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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}
\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}
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\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}
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\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}
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\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}
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\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}
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\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}
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\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---------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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}
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\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}