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\title{Wave Propagation in Fractured Poroelastic Media}
%in heterogeneous porous media}
%\subtitle{in Fluid Saturated  Poroviscoelastic Media}

\subtitle{{\magenta  \small{\bf {WCCM, MS170: Advanced Computational 
Techniques in Geophysical Sciences, \\
Barcelona, Spain, July 2014}}} }
\author{{\bf Juan E. Santos}${^\dag}$ }
\institution{%
$^\dag$ Instituto del Gas y del 
Petr\'oleo (IGPUBA), UBA, Argentina, and Universidad
Nacional de La Plata (UNLP), Argentina and  Department of Mathematics, Purdue University, Indiana, USA,
\\
{\blue  {\bf  work in collaboration with J. M. Carcione (OGS,  Italy) and S. Picotti, R. Martinez Corredor (UNLP)}.  
}}
\begin{document}
\maketitle
%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. I}
{\bf 
{\small
\begin{itemize}
\item {\blue {}Fractures}{\black{} are common in the earth's crust due  to different factors, for instance, 
tectonic stresses and natural or artificial hydraulic 
fracturing caused by a pressurized fluid. }

\item {\magenta {} Seismic wave propagation}{\black{} through } {\magenta {}fractures and cracks} 
{\black{} is an important subject
in exploration and production geophysics, earthquake seismology and mining.}

\item {\black {} In geophysical prospecting, reservoir
development and CO2 storage in geological formations,  knowledge of} {\blue {}  fracture orientation, densities and sizes} {\black {}is essential
since these factors control hydrocarbon production.}

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


%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. II}
{\bf
{\small
\begin{itemize}
\item {\magenta {} A planar fracture} {\black {} embedded in a  Biot  background is a
particular case of the} {\blue {} thin layer problem}, {\black {} when one of the layers is very thin,
highly permeable and compliant.}

\item  {\black {}  A Biot
medium containing a dense set of }{\magenta {} aligned fractures} {\black {}  behaves as an effective 
transversely isotropic and viscoelastic } {\magenta {}  (TIV) medium at the macroscale} {\black {}  when the 
predominant wavelength is much larger than the average distance between fractures}.

\item {\black {} One important mechanism in Biot media at seismic frequencies is} {\magenta {} wave-induced fluid flow} 
{\black {} generated by fast compressional waves at mesoscopic-scale heterogeneities, 
generating slow diffusion-type Biot waves.} 


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



%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. III}
{\bf
{\small
\vskip-.5cm
\begin{itemize}
\item {\black {} In the context of} {\magenta {} Numerical Rock Physics},{\black {}  we present and analyze a set 
of time-harmonic finite element (FE) experiments that take into account the effects of the
presence of aligned fractures and interlayer fluid flow occurring at the mesoscale}.

\item {\black {} These numerical experiments allow  to determine the complex and frequency dependent 
stiffnesses of the}
 {\magenta {}  effective TIV medium}  {\black {} at the macroscale}. {\black {} They  are defined as} {\magenta {} BVP}  
{\black {}   on representative samples  of the fractured material, with boundary 
conditions associated with compressibility and shear tests, which are solved using the 
FE method.}
\end{itemize} 
}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf Anisotropic poroelasticity. I}}

{\bf
{\small
\begin{itemize}
\item 
{\black {} 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 {\black {}  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} {\black {} complex and frequency-dependent stiffnesses of 
the }{\magenta { }equivalent TIV medium}.  


\item {\black {} To test the model and provide a more general modeling tool, 
we present a} {\blue { } FE  upscaling procedure} {\black {} to obtain the 
complex stiffnesses of the} {\magenta { }equivalent  TIV medium.}


\item {\black {} The samples contain  mesoscopic-scale heterogeneities due to 
patchy brine-CO$_2$ 
saturation and fractal porosity (fractal frame properties).}

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


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf TIV media and fine layering. I}}
\vskip-.5cm
{\bf 
{\small
 {\black {}Let us consider isotropic fluid-saturated poroelastic layers.} 
 
{\blue { }   $ {\mathbf u^s}({\mathbf x}), {\mathbf u^f}({\mathbf x})$ } {\black {} : 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 { }${\mathbf \sigma_{kl}({\mathbf u})}, {\mathbf p_f({\mathbf u})}$} {\black {}: Fourier transform of the total stress 
and the fluid pressure, respectively.

{\magenta { }$\be(\bu_s)$} the  strain 
tensor of the solid phase. }
\vskip.3cm
 {\black {} On each plane layer $n$ in a 
sequence of $N$ layers, the }{\magenta { }frequency-domain  stress-strain relations }
 {\black {} are}
\begin{eqnarray*}\label{st_1}
&{\mathbf \sigma}_{kl}({\mathbf u}) =  2 \mu\,e_{kl}({\mathbf u}^s)+\delta_{kl}
\left( \lambda_{_G} \,\nabla\cdot {\mathbf u}^s  + \alpha M  \nabla\cdot {\mathbf u}^f \right),\\
&{\mathbf p_f({\mathbf u}) }=  - \alpha M \nabla\cdot {\mathbf u}^s -  M  \nabla\cdot {\mathbf u}^f.\label{st_7}
\end{eqnarray*} 

}
}

\end{slide}


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

{\bf 
{\small
\vskip-1.cm
\begin{eqnarray*}\label{mod.2}
&&\alpha = 1 - \dfrac{K_m}{K_s},\quad 
M = \left[\dfrac{\alpha- \phi}{K_s} 
+ \dfrac{\phi}{K_f}\right]^{-1}.\nonumber\nn\\
&& K_G = K_m + \alpha^2 ~M,\quad \lambda_G = K_G  - \dfrac23 \mu,\nonumber
\end{eqnarray*} 
$\mu$: shear modulus of the dry matrix, 
\vskip.1cm
$K_m, Ks, K_f, K_G$: bulk modulus 
of the dry matrix, the solid grains, the saturant fluid, and the saturated bulk material, respectively.

\vskip.1cm
{\magenta { } Biot's equations in the diffusive range:}
\begin{eqnarray*}\label{motion1}
&&\nabla\cdot{\mathbf \sigma}({\mathbf u})  = 0, \\
&& \ii \o  \dfrac{\eta}{\kappa} {\mathbf u}^f({\mathbf x},\o)  + \nabla p_f( {\mathbf u} )=0,\label{motion2}
\end{eqnarray*}
\vskip-.1cm
$\eta, \kappa$ {\black {}: fluid viscosity and frame permeability, respectively, }

$\o=2\pi f$ {\black {}: angular frequency}
\vskip.1cm

%\hskip1cm
%$\kappa$ {\black {}: frame permeability}


}
}

\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf TIV media and fine layering. III}}
\vskip-.5cm
{\bf 
{\small
$\tau_{ij}(\widetilde \bu_s), \epsilon_{ij}(\widetilde \bu_s)${\black {}:  stress and strain tensor components 
of the equivalent TIV medium,} 

$\widetilde \bu_s${\black {}: solid displacement vector at the macroscale.}

{\black {}The stress-strain relations (assuming  a closed system ($\nabla\cdot \bu_f = 0$)) }
\begin{eqnarray*}                                                           
&&\tau_{11}(\widetilde \bu_s) = {p}_{11} ~\epsilon_{11}(\widetilde \bu_s) + p_{12} ~\epsilon_{22}(\widetilde \bu_s) 
+ p_{13} ~\epsilon_{33}(\widetilde \bu_s), \label{st_GL_1}\\
&&\tau_{22}(\widetilde \bu_s) = p_{12} ~\epsilon_{11}(\widetilde \bu_s) + p_{11} ~\epsilon_{22}(\widetilde \bu_s)    
 + p_{13} ~\epsilon_{33}(\widetilde \bu_s), \label{st_GL_2}\\  
&&\tau_{33}(\widetilde \bu_s) =  p_{13} ~\epsilon_{11}(\widetilde \bu_s) + p_{13} ~\epsilon_{22}(\widetilde \bu_s)      
+ p_{33} ~\epsilon_{33}(\widetilde \bu_s), \label{st_GL_3}\\   
&&\tau_{23}(\widetilde \bu_s) =  2~  p_{55} ~\epsilon_{23}(\widetilde \bu_s), \label{st_GL_4}\quad                     
\tau_{13}(\widetilde \bu_s) = 2 ~  p_{55} ~\epsilon_{13}(\widetilde \bu_s),\label{st_GL_5} \\                    
&&\tau_{12}(\widetilde \bu_s) = 2 ~  p_{66} ~\epsilon_{12}(\widetilde \bu_s). \label{st_GL_6}  
\end{eqnarray*}   

{\small   
{\black {} 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}}
\vskip.5cm
{\bf 
{\small
{\black {} To determine the complex stiffness we solve Biot's equation   
 in the 2D case on a reference 
square} $\O=(0,L)^2$ {\black {} with boundary} $\G$ {\black {} in the} $(x_1, x_3)${\black {}-plane}. 
\vskip.5cm

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


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine the 
stiffness coefficients. IV}}
\vskip-.5cm
{\bf 
{\small
{\blue { }$p_{55}(\omega)$:} {\black {} the
 boundary conditions are }
\begin{eqnarray*}\label{s_bc1a}
&&-\bsig(\bu) \nu  = {\mathbf  g}, \quad (x_1, x_3)\in \G^T\cup\G^L\cup\G^R,\label{s_bc1b}\\
&&\bu^s = 0, \quad (x_1, x_3)\in \G^B,\label{s_bc1f}\\
&&\bu^f\cdot\bnu=0, \quad (x_1, x_3)\in \G\label{s_bc1g}, 
\end{eqnarray*}
{\black {} where }
\begin{eqnarray*}
{\mathbf 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
{\black {} The change in shape suffered by  the sample  is } 
\begin{eqnarray}\label{shear}
\tan [ \theta(\omega) ] =\frac{\Delta G}{{\blue { } p_{55}(\omega)}}.
\end{eqnarray}
$\theta(\omega)${\black {} : the angle between the original positions of the 
lateral boundaries and the location after applying the shear stresses. 

Since } 
\[
\tan [\theta(\omega)] \approx u^{s,T}_1(\omega)/L,
\]
{\black {} where }
$u^{s,T}_1(\omega)$ {\black {}  is the  average horizontal 
displacement at }$\Gamma^T$,
$p_{55}(\omega)$ {\black {} can be determined  from } (1).

\vskip.1cm
{\black {} To determine }{\magenta { }$p_{66}(\omega)$} {\black {}  (shear waves traveling in 
the} $(x_1, x_2)${\black {} -plane), we 
rotate the sample } 90$^{\rm o}${\black {}  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 { }$p_{13}(\omega)$}{\black {} : the  boundary conditions are} 
\begin{eqnarray*}\label{bc1a_13}
&&\bsig(u) \bnu \cdot \bnu = -\Delta P, \quad (x_1, x_3)\in \G^R\cup \G^T,\label{bc1b_13}\\
&&\bsig(u) \bnu \cdot \bchi = 0, \quad (x_1, x_3)\in \G,\label{bc1c_13}\\
&&\bu^s\cdot\bnu = 0, \quad (x_1, x_3)\in \G^L\cup\G^B,\quad 
\bu^f\cdot\bnu=0, \quad (x_1, x_3)\in \G\label{bc1g_13}.
\end{eqnarray*}
{\black {} In this experiment} $\epsilon_{22}= \nabla\cdot u^f = 0${\black {} , 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*}
$\epsilon_{11}, \epsilon_{33}${\black {} :  the 
strain components at the  right lateral side and  top side 
 of the sample, respectively. Then, since} \,\,\,$\tau_{11} = \tau_{33}= -\Delta P$, 
\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  Numerical Examples}. }
\vskip-.8cm
{\bf 
{
{\black {} 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{\black {}  Case 1:  A brine-saturated sample with fractures.}

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

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

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


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

%----------------------------------------------- SLIDE -
\begin{slide}{\small{\bf Material Properties of background and fractures.}}
%\vskip-.8cm
{\bf 
{
\begin{itemize}
\item {\black {} Background and fractures: }

{\black {} grain density  is} $\rho_s = 2650$ kg/m$^{3}$, 

{\black {}bulk modulus is} $K_s = 37$ {\black {} GPa,}

{\black {}shear modulus is} $\mu_s = 44$ {\black {}GPa. }

\item{\black {} Background:}

{\black {} Porosity is } $\phi = 0.25$, {\black {}  Permeability is } $\kappa = 0.247${\black {} Darcy},

{\black {} dry bulk modulus is} $K_m =1.17$ {\black {}  GPa},

{\black {} shear modulus is} $\mu= 1.45 ${\black {}  GPa.}




\item {\black {}Fractures:}

{\black {} Porosity is } $\phi = 0.5$, {\black {}  Permeability is } $\kappa = 4.44${\black {} Darcy},

{\black {} dry bulk modulus is }$K_m =0.58${\black {} GPa},  

{\black {} shear modulus is }$\mu=0.68 ${\black {}  GPa}. 

\end{itemize}

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


%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Validation. Analytical and numerical solutions for case 1.
Dissipation factor of qP waves and qSV waves at 300 Hz}}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6.cm,height=6.cm]
{Figura_1_color.eps}\\
\vspace*{0.3cm}
{\tiny A very good match between the theoretical and 
numerical results is observed.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Validation. Analytical and numerical solutions for case 1.
Energy velocity  of qP waves and qSV waves at 300 Hz}}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6.cm,height=6.cm]
{Figura_2_color.eps}\\
\vspace*{0.3cm}
{\tiny A very good match between the theoretical and 
numerical results is observed.}
\end{center}
\end{slide}



%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Dissipation factors of qP waves  at 50 Hz and 300 Hz. Cases 1, 2 and 3}}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_3_color.eps}\\
\vspace*{-0.4cm}
{\tiny Note strong  Q anisotropy, with higher attenuation at  300 Hz  
and patchy brine-CO$_2$ saturation. Energy losses 
are much higher for angles between  60 and 90 degrees (waves incident normal to the fracture layering)}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Energy velocity of qP waves  at 50 Hz and 300 Hz. Cases 1, 2 and 3 }}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_4_color.eps}\\
\vspace*{-0.4cm}
{\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. Velocity behaves isotropically in case 2}
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Fluid pressure distribution 
at 50 Hz and 300 Hz for  case  3 and  
compressions normal to the fracture layering.}}}

\begin{center}
\includegraphics[width=6.cm]{Figura_5_color.eps}
\includegraphics[width=6.cm]{Figura_6_color.eps}\\
\end{center}
{\small Pressure gradients take their highest values at the fractures (mesoscopic losses), and at 300 Hz remain  always higher than at 50 Hz.  
This explains the higher losses for qP waves at 300 Hz as compared with the 50 Hz experiment
observed  before.}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Relation among different attenuation mechanisms  for cases 1, 2 and 3  at 300 Hz.}}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=7.5cm]
{Figura_7_color.eps}\\

{\tiny $Q_{\rm P1}, Q_{\rm P2}$ and $Q_{\rm P3}$: qP-quality factors associated 
with cases  1 (brine with fractures), 2 (patchy without fractures)  and 3 (patchy with fractures). 
Approximate validity of 
the commonly used approximation 
$$Q_{\rm P3}^{-1} = Q_{\rm P1}^{-1} + Q_{\rm P2}^{-1}.$$


%Numerical experiments have shown that this equation cannot be used as 
%an approximation in Case 4 (fractal frame and fractures).
}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Dissipation factors of qSV waves  at 50 Hz and 300 Hz. Cases 1, 2 and 3}}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_8_color.eps}\\
\vspace*{-0.4cm}
{\tiny Case   2 is lossless, while for a fractured sample brine 
or patchy saturated (cases  1 and 3 ), Q anisotropy is 
strong for angles between 30 and  
60 degrees, with  about a  50 \% increase in attenuation at 300 Hz with respect to 50 Hz }
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Energy velocity of qSV and SH  waves  at 50 Hz. Cases 1, 2 and 3 }}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_9_color.eps}\\
\vspace*{-0.4cm}
{\tiny Case  2 shows isotropic velocity for both waves. Velocity 
anisotropy is observed to be induced by fractures (cases 1 and 3 ). Patchy saturation 
does not affect the anisotropic behavior of the  velocities. At 300 Hz the behavior is almos identical for qSV was and identical for SH waves.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factors of qP and qSV waves for case 3 at 300 Hz
 as function of  
CO$_2$ saturation.}}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_10_color.eps}\\
\vspace*{-0.4cm}
{\tiny For qP waves,  an increase of  CO$_2$ saturation from 10$\%$ to 50$\%$  induces 
a strong decrease in attenuation for angles 
close to the normal orientation of the fractures. For qSV waves  the 
same decrease in attenuation is observed, 
but for angles between 30 and 60 degrees.}
\end{center}
\end{slide}

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

\begin{slide}{\tiny{\bf {Energy velocities of qP and qSV waves for case 3 at 300 Hz
 as function of   
CO$_2$ saturation.}}}
{\bf 
\begin{itemize}
\item
{\black {} qP energy velocities decrease for increasing CO$_2$ saturation, with the greater decreases for 
angles closer to the normal layering of the fractures.} 

\item
{\black {} For qSV and SH waves, energy velocities 
show almost no change between the 10\% and 50\%  CO$_2$ saturations.}

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

\begin{slide}{\tiny{\bf {Dissipation factors for cases  1, 2 and 3  for waves 
parallel (`11' waves ) and normal (`33' waves) to the fracture layering}}}
\begin{center}
\includegraphics[width=9.cm]
{Figura_11_color.eps}\\
{\tiny `11' waves for  case  1 (brine-saturated homogeneous background with fractures) 
are lossless, while the cases of patchy saturation with and without fractures suffer similar attenuation. }
\end{center}
\end{slide}

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

\begin{slide}{\tiny{\bf {Dissipation factors for cases  1, 2 and 3 for waves 
parallel (`11' waves ) and normal (`33' waves) to the fracture layering}}}
\begin{center}
\includegraphics[width=9.cm]
{Figura_11_color.eps}\\
{\tiny `11' waves for  case  1 (brine-saturated homogeneous background with fractures) 
are lossless, while the cases of patchy saturation with and without fractures suffer similar attenuation. 
`33' show much higher attenuation than those for `11' waves 
for the three cases. }
\end{center}
\end{slide}
%----------------------------SLIDE -------------------%----------------------------

\begin{slide}{\tiny{\bf {Velocities for cases  1, 2 and 3 for waves 
parallel (`11' waves ) and normal (`33' waves) to the fracture layering}}}
\begin{center}
\includegraphics[width=9.cm]
{Figura_12_color.eps}\\
{\tiny  In case  1,  `11' velocities  are essentially independent of frequency.  
In the case of  patchy saturation with fractures (case 3),
velocities are always smaller than in case  1. For `33' waves the presence of fractures induces a noticeable reduction of velocities normal to the fracture plane, either for brine or patchy saturation.  }
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Lam\'e coefficient $\lambda_G$ of the brine saturated fractal 
sample used in case 4}}}

\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm]{Figura_13_color.eps}
\end{center}
{\tiny A binary fractal permeability   
is obtained starting  with the  relation 
$$
\text{log} ~ \kappa(x,z) = \left\langle \text{log} ~\kappa \right\rangle + f(x,z)  
$$
$f(x,z)$:  fractal  spatial 
fluctuation of the permeability  field, of fractal dimension $D = $ 2.2, 
correlation length 2 cm,  
and average permeability 0.25 Darcy  in the background and 4.44 Darcy in the fractures.
{\magenta {}  Porosity was obtained using the Kozeny-Carman relation.} }
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Dissipation factors of qP and qSV waves  at 50 Hz for cases 1 and 4}}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_14_color.eps}\\
\vspace*{-0.4cm}
{\tiny Frame heterogeneities induce a noticeable 
increase  in Q anisotropy for qP waves for angles normal to the fracture plane
and for qSV waves for angles between 30 and 60 degrees.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {Energy velocity of qP and qSV waves  at 50 Hz for cases 1 and 4 }}}
\begin{center}
\vspace*{-.8cm}
\includegraphics[width=7.cm,height=7.cm]
{Figura_15_color.eps}\\
\vspace*{-0.4cm}
{\tiny  Observe the expected energy velocity reduction 
 in the heterogeneous case, and that velocity anisotropy is less affected 
 by  frame heterogeneities  than Q anisotropy.}
\end{center}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small{\bf  Approximate representation of fractures  using  boundary conditions. I}}
%\vskip-.8cm
{\bf 
{\small

{\black {} Bakulin and Molotov (SEG, 1997)  first and later Nakawa and Schoenberg (JASA, 2007) 
presented a boundary condition to 
represent  approximately fractures in poroelastic media.  
}
{\black {}  Consider an horizontal   fracture } $\G$ {\black {} in the} $(x_1, x_3)$-{\black {} plane  separating two half spaces } 
$\O^{(1)}, \O^{(2)}$ in $R^2$.   

$\bnu_{1,2},\bchi_{1,2}$:  
{\black {} be the  unit outer  normal and a unit tangent (oriented counterclockwise)  
on} $\G$ {\black {}  from } $\O^{(1)}$  to $\O^{(2)}$ 
%such that $\{\nu_{1,2}, \chi_{1,2} \}$ is an orthonormal system on $\G^{(f,1)}$.

$[\bu^s], [\bu^f]$: {\black {} jumps of the solid and fluid displacement vectors
 at $\G$:}
\[
[\bu^t] = \left(u^{(t,2)} - \bu^{(t,1)}\right)|_{\G}, \,\, t=s,f 
\]
$\bu^{(t,1)}$: {\black {}  displacement vector restricted to} $\O^{(1)}$,{\black {} and similarly for }$u^{(t,2)}$.




}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small{\bf Approximate representation of fractures  using  boundary conditions. II}}
%\vskip-.8cm
{\bf 
{\small



Boundary conditions on $\G$: 
%(here we use $Z_N$ and $Z_T$ for $\eta_N$ and $\eta_T$ in Nakawa and Schoemberg JASA07)

\begin{eqnarray*}
&&[\bu_s]\cdot \bnu_{1,2} = \eta_{N_d} \bsig(\bu)\bnu_{12} \cdot\bnu_{12} 
+  \alpha \eta_{N_d} p_f(\bu), \quad \G,\label{bc_11}\\
&&[\bu_s]\cdot \bchi_{1,2} =  \eta_T \bsig(u)\bnu_{12} \cdot\bchi_{12},\quad \G,\label{bc_12}\\
&&[\bu_f]\cdot \bnu_{1,2} = - \alpha \eta_{N_d} \bsig(u)\bnu_{12} \cdot\bnu_{12}
- \dfrac{\alpha \eta_{N_d}}{\widetilde B} p_f(\bu)\quad \G.\label{bc_13}
\end{eqnarray*}
$\eta_{N_d}$ : Dry normal compliance (m/Pa)
\vskip.4cm
$\eta_{T}$ :  shear compliance (m/Pa)
\vskip.4cm
$ \widetilde B = \alpha \dfrac{M}{H_G}$, \hskip1.cm  $H_G = K_G + \dfrac43 \mu$


}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Magnitude of reflection coefficients. Fractures as fine layers and as boundary condition}}}

\begin{center}
\includegraphics[width=6.cm]{Figura_4a.eps}
\includegraphics[width=6.cm]{Figura_4b.eps}\\
\end{center}
{\small P1-reflection coefficient (left), P2-reflection coefficient (right).  
Incident wave is a P1-wave of normal incidence  to the fracture layering. Fracture aperture   $h$ is , $h=$0.001 m and  $h=$0.00001 m.}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf { EQ 53}}}

\begin{center}
\includegraphics[width=6.cm]{Figura_4c.eps}
\includegraphics[width=6.cm]{Figura_4d.eps}\\
\end{center}
{\small  P1-transmission coefficient (left), P2-transmission coefficient (right). 
Incident wave is a P1-wave of normal incidence  to the fracture layering. Fracture aperture   $h$ is , $h=$0.001 m and  $h=$0.00001 m.  }
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{ {\bf Conclusions} }
\vskip-.7cm
{\bf 
{\small
\begin{itemize}

\item We employed the  FEM  to determine the complex and 
frequency-dependent 
stiffnesses of a VTI homogeneous  medium equivalent to 
a  fluid-saturated poroelastic material containing a dense set of planar 
fractures. 
\item Fractures induce anisotropy and the P-S coupling generates 
losses in the shear waves  as well.

\item Fractures were modeled as very thin highly permeable 
poroelastic layers of negligible frame moduli.

\item  Velocity and attenuation anisotropy is observed in the qP 
and qSV wave modes, with  attenuation stronger for the patchy saturated cases. 
the SH wave 
is lossless because it is a pure mode in TI media, but  shows velocity anisotropy.

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

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




\end{document}

