\small
{\blue { }$p_{55}(\omega)$:} the
 boundary conditions are 
\begin{eqnarray*}\label{s_bc1a}
&&-\sigma(u) \nu  = g, \quad (x_1, x_3)\in \G^T\cup\G^L\cup\G^R,\label{s_bc1b}\\
&&u^s = 0, \quad (x_1, x_3)\in \G^B,\label{s_bc1f}\\
&&u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{s_bc1g}, 
\end{eqnarray*}
where 
\begin{eqnarray*}
g = \begin{cases} (0, \Delta G),\quad (x_1, x_3)\in \G^L,\\
(0, -\Delta G),\quad (x_1, x_3)\in \G^R,\\
(-\Delta G, 0), \quad (x_1, x_3)\in \G^T.\end{cases} 
\end{eqnarray*}


}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine 
the stiffness coefficients. V}}
\vskip-.5cm
{\bf 
{\small
The change in shape suffered by  the sample  is  

\begin{equation}\label{shear}
\tan [ \theta(\omega) ] =\frac{\Delta G}{{\blue { } p_{55}(\omega)}}.
\end{equation}
$\theta(\omega)$: the angle between the original positions of the 
lateral boundaries and the location after applying the shear stresses. 

Since  

$\tan [\theta(\omega)] \approx u^{s,T}_1(\omega)/L$, where 
$u^{s,T}_1(\omega)$  is the  average horizontal 
displacement at $\Gamma^T$,
$p_{55}(\omega)$ can be determined  from 

\vskip.2cm
to determine {\magenta { }$p_{66}(\omega)$}  (shear waves traveling in 
the $(x_1, x_2)$-plane), we 
rotate the layered sample  90$^{\rm o}$ and apply the shear test as 
indicated for {\blue { }$p_{55}(\omega)$}.

}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf  The harmonic experiments to determine the 
stiffness coefficients. VI}}
\vskip-.5cm
{\bf 
{\small
{\blue { }$p_{13}(\omega)$}: the  boundary conditions are 
\begin{eqnarray*}\label{bc1a_13}
&&\sigma(u) \nu \cdot \nu = -\Delta P, \quad (x_1, x_3)\in \G^R\cup \G^T,\label{bc1b_13}\\
&&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G,\label{bc1c_13}\\
&&u^s\cdot\nu = 0, \quad (x_1, x_3)\in \G^L\cup\G^B,\quad 
u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{bc1g_13}.
\end{eqnarray*}
In this experiment $\epsilon_{22}= \nabla\cdot u^f = 0$, so that 
\begin{eqnarray}\label{st_backus_2d}
&&\tau_{11} =  p_{11} \epsilon_{11} + {\magenta { } p_{13}} \epsilon_{33},\quad 
\tau_{33} = {\magenta { }p_{13} }\epsilon_{11} +  p_{33} \epsilon_{33},
\end{eqnarray}
$\epsilon_{11}, \epsilon_{33}$:  the 
strain components at the  right lateral side and  top side 
 of the sample, respectively. Then,  
\vskip-.2cm
%Then, once {\green { }$p_{11}$} and {\green { }$p_{33}$} have been determined, 
%\begin{equation} \label{p13}
%p_{13}(\o) = \frac {p_{11} 
%\epsilon_{11} - p_{33} \epsilon_{33}}{\epsilon_{11} - \epsilon_{33}}.\nonumber
%\end{equation} 
\[
p_{13}(\o) = \left(p_{11} \epsilon_{11} - p_{33} \epsilon_{33}\right)/
\left(\epsilon_{11} - \epsilon_{33}\right).
\]
%\end{equation} 

}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Schematic representation of 
the  oscillatory compressibility and shear  tests in $\O$ }}
\begin{center}
\vskip-.3cm
%\vspace*{0.1cm}
\includegraphics[width=7cm,height=6.5cm]{fig1.eps}\\
%\vspace*{-0.2cm}
{\small a): $p_{33}$,\hskip.3cm  b): $p_{11}$, \hskip.3cm c):
 $p_{55}$,\hskip.3cm  d): $p_{13}$ \hskip.3cm  e): $p_{66}$}
\end{center}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small{\bf  Examples}. I  }
\vskip-.8cm
{\bf 
{
A set of numerical examples consider the following  cases for a 
square poroelastic sample of 160 cm side length and 
10 periods of 1 cm fracture, 15 cm background:
\begin{itemize}

\item Case 1:  A brine-saturated sample with fractures.

\item Case 2:  A brine-CO$_2$ patchy saturated sample without fractures.

\item Case 3:  A brine-CO$_2$ patchy saturated sample with fractures.

\item Case 4: A brine saturated sample with a fractal frame and fractures. 
                  
\end{itemize}
The discrete boundary value problems to determine the complex stiffnesses $p_{IJ}(\o)$ 
 were solved for 30 frequencies using a public domain sparse matrix solver package. 
\vskip.01cm
Using relations  not included for brevity, the $p_{IJ}(\o)$'s determine  
in turn the energy velocities and  dissipation coefficients shown in the next figures. 


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


%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qP energy velocity vector 
at 50 Hz for cases  1,   2 and 3  }}}
\begin{center}
\vspace*{-.7cm}
\includegraphics[width=6.cm,height=6.cm]
{fig5_VqP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
\vspace*{0.3cm}
{\tiny Velocity anisotropy caused by the 
fractures in cases 1 and 3  is enhanced for the case of patchy saturation, 
with lower velocities when patches are present. The velocity behaves isotropically in case   2.
}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of the qP waves
at 50 Hz for cases  1,   2 and 3  }}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6.cm,height=6.cm]
{fig3_1000_over_qP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
\vspace*{0.3cm}
{\tiny Fractures induce strong Q anisotropy for angles normal to the fracture plane, enhanced by patchy saturation.}
\end{center}
\end{slide}


%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Fluid pressure distribution at 50 and 300  Hz. 
Compressibility test for $p_{33}$ for case 3.}}}
\vspace*{0.9cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=5.9cm]
{fig7_presion_fluido_p33_relation_1-15_10percent_sat_color_50Hz.eps} 
\includegraphics[width=5.9cm]{fig8_presion_fluido_p33_relation_1-15_10percent_sat_color_300Hz.eps}
%\vspace*{0.3cm}
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
{\small Compression is normal to the fracture plane. 
Attenuation is stronger at 300 Hz.}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qSV energy velocity vector 
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.7cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig12_VqSV_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny Velocity anisotropy is  induced by fractures (cases  1 and 3 ). Patchy saturation  
does not affect the anisotropic behavior of the  qSV velocities. Case  2 shows isotropic velocity, 
with higher velocity values than for the fractured cases}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of  qSV waves
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.5cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig10_1000_over_qsv_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny qSV anisotropy is 
strong for angles between 30 and  
60 degrees. The lossless case 2 is represented by a triangle at the
origin }
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qSH energy velocity vector 
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.7cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig13_VSH_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny SH velocity anisotropy  
is observed to be induced by fractures. Cases 1 and 3  are almost 
indistinguishable. Velocity for case 2 is isotropic. SH waves are lossles, since $p_{55}$ and $p_{66}$ are real.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Lam\'e coefficient (GPa) for the brine-saturated 
fractal porosity-permeability sample of case 4.}}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=8cm]
{fig20_lambda_fractal_relation_1-15_160x160_color.eps} 
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
{\tiny $\text{log} ~ \kappa(x,z) = \left\langle \text{log} ~\kappa \right\rangle + f(x,z)$,  $f(x,z)$: fractal representing the spatial 
fluctuation of the permeability  field $\kappa(x,z)$.}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of  qP waves
at 50 Hz for cases  1 and 4.  }}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.cm,height=6.cm]
{fig21_1000_over_qP_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny Note the increase  in Q anisotropy for qP waves for angles normal to the fracture plane.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of  qSV waves
at 50 Hz for cases  1 and 4.}}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.cm,height=6.cm]
{fig22_1000_over_qsv_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny Note the increase  in Q anisotropy for angles between 30 and 60 degrees}
\end{center}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Macroscale. I} }
\vskip-.5cm
{\bf 
{\small
The model:  $\O$ consists of   an
anisotropic rotated TIV layer between two isotropic half-spaces.

The upper half space has  P- and S-wave velocities equal to 1890 m/s and 592
m/s, respectively, while the lower half-space has P- and
S-wave velocities of 2320 m/s and 730 m/s.  

The anisotropic layer, with a thickness
of 300 m, is an  viscoelastic medium with five complex and frequency dependent 
stiffness obtained  using the upscaling procedure. The medium is rotated by 20 degrees , i.e., the angle between
the symmetry axis and the vertical direction. 

The average density in $\O$ is  2300  kg/m3.

The  mesh consists of square cells having  side length 3.7 m.

The source is a dilatational 
perturbation of central  frequency is 30 Hz,  located 150 m below the first interface (an asterisk in the figures).

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {The Macroscale. II.}}}
\begin{center}
\vspace*{-.4cm}
\includegraphics[width=7cm]
{model.eps} 
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
\vskip.2cm
{\small The Macroscopic TIV model.}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{\small{The Macroscale. III. Seismic modeling.}}

{\bf 
{\small
We solve the following boundary value problem at the macroscale (in $\O$):  
\begin{eqnarray*}\label{mod3}
&&\hskip-.5cm-\o^2  \rho u  -  \nabla\cdot\tau(u) = F,\quad \O\\
&&-\tau(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\text{(absorbing bounday condition, $D>0$)}\label{mod4}
\end{eqnarray*}
$u =(u_x, u_z)$: displacement vector, \hskip.2cm 
$ \rho$: average  density.
\vskip.1cm
\noi
$\tau(u)$:  stress-tensor of our {\magenta { } equivalent viscoelastic material}, defined in terms of the $p_{IJ}'s$.
\vskip.2cm   
Instead of solving the global problem associated with the above model, we obtained the solution using a parallel FE  iterative hybridized domain decomposition procedure employing a nonconforming FE space. 
\vskip.1cm 
A run in santalo under MPI with 100 processors required 10 hours of CPU time  for 1000x1000 mesh, 6.000.000 unknowns and  900 frequencies.
}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {The Macroscale. IV. Snapshots of the vertical displacement at 200 ms (a and c) and 500 ms (b and d ).}}}
\begin{center}
\vspace*{-.4cm}
\includegraphics[width=7.cm]
{fv6.eps}\\
{\tiny  Ideal isotropic case (left, a and b) and real (rotated TIV) case (right, c and d). }
\end{center}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small{The Macroscale. IV. CO$_2$ injection and seismic monitoring.}}
{\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 Example of injection  into the Utsira Sand, 
a saline aquifer at the Sleipner field, North Sea.

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

\end{itemize} 
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small {The Macroscale. V. 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}
{\tiny The thin lines represent mudstone  layers within the formation.
}
}
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------

\begin{slide}{\small{The Macroscale. VI. Injection Modeling. Black-Oil simulator.}}

%\vskip -.6cm
\begin{center}
\includegraphics[width=5.6cm,height=6.cm]{sat_CO2_600x400_frac_2a_10km_dt005_test3.gnu.eps} 
\includegraphics[width=5.6cm,height=6.cm]{sat_CO2_600x400_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}{\tiny{The Macroscale. VII. P-wave phase velocity $v_p(\o)$(m/s)
and attenuation coefficient $Q_p(\o)$   
at 50 Hz.}}

\vskip -.5cm
\begin{center}
\includegraphics[width=5.9cm,height=5.2cm]{vp_map_2a_gnu_ve_con_base.eps} 
\includegraphics[width=5.9cm,height=5.2cm]{qp_map_2a_gnu_ve_con_base.eps}\\

{\tiny Observe a 
decrease in  P- wave velocity  (left) and a  decrease in  attenuation coefficient $Q_p$ (right)  in  zones  
of CO$_2$ accumulation. Mesoscopic attenuation adn dispersion effects are taking into account using a White model.
}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Traces before and after 2 years of CO$_2$ injection. Line Source. Both figures shown in the SAME SCALE. }}
\vskip-.2cm
{\bf
{\small

%\vspace*{-.4cm}
\includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base_solo_brine.ps}\quad 
\includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base.ps}\\
%\vspace*{-0.2cm}
\tiny{ Traces of the  z-component of the particle velocity before (left) and after 
2 years (right) CO$_2$ injection. The two arrivals seen 
on the left Figure  come from the  
base and top of the Utsira. The strong later arrivals seen on the right Figure are due to P-waves reflected at  CO$_2$ accumulations and the base of the Utsira.}
}
}

\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Traces  before and after 2  and 6 years of CO$_2$ injection.  Line Source.}}
%\vskip-.7cm
{\bf
{\small

%\vspace*{-.4cm}
\includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base.ps}\quad 
\includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_6a_con_base.ps}\\
%\vspace*{-0.2cm}
\tiny{Traces of the  z-component of the particle velocity  after 
2 years (left) and 6 years (right) of CO$_2$ injection. The earlier 
arrivals on the right Figure  are due to waves reflected at the CO$_2$ 
accumulations below the shallow mudstone layers}.
}
}

\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf Conclusions} }
\vskip-.5cm
{\bf 
{\small
\begin{itemize}

\item Numerical upscaling procedures allow to represent mesoscopic scale heterogeneities in the solid frame and saturant fluids, fractures  and craks 
affecting observations at the macroscale.

\item THE FEM is a useful tool to solve  local problems in the context of Numerical Rock Physics and global problems at the macroscale. 

\item The techniques presented here to model acoustics of porous media can be extended to other fields, like ultrasound testing of quality of foods,  
groundwater flow and contamination  among others. 

\item Thanks for your attention !!!!!.

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

\end{document}

\end{document}

