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\title{
P-wave attenuation in thin-layered  non-isothermal fluid-saturated  poroelastic solids 
%Numerical experiments to characterize  P-wave attenuation in partially saturated 
%non-isothermal porous media
}
%in heterogeneous porous media}

\subtitle{SEG Rock Physics and Geofluid Detection Workshop, 8-10 December 2023, Hohai University, 
 Nanjing, China
 } 
\author{{\bf Juan E. Santos},}%${^\dag}$ }
\institute{% 
Universidad de Buenos Aires (UBA),  Argentina, Hohai University, Nanjing, China and Purdue University, Indiana, USA.\\
 {\color{blue} In collaboration with Naddia  D. Arenas (PhD student).    G. B. Savioli (UBA) and 
 J. M. Carcione (OGS, Italy)  } 
} 

%\author{Pepe}
%\institute{% 
%UBA. }
\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Sketch of the talk}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}

\begin{document}

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


%\section{Wave propagation  in non-isothermal poroelastic media}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Waves in non-isothermal poroelastic media. I}}
{\small
\begin{itemize}

\item This work uses a model that  combines the Biot (J.App.Phys., 1957) and Lord-Shulman theories 
(J.Mech.Phys.Sol., 1967)   to study  {\color {blue} mesoscopic P-wave attenuation} 
in  a  thin-layered {\color {magenta} non-isothermal poroelastic medium} alternately saturated by gas and water.


\item The model  predicts the existence  of {\color {magenta} four waves}: two compressional P waves, {\color {blue}  one fast (P1)} 
and one  {\color {blue} slow (diffusive)  (P2)}, a {\color {magenta}  slow (diffusive) 
thermal  (T)} wave, and a {\color {magenta} shear (S)} wave.

\item
The T wave is  {\color {magenta} coupled} with  both P-waves (Sharma (J.Earth.Sys.Sci, 2008), 
Carcione et al. (J.Geophys.Res., 2019). S-waves are uncoupled with T-waves.

\item The model assumes    that the temperature in the solid and  fluid is the same.

%\item The two slow waves exhibit diffusive behavior at low frequencies, depending on the viscosity %and thermoelasticity constants. 


\end{itemize} 
}
\end{frame}
\begin{frame}
\frametitle{{\small Waves in non-isothermal poroelastic media. II}}
\vskip-.1cm
{\small
\begin{itemize}

\item The mesoscopic loss mechanism is due to  {\color {magenta} mode conversion} at the gas-water interfaces.


\item The   {\color {blue} objective } is to study  {\color {blue}  P-wave attenuation} associated with  the presence of   {\color {magenta} T-waves } and its relation  
with the wave induced fluid flow  {\color {blue} (WIFF) mechanism.}  

\item  A Finite Element (FE) procedure is used to solve an  initial boundary value problem   {\color {blue}(IBVP) }
for the Biot/Lord-Schulman equations in  layered media.

\item The solution of the  {\color {blue} IBVP } yields the  seismic response in terms of displacements of the solid and fluid phases and temperature, which are recorded to  observe and quantify attenuation effects of P-waves.

\item The  {\color {blue} spectral ratio and frequency shift methods} were used to estimate the 
 {\color {magenta} quality factor   Q}.

\end{itemize} 
}
\end{frame}


% -------------------------------SLIDE---------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small   Mathematical model. I}}
{\small
Consider a fluid-saturated  poroelastic  medium,  
%saturated by a single phase, 
%compressible viscous fluid  
and assume 
that whole aggregate is isotropic. 
 \vskip0.4cm
 
 {\color {blue} $\theta$:   }
increment of temperature above a 
reference absolute temperature {\color {magenta}$T_0$ } for the state of zero stress and strain.

\vskip0.4cm
{\color {blue} $\bu^{s} = (u_i^s)$,  
$\bu^{f} = (u^{f}_i)$: }   
average particle displacement vectors  of the solid and relative fluid phases, 
respectively. 

{\color {blue} $\bu =(\bu^s, \bu^f)$ }
 \vskip0.4cm
{\color {magenta} $\bvarep({\color {blue}\bu^s}) = (\varepsilon_{ij}({\color {blue}\bu^s}))$, }  
{\color {magenta} $\bsig({\color {blue} \bu,\theta})=(\sigma_{ij}({\color {blue} \bu,\theta}))$:}   strain and stress tensors of the solid and  bulk material,  respectively.

\vskip0.4cm
{\color {magenta} $p_f= p_f({\color {blue} \bu,\theta})$: } fluid pressure. 
 
 
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Dynamical equations in thermo-poroelastic media}}
{\small

\begin{eqnarray*}
&&\rho_b {\color {blue}\bu^s} + \rho_f {\color {blue}\bu^f } 
- \nabla\cdot {\color {magenta}\bsig}({\color {blue} \bu,\theta}) = {\mathbf f}^s,\label{biot1}\\
&& \rho_f {\color {blue} \bu^s} + g {\color {blue}\bu^f} 
+ \dfrac{\eta}{\kappa} {\color {blue}\bu^f} 
+ \nabla {\color {magenta} p_f}({\color {blue}\bu,\theta}) = {\mathbf f}^f.\nonumber
%&&{\mathcal {P}} \ddot \bu +  
%{\mathcal {B}} \dot \bu^f - {\mathcal {L}}(\bu,\theta)= {\mathbf f}.\label{mod3a}
\end{eqnarray*}

$\rho_b, \rho_f$:  mass density of the bulk material and  the fluid 

\vskip.5cm
$\eta$: fluid viscosity \qquad 
$\kappa$:permeability

\vskip.5cm
$g$: mass coupling parameter. 
}

\vskip.5cm
To represent thermal effects, Biot's original constitutive equations are modified by 
adding  {\color {magenta} positive coupling  
 coefficients} {\color {blue} $\beta$ and $\beta_f$} of the bulk material   and fluid, respectively.
\end{frame}
%\end{document}

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

\begin{frame}
\frametitle{{\small Constitutive relations}}
\vskip-.6cm
{\small

\begin{eqnarray*}
&&
%\sigma_{ij}(\bu,\theta)
{\color {magenta}\sigma_{ij}}({\color {blue} \bu,\theta})=2 \mu\,\varepsilon_{ij}(\bu^s)
%\nonumber \\ 
%&& 
+\delta_{ij}
(\lambda_u \,\nabla\cdot \bu^s + B \nabla\cdot \bu^f- \beta \, \theta),\label{mod.2a}\\
&&
%p_f(\bu,\theta)
 - {\color {magenta}  p_f({\color {blue} \bu,\theta})} =   B \,\nabla\cdot \bu^s + M \nabla\cdot \bu^f - \beta_f \theta\label{mod.2b}. 
\end{eqnarray*}
 $\mu$: dry-material  shear modulus, 
 
\vskip.4cm
$M = \left(\dfrac{\alpha- \phi}{K_s} 
+ \dfrac{\phi}{K_f}\right)^{-1}$,  \, $\phi$: porosity \, $\alpha = 1 - K_m/K_s$ 

\vskip.4cm 
$B = \alpha ~ M,$ \,\,  $\lambda_u= \lambda + \alpha^2 M$

\vskip.4cm 
 $K_s, K_m$ and $ K_f$: bulk moduli of the grains, solid frame and fluid,  
respectively.


}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Generalized heat equation}}
\vskip-.5cm
{\small


\begin{eqnarray*}\label{heat1}
&&\tau ~ c ~ {\color {blue}\ddot \theta} + c ~  {\color {blue}\dot \theta} -\nabla\cdot (\gamma\nabla {\color {blue}\theta})
+  \beta T_0 \nabla\cdot {\color {blue}\dot\bu^s}\nonumber \\
&& + \beta T_0 \nabla\cdot {\color {blue}\dot\bu^f} 
+ \tau \beta T_0 \nabla\cdot {\color {blue}\ddot \bu^s}  
%\\
%&&
+ \tau  \beta T_0 \nabla\cdot {\color {blue}\ddot \bu^f}= -q. \nonumber
\end{eqnarray*}


{\color {magenta} $\gamma $}: thermal conductivity


\vskip.3cm
{\color {magenta} $c$}: bulk specific heat of the unit volume in the absence of deformation

\vskip.3cm
{\color {magenta} $\tau$}: relaxation time,\,\, $q$: heat source.

These equations assume that the temperature in both phases is the same. 

\vskip.2cm
{\color {blue} $\beta$, $\beta_f$, $\gamma$ and $c$} are considered as parameters, obtained 
from experiments or from a specific theoretical model.
}

\end{frame}


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}
\frametitle{{\small Numerical experiments. Quality factor  {\color {blue} $Q$}. }}

{

 Attenuation is defined as the decrease in amplitude of a wave as it propagates over a distance.
 
 
 \vskip1cm 
 
 The {\color {blue} quality factor $Q$} estimates the material dissipation and quantifies the attenuation. 
 
%It is defined  as a function of frequency $f$  in terms of the complex velocity $v_c(2 \pi f)$  by the relation
%\begin{eqnarray*}
%{\color {blue} Q(f)} = \dfrac{{\rm Re} (v_c(2 \pi f))}{ {\rm Im} (v_c(2 \pi f))}.
%\end{eqnarray*}

}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Estimation  of the quality factor  {\color {blue} Q}. Spectral-ratio (SR) method  }}
The   Spectral-ratio method  estimates  {\color {blue} $Q$} from the relation: 


\begin{eqnarray}\label{def_ampl}
{\rm ln}\left[\dfrac{A(f,r_s)}{A(f,r_t)}\right] = \dfrac{\pi (d_t-d_s)}{c_pQ} f 
\label{spectral}
\end{eqnarray}

$A(f,r_s)$, $A(f,r_t)$: amplitude spectrum   
of receivers $r_s$,  $r_t$.

\vskip.3cm
$r_s$  considered as a source  for the receiver  $r_t$
\vskip.3cm
$d_s, d_t$: the distances of  $r_s$ and   $r_t$   from the source

\vskip.3cm
$c_p$: average compressional phase velocity {\color {magenta}in a 
region containing the receivers $r_s$ and $r_t$} estimated from the corresponding {\color {blue} arrival times}.

 {\color {blue} $Q$} is computed from the slope of the semi-log relationship \eqref{def_ampl}
\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Estimation  of the quality factor  {\color {blue} Q}. Frequency-shift (FS) method}}

The Frequency-shift method   relates   $Q$ with the {\color {magenta} centroid 
 frequencies $f_s$ and $f_t$ of  $r_s$ and $r_t$} using the relation

\begin{eqnarray*}
\pi\dfrac{(d_t-d_s)}{c_pQ} = \dfrac{(f_s - f_t)}{{\color {blue}\sigma_s}^2}
\end{eqnarray*}

\begin{eqnarray*}
 {\color {magenta}f_j }= \dfrac{ \int_{0}^{\infty} fA(f,r_j) df}{\int_{0}^{\infty} A(f,r_j) df}, \quad j = s, t
\end{eqnarray*}
A wave  propagating through this layered  medium loses high frequencies, thus the centroid decreases. 
This effect is measured by the downshift $\Delta f = f_s - f_t$.

${\color {blue}\sigma_s}$:  obtained by fitting a Gaussian curve to the amplitude spectrum $A(f, r_s)$.

\end{frame}




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

\begin{frame}
\frametitle{{\small Numerical experiments. White model.}}

{\small
To analyze the amplitude damping of a wave propagating in a  thin-layered  {\color {blue} non-isothermal }
poroelastic medium alternatively saturated by gas or water, the results are compared  with those
 of  {\color {magenta}  White's isothermal theory}. 

\vskip.5cm

{\color {magenta}  White's isothermal theory}  describes dissipation in this type of medium, 
which is induced by  the interaction generated by the presence of two fluids with different compressibilities.


\vskip.5cm

Values of  White's  model  are obtained using the material properties in Table 1.

}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments. Material properties.}}

{\small

 In the examples we consider a periodic layered medium of material properties as in Table 1 of
 layer thicknesses  15, 20 and 30 cm,  alternatively saturated with either gas or water.
 
{\tiny
\begin{center}
{\bf Table 1. Material Properties}
\[
\begin{tabular}{cc}
\hline
\hline

Grain bulk modulus, $K_s$   &  37  GPa  \\ 
   \   density, $\rho_s$   &  2650  kg/m$^3$ \\
\hline   


Frame bulk modulus, $K_m$   &  8  GPa  \\ 
      shear modulus, $\mu_m$   &  9.5  GPa  \\ 
      porosity, $\phi$  &  0.3  \\ 
    \ \  permeability, $\kappa$ &  1 darcy\\ 
\hline

Water bulk modulus, $K_w$    &  2.25  GPa  \\ 
      Water density, $\rho_w$    &      1040  kg/m$^3$ \\
	  Water viscosity, $\eta_w$  &      0.003    Pa $\cdot$ s  \\
 
\hline

Gas bulk modulus, $K_g$   & 0.012  GPa  \\ 
    Gas  density, $\rho_g$   &    78  kg/m$^3$ \\
	 Gas viscosity, $\eta_g$ &         0.000015 Pa $\cdot$ s  \\
	 
\hline	 

 
Bulk specific heat, $c$    &   820 kg/(m s$^2$ K)   \\
     thermoelasticity coefficient, $\beta$ &     90000 kg/(m s$^2$ K)  \\
\ \	  thermoelasticity coefficient, $\beta_f$ &    50000 kg/(m s$^2$ K)  \\
     absolute temperature, $T_0$ &    300  K \\
     thermal conductivity, $\gamma$  &   4.5 $\times 10^6$ kg/m$^3$ \\
     \ \    relaxation time, $\tau$ &     1.5 $\times 10^{-2}$ s  \\		

\hline
\hline 
\end{tabular}
\]
\end{center}
}
}
\end{frame}

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

\begin{frame}
\frametitle{{\small Numerical experiments. White model.}}
\vskip-.1cm
{\tiny

 {\color {blue} $Q^{-1}$}  determined by {\color {blue} using the White model}  as a function of frequency for a periodic layered medium  of thicknesses 
15 cm, 20 cm and 30 cm. The minimum value of the quality factor for all thicknesses is   $\approxeq$ 28.
%{\color {blue} using the White model}. 
\vskip.2cm
This  {\color {blue}  minimum value of $Q$ is  attained }  
at {\color {blue} $f_0=$ 140 Hz},  {\color {blue} $f_0=$ 77 Hz} and  {\color {blue} $f_0=$ 34 Hz} for {\color {magenta} layer thickness 15 cm, 20 cm and 30 cm}, respectively.
{\tiny These frequencies 
 are used as dominant frequency of the external source in the next  experiments}.


\begin{figure}
\label{fig1}
{
%\centering
\begin{center}
\includegraphics[scale=0.24,angle=0]{white_3espesores.eps}
\end{center}
%\vskip-.2cm
{\tiny  {\color {blue} $Q^{-1}$ }:  periodic layered  material alternatively saturated with either
{\color { magenta} gas}  or  {\color {blue} water}. Layer thicknesses: 15,  20, 30  cm}
}
\end{figure}

}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments. The IBVP}}

{\small

The seismic response (time histories) of a periodic seismic non-isothermal poroelastic sequence
$\Omega $
is obtained by exciting the medium  with a point source.  

\vskip.2cm 

Thus  an {\color {blue} IBVP } for the Biot/Lord-Shulman equations is  formulated 
and  solved by using an explicit  
 conditionally stable (i.e., satisfying a CFL stablity constrain) Finite Element Method (FEM) in an open interval  
$\Omega$  = (0, 400 m).  

\vskip.2cm 

The FEM uses  a uniform  partition ${\mathcal T}^h(\Omega)$ of $\Omega$ 
into subintervals $\Omega_j$  of size  $h$ . 

\vskip.1cm
The solid and fluid phases and the temperature 
are represented by using   globally continuous 
piecewise-linear  polynomials over  ${\mathcal T}^h(\Omega)$.
}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments. The IBVP. Layer thickness 20 cm}}

{\small

 The medium  $\Omega_j$,   initially at rest, is   excited  with a {\color {blue} dilatational point source} located at $x_s = $4 m, of time history,
 \begin{eqnarray*}
g(t) = -16 f_0^2 (t - t_0)  e^{-8 f_0^2 (t - t_0)^2}
\end{eqnarray*}
with $t_0$ = 1.25/$f_0$, {\color {magenta}   $f_0$ = 77 Hz being the dominant frequency. }
 This source generates {\color {blue}   P-waves with wavelenghts much larger} than the layer thickness.

 \vskip.1cm 
Time histories  of 
the frame  particle velocities are recorded at equally spaced  receivers r1, r2, r3 and r4, located at 
$x_1= 70$ m, $x_2= 100$ m, $x_3= 130$ m and $x_4= 160$ m. 

\vskip.1cm 
The time histories amplitudes and their amplitude spectrums 
are normalized to the maximum amplitude 
of the signal at  r1  in the uncoupled case. 

\vskip.1cm 
This normalization provides a better representation of the relative changes 
in wave propagation and attenuation between the uncoupled and coupled cases. 


}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments.  Time histories  at  receivers r1 and r4. Layer thickness 20 cm.}}
\vskip.5cm
\begin{figure}
\label{fig2}
{
%\centering
\begin{center}
\includegraphics[scale=0.29,angle=0]{figure2_jasa.eps}
\end{center}

 {\tiny  Figure 2: Time histories of the frame particle velocity at $r_1$  and $r_4$.
 P-Waves {\color {magenta}arrive faster} and have {\color {blue}lower amplitude} in the {\color {magenta} coupled case than} in the {\color {blue} uncoupled} one, with wave {\color {magenta} amplitudes being reduced} as {\color {blue} source-receiver distance increases} 
 }

}
\end{figure}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments.  Amplitude spectrums at  r1 and r4. Layer thickness 20 cm.}}
\vskip.5cm
\begin{figure}
\label{fig3}
{
%\centering
\begin{center}
\includegraphics[scale=0.29,angle=0]{figure3_jasa.eps}
\end{center}

 {\tiny  Figure 3: amplitude spectrums  of the  time histories at  receivers  r1  and r4.
 

   
 }

}
\end{figure}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments.  Amplitude spectrums for  receivers 
$r_1, r_2, r_3, r_4$. Coupled case.Layer thickness 20 cm. }}
{\small
 
Amplitude spectrums decrease as the distance between the receivers and the source increases.}
\vskip.5cm
\begin{figure}
\label{fig4}
{
%\centering
\begin{center}
\includegraphics[scale=0.29,angle=0]{figure4_jasa.eps}
\end{center}

 {\tiny  Figure 4: amplitude spectrums  of  time histories of the receivers  r1, r2, r3 and r4.
 Coupled case.
 

   
 }

}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments. {\color{blue} $Q$} estimates. Layer thickness 20 cm. }}

{\small

{\tiny
\begin{center}
{\bf Table 2. Estimated {\color{blue} $Q$} computed with  {\color{magenta} SR} and {\color{magenta} $FS$}  
  methods. Coupled and uncoupled cases }
\[
\begin{tabular}{cccccccc}
\hline
\hline
source   &  receiver  &  SR coupled  &  FS coupled & SR uncoupled &  FS uncoupled\\
     r1  &  r2  &   24.53 &  23.9  & 29.17 &  27.47 \\
\hline     
     r1  &  r3  &   24.51 &  24.12 & 28.48 &  27.1\\
     
\hline  

     r1  &  r4  &   24.07 &  23.91 & 28.72 &  27.68\\
     
\hline      
     r2  &  r3  &   24.53 &  24.35 & 27.79 &  26.72\\
     
 \hline      
     r2  &  r4  &   23.84 &  23.91 & 28.49 &  27.78\\
     
\hline       
     r3  &  r4  &   23.15 &  24.46 & 29.22 &  28.85\\
\hline
\hline 
\end{tabular}
\]
\end{center}
}
{\tiny  As expected, {\color{blue} $Q$}-values in the uncoupled case are close to that given by  White's theory 
({\color{blue} $Q$} = 28) and higher than the estimates for the coupled case. 

\vskip0.5cm
{\large In  the {\color{magenta} coupled  case attenuation is higher}, thus {\color{blue} $Q$-values  decrease}.}
}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Second example. Comparison of {\color{blue} Q}-estimates for  different layer thickness. 
Coupled case}}

{\small

The second example  considers an interval  $\Omega$ = (0, 400 m) and two sequences of periodic 
 layers of {\color{blue} layer thickness  15 and  
30 cm,} respectively.

\vskip.6cm

The  source of dominant frequency is {\color{magenta} $f_0$ = 34 Hz} for layer thickness 30 cm and {\color{magenta} $f_0$ = 140 Hz} for layer thickness 15 cm, respectively.  The frame particle velocities  are   recorded 
at   receivers  R1 and R2 located at $x_1=$ 200 m, $x_2= $250 m, respectively.
%$x_1= 50$ m, $x_2= 100$ m, $x_3= 150$ m, 
%$x_4= 200$ m, $x_5= 250$ m.


}

   


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Amplitude spectrums for receiver  R2 and two layer thickness. 
Coupled case}}

{\tiny
.

 The higher  amplitude spectrum  at receiver  R2 for layer thickness  30 cm as compared with that of 15 cm 
  is  consequence of  less mode conversion and consequently, larger amplitude spectrum  and lower  
  attenuation.
}

\vskip.5cm
\begin{figure}
\label{fig5}
{
%\centering
\begin{center}
\includegraphics[scale=0.25,angle=0]{fig5_macro.eps}
\end{center}

 {\tiny  Figure 5: Amplitude spectrums corresponding to the time history of receiver R2. Layer thickness 15 cm and 30 cm .
 

   
 }

}
\end{figure}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% slide %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Q estimates for different layer thickness. Coupled case }}
{\small

{
\begin{center}
{\bf {\tiny Table 3. Estimated {\color{blue}$Q$} for layer thickness 15 cm and 30 cm  computed with the {\color{magenta} SR} and {\color{magenta} FS}
  methods. Coupled case.}}
\[
\begin{tabular}{cccc}
\hline
\hline
15 cm layer thickness \\ 
\hline
source   &  receiver  &  SR  &  FS\\
  R1  &  R2  &   24.16 &  23.56 \\
\hline
\hline
30 cm layer thickness \\ 
\hline
  R1  &  R2  &   25.23 &  24.98\\

\hline
\hline
\end{tabular}
\]
\end{center}
}
\vskip 1cm 

 {\tiny {\color{blue}$Q$}-values for a layer thickness of 30 cm are  higher than those of 15 cm, since there is less mode conversion. 
 
 Thus {\color{magenta} for 30 cm we have  larger amplitude spectrum   and less attenuation than for 15 cm},} 
 

}


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{ CONCLUSIONS}}

{\small
\begin{itemize}
\item 
We study   {\color {blue} attenuation and dispersion effects } 
of {\color {magenta}  P-waves } of wavelenghts much larger than the layer thickness in  non-isothermal poroelastic layered media alternatively saturated with either  gas or water.


\item Time histories of frame  particle displacements were  recorded at equally spaced receivers.

\item  {\color {magenta} P-waves attenuation }  was measured by estimating the quality factor {\color {blue} Q} using two different procedures. 

\item The experiments clearly show the {\color {blue}  additional energy losses in non-isothermal media}, associated with the presence of {\color {magenta} T- waves}, in addition to  {\color {magenta}  WIFF  loss effects} present 
in the isothermal case. 


\end{itemize}

}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{P-wave attenuation in thin-layered  non-isothermal fluid-saturated  poroelastic solids  }}

\vskip1cm 
\Large THANKS FOR YOUR ATTENTION !!!!!

\end{frame}

\end{document}





\end{document}






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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments VI.Time histories at receivers $r_1$ and $r_3$.}}

\begin{figure}
\label{fig2}
{
%\centering
\begin{center}
\includegraphics[scale=0.2,angle=0]{fig2_r1_uncoupled_coupled.eps}\quad 
\includegraphics[scale=0.2,angle=0]{fig4_r3_uncoupled_coupled.eps}
\end{center}
\vskip-.4cm
 {\tiny Figure 2: Time histories of the frame particle velocity at $r_1$ (top) and $r_3$ (bottom).
 P-Waves {\color {magenta}arrive faster} and have {\color {blue}lower amplitude} in the {\color {magenta} coupled case than} in the {\color {blue} uncoupled} one, with wave {\color {magenta} amplitudes being reduced} as {\color {blue} source-receiver distance increases} }
}
\end{figure}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small Numerical experiments. VII. Amplitude spectrums at  $r_1$,$r_2, r_3$.}}
\vskip.5cm
\begin{figure}
\label{fig3}
{
%\centering
\begin{center}
\includegraphics[scale=0.29,angle=0]{fig5_nueva.eps}
\end{center}

 {\tiny Figure 3: Amplitude spectrums at   $r_1$, $r_2$ and $r_3$. The spectrums  {\color {magenta} shift 
 to lower frequencies} as the  source-receiver distance  {\color {blue}increases}. {\color {magenta}  Coupled Case 
 exhibits lower amplitudes than  uncoupled one},  due to the combined {\color {magenta}  attenuation  effects 
of  WIFF and T-waves}}

}
\end{figure}
\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{\small  {\color {blue} Q} estimates. Frequency-Shift and Spectral-Ratio methods}}
\vskip -.9cm

{\small

\begin{center}
{\bf {\color {blue} Q} estimates. {\color {magenta} Uncoupled Case}, WIFF effect.}
\[
\begin{tabular}{cccc}
\hline
\hline
Source   &  Receiver  &  Spectral-Ratio &  Frequency-Shift \\ 
    r1  &  r2  &   27.97  &  29.44\\
    r1  &  r3  &   27.97  &  29.80\\
     r2  &  r3  &   28.01  &  30.17\\
\hline
\hline
\end{tabular}
\]
\end{center}


\begin{center}
{\bf  {\color {blue} Q} estimates. {\color {magenta} Coupled Case}, WIIF $+$ T-waves effect}
\[
\begin{tabular}{cccc}
\hline
\hline
Source   &  Receiver  &  Spectral-Ratio &  Frequency-Shift \\ 
    r1  &  r2  &   24.00  &  26.21\\
    r1  &  r3  &   23.91 &  26.15\\
     r2  &  r3  &   23.82  &  26.09\\
\hline
\hline
\end{tabular}
\]
\end{center}
{\tiny 
Values of {\color {blue} Q} in the  
{\color {magenta} uncoupled  Case} being are {\color {blue} close} to the theoretical  value {\color {blue} 28 given by White's theory}, with decreasing agreement 
as the distance receiver-source increases. The {\color {magenta}lower  values of Q in the coupled Case} as compare with the uncoupled one correspond to the  {\color {blue} higher attenuation} in the receiver time histories {\color {blue} for the coupled Case}.
}
}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{ CONCLUSIONS}}

{\small
\begin{itemize}
\item 
We study   {\color {blue} attenuation and dispersion effects } 
of {\color {magenta}  P-waves } of wavelenghts much larger than the layer thickness in  non-isothermal poroelastic layered media alternatively saturated with either  gas or water.


\item Time histories of frame  particle displacements were  recorded at equally spaced receivers.

\item  {\color {magenta} P-waves attenuation }  was measured by estimating the quality factor {\color {blue} Q} using two different procedures. 

\item The experiments clearly show the {\color {blue}  additional energy losses} associated with the presence of  T- waves. 


\end{itemize}

}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  slide %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{Numerical experiments to characterize  P-wave attenuation in partially saturated 
non-isothermal porous media}}

\vskip1cm 
\Large THANKS FOR YOUR ATTENTION !!!!!

\end{frame}

\end{document}
