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\title{Seismic response of fractures and induced  anisotropy in poroelastic media}
%in heterogeneous porous media}
%\subtitle{in Fluid Saturated  Poroviscoelastic Media}

\subtitle{SEG Rock Physics Workshop, Quindao,   October 25-27, 2019
 } 
\author{{\bf Juan E. Santos}}%${^\dag}$ }
\institute{%
Instituto del Gas y del 
Petr\'oleo (IGPUBA), Universidad de Buenos Aires (UBA), Argentina,
 Department of Mathematics, Purdue University,  West Lafayette, Indiana, USA,
 %and Universidad
%Nacional de La Plata (UNLP), Argentina.
 \\
%{\green  {\bf  
{{
%\color {magenta}  In collaboration with P. M. Gauzellino (UNLP), G. B. Savioli (UBA), 
%A. Sanchez Camus (UNLP) and  R. 
%Martinez Corredor, (UNLP)
}}.  
}
%}}


\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Sketch of the talk}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}



\section{Introduction}
\begin{frame}
\frametitle{{\small Seismic waves in fractured fluid-saturated poroelastic materials. I}}
{\small
\begin{itemize}
%{\color {black}Fractures}

\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 
{\color {blue}   viscoelastic transversely isotropic 
  medium (VTI) }
 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{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{{\small Seismic waves in fractured fluid-saturated poroelastic materials. II}}
{\small
\begin{itemize}
%{\color {black}Fractures}

\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 {\color {blue}(mesoscopic loss).}


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

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


\end{itemize} 
}
\end{frame}
%\end{document}


 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{{\small Seismic waves in fractured fluid-saturated poroelastic materials. III}}
{\small
\begin{itemize}
%{\color {black}Fractures}

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


\item To determine the  
 complex stiffness coefficients of the 
{\color {magenta} equivalent VTI  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 {\color {blue}(FEM)}.



\end{itemize} 
}
\end{frame}






\begin{frame}
\frametitle{{\small Seismic waves in fractured fluid-saturated poroelastic materials. IV}}
{\small
\begin{itemize}

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

\item  {\color {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  {\color {blue}  repeatable, essentially
free from experimental errors}, and may easily be run using 
alternative models of the rock and  fluid properties.

\end{itemize} 
}
\end{frame}

\begin{frame}
\frametitle{{\small The Mesoscale. Anisotropic poroelasticity. I}}
{\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  {\color {blue} five} complex and frequency-dependent stiffnesses of 
the  {\color {magenta} equivalent VTI medium}.  

\item To provide a more general modeling tool, 
we present a {\color {magenta} numerical  upscaling procedure} to obtain the 
complex stiffnesses of the effective  {\color {blue}   VTI} medium.

\end{itemize} 
}
\end{frame}



\begin{frame}
\frametitle{{\small The Mesoscale. Anisotropic poroelasticity. II}}
{\small
\begin{itemize}

\item We employ the FEM 
to solve in  the space-frequency  
domain  {\color {blue} Biot's equations in the diffusive range} 
with boundary conditions representing 
{\color {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{frame}



\begin{frame}
\frametitle{{\small The Mesoscale. Anisotropic poroelasticity. III}}
{\bf \small

Consider isotropic fluid-saturated poroelastic layers. 

 
{\color {blue}  $ u^s(x),  u^f(x)$ } : time Fourier transform of the 
displacement vector of the solid and fluid relative to the solid   
frame, respectively.

 
\smallskip

{\color {blue} $u  = (u^s, u^f)$}
\vskip.2cm
 
{\color {magenta} $ \sigma_{kl}(u),  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 {\color {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{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SLIDE


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



\begin{frame}
\frametitle{{\small The Mesoscale. Anisotropic poroelasticity. IV}}
{\bf \small
${\mathbf \tau}_{{\mathbf i} {\mathbf j}}$: stress tensor of the equivalent VTI  medium

For a {\color {blue} closed system( $\nabla\cdot u^f  = 0) $}, 
the corresponding {\color {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 {\color {magenta} interlayer flow effects}.}

}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small The experiments to determine the five $p_{IJ}$ VTI stiffness}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
%\includegraphics[width=7.3cm,height=7.3cm]
%{fig5_VqP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}
\end{center}
\vskip-.5cm
{\tiny (I) : Figs (a) and (b) show how to determine  $p_{33}$ and $p_{11}$, 
(c) determines $p_{55}$,  (e) determines   $p_{66}$ and (d)  determines   $p_{13}$.
}
}
\end{frame}



%----------------------------------------------- SLIDE -
\begin{frame}
\frametitle{{\small Local degrees of freedom for the FEM 
solution of the harmonic  tests. }}
{\small
\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 {\color {blue} FEM}. The figure displays the local 
degrees of freedom {\color{magenta} (DOFs)} associated with each component 
of the solid displacement and the fluid displacement vectors. 

}


}
\end{frame}

\section{Numerical Experiments}
%----------------------------------------------- SLIDE -
\begin{frame}
\frametitle{{\small Numerical Experiments. I}}
{\small

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 complex stiffnesses $p_{IJ}(\omega)$ 
 were determined for 30 frequencies using a public domain sparse matrix solver. 
\vskip.01cm
The $p_{IJ}(\omega)$'s determine  
 the energy velocities and  dissipation coefficients shown in the next figures. 
}
\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of energy velocity vector   of qP waves 
at 50 Hz for Cases 1, 2 and 3}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig5_VqP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}
\end{center}
\vskip-.5cm
{\tiny Cases 1 and 3  show velocity anisotropy caused by  
fractures. Case 3 (patchy saturation) exhibits lower velocities. Velocity behaves isotropically in Case 2.
}
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of  dissipation factor  of qP waves 
at 50 Hz for Cases 1, 2 and 3}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig3_1000_over_qP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}
\end{center}
\vskip-.5cm
{\tiny Fractures induce strong Q anisotropy for angles normal to the fracture plane, enhanced by patchy saturation.
}
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Fluid pressure distribution at 50 Hz. 
Compressibility test for $p_{33}$ for Case 3.}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=9.cm]
{fig7_presion_fluido_p33_relation_1-15_10percent_sat_color_50Hz.eps}
\end{center}
\vskip-.5cm
{\tiny  Compression is normal to the fracture plane. This Figure illustrates the mesoscopic loss effect.
}
}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of energy velocity vector   of qSV waves 
at 50 Hz for Cases 1, 2 and 3}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig12_VqSV_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}
\end{center}
\vskip-.5cm
{\tiny  Velocity anisotropy is  induced by fractures (Cases  1 and 3 ). Patchy saturation  
does not affect the anisotropic behavior of  qSV velocities. Case  2 is isotropic  
with higher velocity values than for the fractured cases.
}
}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of dissipation factor  of qSV waves 
at 50 Hz for Cases 1, 2 and 3}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig10_1000_over_qsv_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}
\end{center}
\vskip-.5cm
{\tiny  qSV attenuation anisotropy is 
strong for angles between 30 and  
60 degrees. The lossless Case 2 is represented by a triangle at the
origin.
}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of energy velocity of SH waves  
at 50 Hz for Cases  1,   2 and 3  }}
\vskip-.5cm
{\small
%\centering
\begin{center}
\includegraphics[width=7cm]{fig13_VSH_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps} 
\end{center}
\vskip-.4cm
{\tiny {\bf SH velocity anisotropy  is induced by fractures. 
(Cases 1 and 3). Velocity for Case 2 is isotropic. SH waves are lossles since $p_{55}$ and $p_{66}$ are real.}
}
}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Lam\'e coefficient (GPa) for the brine-saturated 
fractal porosity-permeability sample of Case 4. }}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.3cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=9.cm]
{fig20_lambda_fractal_relation_1-15_160x160_color.eps}
\end{center}
\vskip-.5cm
{\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{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of energy velocity vector of qP waves  
at 50 Hz for Cases  1 and 4.}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.5cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{VP_relation_1-15_fractal_por_perm_50Hz_color.eps}
\end{center}
\vskip-.7cm
{\tiny {\bf  Note the decrease  in velocity of qP waves  for angles normal to the fracture plane.}
}
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of dissipation factor of qP waves  
at 50 Hz for Cases  1 and 4.}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.5cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig21_1000_over_qP_relation_1-15_fractal_por_perm_50Hz_color.eps}
\end{center}
\vskip-.7cm
{\tiny {\bf  Note the increase  in Q anisotropy of  qP waves for angles normal to the fracture plane.}
}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation of energy velocity vector of qSV waves  
at 50 Hz for Cases  1 and 4.}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.5cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{VqSV_relation_1-15_fractal_por_perm_50Hz_color.eps}
\end{center}
\vskip-.7cm
{\tiny {\bf  There is decrease in qSV velocity for Case 4}
}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Polar representation ofdissipation factor  of qSV waves  
at 50 Hz for Cases  1 and 4.}}
%\vskip-2.5cm
{\tiny
%\centering
\begin{center}
\vskip-.5cm
%\includegraphics[width=7.3cm,height=7.3cm]{fig1_5exper.eps}
\includegraphics[width=7.3cm,height=7.3cm]
{fig22_1000_over_qsv_relation_1-15_fractal_por_perm_50Hz_color.eps}
\end{center}
\vskip-.7cm
{\tiny {\bf   Attenuation  of qSV waves is strong for angles between 30 and 60 degrees, and higher in Case 4.}
}
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small The Macroscale I. Seismic modeling}}
%\vskip-2.5cm
{\small
We solve the following boundary value problem at the macroscale in a domain $\Omega$ with boundary $\p\Omega$:  
\begin{eqnarray*}\label{mod3}
&&\hskip-.5cm \omega^2  \rho u  +  \nabla\cdot\tau(u) = F,\quad \Omega\\
&&-\tau(u) \nu = i \omega {\mathcal D} u,\quad \partial \Omega,\text{(absorbing B. C., $D>0$)}\label{mod4}
\end{eqnarray*}
$u =(u_x, u_z)$: macroscopic displacement vector, \hskip.2cm 
$ \rho$: average  density.

\noi
$\tau(u)$:  stress-tensor of our {\color{magenta} equivalent VTI medium}, defined in terms of the $p_{IJ}'s$.
\vskip.2cm  

The solution of the above wave equation was computed at a range of frequencies of interest 
using an iterative FE domain decomposition procedure.

}
\end{frame}
%----------------------------------------------- SLIDE -
\begin{frame}
\frametitle{{\small  The Macroscale Model. I}}
{\small

\begin{itemize}
\item
The macroscopic model  consists of an isotropic square 
background $\Omega$  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{frame}

%----------------------------------------------- SLIDE -
\begin{frame}
\frametitle{{\small  The Macroscale Model. II}}
{\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{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small The macroscopic model. III }}
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\hskip-0.6cm\includegraphics[width=5.cm,height=6.5cm]{modelo_hti.eps} & \includegraphics[width=5.cm,height=6.5cm]{modelo_vti.eps}\\
\end{tabular}

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

}


\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small Snapshots  of z-component of velocity at 280ms (left) and 300 ms (right). HTI  brine saturated fractures. }}
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\hskip-0.6cm\includegraphics[scale=0.4]{vsz_hti_brine_280ms_dado_vuelta.eps} & 
\includegraphics[scale=0.4]{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{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small Synthetic traces. HTI and VTI brine  saturated fractures. }}
{\small
\begin{center}
\begin{table}[H]
\begin{tabular}{c }
\includegraphics[scale=0.35]{trace_80_hti_vti_brine_point_source.eps}
\end{tabular}

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

}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\small Synthetic traces. HTI brine  and patchy brine-CO$_2$ saturated fractures. }}
{\small

\begin{center}
\begin{table}[H]
\begin{tabular}{c }
\includegraphics[scale=0.31]{trace_80_hti_brine_10co2_point_source.eps}
\end{tabular}

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

\end{center}

}


\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny Synthetic seismograms. HTI (left) and VTI (right) brine saturated fractures. }}

{\small

%\begin{center}
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.28]{brinehti.eps} & \includegraphics[scale=0.28]{brinevti.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 and VTI layers, and 
later arrivals from  conversions from incident P to qP and qSV waves at the top and bottom of the layers. It is clearly seen the {\color{blue} late } {\color{magenta} P-qP-P arrival} of the VTI case as compared with 
the corresponding one in the HTI case. qP-velocities in the HTI and VTI layers are about 3800 m/s and 3000m/s, respectively.}
%\end{center}

}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{\bf {\tiny  Synthetic seismograms. HTI brine (left) and 10 \% CO$_2$ (right)  saturated fractures.}}
\vskip-.9cm
{\small
%\centering
\begin{center}
\vskip-.7cm
%\hskip-.5cm
\begin{table}[H]
\begin{tabular}{c c}
\includegraphics[scale=0.28]
{brinehti.eps} &
\includegraphics[scale=0.28]{htico2.eps}
\end{tabular}

\end{table}
\end{center}
\vskip-.7cm
{\tiny
Seismograms show  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. Note that the  {\color{magenta} P-qP-qP-qP-qP-qP-P arrival}  in the
brine-saturated case is not seen in the patchy brine-CO$_2$ case (strong attenuation $Qp\approx 10$ at normal incidence). Instead qSV-waves are less attenuated when CO$_2$ is present and the {\color{magenta} P-qP-qSV-S-arrival} is still observed.
}
}
\end{frame}


\begin{frame}
\frametitle{{Conclusions }}
{\small
\begin{itemize}
%{\color {black}Fractures}

\item
Numerical upscaling procedures allow to represent  at the {\color{blue} macroscale }anisotropy 
induced by  a dense set of {\color{magenta}  aligned fractures} at {\color{magenta} mesoscopic scales} in   
fluid-saturated poroelastic samples.


\item
The {\color{blue} Finite Element Method } is a useful tool to solve  local problems at the mesoscale  in the context of Numerical Rock Physics 
 and {\color{blue} 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

 {\color{magenta} Thanks for your attention !!!!!.}

\end{itemize} 
}
\end{frame}

\end{document}

