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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, 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}
{\small
\begin{itemize}
\item {\green {}Fractures} 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} through {\green {}fractures and cracks} 
is an important subject
in exploration and production geophysics, earthquake seismology and mining.

\item {\green {}Fractures} constitute the sources of earthquakes,  and 
hydrocarbon and geothermal reservoirs are mainly composed of 
{\green {}fractured rocks}.


\end{itemize} 
}
\end{slide}

\end{document}
%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. II}
{\small
\begin{itemize}
\item Modeling fractures requires a suitable interface model.
Schoenberg (JASA (1980), GP (1983)) proposed the 
so-called linear-slip boundary condition model {\green {}(LSBC)}, based 
on the discontinuity of the displacement across the fractures. 
{\green {}(Schoenberg's model)}. 

\item  A generalization of the  {\green {}(LSBC)} (Carcione, JGR (1996)) 
 states that across a fracture  
 {\magenta {}stress components} are proportional to 
the displacement and velocity 
discontinuities through the {\green {} specific stiffnesses and specific (
viscosities,} 
respectively.



\end{itemize} 
}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. III}
{\small
\begin{itemize}
\item Displacement discontinuities {\green {} conserve energy}, while 
velocity discontinuities generate {\green {} energy loss} at the fractures. 
The specific viscosity accounts for the presence of a {\magenta {}liquid} 
 under saturated conditions, introducing  
a viscous coupling between both sides of a fracture.

\item Schoemberg's theory predicts that  {\green {} a dense set of 
parallel plane fractures} behaves  
as a {\magenta{} Transversely Isotropic Viscoelastic (TIV) medium}  if 
the dominant wavelength
of the traveling waves is much larger than the distance between the fractures. 



\end{itemize} 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. IV}
{\small
\vskip-.5cm
\begin{itemize}
\item {\green {}Schoenberg's model} has never been simulated with a 
numerical method. 

\item To test the theory, in the 
context of {\green {}Numerical rock physics} we  developped  
a {\magenta {} novel numerical solver} 
that can be used in more  general situations.  

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

\item {\green{} Numerical  experiments} are {\magenta{} inexpensive, repeatable},  essentially free  from experimental errors and 
can easily be run using alternative models of the materials being analized.


\end{itemize} 
}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small Fractured media. V}
{\small

\begin{itemize}

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

\item The BVP's  represent {\green{} harmonic tests} at a finite number of 
frequencies on a sample having a dense set of fractures, modeled using
 the  {\green{} LSBC}.



\end{itemize} 
}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{{\small  {\bf The equivalent TIV medium. I}}}
\vskip-.2cm
{\bf 
{\small
Consider a {\green{} viscoelastic isotropic background medium} having a 
 {\magenta{} set of parallel (horizontal) fractures}   
and its description in the space-frequency domain.

${\bf u}$, $e_{ij}({\bf u}), \sigma_{ij}({\bf u})$:  frequency domain 
displacement vector, strain components and stress components of 
the background medium.

The {\green{} stress-strain relations}  and {\magenta{} equations of motion}:
\begin{eqnarray}\label{st_layer}\nonumber
&&\sigma_{jk}({\bf u})= \lambda \delta_{jk} \nabla\cdot {\bf u} + 2 \mu 
e_{jk}({\bf u})\\\nonumber\\
&&\rho \omega^2 {\bf u} (x, z, \omega) + \nabla \cdot {\BM \sigma} 
[ {\bf u}(x, z, \omega)] = 0 \nonumber
\end{eqnarray}
$\delta_{jk}$: Kroenecker delta \hskip.1cm 
$\lambda, \mu$: complex Lam\'e constants
\hskip.1cm$\rho$: mass density.

}
}
\end{slide}



%--------------------------------------------------------Slide
\begin{slide}{{\small  {\bf The equivalent TIV medium. II}}}
\vskip-.2cm
{\bf 
{\small
$x_1$ and $x_3$: horizontal and vertical coordinates,
respectively.

When a {\green{} dense set of parallel fractures} is present, 
the medium behaves 
as a {\magenta {}TIV medium} with $x_3$-axis of symmetry at long wavelengths.

${\mathbf \tau_{ij}}, {\mathbf\epsilon_{ij}}$:  stress and strain tensors 
of the equivalent TIV medium  
{\magenta{}
Stress-strain relations:} 
\begin{eqnarray}                                                           
&&\tau_{11}({\bf u}) = {p}_{11} ~ \epsilon_{11}({\bf u}) + p_{12} ~ \epsilon_{22}({\bf  u}) 
+ p_{13} ~ \epsilon_{33}({\bf u}), \label{st_GL_1}\\
&&\tau_{22}({\bf u}) = p_{12} ~ \epsilon_{11}({\bf u}) + p_{11} ~ \epsilon_{22}({\bf u})    
 + p_{13} ~ \epsilon_{33}({\bf u}), \label{st_GL_2}\\  
&&\tau_{33}({\bf u}) =  p_{13} ~ \epsilon_{11}({\bf u}) + p_{13} ~ \epsilon_{22}({\bf u})      
+ p_{33} ~  \epsilon_{33}({\bf u}),\label{st_GL_3}\\   
&&\tau_{23}({\bf u}) =  2~  p_{55} ~ \epsilon_{23}({\bf u}), \label{st_GL_4}\\                    
&&\tau_{13}({\bf u}) = 2 ~  p_{55} ~ \epsilon_{13}({\bf u}),\label{st_GL_5} \quad                     
\tau_{12}({\bf u}) = 2 ~  p_{66} ~ \epsilon_{12}({\bf u}) . \label{st_GL_6}           
\end{eqnarray}                                                                         
%}


}
}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{{\small  {\bf The equivalent TIV medium. III}}}
\vskip-.2cm
{\bf 
{\small
Schoenberg's theory predicts that if the background medium is homogeneous, 
the stiffnesses {\green{}${\bf p_{IJ}}$'s} are given by 
\begin{eqnarray}\label{def_pIJ} 
&&p_{11} = p_{22} = E - \lambda^2 Z_N  c_N,\quad 
p_{12} =  \lambda - \lambda^2 Z_N  c_N \quad
p_{13} = \lambda c_N, \\
&&p_{33} = E c_N, \quad
p_{55}= \mu c_T, \quad
p_{66}= \mu. \nonumber
\end{eqnarray}   
where
\begin{equation}\label{PTI1_1}\nonumber
c_N = (1+ E Z_N)^{-1} \ \ \ {\rm and} \ \ \ c_T = (1+ \mu Z_T)^{-1} , 
\end{equation}
{\magenta {}${\bf Z_N}$} and {\magenta {}${\bf Z_T}$}: normal and tangential complex compliances of 
the fractures 

$E = \lambda + 2 \mu$.  

}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{\small  {\bf The equivalent TIV medium. IV}}}
\vskip-.2cm
{\bf 
{\small
{\green {}The theory} assumes that that distance 
between fractures 
is much smaller than the wavelength of the signal and that 
the {\magenta {}boundary condition is the same for all the fractures}. 

Moreover, the theory assumes that the {\magenta {}fracture distance is constant}.
\vskip.4cm
The {\green{}numerical solver} may consider an 
inhomogeneous background medium, unequal fracture distances and dissimilar 
boundary conditions at the fractures surfaces.   
\vskip.4cm
The {\green{}${\bf p_{IJ}}$'s} are the complex and frequency-dependent 
stiffnesses to be determined numerically with the harmonic experiments.

}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components ${\bf p_{IJ}}$}}}
\vskip-.2cm
{\bf 
{\small
$\O= (0,D)^2$: a square  sample  of  boundary 
 $\G = \G^L\cup\G^R\cup\G^B\cup\G^T$, where
\begin{eqnarray}
\G^L = \{ (x,z)\in \G: x=0\},\quad \G^R = \{ (x,z)\in \G: x= D \},\nonumber\\
\G^B = \{ (x,z)\in \G: z=0\},\quad \G^T = \{ (x,z)\in \G: z= D \}.\nonumber 
\end{eqnarray}
$\G^{(f,l)}, l=1,\cdots,J^{(f)}$: a set   of $J^{(f)}$
horizontal fractures each one of length $D$ 
 in our domain $\O$. 

This set of fractures  divides $\O$ in a 
set  of nonoverlapping rectangles $R^{(l)}, l=1,\cdots, J^f + 1$, so that 
\[
\O = \cup_{l=1}^{J^{(f)} + 1} R^{(l)}.
\] 

}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Boundary conditions at the fractures. I}}}
\vskip-.2cm
{\bf 
{\small


Consider a  fracture $\G^{(f,l)}$ and the two rectangles $R^{(l)}$ and 
$R^{(l+1)}$ having as a common side $\G^{(f,l)}$. 
\vskip.5cm
${\bf n}_{l,l+1},{\bf m}_{l,l+1}$: the  unit outer  
normal and a unit tangent (oriented counterclockwise)  
on $\G^{(f,l)}$  from  $R^{(l)}$  to $R^{(l+1)}$, 
 such that $\{{\bf n}_{l,l+1}, {\bf m}_{l,l+1} \}$ is an 
orthonormal system on $\G^{(f,l)}$. 
\vskip.5cm
Set ${\bf u}^{(l)} = {\bf u}|_{R^{(l)}}$: restriction of ${\bf u}$ to $R^{(l)}$, 
and let
\[
[{\bf u}] = \left({\bf u}^{(l)} - {\bf u}^{(l+1)}\right)|_{\G^{(f,l)}} 
\]
denote the jump of ${\bf u}$  at $\G^{(f,l)}$  


}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Boundary conditions at the fractures. II}}}
\vskip-.2cm
{\bf 
{\small
The boundary conditions (B.C.) at the  fractures  $\G^{(f,l)}$ are {\green{} the stress continuity}  and  the  condition that   
{\green{} stress components  be proportional 
to the displacement and velocity discontinuities} through  {\magenta{}specific stiffnesses 
and viscosities}, respectively. Thus, 
\begin{eqnarray}\label{bc_frac}
&&\BM \sigma({\bf u}^{(l)}){\bf n}_{l,l+1} = \BM \sigma({\bf u}^{(l+1)}){\bf n}_{l,l+1}
\quad \G^{(f,l)},\\ \nonumber\\
&& \left(\BM \sigma({\bf u}^{(l)}) {\bf n}_{l,l+1}\cdot {\bf n}_{l,l+1}, 
   \BM \sigma({\bf u}^{(l)}) {\bf n}_{l,l+1}\cdot {\bf m}_{l,l+1}\right)^T \nonumber\\ \nonumber\\
&&\quad =   
{\green{}{\mathbf S^{(l)}}(\o)}\left([{\bf u}]\cdot {\bf n}_{l,l+1}, [u]\cdot {\bf m}_{l,l+1} 
\right)^T
\quad \G^{(f,l)},\,\, l=1,\cdots,J^{(f)}.
\label{bc_frac1}
\end{eqnarray}
where $T$ indicates  the transpose. 


}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Boundary conditions at the fractures. III}}}
\vskip-.2cm
{\bf 
{\small
If $L$ denotes the average distance between the fractures, the matrix 
{\green{}${\mathbf S^{(l)}}(\o)$} has the form 
\begin{eqnarray}\label{def_matrix_S}
{\green{} {\mathbf S^{(l)}}(\o)}= 
\begin{pmatrix}
(L Z_N^{l})^{-1}& 0 \\
0 &(L Z_T^{(l)})^{-1}
\end{pmatrix} 
\end{eqnarray}
The {\magenta{} compliances $Z$ ($Z_N$ or $Z_T$)}
 are complex and frequency-dependent and can be expressed as 
\begin{equation}\label{Z1}\nonumber
Z^{-1} = L ( \kappa + \ii \omega \eta ) , 
\end{equation}
where {\green{} $\kappa$ is a specific stiffness} and {\magenta{}$\eta$ is a
specific viscosity}, having dimensions of 
stiffness and viscosity per unit length, respectively. 

}
}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. I}}}
\vskip-.2cm
{\bf 
{\small
{\green{} For $p_{33}$}  we solve solve the wave equation  in $\O$ using the 
fracture B. C.'s   with the additional B. C.'s 
\begin{eqnarray} 
&& \BM \sigma({\bf u}) {\bf n} \cdot {\bf n} = -\Delta P, \quad  \G^T,  \label{bc1a}  \\
&& \BM\sigma({\bf u}) {\bf n}  \cdot {\bf m} = 0, \quad \G,   \label{bc1b} \\
&& {\bf u}\cdot\nu = 0, \quad  \G^L\cup\G^R\cup \G^B.  \label{bc1c}  
\end{eqnarray}
In this experiment $~ \epsilon_{11}({\bf u})= ~ \epsilon_{22}({\bf u})= 0$.
}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
$\bf{  p_{IJ}}$. II}}}
\vskip-.2cm
{\bf 
{\small
Denoting by $V$ the original volume of the sample, 
its (complex) oscillatory volume change, 
$\Delta V(\omega)$, we note that in the quasistatic case  

\begin{equation}\label{p33-dv}\nonumber
\frac{\Delta V(\omega)}{V}=-\frac{\Delta P}{p_{33}(\omega)},
\end{equation}

The computed average vertical displacement $u^{s,T}_3(\omega)$ suffered by the 
boundary
$\Gamma^T$ allows us to use  the estimate  
\[
\Delta V(\omega)\approx D u^{s,T}_3(\omega), 
\]
from where we can determine $p_{33}(\o)$.
 
}
}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. III}}}
\vskip-.2cm
{\bf 
{\small
{\green{}For  $p_{11}$}  we solve the wave equation with the fracture B. C.'s 
and the additional B. C.'s
\begin{equation}\label{bcp11}\nonumber
\begin{array}{l}
{\BM \sigma} ({\bf u}) \cdot {\bf n} \cdot {\bf n} = - \Delta P, \quad  \G^R,  \\ 
{\BM \sigma} ({\bf u}) \cdot {\bf n} \cdot {\bf m} = 0, \quad \G,  \\
{\bf u} \cdot {\bf n} = 0, \quad  \G^L\cup\G^B\cup\G^T.
\end{array}
\end{equation}
In this experiment, $\epsilon_{33}$ = $\epsilon_{22}$ =  0
and this experiment determines $p_{11}$ computing the volume change in the same way
indicated for $p_{33}$.

}
}
\end{slide}


%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. IV}}}
\vskip-.2cm
{\bf 
{\small
{\green{} For $p_{55}$} we solve the wave equation with the fracture B. C.'s and the 
additional B. C.'s
\begin{equation}\label{bcp55}
\begin{array}{l}\nonumber
{\BM \sigma} \cdot {\bf m} = {\bf g},  \quad  \G^T \cup \G^L \cup \G^R ,  \\
{\bf u}  = 0, \quad  \G^B ,   
\end{array}
\end{equation}
where
\begin{eqnarray*}
{\bf g} = 
\begin{cases} (0,  \Delta G),\quad  \G^L,\\
(0, - \Delta G),\quad  \G^R,\\
(\Delta G, 0), \quad \G^T.
\end{cases}
\end{eqnarray*}

}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. V}}}
\vskip-.2cm
{\bf 
{\small
Let $\theta (\omega)$: angle between the original positions of the
lateral boundaries and the location after applying the shear stresses.
 
To estimate $\theta (\omega)$, we  compute  the  average horizontal
displacement  ${\bf u}^{T}_1(\omega)$ at the
boundary $\Gamma^T$ and use that    
\[
\tan [\theta(\omega)] \approx {\bf u}^{T}_1(\omega)/{D}.
\]
Thus, {\magenta {}the change in shape of the rock sample} allow us to  determine 
 {\green {}$p_{55}(\omega)$} from the relation (Kolsky, 1963)
\begin{equation}\label{tg}\nonumber
\tan [ \theta(\omega) ] =\frac{\Delta G}{p_{55}(\omega)}.
\end{equation}

}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. VI}}}
\vskip-.2cm
{\bf 
{\small
{\green{} For $p_{13}$} we solve the wave equation with the fracture B. C.'s and the additional 
 B. C.'s 
\begin{equation}\label{bcp13}
\begin{array}{l}\nonumber 
 {\BM \sigma} ({\bf u}) \cdot {\bf n} \cdot {\bf n} = - \Delta P, \quad \G^R \cup \G^T,  \\ 

 {\BM \sigma} ({\bf u}) \cdot {\bf n} \cdot {\bf m} = 0, \quad  \G,  \\
{\bf u} \cdot {\bf n} = 0, \quad \G^L\cup\G^B . 
\end{array}
\end{equation}
In this experiment $\epsilon_{22}$ = 0, and  from the  stress-strain relations
at the macroscale we get 
}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{ {\bf Determination of the stiffness components 
${\bf p_{IJ}}$. VII}}}
\vskip-.2cm
{\bf 
{\small
\begin{equation}\label{st13}
\begin{array}{l}\nonumber
\tau_{11} =   p_{11} \epsilon_{11} +  p_{13} \epsilon_{33}, \\
\tau_{33} =   p_{13} \epsilon_{11} +  p_{33} \epsilon_{33},
\end{array}
\end{equation}
$\epsilon_{11}, \epsilon_{33}$:  macroscopic  
strain components at the  right lateral side and top side
of the sample, respectively. 
\vskip.5cm
Then  using that 
 $\tau_{11}=\tau_{33}= - \Delta P$  we obtain
\begin{equation} \label{p13}\nonumber
p_{13}(\o) = \frac {p_{11} \epsilon_{11} - p_{33} \epsilon_{33}}{\epsilon_{11} - \epsilon_{33}}.
\end{equation}


}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf A variational formulation}}}
\vskip-.2cm
{\bf 
{\tiny
Test space fo {\green{} ${\bf p_{33}(\o)}$}:
\[
{\mathcal W}_{33}(\O)= \{v\in [L^2(\O)]^2: v|_{R^{(l)}}\in [H^1(R^{(l)})]^2,\,\,
v\cdot\nu = 0 \,\,\,  \text{on} 
\, \,\G^L\cup \G^R\cup \G^B \},
\]
To determine {\green{} ${\bf p_{33}(\o)}$}: 
find ${\bf u^{(33)}}\in  {\mathcal W}_{33}(\O)$  such that: 
\begin{eqnarray}\label{def_bilinear}
&&- \o^2 (u, v) +  
  \sum_{l=1}^{J^{(f)}+1}\sum_{s,t=1,3}\left(\sigma_{st}(u),\epsilon_{st}(v)
\right)_{R^{(l)}}\\
&&+\sum_{l=1}^{J^{(f)}} \left[ \left\langle [L Z_N^{(l)}]^{-1}  [u]_3, 
[v]_3\right\rangle_{\G^{(f,l)}} 
+ \left\langle [L Z_T^{(l)}]^{-1} [u]_1, [v]_1\right\rangle_{\G^{(f,l)}}\right]\nonumber\\
&&= -\left\langle \Delta P, v \cdot\nu\right\rangle_{\G^T}, \qquad  
\forall  v \in {\mathcal W}_{33}(\O).
\end{eqnarray}

Similar formulations hold for the other $\bf{ p_{IJ}}$'s 


}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Finite element implementation}}}
\vskip-.2cm
{\bf 
{\small
The FE variational formulation uses {\magenta {} bilinear elements} to compute approximate solution of the BVP, 
and  is explained in Santos et al. (CMAME, 2012, submittted), 
where apriori error estimates which are optimal for the regularity of the solution are given.

\vskip.5cm
Results on the existence and uniqueness of the continuous and discrete BVP's  
are also given in that reference.  

\vskip.5cm
The {\green{} error} is of the order of {\magenta {} ${\bf h}$} in the {\magenta {}${\bf L^2}$-norm }and of the
order of {\magenta {}${\bf h^{1/2}}$} both in the {\magenta {}interior energy norm} and in the {\magenta {}${\bf L^2}$-norm} on
the set of fractures, where  {\magenta {} ${\bf h}$ is the diameter} of the elements.


}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Numerical experiments.}}}
\vskip-.4cm
{\bf 
{\small
We consider the data provided by the laboratory experiments of Chichinina et al. (TPM, 2009).
The  background medium is isotropic with  $\lambda$ = 10 GPa, $\mu$ = 3.9 GPa and $\rho$ = 2300 kg/m$^3$. 

The simulations to determine the {\magenta {}${\bf p_{IJ}}$'s} have fracture distance {\green {}$L$ =1 cm},  grid spacing 
{\green {}${\bf h}$ = 0.5 cm} and a frequency {\green {}$f_0$ = 25  Hz}. 

{\magenta {} ${\bf p_{11}}$, ${\bf p_{13}}$ and ${\bf p_{33}}$:} 29 equally spaced fractures, $60 \times 60$ mesh. 

{\magenta {}${\bf p_{55}}$}:  14 fractures, $30 \times 30$ mesh. 


Experimental values of {\magenta {}${\bf Z_N}$ } and {\magenta {}${\bf Z_T}$ }for {\green {}wet fractures } scaled to seismic frequencies:

{\magenta {}${\bf Z_N^{-1}}$} = [ 34 + $\ii (f/f_0)$ 24.7 ] GPa \hskip.5cm 
{\magenta {}${\bf Z_T^{-1}}$ }= [ 15.5 + $\ii (f/f_0)$ 11.3 ] GPa 

Determination of reliable fracture parameters needs measurements at the seismic range (experimental data was obtained at 100 kHz).
   



}
}
\end{slide}



%--------------------------------------------------------Slide
\begin{slide}{\small{{\bf Validation of the FE method. Phase velocities as function of frequency 
for wet fractures.}}}
{\bf
{\small
\begin{center}
\includegraphics[width=5.cm,height=4cm]{validation_vp_11_33_fe_and_analytical_wet.eps}\hskip1cm 
\includegraphics[width=5.cm,height=4cm]{validation_vs_55_fe_and_analytical_wet.eps}\\
\end{center}
\vskip.3cm
{\tiny ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane.
 ``55'' refers to the qS wave  perperdicular to the fracture plane. 
A very good fit is observed. qP waves along the fracture plane (``11'')  travel faster 
  than  qP waves travelling perpendicular to the fractures (``33''). 
}
}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{\small{{\bf Validation of the FE method. Dissipation factor as function of frequency 
for wet fractures.}}}
{\bf
{\small
\begin{center}
\includegraphics[width=5.cm,height=4cm]{validation_inv_qp_11_33_fe_and_analytical_wet.eps}\hskip1cm 
\includegraphics[width=5.cm,height=4cm]{validation_inv_qs55_fe_and_analytical_wet.eps}\\
\end{center}
\vskip.3cm
{\tiny ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane.
 ``55'' and ``66'' refer to the qS and SH  waves perperdicular 
and along the fracture plane. The SH wave is lossless. A very good fit is observed. qP waves along the fracture plane (``11'') suffer  lower attenuation than  qP waves travelling perpendicular to the fractures (``33''). 
}
}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Fractures at varying pore fluid pressure .}}}
\vskip-.4cm
{\bf 
{\small
Daley et a. (GPY, 2006) suggest to take high values of fracture 
compliance at low normal {\magenta {}effective stress $\sigma = pc - pp$,} where $pc$ is the confining pressure and $pp$ the prore presure. 
  
For  a contant $pc$ = 30 MPa, we  consider two  pore pressures 5 MPa and 25 MPa, normal and 
overpressure values, respectively. Using their model, we  obtain, at 25 Hz, 
\begin{equation}\label{Zm}\nonumber
\begin{array}{lll}
{\magenta {}pp = 5 \ {\rm MPa}}, &  Z_N^{-1} = (23.1 +  5.9 \ii )  \ {\rm GPa} , & Z_T^{-1} = (75 +  9.4 \ii )  \ {\rm GPa} , \\
{\magenta {}pp = 28  \ {\rm MPa}}, & Z_N^{-1} = (14.4 + 3.6 \ii )  \ {\rm GPa} , & Z_T^{-1} = (21 +  2.6 \ii ) \ {\rm GPa} ,
\end{array}
\end{equation}

We consider a set of equispaced fractures with $L$ = 1 cm and 80 
\% binary fractal variations of $Z_N$ and $Z_T$ around these mean values.
}
}
\end{slide}


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

\begin{slide}{{{\bf{\small Real part of fractal ${\bf Z_N^{-1}}$ at pore pressure 28 MPa. 
}}} }
\begin{center}
\vspace*{-0.3cm}
%\hskip-.2cm
\includegraphics[width=7.cm,height=5.cm]{frac_LcapaN_gnu_aa_0.3_binary_80_percent_28MPa.eps}
\vskip.5cm
{\small 80 \% binary fractal variations of $Z_N$ around the mean value  23.1 GPa}
\end{center}
\end{slide}


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

\begin{slide}{\tiny {{\bf Phase velocity as function of frequency for   
fractal ZN ZT, wet fractures. Confining pressure is 30 MPa. Pore pressures are 5 MPa (normal) and 28 MPa (overpressure).}} }
\begin{center}
%\vspace*{-0.2cm}
%\hskip-.2cm
\includegraphics[width=7.cm,height=4.3cm]{vp_11_33_fe_fractal_ZN_ZT_5MPa_28_MPa_wet_logfreq.eps}
\vskip.4cm
{\tiny $\sigma = pc- pp$ is the effective normal stress, $pc$ = confining pressure, $pp$= pore pressure. ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane.  Higher pore pressure (circles) implies lower phase velocity. The ``33'' qP wave is the one more affected by overpressure.}
\end{center}
\end{slide}


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

\begin{slide}{\tiny {{\bf Dissipation factor as function of frequency for   
fractal ZN ZT, wet fractures. Confining pressure is 30 MPa. Pore pressures are  5 MPa (normal) and 28 MPa (overpressure).}} }
\begin{center}
%\vspace*{-0.2cm}
%\hskip-.2cm
\includegraphics[width=7.cm,height=4.5cm]{inv_qp_11_33_fe_fractal_ZN_ZT_5MPa_28_MPa_wet_loggfreq.eps}
\vskip.5cm
{\tiny $\sigma = pc- pp$ is the effective normal stress, $pc$ = confining pressure, $pp$= pore pressure. ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane. Attenuation is stronger in the overpressured case (circles) for  ``33'' waves.}
\end{center}
\end{slide}
%-----------------------------------------Slide

\begin{slide}{\tiny {{\bf Phase velocity as function of frequency for   
fractal and uniform ZN ZT, wet fractures. Confining pressure is 30 MPa. Pore pressure is 28 MPa (overpressure).}} }
\begin{center}
%\vspace*{-0.2cm}
%\hskip-.2cm
\includegraphics[width=7.cm,height=4.3cm]{vp_11_33_fe_fractal_and_uniform_ZN_ZT_28MPa_wet_logfreq.eps}
\vskip.3cm
{\tiny ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane. Phase velocities in the fractal case are lower than those obtained with the mean values. The ``33'' qP wave  is the one more affected by the heterogeneities.}
\end{center}
\end{slide}


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

\begin{slide}{\tiny {{\bf Dissipation factor as function of frequency for   
fractal and uniform ZN ZT, wet fractures. Confining pressure is 30 MPa. Pore pressure is 28 MPa (overpressure).}} }
\begin{center}
%\vspace*{-0.2cm}
%\hskip-.2cm
\includegraphics[width=7.cm,height=4.5cm]{inv_qp_11_33_fe_fractal_and_uniform_ZN_ZT_28MPa_wet_logfreq.eps}
\vskip.5cm
{\tiny ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane. Dissipation factor of the ``33'' qP wave  is more affected by the heterogeneities, showing lower values in the fractal case. }
\end{center}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf Fractal $\lambda$ and $\mu$ background.}}}

{\bf 
{\small
{\tiny
Next we consider 50 \% binary fractal variations of the background Lam\'e constants $\lambda$ and $\mu$ with respect to the  mean values 10 GPa and 3.9 GPa, respectively.
}
\begin{center}
\includegraphics[width=5.cm,height=5cm]{frac_lambda_gnu_mean_10_GPa_aa_0.7_binary_50_percent.eps}\hskip.1cm 
\includegraphics[width=5.cm,height=5cm]{frac_mu_gnu_mean_3.9_GPa_aa_0.7_binary_50_percent.eps}\\
\end{center}

}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{\tiny{{\bf Phase velocities and dissipation as function of 
frequency. Uniform and fractal brackground, dry  fractures: $Z_N^{-1}$ =  9.6 + $\ii (f/f_0)$ 4.8 GPa, 
$Z_T^{-1}$ = 3.1+ $\ii (f/f_0)$ 0.12 GPa}}}
{\bf
{\small
\begin{center}
\includegraphics[width=5.cm,height=4cm]{vp11_vp33_fe_fractal_and_uniform_lambda_mu.eps}\hskip1cm 
\includegraphics[width=5.cm,height=4cm]{inv_qp11_33_fe_fractal_and_uniform_backgound.eps}\\
\end{center}
\vskip.3cm
{\tiny ``11'' and ``33'' refer to the qP wave along and perpendicular to the fracture plane. Phase velocities are lower for the fractal case for both ``11'' and ``33'' qP waves.  Concerning attenuation,  for qP ``33'' waves  is  lower than in the uniform background case, while attenuation   for qP ``11'' waves is not affected by the fractal background.
}
}
}
\end{slide}

%--------------------------------------------------------Slide
\begin{slide}{{  {\bf CONCLUSIONS.}}}

{\bf 
{\small
\begin{itemize}
\item 
{\magenta {}Schoenberg's theory} predicts that an homogeneous background
containing a dense set of horizontal parallel fractures behaves like a
{\green {}TIV  medium} at long wavelengths.

\item We presented a collection of {\green {}novel FE harmonic experiments} to test and validate the theory. 

\item The methodology was applied 
 to a case  where there is no analytical solution, such as fractal 
 variations of the fracture compliances at different pore pressures and fractal Lam\'e parameters. 

\item In particular, it is shown that {\magenta {}attenuation} can be an {\magenta {}indicator of overpressure} with higher values at high pore pressures. 

\end{itemize}
}
}
\end{slide}
%--------------------------------------------------------Slide
\begin{slide}{{  {\bf AKNOWLEDGEMENTS.}}}

{\bf 
{

I give special thanks to professor Yalchin Efendiev for inviting me to present this work in the 2012  KAUST-IAMCS WORKSHOP.
\vskip.5cm
I also  mention the great support of   all the people at ISC-TAMU and KAUST  in dealing  
with all the paperwork needed to get me here.
\vskip.5cm
{\magenta {}THANKS FOR YOUR ATTENTION !!!! }
}
}
\end{slide}

\end{document}
%-----------------------------------------Slide

\begin{slide}{ {\small{\bf Illustration of the harmonic experiments to determine the complex stiffness ${\bf p_{IJ}}$'s }} }
\begin{center}
\vspace*{-0.2cm}
\hskip-.2cm
\includegraphics[width=6.cm,height=6.cm]{fs1box.eps}
%\\ 
%{\tiny Figure  2. Top left: gas, Top right: water, Bottom: periodic gas-water }
\end{center}
\end{slide}