%\documentclass[pdf,autumn,slideColor,colorBG]{prosper} % 2
%\documentclass[pdf,contemporain,slideColor,colorBG]{prosper}
%\documentclass[pdf,lignesbleues,slideColor,colorBG]{prosper}
%\documentclass[pdf,azure,slideColor,colorBG]{prosper}
%\documentclass[pdf,blends,slideColor,colorBG]{prosper}
%\documentclass[pdf,capsules,slideColor,colorBG]{prosper}
%\documentclass[pdf,nuancegris,slideColor,colorBG]{prosper}  
\documentclass[pdf,serpaggi,slideColor,colorBG]{prosper}  % 1
\usepackage[latin1]{inputenc}
\usepackage{amsmath,subeqnarray}
\usepackage{epsfig}
\def\noi{\noindent}
\def\cl{\centerline}
\def\bs{\bigskip}
\def\ms{\medskip}
\def\sm{\smallskip}
\bibliographystyle{plain}
\def\nn{\nonumber}
\renewcommand\a{\alpha}
\def\<{\left\langle}
\def\>{\right\rangle}
\def\b{\beta}
\newcommand{\bc}{\rm I\!\!\!C}
\newcommand{\bh}{\rm I\!H}
\newcommand{\br}{\rm I\!R}
\newcommand{\brn}{{\rm I\!R}\,^n}
\def\ai{\left\langle\langle}
\def\ad{\right\rangle\rangle}
\def\bus{{\mathbf{u}}^s}
\def\bun{{\mathbf{u}}^n}
\def\buw{{\mathbf{u}}^w}
\def\bnab{\mathbf \nabla}
\def\d{\delta}
\def\cV{\mathcal{V}}
\def\D{\Delta}
\def\sD{{\mathcal{D}}}
\def\dfrac{\displaystyle\frac}
\def\diag{\mathrm{diag\,}}
\def\dint{\displaystyle\int}
\def\doint{\displaystyle\oint}
\def\dist{\mathrm{dist\,}}
\def\Div{\mathrm{div\,}}
\def\ds{\displaystyle}
\def\dsum{\displaystyle\sum}
\def\e{\varepsilon}
\def\g{\gamma}
\def\G{\Gamma}
\def\hp{\widehat p}
\def\k{\kappa}
\def\lam{\lambda}
\def\Lam{\Lambda}
\def\Lie{\mathrm{Lie}}
\def\nab{\nabla}
\renewcommand\o{\omega}
\renewcommand\O{\Omega}
\def\og{\overline{g}}
\def\OS{\overline{S}}
\def\p{\partial}
\newcommand{\pp}[2]{\dfrac{\partial {#1}}{\partial {#2}}}
\newcommand{\ppt}{\frac{\partial}{\partial t}}
\def\rank{\mathrm{rank}}
\def\lma{\left\langle}
\def\rma{\right\rangle}
\def\s{\sigma}
\def\sC{{\rho}}
\def\V{{\mathcal{V}}}
\renewcommand\t{\theta}
\def\Tau{\mathcal T}
\def\tK{\widetilde K}
\def\tp{\widetilde p}
\newcommand\var{\varphi}
\def\wse{\widetilde {E}}
\def\wsa{\widetilde {A}}
\def\wsc{\widetilde {C}}
\def\z{\zeta}
\def\wh{\widehat}
\def\vx{{\bf x}}
\newcommand{\dd}[2]{\frac{\mbox{d} {#1}}{\mbox{d} {#2}}}
\def\du{\dot u}
\def\dU{\dot U}
\newcommand{\DD}[2]{\frac{\mbox{D} {#1}}{\mbox{D} {#2}}}
\def\stn{\dot{R}_{N}^S(\o)}
\def\ctn{\dot{R}_{N}^C(\o)}
\def\stlam{\dot{R}_{\lam_c}^S(\o)}
\def\ctlam{\dot{R}_{\lam_c}^C(\o)}
\def\stbu{\dot{R}_{B_1}^S(\o)}
\def\ctbu{\dot{R}_{B_1}^C(\o)}
\def\stbd{\dot{R}_{B_2}^S(\o)}
\def\ctbd{\dot{R}_{B_2}^C(\o)}
\def\stmu{\dot{R}_{M_1}^S(\o)}
\def\ctmu{\dot{R}_{M_1}^C(\o)}
\def\stmd{\dot{R}_{M_2}^S(\o)}
\def\ctmd{\dot{R}_{M_2}^C(\o)}
\def\stmt{\dot{R}_{M_3}^S(\o)}
\def\ctmt{\dot{R}_{M_3}^C(\o)}
\def\deps{\dot\varepsilon}
\def\dxi{\dot\xi}

\title
{
%Finite Element Methods for the Approximation of Waves 
%in Fluid Saturated  Poroviscoelastic Media}
%\subtitle{in Fluid Saturated  Poroviscoelastic Media
%Ondas en Medios Porosos Saturados. Teor\'\i a y Aplicaciones.  
Acoustics in Porous Media.}
%\subtitle{with Applications from the Seismic to the Ultrasonic Range}
\author{{\bf JUAN E. SANTOS}$^\dag$ 
\\ work in collaboration with\\ J. M. Carcione, S. Picotti, P. M. Gauzellino  and R. Martinez Corredor.
}

\institution{%
$^\dag$ Department of Mathematics, Purdue University, W. Lafayette, Indiana, and
 CONICET, Instituto del Gas y del Petr\'oleo,  { Facultad de Ingenier\'\i a Universidad de Buenos Aires}, 
 and Universidad Nacional de La Plata, Argentina\\
%Department of Mathematics, Purdue University, W. Lafayette, Indiana\\

$ $ \\ 
{\blue {} Institute of Acoustics Orso Mario Corbino July 19th, 2013 } 
 }

%\begin{figure}[h]
%\centerline{\psfig{figure=logo-completo.ps,width=4cm,angle=0}}
%\end{figure}

\begin{document}

\maketitle

%----------------------------------------------- SLIDE -
\begin{slide}{Introduction.}
\vskip-.5cm
{\small
\begin{itemize}
\item The acoustics of fluid-saturated porous media is an active area of research 
  in many fields, such as seismic exploration, petroleum reservoir 
characterization and 
monitoring, CO$_2$ sequestration   
 and nondestructive testing of materials by ultrasonic methods.
\item
M. A. Biot (JASA, 1956; J. Appl. Physics, 1962) 
developped a theory for wave propagation in a  
porous medium  saturated by a single-phase fluid, showing the existence of 
two compressional waves (P1, P2 waves) and one shear wave. Plona (Appl. Phys. Lett., 1980)
was the first to observe the P2 wave in the laboratory.
\item 
Numerical simulation based on  Biot's theory  is a useful tool 
to characterize this multiphasic materials.

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

\begin{slide}{\small {A representative sample of a porous medium } }
\begin{center}
\vspace*{-0.5cm}
%\vskip.2cm
\includegraphics[width=7cm]{fig0_m_poroso_1.eps} \\
{\small{\bf Blue regions represent water, yellow regions hydrocarbon and  the void spaces 
 the  solid matrix.}}
\end{center}
\end{slide}

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

\begin{slide}{The Biot Model. } 
{\bf 
{\small
\vskip-.2cm 
{$V_b$} : volume of homogeneous {\it bulk material} 
of {\magenta{}effective}   porosity $\phi$ 
containing a {\it pore volume} {$V_p$} completely saturated 
by a single-phase fluid: 
\[
\phi = \frac{V_p}{ V_b}
\]

}
}
{\bf {\small 

\noi
$u^{(s)}$, $\widetilde u^{(f)}$, : \ time Fourier transform of the 
 averaged  displacement vectors of the solid and fluid phases, respectively.

\begin{equation}\nn
u^{(f)}=\phi(\widetilde u^{(f)} - u^{(s)}), 
%\quad \xi =-\nabla\cdot u^{(f)},  
\qquad u=(u^{(s)},u^{(f)}).
\end{equation}

\noi
$\sigma_{ij}$:  time Fourier transform of the total stress 
tensor of the bulk material,

\noi
$p_f$:  time Fourier transform of the fluid pressure, 

$\varepsilon_{ij}$  
: time Fourier transform of the  strain 
tensor of the solid. 
 
}
}
\end{slide}
% --------------------------------------------Slide

\begin{slide}{Constitutive Relations.}
\vskip -0.8cm
{\bf 
{\small
\begin{eqnarray}\label{mod.2}
&{\mathrm{ }}&\sigma_{ij}(u)=2 \mu\,\varepsilon_{ij}(u^{(s)})+\delta_{ij}
(\lambda_c  \nabla\cdot u^{(s)} + \alpha K_{av} \, \nabla\cdot u^{(f)}),
\nonumber\\\nn\\
&{\mathrm{ }}&p_f(u) = - \alpha K_{av} \, \nabla\cdot u^{(s)} -
 K_{av} \, \nabla\cdot u^{(f)},\nonumber\\
&\hskip-1.2cm\text{where}\nn\\ 
&&\alpha = 1 - \dfrac{K_m}{K_s},\quad 
K_{av} = \left[\dfrac{\alpha- \phi}{K_s} 
+ \dfrac{\phi}{K_f}\right]^{-1}.\nonumber\nn\\
&& K_G = K_m + \alpha^2 K_{av}\quad \lambda_G = K_G  - \dfrac23 \mu,\nonumber
\end{eqnarray} 
$\mu$: shear modulus of the dry matrix, 
\vskip.2cm
$K_m, Ks, K_f, K_G$: bulk modulus 
of the dry matrix, the solid grains, the saturant fluid, and the saturated bulk material, respectively.


%}
%}

}
}
\end{slide}


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

\begin{slide}{Biot's Equations of Motion. }
\vskip-.8cm
{\bf
{\small
\begin{eqnarray}\label{mod10}
&&-\o^2 \rho u^{(s)} - \o^2 \rho_f u^{(f)} - \nabla\cdot\sigma(u) = F^{(s)}\nn\\\nn\\
&&-\o^2 \rho_f u^{(s)} - \o^2 g u^{(f)} + i \o \dfrac{\eta}{\kappa} 
u^{(f)} + \nabla p_f(u)= F^{(f)}.\nn
\end{eqnarray}
}
\noindent
$\o$: angular frequency 

\vskip.3cm
\noindent
$\rho = (1 - \phi) \rho_s + \phi \rho_f$: \quad \text{bulk modulus}
\vskip.3cm
\noindent
$\rho_f, \rho_s$:  mass densities of the  fluid  and the solid  grains, 
\vskip.3cm
\noindent
$ g$ : mass coupling parameters
\vskip.3cm
\noindent
$\eta$: Fluid viscosity, 
\hskip.3cm
$\kappa$: Permeability,



}
\end{slide}


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


\begin{slide}{Plane Wave Analysis}
\vskip-.5cm
{\bf
In these type of media, two  compressional waves 
({\blue {} $P_1, P_2$}) and one shear wave 
(or {\blue {} $S$})-wave) can propagate.
 
\vskip .3cm
\noindent
${\blue {}P_1}$ and ${\blue {}S}$ waves: analogues of the classical 
fast waves propagating in elastic or viscoelastic isotropic solids.

\vskip .3cm
\noindent
${\blue {}P_2}$ wave: {\magenta{} slow diffusion type } wave in the low frequency range and 
a  truly propagating mode in the ultrasonic range.
\vskip .3cm
\noindent
The ${\blue {}P_2}$ wave is due to the  motions out of phase of the solid and fluid phases. 
\vskip.2cm
${\blue {}P_2}$ waves are generated by  conversion of the fast ${\blue {}P_1}$ 
and ${\blue {}S}$ waves at interfaces within heterogeneous poroelastic materials,   causing  
attenuation and dispersion on the  fast modes.  
 

}


\end{slide}




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


\begin{slide}{\small{A Variational Formulation. 2D Case.}}
\vskip-.5cm
{\bf
{\small
{
$\O: \mbox{rectangular
domain}, \quad {\mathcal V} = \left[ H^1(\O)\right]^2\times H(\text{div};\O)$. 
 
Testing   Biot's equation of motion   with $v\in {\mathcal V}$ we get the  
 {\it  weak form}:
 \begin{eqnarray*}\label{weak1}
&&-\o^2 \left({\mathcal P} u,v\right) + i \o \left(\dfrac{\eta}{\kappa} u^{(f)}, v^{(f)}\right) 
+ {\mathcal A}(u,v)
+ i \o \left\langle {\mathcal D} u, v\right\rangle= (F,v),\nn
\\
&&\hskip 6cm   v=\left(v^{(s)},v^{(f)}\right)^t\in {\mathcal V},\nonumber\\
\nn\\
%\end{eqnarray}
%\begin{eqnarray}\label{bilinear1}
&&{\mathcal A}(u,v)=\sum_{l,m}\left(\sigma_{l m}(u),\e_{l m}(v^{(s)})\right)
 - \left(p_f(u),\nabla\cdot v^{(f)})\right), \  u,v \in {\mathcal V}.\nn\\
%\end{eqnarray}
%
%\begin{eqnarray*}
&&{\mathcal P} =
\begin{pmatrix}
\rho I&\rho_f I\\
\rho_f I& g I
\end{pmatrix},
%\quad
%{\mathcal B}=\begin{pmatrix}
%0 I&0 I\\
%0 I&b I
%\end{pmatrix},
\qquad F = (F^{(s)},F^{(f)}). 
\end{eqnarray*}
$I$:identity matrix in $R^{2\times 2}$ \hskip2cm 
 ${\mathcal D}$ : a positive definite matrix .
} 
}
}


\end{slide}
% ------------------------------------Slide  ---------------------


\begin{slide}{\small{The Finite Element Method (FEM). Rectangular Elements.}}
\vskip-.7cm
{\bf
{\small


$ {\mathcal V}^h = \left[{\mathcal NC}^h\right]^2\times {\mathcal W}^h$

${\mathcal NC}^h$: {\tiny a nonconforming FE  space   used to  
approximate each component of the {\magenta {} solid displacement vector}. }

${\mathcal W}^h:$ {\tiny conforming FE  space  used to aproximate 
 the {\magenta { }fluid displacement}}. 
\begin{center}
\includegraphics[width=5.5cm,height=5.5cm]{fig1_noconf_DOF.eps}\\ 
{\tiny  Local degrees of freedom (DOF's)}
\end{center}


}
}


\end{slide}


%----------------------------------------------Slide
\begin{slide}{\small{Applications. The Mesoscopic Loss Mechanism. I}}
%\vskip-.6cm
{\bf
{
\begin{itemize}
\item
A  major cause of seismic attenuation in
porous media is {\magenta {}wave-induced fluid flow}, which
occurs at mesoscopic scales. 

\item
A fast  $P_1$ wave induces a fluid-pressure
difference at mesoscopic-scale inhomogeneities (larger than the
pore size but smaller than the wavelength, typically tens of cen-
timetres), generating fluid flow and slow (diffusion) $P_2$ Biot waves
\item
White (1975) was the first to show that {\magenta {}  mesoscopic loss }  can be observed in the seismic range in a  
  sandstone with  partial gas saturation.
\item
Next we show a numerical verification of the {\magenta {}mesoscopic loss} mechanism using the FEM.
\end{itemize}  
}
}
\end{slide}





%----------------------------------------------Slide
\begin{slide}{\small{Numerical Simulations to illustrate the mesoscopic loss mechanism.}}
\vskip-.5cm
{\bf
{\small
The  computational domain is a square of side length  $800$~m representing  a poroelastic rock alternately saturated
with  gas and water.   

\vskip.4cm
\noi
The perturbation  $F=(F^{(s)}, F^{(f)})$  is a compressional point source applied
to the matrix, located inside the region at $(x_s,y_s)$ = (400 m, 4 m) 
with (dominant) frequency  20 Hz.

\vskip.4cm
\noi
Biot's equations of motion  were  solved for 
110  temporal frequencies in the interval $(0,60 Hz)$ employing a FE domain decomposition iterative procedure.

\noindent
The domain was discretized into square cells 
of side length $h = 40$ cm.
\vskip.4cm
\noi
The time domain solution 
was obtained performing an approximate inverse
Fourier transform. 
}
}
\end{slide}







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

\begin{slide}{\tiny {Time histories for a porous rock 
 saturated by 
gas (top left), water (top right) and periodic gas-water (bottom)} }
\begin{center}
\vspace*{-0.3cm}
\hskip-.2cm
\includegraphics[width=5.cm,height=3.3cm]{fig2_gas_mesoscopic_loss.eps}\quad 
\includegraphics[width=5cm,height=3.3cm]{fig2_water_mesoscopic_loss.eps}\\ 
\vskip.35cm
\includegraphics[width=5.5cm,height=3.3cm]{fig2_gas_water_mesoscopic_loss.eps}
%\\ 
%{\tiny Figure  2. Top left: gas, Top right: water, Bottom: periodic gas-water }
\end{center}
\end{slide}


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

\begin{slide}{\tiny {Time histories observed at a receiver for a porous rock 
 saturated by gas, water and periodic gas-water.} }
\begin{center}
\vspace*{0.1cm}
%\vskip.2cm
\includegraphics[width=9cm,height=5.5cm]{fig3_mesoscopic_loss.eps}\\ 
{\tiny{  The {\magenta {} delay in in the arrival
time} in the periodic gas-water case is due to the velocity dispersion caused by the mesoscopic scale heterogeneities. The associated {\magenta {} attenuation} 
is in agreement with that predicted by White's theory.
}
}\end{center}
\end{slide}
%----------------------------------------------Slide
\begin{slide}{\small{Extensions of Biot theory.}}
{\bf
{\small
\begin{itemize}

\item A porous medium saturated by immiscible fluids. The theory was presented  by Santos et al. (J. Acoust. Soc. Am.,  1990a, b).

In these type of media, three  compressional waves 
(${\blue {} P_1, P_2, P_3}$ waves)  and one shear wave (${\blue {}S}$ wave) can
propagate. 


\item A porous composite matrix saturated by a single-phase fluid. 
The theory was developped by  Leclaire et al. (J. Acoust. Soc. Am., 1995) for the uniform porosity case and then extended to the more realistic variable porosity case by  Carcione et al. (J. Appl. Physics, 2003) and 
Santos et a. (J. Acoust. Soc. Am., 2004).

In this case 3 compressional waves (${\blue {} P_1, P_2, P_3}$ waves) and 2 shear waves (${\blue {} S_1, S_2}$ waves) can propagate. Observation of the slow  waves were presented in Leclaire et al. (J. Acoust. Soc. Am., 1995).



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

\begin{slide}{\tiny {The immiscible fluids model. Saturation interface. 
Snapshot of the solid phase z-velocity at $t=0.007$ ms.}}
\begin{center}
\vspace*{-0.7cm}
\includegraphics[width=6.cm]{fig4_type3_waves.eps}\\
\vspace*{-0.1cm}
{\tiny {  The interface divides the domain in two layers of equal thickness (3 cm). The lower layer (where the source is located) has nonwetting saturation $S_{n} = $0.9, 
while in the upper layer  $S_{n} = $ 0.1. A compressional point source of dominant   frequency $500$ ~ kHz  is located  at $x = $ 3 cm, $z = $ 2 cm.}}
\end{center}
\end{slide}


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

\begin{slide}{\tiny {Saturation interface. Snapshot of the solid phase z-velocity at $t = 0.012$ ms}}
\begin{center}
\vspace*{-0.7cm}
\includegraphics[width=6.cm]{fig5_type3_waves.eps}\\
%\vspace*{-0.2cm}
{\tiny { Amplitude of wavefronts measured with respect to amplitude of incident $P_{1d}$ wavefront: $A(P_{1r})\approx 0.06$, $A(S_{r})\approx 0.09$, $A(S_{t})\approx 0.29$, $A(P_{1t})\approx 0.2$}}
\end{center}
\end{slide}

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

\begin{slide}{\tiny {Saturation interface. Snapshot of the solid phase z-velocity at $t = 0.045$ ms}}
\begin{center}
\vspace*{-0.7cm}
\includegraphics[width=6.cm]{fig6_type3_waves.eps}\\
%\vspace*{-0.2cm}
{\tiny { Amplitude of wavefronts measured with respect to amplitude of incident $P_{3d}$ wavefront: $A(P_{1r})\approx 0.1$, $A(P_{3r})\approx 0.122$, $A(S_{r})\approx 0.044$, $A(P_{1t})\approx 0.09$, $A(P_{2t})\approx 0.055$, $A(S_{t})\approx 0.006$}}
\end{center}
\end{slide}

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

\begin{slide}{\tiny{Patchy saturation map. White  regions correspond to 
almost full gas saturation.}}
\begin{center}
%\vspace*{-0.7cm}
\includegraphics[width=6.cm]{fig7_type3_waves.eps}\\
%\vspace*{-0.2cm}
{\small {\bf Macroscopic nonwetting  saturation $S_{n} = $ 0.1 } }
\end{center}
\end{slide}


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

\begin{slide}{\tiny {Patchy saturation. Snapshot of the solid phase z-velocity at $t = 0.007$ ms} }
\vskip-.2cm
\begin{center}
\vspace*{-0.4cm}
\includegraphics[width=6.cm]{fig8_type3_waves.eps}\\
%\vspace*{-0.2cm}
{\small {\bf Irregular wavefronts are associated with compressional 
and shear waves  generated at the gas patches. Thin wavefronts are associated with slow ${\blue P_2, P_3}$ waves.}}
\end{center}
\end{slide}


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

\begin{slide}{\tiny {Patchy saturation. Curl of the velocity of the solid phase at $t = 0.007$ ms}}
\begin{center}
\vspace*{-0.3cm}
\includegraphics[width=6.cm]{fig9_type3_waves.eps}\\
%\vspace*{-0.2cm}
{\small {\bf Shear waves are generated at the  gas-water interfaces by
mode conversion.}}
\end{center}
\end{slide}





%%%%%%%%%%%%%%%%%%%%%%%%

%----------------------------------------------Slide
\begin{slide}{\tiny {The composite matrix model. Application to wave propagation in gas hydrate bearing sediments.} }
{\bf
{\small
\noi
\begin{itemize}
\item{\magenta {} Gas hydrates (GH)} are crystalline molecular 
 complexes composed of water and
natural gas, mainly methane. They form under conditions of low temperature, 
high pressure and gas concentration and  are observed in 
permafrost regions and seaflor 
sediments along the continental margins. 

%\vskip.3cm
%\noi
\item
{\magenta {} GH } are ice-like structures within the pore space that cause strong
changes in the physical and mechanical properties of the rock, 
making possible  to detect its presence  using seismic data.
%\vskip.3cm
%\noi 
\item
The interest in {\magenta {} GH } bearing sediments  GH is mainly due to their potential as future 
energy resources.
\end{itemize}
\noi 
}
}
\end{slide}

%----------------------------------------------Slide
\begin{slide}{\small{ Wave  Propagation in GH Bearing Sediments.}}
{\bf
{\small
\begin{itemize}
\item
Borehole sonic log 
and cross-hole tomography 
data from the Mallik field (Mackenzie Delta, Canada), indicate 
that attenuation {\blue {} increases} with {\magenta {} GH content}.
\item
Slow-waves energy conversions take place when a fast (classical) seismic  
wave strikes 
an heterogeneity within the {\magenta {} GH } bearing sediments, generating slow compressional (${\blue P_2, P_3}$) and shear  (${\blue S_2}$) waves.

\item
These scattering-type effects can  explain the observed levels of 
attenuation in {\magenta {} GH-bearing sediments}, which is tested using numerical simulations. 

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

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

\begin{slide}{\tiny{GH saturation distribution. The domain is a square of 
side length  1.5 m. Overall GH saturation $S_{gh}= $ .58}}
\begin{center}
%\vspace*{-0.7cm}
\vspace*{-1.cm}
%\includegraphics[width=6cm,height=6cm]{figure4.ps}\\ 
\includegraphics[width=5cm]{fig10_GH_map.ps}\\ 
\vspace*{-0.5cm}
{\small {\bf GH saturation distribution. 
White zones correspond to $ S_{gh}= $0.9,  while  black zones represent 
cells with $S_{gh} = $ 0.25.}}
\end{center}
\end{slide}
%-----------------------------------------Slide

\begin{slide}{\tiny{Snapshots of z-component of the solid matrix. 
The domain is a square of 
side length  150  cm.}}
\vskip-.7cm
\begin{center}
%\vspace*{1.cm}
\hskip-.2cm
%\includegraphics[width=3cm,angle=-90]{figure5.ps}\\
\includegraphics[width=5.5cm,height=6.9cm,angle=-90]{fig11a_GH_snap.ps}\hskip-1.9cm 
\includegraphics[width=5.5cm,height=6.9cm,angle=-90]{fig11b_GH_snap.ps}
\\
{\tiny { Snapshots of the z-component of the  velocity
of the solid matrix  at $t=0.25\mbox{ms}$ and $t=0.5\mbox{ms}$. 
A {\magenta {}compressional point source of dominant   frequency $10$ ~ kHz}  
is located  at $x = $ 75 cm, $z = $ 10 cm. Strong amplitude decays and 
conversions to {\magenta {} slow}  compressional (${\blue P_2, P_3}$) and shear (${\blue S_2}$)
waves  are observed.}}
\end{center}
\end{slide}
%-----------------------------------------Slide


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

\begin{slide}{\tiny{Snapshots of the divergence and  curl of the solid matrix.  
The domain is a square of 
side length  150 cm. }}

\begin{center}
%\vspace*{1.cm}
\hskip-.2cm
%\includegraphics[width=3cm,angle=-90]{figure5.ps}\\
\includegraphics[width=5.5cm,height=6.9cm,angle=-90]{fig12a_GH_div_snap.ps}\hskip-1.9cm
\includegraphics[width=5.5cm,height=6.9cm,angle=-90]{fig12b_GH_curl_snap.ps}
\\ 
{\tiny{  Snapshots of the divergence (left) and  curl (right) 
of the velocity of the solid matrix at $t=0.5\mbox{ms}$. 
The figures display the  generation of {\magenta {} slow}  compressional (${\blue P_2, P_3}$) and shear (${\blue S_2}$)
waves  due to energy conversions at the heterogeneities.}}
\end{center}
\end{slide}

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

\begin{slide}{\tiny {Time histories 
for the homogeneous (constant $S_{gh}=0.58$)  and 
heterogeneous cases.} }
\begin{center}
\vspace*{-1.cm}
%\vskip.2cm
\includegraphics[width=6cm,angle=-90]{fig13_traces_GH_and_homo.ps}\\ 
{\tiny { Time histories  of the z-component of the particle velocity 
of the solid matrix for the homogeneous and heterogeneous cases 
at a receiver located
at $x = 75\mbox{cm}$, $z = 70\mbox{cm}$. 
The amplitude peak corresponding to the arrival of the fast  
compressional ${\blue P_1}$ wave in the heterogeneous case 
is much lower than in the homogeneous case due to energy conversions.}}
\end{center}
\end{slide}
%-----------------------------------------Slide

\begin{slide}{\tiny {Time histories of waves travelling in highly heterogeneous 
GH bearing sediments} }
\begin{center}
\vspace*{-0.7cm}
%\hskip-.2cm
\includegraphics[width=6.5cm,height=6.6cm,angle=-90]{fig14_traces_vz_GH_3receivers.ps}
\hskip-1.5cm%\\
%\vspace*{-.5cm}
\includegraphics[width=6.5cm,height=6.6cm,angle=-90]{fig15_traces_vx_GH_3receivers.ps}
\\ 
{\tiny Time histories  of the vertical (left) and horizontal (right) 
components of 
the velocity of the solid matrix at  $3$ receivers. The  average 
quality factors: $Q_p$ =
10.5 (from left) and $Q_s$ = 5.5 (from right)  
are in excellent agrement with those measured at 
the Mallik 5L-38 {\magenta {}GH}  Research Well.}
\end{center}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%----------------------------------------------- SLIDE -
\begin{slide}{\small Numerical simulations at the  Mesoscale. I}
{\bf 
{\small
\begin{itemize}

\item Due to the {\magenta { }extremely fine meshes} 
needed to properly represent very thin layers and other type of mesoscopic-scale heterogeneities, 
numerical simulations at the macroscale using Biot's equations 
are  very expensive or even not feasible.

\item
{\blue { }In the contex of Numerical Rock Physics}, we   
use a  numerical upscaling procedure to determine 
the complex and frequency dependent stiffness of an {\magenta { } equivalent  TIV  medium} at the macroscale   including  the 
mesoscopic-scale effects. 

\item 
The methodology is illustrated for the case of a fluid-saturated porous 
rock containing parallel fractures.

%in the case of To determine the {\blue{} five 
% complex stiffness} coefficients of the 
%{\magenta{}equivalent  TIV medium}, we 
%solve a set  of 
% boundary value problems (BVP's) for  Biot's equation in the diffusive range   
% using  the FEM.


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

%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured induced anisotropy in porous media. The Mesoscale. II}
\vskip-.3cm
{\bf 
{\small
\begin{itemize}
\item
Seismic wave propagation through fractures is an important subject in 
hydrocarbon exploration geophysics, mining and reservoir characterization and
production. 

\item
Naturally fractured reservoirs have received interest in recent years, since, generally, natural fractures control the permeability
of the reservoir. 

\item
In geophysical prospecting and reservoir development, knowledge of fracture orientation, densities and sizes is essential since these factors
control hydrocarbon production. 

\item
This is also important in CO$_2$ storage
in geological formations to monitor the injected plumes as faults and fractures are generated, where CO$_2$ can leak to the surface.
\end{itemize} 
}
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured induced anisotropy in porous media. The Mesoscale. III.}
\vskip-.3cm
{\bf 
{\small
\begin{itemize}
\item 
A planar fracture embedded in a fluid-saturated poroelastic background is
a particular case of the {\magenta {} thin layer problem}, when one of the layers is {\magenta {}  very
thin and compliant}. 

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

\item
Wave anelasticity and 
anisotropy are significant in fractured poroelastic rocks due to conversion of   
fast compressional waves into diffusion-type Biot slow waves.

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

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


%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The constitutive relations of a TIV media.}}
\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 relations provide 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 $(x, z)$-plane. 
Set {\blue { }$\G = \G^L\cup\G^B\cup\G^R\cup\G^T$}, where  
\begin{eqnarray}
{\blue { }\G^L} = \{ (x, z)\in \G: {\blue { }x = 0}\}, 
\quad {\blue { }\G^R} = \{ (x , z)\in \G: {\blue { }x = L}\},
\nonumber\\
\nonumber\\
{\blue { }\G^B} = \{ (x, z)\in \G: {\blue { } z = 0}\},
\quad {\blue { }\G^T }= \{ (x, z)\in \G:{\blue { } z = 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, z)\in \G^T,\label{bc1b}\\
&&\sigma(u) \nu \cdot \chi = 0, \quad (x, z)\in \G^T\cup\G^L\cup\G^R,\label{bc1c}\\
%&&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G^L\cup\G^R,\label{bc1d}\\
&&u^s\cdot\nu = 0, \quad (x, z)\in \G^L\cup\G^R,\label{bc1e}\\
&&u^s = 0, \quad (x, z)\in \G^B,\quad 
u^f\cdot\nu=0, \quad (x, z)\in \G\label{bc1g}.
\end{eqnarray*}
{\magenta { } $\nu$}:  the unit outer normal on $\G$
   
{\magenta { } $\chi$}:  a   unit tangent  on $\G$  so that ${\blue { } \{\nu,\chi\}}$ is 
an orthonormal system on $\G$. 
Denote by $V$ the original volume of the sample and by  
$\Delta V(\omega)$
its (complex) oscillatory volume change. 
}
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {\bf The harmonic experiments to determine 
the stiffness coefficients. III}}
\vskip-.5cm
{\bf 
{\small
In the quasistatic case  
\begin{eqnarray*}\label{c33-dv}
\frac{\Delta V(\omega)}{V}=-\frac{\Delta P}{{\magenta { } p_{33}(\omega)}},
\end{eqnarray*}
Then after computing the  average $u^{s,T}_z(\omega)$ of the  vertical displacements 
on  $\Gamma^T$, we approximate
\[
\Delta V(\omega)\approx L u^{s,T}_z(\omega)
\]
which enable us to compute  ${\magenta { } p_{33}(\omega)}$
\vskip.5cm
The other coefficients are determined measuring changes in volume or shape of the sample.
}
}
\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]{fig16_5_harmonic.eps}\\
%\vspace*{-0.2cm}
{\small a): $p_{33}$,\hskip.3cm  b): $p_{11}$, \hskip.3cm c):
 $p_{55}$,\hskip.3cm  d): $p_{13}$ \hskip.3cm  e): $p_{66}$}
\end{center}
\end{slide}

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

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

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

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

\item Case 4: A brine saturated sample with a fractal frame and fractures. 
                  
\end{itemize}



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

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

{\bf 
{

\begin{itemize}

\item 
The discrete BVP's to determine the complex stiffnesses $p_{IJ}(\o)$ 
 were solved for 30 frequencies using a public domain sparse matrix solver package.
\item  
Using relations  not included for brevity, the $p_{IJ}(\o)$'s determine  
in turn the energy velocities and  dissipation coefficients shown in the next figures. 

\end{itemize}


}
}
\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]
{fig17_VqP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
\vspace*{0.3cm}
{\tiny { Velocity anisotropy caused by the 
fractures in cases 1 and 3  is enhanced for the case of patchy saturation, 
with lower velocities when patches are present. The velocity behaves isotropically in case   2.
}}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of the qP waves
at 50 Hz for cases  1,   2 and 3  }}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6.cm,height=6.cm]
{fig18_1000_over_qP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
\vspace*{0.3cm}
{\tiny Fractures induce strong Q anisotropy for angles normal to the fracture plane, enhanced by patchy saturation.}
\end{center}
\end{slide}


%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Fluid pressure distribution at 50 and 300  Hz. 
Compressibility test for $p_{33}$ for case 3.}}}
\vspace*{0.9cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=5.9cm]
{fig19_presion_fluido_p33_relation_1-15_10percent_sat_color_50Hz.eps} 
\includegraphics[width=5.9cm]{fig20_presion_fluido_p33_relation_1-15_10percent_sat_color_300Hz.eps}
%\vspace*{0.3cm}
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
{\small Compression is normal to the fracture plane. 
Attenuation is stronger at 300 Hz.}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qSV energy velocity vector 
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.7cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig21_VqSV_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny Velocity anisotropy is  induced by fractures (cases  1 and 3 ). Patchy saturation  
does not affect the anisotropic behavior of the  qSV velocities. Case  2 shows isotropic velocity, 
with higher velocity values than for the fractured cases}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of  qSV waves
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.5cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig22_1000_over_qsv_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny qSV anisotropy is 
strong for angles between 30 and  
60 degrees. The lossless case 2 is represented by a triangle at the
origin }
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Polar representation of the qSH energy velocity vector 
at 50 Hz for cases  1,   2 and 3  }}}
\vspace*{-0.7cm}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.2cm,height=6.2cm]
{fig23_VSH_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\
%\vspace*{0.3cm}
{\tiny SH velocity anisotropy  
is observed to be induced by fractures. Cases 1 and 3  are almost 
indistinguishable. Velocity for case 2 is isotropic. SH waves are lossles, since $p_{55}$ and $p_{66}$ are real.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Lam\'e coefficient (GPa) for the brine-saturated 
fractal porosity-permeability sample of case 4.}}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=8cm]
{fig25_lambda_fractal_relation_1-15_160x160_color.eps} 
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
{\tiny $\text{log} ~ \kappa(x,z) = \left\langle \text{log} ~\kappa \right\rangle + f(x,z)$,  $f(x,z)$: fractal representing the spatial 
fluctuation of the permeability  field $\kappa(x,z)$. Porosity was obtained using the Kozeny-Carman relation. The Lam\'e coefficients were computed using a Krief model.
}
\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]
{fig26_1000_over_qP_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny Note the increase  in Q anisotropy for qP waves for angles normal to the fracture plane.}
\end{center}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {Dissipation factor of  qSV waves
at 50 Hz for cases  1 and 4.}}}
\begin{center}
%\vspace*{.4cm}
\includegraphics[width=6.cm,height=6.cm]
{fig27_1000_over_qsv_relation_1-15_fractal_por_perm_50Hz_color.eps}\\
\vspace*{0.3cm}
{\tiny Note the increase  in Q anisotropy for angles between 30 and 60 degrees}
\end{center}
\end{slide}
%----------------------------SLIDE -------------------------------
\begin{slide}{\small{\bf {The Macroscale. I.}}}
\begin{center}
\vspace*{-.4cm}
\includegraphics[width=6cm]
{fig28_model.eps} 
%{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)}
\end{center}
\vskip.2cm
{\small The Macroscopic TIV model: an 
anisotropic rotated TIV layer between two isotropic half-spaces.
The source is a dilatational 
perturbation of central  frequency is 30 Hz}
\end{slide}



%--------------------------------------------------------Slide
\begin{slide}{\small{The Macroscale. II. Seismic modeling.}}
{\bf 
{
We solve the following boundary value problem at the macroscale (in $\O$):  
\begin{eqnarray*}\label{mod3}
&&\hskip-.5cm-\o^2  \rho u  -  \nabla\cdot\tau(u) = F,\quad \O\\
&&-\tau(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\text{(absorbing bounday condition, $D>0$)}\label{mod4}
\end{eqnarray*}
$u =(u_x, u_z)$: displacement vector, \hskip.2cm 
$ \rho$: average  density.
\vskip.5cm
\noi
$\tau(u)$:  stress-tensor of our {\magenta { } equivalent viscoelastic material}, defined in terms of the $p_{IJ}'s$, obtained  using the upscaling procedure.\vskip.5cm
\noi
We solved this problem employing a FE  iterative  
domain decomposition technique formulated in the space-frequency domain. 
}
}
\end{slide}

%----------------------------SLIDE -------------------------------
\begin{slide}{\tiny{\bf {The Macroscale. III. Snapshots of the vertical displacement at 200 ms (a and c) and 500 ms (b and d ).}}}
\begin{center}
\vspace*{-.4cm}
\includegraphics[width=7.cm]
{fig29_snapshots_macoscale_sleipner.eps}\\
{\tiny  Ideal isotropic case (top, a and b) and real (rotated TIV) case (bottom, c and d). }
\end{center}
\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{ {\bf Conclusions} }
\vskip-.7cm
{\bf 
{\small
\begin{itemize}

\item Numerical simulation   
is a useful tool in   
hydrocarbon exploration geophysics, mining and reservoir  characterization and production.


\item  The FEM allows to analyze and represent 
mesoscopic scale heterogeneities in the solid frame and saturant fluids, fractures  and craks  affecting observations at the macroscale.

\item Wave propagation at the macroscale can be efficiently performed 
employing the  FEM combined with  iterative  domain decomposition techniques formulated in the space-frequency domain. 


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

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

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


\end{document}


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

{\small
\begin{itemize}

\item The methodology is  
applied to saturated isotropic 
poroelastic samples having a dense set 
of horizontal  fractures modeled as very thin and compliant layers.

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


\end{itemize} 
}
\end{slide}




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

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

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


\item The numerical simulators were validated using  the theories of  
Gelinsky and  Shapiro (GPY, {\bf 62}, 1997) and  
Krzikalla and  M\"uller (GPY, {\bf 76}, 2011) allow to  
obtain the {\blue { }five} complex and frequency-dependent stiffnesses of 
the {\magenta { }equivalent VTI medium}.  
%----------------------------------------------Slide
\begin{slide}{\small{Applications. The Mesoscopic Loss Mechanism. I}}
%\vskip-.6cm
{\bf
{
Attenuation of (fast) ${\blue P_1}$ waves by  {\magenta {} mode conversion to slow 
${\blue P_2}$ waves}  at mesoscopic scale heterogeneities is a significant loss mechanism 
at seismic frequencies (1-100 Hz). 
\vskip.3cm
Mesoscopic scale heterogeneities  are larger than the pore size 
 but much smaller than ${\blue P_1}$ wavelengths, typically tens of centimeters,  
  and can be due  to local variations in lithological properties (porosity, permeability, etc)  or  to patches of immiscible fluids.
 \vskip.3cm  
White (1975) was the first to show that {\magenta {}  mesoscopic loss }  can be observed in the seismic range in a  
  sandstone with  partial gas saturation.  
}
}
\end{slide}

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

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

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

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

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

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

}
}
\end{slide}

