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\title{Fractures and induced  anisotropy in poroelastic media. From the Mesoscale to the Macroscale}
\vskip-.8cm
\author{{\bf \Large {Juan E. Santos}}
}
\vskip-.3cm
\institution{%
Instituto del Gas y del Petr\'oleo,  {{\bf \blue Facultad de Ingenier\'\i a UBA, CONICET}}    
and Department of Mathematics, {{\bf \blue Purdue University}} and \\ 
Universidad Nacional de La Plata {{\bf\blue  (UNLP)}} \\ 
{Work  in collaboration with J.M. Carcione {{\bf\blue(OGS)}}, P. M. Gauzellino {{\bf\blue(UNLP)}}
\\
and R. Mart\'\i  nez Corredor {{\bf\blue(UNLP)}}
}
\vskip-.3cm
{{\bf \magenta IV International Conference on Approximation Methods for Design and Control, November 5th 2015 }}}
\begin{document}
\maketitle



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

\begin{slide}{\small Introduction. I}
{\bf 
{\small
\begin{itemize}

\item  A planar fracture embedded in a fluid-saturated poroelastic  -- Biot - medium 
can be modeled  as
a extremely thin,  highly permeable and  compliant 
porous layer.

\item
A Biot medium containing a dense set of aligned fractures behaves as an effective 
transversely isotropic 
and viscoelastic {\magenta (TIV)} 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 Introduction. II}
{\bf 
{\small
\begin{itemize}


\item
P-waves traveling in this type of medium 
induce fluid-pressure gradients at fractures and mesoscopic-scale heterogeneities, 
generating fluid flow and slow (diffusion) Biot waves, 
causing attenuation and dispersion of the fast modes (mesoscopic loss).


\item
A poroelastic medium with embedded aligned fractures exhibits 
significant attenuation and dispersion effects due to this mechanism.

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




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



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

\begin{slide}{\small Introduction. III}
{\bf 
{\small
\begin{itemize}




\item {\magenta { } Our approach:} In the context of {\blue Numerical Rock Physics},  
we use a {\magenta { } numerical upscaling procedure } to determine 
the complex and frequency dependent stiffness at the macroscale of a TIV   
 medium  equivalent to a Biot medium with aligned fractures.


\item To determine the {\blue{} 
 complex stiffness} coefficients of the 
{\magenta{}equivalent TIV  medium at the macroscale}, we 
solve a set  of 
 boundary value problems (BVP's) for Biot's equation in the diffusive range.
 
\item  The BVP's are stated  and solved in the space-frequency-domain using  the finite-element method {\magenta(FEM)}.


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



%----------------------------------------------- SLIDE -
\begin{slide}{\small Introduction. IV}
{\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 { \blue 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 {\bf The Mesoscale. Anisotropic poroelasticity. I}}
\vskip-.5cm
{\small
\begin{itemize}
\item 
For 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,  62, 1997) and  
Krzikalla and  M\"uller (GPY, 76, 2011) allow to  
obtain the {\blue { }five} complex and frequency-dependent stiffnesses of 
the {\magenta { }equivalent TIV medium}.  


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

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Mesoscale. Anisotropic poroelasticity. II}}
\vskip-.5cm
{\small
\begin{itemize}

\item We employ the FEM 
to solve in  the space-frequency  
domain  {\magenta { } Biot's equations in the diffusive range} 
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.

\item Applying a  Bond transfomation  matrix, the macroscale stiffness coefficients can be used to represent 
fractures at any orientation.

\end{itemize} 
}
\end{slide}



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

{\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 The TIV medium at the Macroscale.}}
\vskip-.5cm
{\bf 
{\small

${\mathbf \tau}_{{\mathbf i} {\mathbf j}}$: stress tensor of the equivalent TIV  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,\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\setminus \G^T,\label{bc1e}\\
&& u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{bc1g}.
\end{eqnarray*}
\noindent
{\blue { } $\nu$}:  the unit outer normal on $\G$
   
{\blue { } $\chi$}:  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} 
\[
{\magenta { }p_{13}}(\o) = \left(p_{11} \epsilon_{11} - p_{33} \epsilon_{33}\right)/
\left(\epsilon_{11} - \epsilon_{33}\right).
\]
%\end{equation} 

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Schematic representation of 
the  oscillatory compressibility and shear  tests.}}
\vskip-.3cm
\begin{center}
\vskip-.3cm
%\vspace*{0.1cm}
\includegraphics[scale=0.3]{5_experiments.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{Local degrees of freedom for the FEM 
solution of the harmonic  tests.}}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=7.cm]{apreton_dofs.eps}\\
\end{center}
\small{\bf The solution of the oscillatory tests was computed 
using the {\magenta FEM}. The figure displays the local 
degrees of freedom {\magenta (DOFs)} associated with each component 
of the solid displacement and the fluid displacement vectors. 
Once we computed the $p_{IJ}$ coefficients, we can determine the velocities and attenuation factors
of the equivalent TIV medium.
}
\end{slide}




%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf Velocities and attenuation in TIV media. I}}
\vskip-.5cm
{\bf 
{\small
Complex velocities  of the equivalent TIV anisotropic medium:
\begin{eqnarray*}\label{vpPS}
%\begin{array}{l}
&&v_{\rm qP} =  (2 \bar \rho )^{-1/2} \sqrt{ p_{11} l_1^2 + p_{33} l_3^2 + p_{55} + A }, \\
&&v_{\rm qSV} =  (2 \bar \rho )^{-1/2} \sqrt{ p_{11} l_1^2 + p_{33} l_3^2 + p_{55} - A }, \\
&&v_{\rm SH} =   \bar \rho^{-1/2}\sqrt{p_{66} l_1^2 + p_{55} l_3^2}, \\
&&\hskip-.6cm A = \sqrt{ [(p_{11} - p_{55}) l_1^2 + (p_{55} - p_{33}) l_3^2]^2 + 4 [(p_{13} + p_{55}) l_1 l_3]^2}.
\end{eqnarray*}
%\end{equation}
$\bar \rho = \langle \rho \rangle$: average bulk density,
\vskip.5cm
$l_1 = \sin \theta$,  $l_3 = \cos \theta$ are the directions
cosines, $\theta$ is the propagation angle between 
the wavenumber vector and the $x_3$-symmetry axis 
and the three velocities correspond to the qP, qS and SH waves, respectively.

}
}
\end{slide}

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

\begin{slide}{\small {\bf Velocities and attenuation in TIV media. II}}
\vskip-.5cm
{\bf 
{\small
The seismic phase velocity and quality factors are:
\begin{eqnarray*} \label{vph}
v_p = \left[{\rm Re} \left( \frac { 1 }{ v } \right) \right]^{-1} \ \ \ {\rm and} \quad Q = \frac { {\rm Re} (v^2) }{ {\rm Im} (v^2) }, 
\end{eqnarray*}
where $v$ represents either $v_{\rm qP}$, $v_{\rm qSV}$ or   $v_{\rm SH}$.

The energy-velocity vector ${\bf v}_e$ of the qP and qSV waves is 
\begin{eqnarray*} \label{57n}
\frac{{\bf v}_e}{v_p} = ( l_1 + l_3 \cot \psi )^{-1} \hat {\bf e}_1 +
( l_1 \tan \psi + l_3 )^{-1} \hat {\bf e}_3, 
\end{eqnarray*}
$\psi$: angle between the energy-velocity vector 
and the $x_3$-axis.

The energy velocity of the SH wave is
\begin{eqnarray*}\label{evelSHav}
{\bf v}_e =
\frac{1}{\bar \rho v_p} \left( l_1 p_{66} \hat {\bf e}_1+
l_3 p_{55} \hat {\bf e}_3 \right).
\end{eqnarray*}


}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small{\bf  The Mesoscale. 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
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 {\bf 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 {\bf 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 {\bf  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 {\bf 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 {\bf 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 {\bf 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 {\bf  $\text{log} ~ \kappa(x,z) = \left\langle \text{log} ~\kappa \right\rangle + f(x,z)$,  $f(x,z)$: fractal representing the spatial 
fluctuation of the permeability  field $\kappa(x,z)$.}}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qP energy velocity vector 
at 50 Hz for cases  1 and 4.  }}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.cm,height=6.cm]
{VP_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny {\bf 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 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 {\bf 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 {Polar representation of the qSV energy velocity vector 
at 50 Hz for cases  1 and 4.  }}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.cm,height=6.cm]
{VqSV_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny {\bf Note the energy velocity reduction in the   heterogeneous case. 
}}
\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 {\bf  Note the increase  in Q anisotropy for angles between 30 and 60 degrees. 
Also, velocity anisotropy is less affected by frame heterogeneities than Q-anisotropy.}}
\end{center}
\end{slide} 



%------------------------ --------------------------------Slide
\begin{slide}{\small{The Macroscale I. Seismic modeling. Variational formulation}}
\vskip-.5cm
{\bf 
{\small
We solve the following boundary value problem at the macroscale in a domain $\O$ with boundary $\p\O$:  
\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.

\noi
$\tau(u)$:  stress-tensor of our {\magenta { } equivalent TIV medium}, defined in terms of the $p_{IJ}'s$.
\vskip.2cm   
A global continuous variational formulation: Find $u(x,\omega) \in [H^1(\Omega)]^2$ such that
\begin{eqnarray*}\label{motion}
-(\rho\omega^2  u,\varphi)+ (\tau(u),
 \varepsilon(\varphi))
 + i    \omega \left\langle  {\mathcal D} u, \varphi \right\rangle_{\partial \O} = (
f, \varphi),\quad \varphi \in [H^1(\Omega)]^2 ,
\end{eqnarray*} 

}
}
\end{slide}

%------------------------ --------------------------------Slide
\begin{slide}{\small{The Macroscale. II. Global continuous and FE variational formulation.}}
\vskip-.7cm
{\bf 
{\small

${\mathcal T}^h$: a  partition of $\bar\Omega$ into rectangles of diameter bounded by $h$: 
\[
\bar\Omega=\cup_{j=1}^J \Omega_j
\]
$\Gamma_j=\partial\Omega\cap\partial\Omega_j$, $\Gamma_{jk}=\Gamma_{kj}=\partial\Omega_j\cap\partial\Omega_k$;
$\xi_{jk}$ the centroids of $\Gamma_j$ and $\Gamma_{jk}$.

Reference rectangular element
\[
R=[-1,1]^2, \qquad S(R)={\rm Span}\left\{1,x,z, \left(x^2-\frac{5}{3}
x^4 \right)- \left( z^2-\frac{5}{3} z^4 \right)\right\} .
\]
The DOF associated with  $S$: values at the mid points of the faces of $R$. 

If $a_1=(-1,0)$, $a_2=(0,-1)$, $a_3=(1,0)$ and $a_4=(0,1)$,

the basis function 

$\psi_1(x,z)=\frac{1}{4}-\frac{1}{2}x-\frac{3}{8}\left[ (x^2-\frac{5}{3}x^4)-(z^2-\frac{5}{3}z^4) \right]$ 

is such that 

$\psi_1(a_1)=1$ and $\psi_1(a_j)=0$, $j=2,3,4$.
}
}
\end{slide}



%------------------------ --------------------------------Slide
\begin{slide}{\small{The Macroscale. III.  Global FE variational formulation.}}

{\bf 
{\small

The  nonconforming FE space  ${\mathcal{NC}^h}$
$$
{\mathcal{NC}^h} =\{\varphi\in [L^2(\Omega)]^2: \varphi_j\in  [S(\Omega_j)]^2,\quad
 \varphi_j(\xi_{jk}) = \varphi_k(\xi_{jk}) \forall {j,k}\},
$$
$\varphi_j$: the restriction of $\varphi$ as seen from $\Omega_j$.
The global nonconforming FE procedure: Find $u^h \in {\mathcal{NC}^h}$ such that
\begin{eqnarray*}\label{nc-fem}
-(\rho\omega^2  u^h,\varphi)+ \sum_{j}(\tau(u^h),
 \varepsilon(\varphi))_{\Omega_j}
 + \ii   \omega \left\langle  {\mathcal D} u^h, \varphi \right\rangle_{\partial \O} = (
f, \varphi),\quad \varphi \in {\mathcal{NC}}^h.
\end{eqnarray*}
Instead of solving the global algebraic problem associated with the global FE procedure above,
 we will employ
the iterative domain decomposition procedure.

}
}
\end{slide}
%------------------------ --------------------------------Slide
\begin{slide}{\small{The Macroscale. IV. FE Domain decomposition.}}
\vskip-.7cm
{\bf 
{\small
Consider the decomposed problem over $\Omega_j$ satisfying  he interface consistency conditions 
\[
\tau_{jk} \nu_{jk} + i \beta_{jk}  u_{j}= -\tau_{kj}
\nu_{kj} + i \beta_{jk} u_{k}, \quad   \quad
x \in \Gamma_{jk}\subset \partial \Omega_j ,
\]
\[
\tau_{kj} \nu_{kj} + i \beta_{jk} 
u_{k}= -\tau_{jk} \nu_{jk} + i \beta_{jk}  u_{j}, \quad 
\quad x \in \Gamma_{kj} \subset \partial\Omega_k ,
\]
$\beta_{jk}$: positive definite iteration matrix functions defined on the interior boundaries $\Gamma_{jk}$.

To localize the calculations, we introduce a set of Lagrange multipliers $\lam^h_{jk}$:  
associated with the stress values $-\tau( u_j)\nu_{jk}(\xi_{jk})$:
\[
\widetilde\Lam^h=\{\lam^h\colon\lam^h|_{\G_{jk}}=\lam^h_{jk}\in[P_0(\G_{jk})]^2
= [\Lam^h_{jk}]^2\}.
\]  
$P_0(\G_{jk})$ are constant functions on $\G_{jk}$.
Note that $\Lam^h_{jk}$ and $\Lam^h_{kj}$ are considered to be distinct.

}
}
\end{slide}


\begin{slide}{\small  The Macroscale. V. FE Domain Decomposition Iteration. }
{\bf
{\small
Given an initial guess $
\left(\widehat u^{h,0}_{j},\lam^{h,0}_{jk},
\lam^{h,0}_{kj}\right)\in [NC^h_{j}]^2\times[\Lam^h_{jk}]^2
\times[\Lam^h_{kj}]^2$, 
compute$
\left(\widehat u^{h,n}_{j},\lam^{h,n}_{jk}\right)
\in [NC^h_{j}]^2\times [\Lam^h_{jk}]^2$  
as the solution of the equations
\begin{eqnarray*}
\label{exp8}
&&-(\rho\omega^2 \widehat u^{h,n}_{j},\var)_j+
\sum_{pq}(\tau_{pq}(\widehat u^{h,n}),\varepsilon_{pq}(\var))_j
+i\omega\left\langle\left\langle
{\mathcal{D}}\widehat u_j^{h,n},\var\right\rangle\right\rangle_{\G_j}\nonumber \\
&&\quad+\sum_{k}\left\langle\left\langle\lam^{h,n}_{jk},
\var\right\rangle\right\rangle_{\G_{jk}}=(\widehat f,\var)_j,
\quad\var\in [NC^h_{j}]^N,\\
&&\lam^{h,n}_{jk}=-\lam^{h,n-1}_{kj}+i\b_{jk}[\widehat u^{h,n}_{j}(\xi_{jk})-
\widehat u^{h,n-1}_{k}(\xi_{jk})],\qquad\mbox{ on }\G_{jk}.
\label{exp9}
\end{eqnarray*}
}
}
It can be shown that if $u^h$ is the solution of the global FE procedure,
\[
\|\widehat u^{h,n}- u^h\|_0\rightarrow0\mbox{ in }[L^2(\O)]^2 
\quad\mbox{when}\quad n\rightarrow\infty,
\]
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Macroscale Model. I} }

{\bf 
{\small
\begin{itemize}
\item
The macroscopic model  consists of an isotropic square 
background $\O$  of 1.6 km side length  
containing an anisotropic  layer of 400m thickness of either horizontal or vertical aligned fractures (HTI or VTI medium).
\item
The  stiffnesses  coefficients of the anisotropic layer  are those 
determined using the harmonic experiments, for the cases of fully brine saturated fractures (Case 1) or 
 10 \%  patchy CO$_2$  saturation (Case 3).   
\item
The isotropic background  has  P- and S-wave velocities at 50 Hz  equal to 2633 m/s and 1270
m/s, respectively. 

\end{itemize}
}
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The Macroscale Model. II} }

{\bf 
{\small
\begin{itemize}

\item
For the HTI-case,  qP- and qSV-wave velocities at 50 Hz are equal to 3808. m/s and 1686 m/s, 
respectively, while for the  VTI-case  qP- and qSV-wave velocities at 50 Hz are equal to 3008. m/s and 1686
m/s, respectively
\item
The  computational mesh consists of square cells having  side length 4 m, and 
The source is a dilatational 
perturbation of central  frequency is 50 Hz,  located at $(x = 800 m, z = 6. m)$. 
\end{itemize}
}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{The Macroscale Model. III}}
\vskip-0.5cm
{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[width=5.5cm,height=6.5cm]{modelo_hti.eps} & \includegraphics[width=5.5cm,height=6.5cm]{modelo_vti.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{}
\end{center}

}
}
\end{slide}


%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Snapshots  of z-component of velocity at 150ms (left) and 200 ms (right). HTI  brine saturated fractures.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.45]{vsz_hti_brine_150ms_dado_vuelta.eps} & \includegraphics[scale=0.45]{vsz_hti_brine_200ms_dado_vuelta.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{At 150 ms the incident P-wave is arriving to the top of the HTI layer. At 200 ms a reflected P-wave travelling upwards 
and a transmitted qP-wave travelling downwards are observed.
The other wavefronts are S-waves generated by the point source at the top boundary. }
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Snapshots  of z-component of velocity at 220ms (left) and 240 ms (right). HTI  brine saturated fractures.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.45]{vsz_hti_brine_220ms_dado_vuelta.eps} & \includegraphics[scale=0.45]{vsz_hti_brine_240ms_dado_vuelta.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{At 220 ms  and 240 ms  a  reflected P-wave and travelling upwards  and a transmitted qP-wave travelling downwards are observed.
Note the deformation of the qP-wavefront because of the anisotropy in the HTI layer. 
The other (thin) wavefronts are S-waves generated by the point source at the top boundary.}
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Snapshots  of z-component of velocity at 280ms (left) and 300 ms (right). HTI  brine saturated fractures.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.45]{vsz_hti_brine_280ms_dado_vuelta.eps} & \includegraphics[scale=0.45]{vsz_hti_brine_300ms_dado_vuelta.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{At 280   and 300 ms we can observe an upgoing P-wavefront  and a tiny upgoing S-wavefront reflected at the top of the HTI layer 
and an upgoing  qP-wavefront  generated by reflection of the downgoing qP-wave at the bottom of the HTI layer. 
At 300 ms the reflected P-wave  generated  at the top of the HTI layer is arriving at the surface geophones. }
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Snapshots  of z-component of velocity at 380ms (left) and 500 ms (right). HTI  brine saturated fractures.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.45]{vsz_hti_brine_380ms_dado_vuelta.eps} & \includegraphics[scale=0.45]{vsz_hti_brine_500ms_dado_vuelta.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{At 380 and 500 ms we can observe the reflected S-wave arriving  at at the surface geophones, together with the qP-P wave 
reflected at the bottom of the  HTI layer. The transmitted qP wave has 
already left the HTI layer and continues travelling downwards as a converted  P-wave. Conversions to S-waves 
can also be seen in both snapshots}
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Synthetic traces. HTI and VTI brine  saturated fractures.}}
%\vskip.3cm
{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c }
\includegraphics[scale=0.38]{trace_80_hti_vti_brine_point_source.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{ }
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Synthetic traces. HTI  brine  and patchy brine-CO$_2$ saturated fractures.}}
{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c }
\includegraphics[scale=0.38]{trace_80_hti_brine_10co2_point_source.eps}\\
\end{tabular}

\end{table}
\vskip-0.5cm
\tiny{}
\end{center}

}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Synthetic seismograms. HTI (left) and VTI (right) brine saturated fractures.}}
\vskip-1.1cm
{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.3]{brinehti.eps} & \includegraphics[scale=0.3]{brinevti.eps}\\
\end{tabular}

\end{table}
\vskip-0.9cm
\tiny{The seismograms show the arrivals of P and S waves reflected at the top of the HTI and VTI layers, as well as 
later arrivals corresponding to conversions from incident P to qP and qSV waves at the top and bottom of the layers. In particular, it is clearly seen the late  {\magenta P-qP-P arrival} of the VTI case as compared with the corresponding one in the HTI case. The qP-velocities in the HTI and VTI layers are about 3800 m/s and 3000m/s, respectively.}
\end{center}

}
}
\end{slide}
%----------------------------------------------- SLIDE -
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Synthetic seismograms. HTI brine (left) and 10 \% CO$_2$ (right)  saturated fractures.}}
\vskip-1.1cm
{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.3]{brinehti.eps} & \includegraphics[scale=0.3]{htico2.eps}\\
\end{tabular}

\end{table}
\vskip-1.cm
\tiny{The seismograms show the arrivals of P and S waves reflected at the top of the HTI layer, and  
later arrivals corresponding to conversions from incident P to qP and qSV waves at the top and bottom of the layer.
In particular, the  {\magenta P-qP-qP-qP-qP-qP-P arrival} in the brine-saturated case is not seen in the patchy brine-CO$_2$ case  due to the strong attenuation of qP waves when CO$_2$ is present ($Qp\approx 10$ at normal incidence). Instead qSV-waves are less attenuated when CO$_2$ is present and the {\magenta P-qP-qSV-S-arrival} is still observed. }
\end{center}

}
}

\end{slide}




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

\item Numerical upscaling procedures allow to represent  poroelastic samples with a dense set of  aligned fractures 
at the mesoscopic-scale saturated by different fluids, bringing  the induced anisotropy  at the macroscale.

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

\item The techniques presented here to model acoustics of porous media in geophysics 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}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Snapshots  of z-component of velocity at 280ms (left) and 300 ms (right). HTI  brine saturated fractures.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.45]{vsz_hti_brine_280ms_dado_vuelta.eps} & \includegraphics[scale=0.45]{vsz_hti_brine_300ms_dado_vuelta.eps}\\
\end{tabular}

\end{table}
\vskip.5cm
\tiny{At 280  and  300 ms P and S waves reflected at the top of the HTI layer and a qP wave reflected from the bottom of the HTI layer 
travelling downwards are observed. At 300 ms the P-wave reflected at the top of the HTI layer has arrived to the geophones located at the surface.
Note the deformation of the qP-wavefront because of the anisotropy in the HTI layer.}
\end{center}

}
}
\end{slide}
%----------------------------------------------- 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  Numerical Solution at the Macroscale - Finite Element Method. }
{\bf 
{\small
\begin{itemize}
\item 
The FE space-frequency solution of the viscoelastic wave equation  was  computed  
at a selected number of frequencies 
in the range of interest  using an {\magenta { }iterative 
 FE  domain decomposition procedure}.  

\item
To approximate each component of the solid displacement vector we employed a 
{\magenta { } nonconforming FE  space} $ NC^h$ defined over a partition of the domain $\O$ 
into rectangles of diameter bonded by $h$. 

\item  The use of the FE space $ NC^h$  generates {\magenta { } less numerical dispersion 
than the standard bilinear elements}. 
The  error of the FE procedure is of order $h^{1/2}$

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

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

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

 
\begin{slide}{\small  Numerical Solution - Finite Element Method. }
{\bf 
{\small
Lagrange multipliers $\lam^h_{jk}$:  
associated with the stress values $-\sigma( u_j)\nu_{jk}(\xi_{jk})$:
\[
\widetilde\Lam^h=\{\lam^h\colon\lam^h|_{\G_{jk}}=\lam^h_{jk}\in[P_0(\G_{jk})]^2
= [\Lam^h_{jk}]^2\}.
\]  
$P_0(\G_{jk})$ are constant functions on $\G_{jk}$.
Note that $\Lam^h_{jk}$ and $\Lam^h_{kj}$ are considered to be distinct.
A discrete domain decomposition (hybridized) iterative algorithm:
}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Reflection coefficients for P1 and P2 waves at a fracture.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.26]{Figura_4a.eps} & \includegraphics[scale=0.26]{Figura_4c.eps}\\
\end{tabular}

\end{table}

\tiny{Absolute value of reflection coefficients of  P1 and P2 waves for a normally incident P1 wave. Permeability $\kappa_0 \rightarrow \infty$.
Fracture aperture $h$  varies from 0.001 m to 0.00001 m.}
\end{center}

}
}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{Transmission coefficients of P1 and P2 waves across a fracture.}}

{\bf
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.26]{Figura_4b.eps} & \includegraphics[scale=0.26]{Figura_4d.eps}\\
\end{tabular}

\end{table}

\tiny{Absolute value of transmission coefficients of  P1 and P2 waves for a normally incident P1 wave. Permeability $\kappa_0 \rightarrow \infty$. 
Fracture aperture $h$  varies from 0.001 m to 0.00001 m.}
\end{center}

}
}
\end{slide}
