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\begin{document}

\title{From Mineralogy to Mechanical Properties: integrating core data and rock physics 
simulations for stiffness tensor estimation in Vaca Muerta organic-rich mudrock}

\renewcommand{\thefootnote}{\fnsymbol{footnote}} 

\author{Juan E. Santos, School of Earth Sciences and Engineering, Hohai University, Universidad de Buenos Aires, Facultad de Ingenier\'\i  a, Instituto del Gas y del Petr\'oleo, Department of Mathematics, Purdue University,  A. Sanchez Camus, Facultad de Ciencias Astron\'omicas y Geofisicas, Universidad Nacional de La Plata, Solaer Ingenier\'\i  a S.A., Gabriela B. Savioli, Universidad de Buenos Aires, Facultad de Ingenier\'\i  a, Instituto del Gas y del Petr\'oleo,    
Patricia M. Gauzellino, Facultad de Ciencias Astron\'omicas y Geofisicas, Universidad Nacional de La Plata, Jing Ba \footnotemark[1], School of Earth Sciences and Engineering, Hohai University }

\footer{}
\lefthead{Santos, Sanchez Camus, Savioli, Gauzellino, Ba}
\righthead{From Mineralogy to Mechanical Properties using core data and rock 
physics for stiffness  estimation in VMF}

\maketitle

\begin{abstract}
 From a mechanical perspective, organic mudrocks are viscoelastic media whose anisotropy can be accurately represented using a vertical transverse isotropy (VTI) model. These reservoirs exhibit ultra-low permeability, typically in the range of nano to microdarcies, and possess a complex pore structure. Such characteristics, along with different oil, gas and water fluids saturations, significantly impact the hydraulic properties and mechanical behavior of the reservoir during production.
 This study introduces a practical methodology to estimate the stiffness tensor of dry rock using theoretical models of rock physics, incorporating mechanical, mineralogical, and petrophysical data from a 
  core extracted from the lower section of the Vaca Muerta Formation (VMF), located in the Neuquén Basin, Argentina. This core was extracted from a block located in the maximum oil generation window of the Vaca Muerta formation at a depth of 3100 m.
  The dry-rock stiffness tensor is initially computed and validated against laboratory phase velocity
measurements at 1 MHz, showing discrepancies below 10 percent. The synthetic sample is subsequently
saturated with hydrocarbon fluids to evaluate attenuation and dispersion effects associated with
wave-induced fluid flow (WIFF) and the resulting mode conversions.
 Fluid-saturated simulations
reveal that phase velocities decrease markedly with increasing gas content,  with
frequency variations depending on the fluid mixture, particularly above 100 Hz. Patchy gas-oil saturation
yields higher velocities than uniformly mixed fluids, while exhibiting significantly greater
attenuation due to enhanced WIFF. The frequency-dependent peak attenuation shifts toward lower
frequencies in patchy cases, in agreement with mesoscopic pressure diffusion theory. These results
emphasize the critical role of fluid composition and spatial distribution in seismic velocity and
attenuation in unconventional reservoir characterization.
\end{abstract}

\section{Introduction}

The objective of this study is to apply the two-dimensional numerical model 
—a physically grounded, non-phenomenological formulation based on Biot’s theory of poroelasticity 
 to investigate the anisotropic poroelastic response of the Vaca Muerta Formation (VMF), an organic-rich marlstone situated in the Neuqu\'en Basin, Argentina, under variable fluid saturation 
 conditions and seismic range of  frequencies.

As a basis for the numerical modeling, we employ the rock physics theory which integrates mineralogical and petrophysical data from a core sample extracted  from the VMF. This approach enables the estimation of the dry bulk modulus K$_m$ and shear modulus $\mu_m$ of the multi-mineral porous matrix. Then, the VMF is assumed to consist of a  sequence of very thin horizontal layers composed of kerogen (as the primary organic constituent)  and mineral aggregates, within which Biot’s theory in the diffusive frequency range is applicable.

At long wavelengths compared 
with the average layer thickness VMF behaves as
a transversely isotropic anisotropy with a vertical axis - VTI.

To determine the p${ij}$ stiffness coefficients of the equivalent VTI medium, a set of five boundary  value problems (BVPs) based on Biot’s diffusive equations are formulated in the space–frequency  domain. Each BVP defines a compressibility or shear test, which is solved numerically using the  Finite Element (FE) method across a range of relevant frequencies. 

These tests are first applied to a dry synthetic sample to compute the stiffness tensor and  derive the corresponding dry-rock phase velocities. The simulated phase velocities exhibit very  good  agreement with laboratory measurements, with discrepancies remaining below 10 percent. 

Following the dry-rock analysis, the synthetic sample is saturated with gas, oil, and water to  
investigate attenuation and dispersion effects associated with
mode conversions caused by wave-induced fluid flow (WIFF).

Additionally, the case  of patchy gas–oil saturation is examined and compared against scenarios 
of uniform saturation, in  order to evaluate the effects of fluid distribution on seismic velocity and attenuation.


\section{Results}

\subsection {Data}

A dry  core from the Vaca Muerta Formation is the source of the measured data, which consist of rock mineralogy and phase velocities, $vp_{11}$, $vp_{33}$, $vp_{55}$ and $vp_{66}$ of the core at 3100 m depth. This core corresponds to the window of maximum oil generation in the formation; its location is marked with a red rectangle in Figure \ref{Fig1}. 

To estimate of the elastic   tensor coefficients $p_{ij}$, 
 we assume that VMF consists of a periodic alternation of two porous
materials, each with a porosity of 6 percent and 2.75 10$^{-18}$ m$^2$ permeability.

Material 1 is  composed of seven minerals, including  23 \% kerogen,  37.27 \%  Clay, 14.61 \% Quartz, 10.68 \% Calcite, 2.57 \% Plagioclase, 2.37 \% Dolomite and 3.5 \%  Pyrite. Using the  solid grains density,  bulk and shear moduli of each mineral and a generalized Krief model (see Carcione et al, Geophysics,2005), we obtained the values  $K_s$, $K_m$,  $\mu_m$ for the dry core as shown in Table 1 as well as its average density 

Besides, the single  layer of Material  2 consists of dry  kerogen. Its properties can be seen in Table 1

The harmonic simulations were performed on a representative dry square sample with a side length of 2
mm, consisting of four repetitions of a periodic sequence composed of 49 layers of Material
1 and one layer of Material 2 per period. Each layer has a uniform thickness of 10$^{-5}$ m.
The computational domain was discretized using a 200×200 mesh.


\begin{figure} 
%\vspace{0.5cm}
\includegraphics[width=8cm,angle=0]{figure1.eps}
\caption {Location of the core area, marked with a red rectangle. The green region corresponds to the  the oil-generating window 
in the Vaca Muerta formation.
}
% \vspace{0.3cm}
\label{Fig1} 
\end{figure}
%\vskip20cm


\begin{center}
{\bf Table 1. Properties of Materials }
%\end{center}
\[
%\begin{tabular}{cccccc}
\begin{tabular}{lccccc}
\hline
Property &  Material 1 (Composite) & Material 2 (kerogen)  \\
\hline 
$K_s$ (GPa)  & 56.14    & 7.0    \\
$K_m$ (GPa)  & 29.19    & 1.29   \\
$\mu_m$ (GPa) & 17.78   & 0.36  \\
$\rho$ ($kg/m^3$)  & 2487    &  1400  \\
\end{tabular}
\]
\end{center}

\subsection {Computed VTI phase velocities  using the dry-core data}

Since the phase velocity data corresponds to a dry sample, the numerical experiment  consider air as the saturating fluid. 
Table 2 summarizes the results of the VTI experiments.

\begin{center}
{\bf Table 2. Phase velocities computed, measured and error  percentage.}
\[
\begin{tabular}{lccccc}
\hline
\hline
Phase velocity vp (m/s)& Computed  & Measured& Percentage error \\
\hline 
v11  & 4644.37    & 4331       &  7.2  \% \\
v33  & 3804.51    & 4217.47    &  9.8 \% \\
v55  & 1974.76    & 2193.61    &  9.9   \% \\
v66  & 2742.64    & 2581 &     6.2 \%   \\
\end{tabular}
\]
\end{center}




Figure \ref{Fig2}-\ref{Fig3} shows polar representation  of  phase  and energy  velocities of $qP$,
$qSV$  and $SH$ waves of the dry core at 1 MHz.
Anisotropy is clearly observed  in the behavior of the velocities of the three wave modes.

%\end{document}
\begin{figure} 
%\vspace{0.5cm}
\hspace{-1.5cm}
\includegraphics[width=7cm,angle=-90]{figure2.eps}
\caption {Polar representation  of  phase  velocities of qP,
qSV  and SH waves at 1 MHz. The  medium consists of a periodic sequence 
of forty nine   dry layers of the composite porous solid including kerogen  as a mineral with 23 \% proportion, and  one dry kerogen layer.
}
% \vspace{0.3cm}
\label{Fig2} 
\end{figure}
\vskip2cm


\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=-90]{figure3.eps}
\caption {Polar representation  of energy velocities of qP,  qSV and SH 
waves  computed using the FE method. Frequency is 1 MHz.  
}
% \vspace{0.3cm}
\label{Fig3} 
\end{figure}





\subsection {Phase velocities and attenuation using using hydrocarbon saturated samples}

Figures 4  and 5 present the phase velocity and attenuation behavior of qP wave waves traveling 
normal to the bedding, denoted as '33' waves' ( p$_{33}$ mode)
under four fluid saturation scenarios: 100 percent  oil, 100 percent  gas, 10 percent gas plus 90 percent  oil, and a ternary fluid
mixture consisting of 40 percent gas, 52 percent oil, and 8 percent  water, obtained from the well's fluid saturation log
at the depth corresponding to the core sample.


The corresponding  attenuation factors  1000/Q33 (Q3 i sthe quelity factor) are displayed in Figure \ref{Fig4}. Note that '33' waves exhibit more attenuation and the attenuation peaks are shifted to higher frequencies when the  saturant fluid is gas. 
% 
% The nature of the saturating fluid affects the behavior of the phase velocity and the attenuation factor of the '11' waves (Figures 2 and 4). The greater the presence of oil, the velocity and attenuation increase. The oil causes the peak attenuation to shift to lower values (Figure 4). On the other hand, the velocity and attenuation of the '33' waves are not influenced by the saturating fluid, as can be seen in Figures 3 and 5. 

 
\section{Patchy saturation}
To consider fractal variations of gas and oil  saturations  we  use the von Karman self-similar
correlation function. 

\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure8.eps}
\caption {Fractal patchy gas-oil spatial distribution. White zones correspond to full gas saturation and black regions to full oil saturation, with overall gas saturation 10 percent. Fractal dimension is 2.4, 
correlation length is 0.09  mm. 
}
% \vspace{0.3cm}
\label{Fig8} 
\end{figure}
\vskip20cm 

%Las figuras patchy se hicieron con este path   

%~/santos2024/papers/paper_con_camus/first_draft_paper_april_9/23_abril_runs_2mm_sample/freq_loop_p33_4periods_dsize_2mm_100oil_100gas10gas_90oil_40_gas_53_oil_7water/p33_90_oil_wetting_10_gas_non_wetting_ex_biot$ xmgrace inv_qp33_fe.freq ~/santos2024/papers/paper_con_camus/first_draft_paper_april_9/patchy_saturation/2025_patchy_saturation_runs/200x200_fractal_runs_4_periods/200x200_fractal_runs_4_periods/freq_loop_p33_patchy_10_gas_version1/inv_qp33_fe.freq



\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure9.eps}
\caption {Phase velocity of '33' waves as function of frequency for  10 \% gas, 90 \% oil 
and uniform and patchy saturation.
}
% \vspace{0.3cm}
\label{Fig9} 
\end{figure}
\vskip20cm
In Figure \ref{Fig9} the
uniform case assumes perfect mixing at the pore scale, producing a single-phase effective fluid with
increased compressibility, resulting in lower velocities. In contrast, the patchy case consists of
spatially segregated gas- and oil-filled regions. Wave propagation is  governed by the stiffer oil-saturated zones, leading to higher velocities
across the frequency spectrum. The divergence between the two models becomes particularly
evident above 100 Hz, consistent with mesoscopic WIFF activation.

%Las figuras patchy se hicieron con este path
%~/santos2024/papers/paper_con_camus/first_draft_paper_april_9/23_abril_runs_2mm_sample/freq_loop_p33_dsize_2mm_100oil_100gas10gas_90oil_40_gas_53_oil_7water/90_oil_wetting_10_gas_non_wetting xmgrace vp33_fe.freq  ../../patchy_saturation/100x100_fractal_runs_2_periods/freq_loop_p33_patchy_10_gas_version1/vp33_fe.freq

\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure10.eps}
\caption {Attenuation factor of '33' waves  as function of frequency for  10 \% gas, 90 \% oil 
and uniform and patchy saturation.
} 
% \vspace{0.3cm}
\label{Fig10} 
\end{figure}
\vskip20cm

Patchy saturation in Figure \ref{Fig10} results in markedly higher attenuation due to enhanced WIFF at gas–
oil interfaces. These effects promote localized pressure gradients and viscous fluid movement,
leading to increased energy dissipation. Additionally, the attenuation peak is shifted to lower
frequencies under patchy saturation, reflecting the longer diffusion times associated with larger-
scale heterogeneities and limited pressure communication between fluid patches.
\begin{figure} 
%\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure11a.eps}
\caption {Fluid pressure  for the case of 10 \% gas, 90 \% oil  and uniform    saturation.} 
% \vspace{0.3cm}
\label{Fig11} 
\end{figure}
\vskip20cm


\begin{figure} 
%\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure11b.eps}
\caption {Fluid pressure  for the case of 10 \% gas, 90 \% oil and patchy  saturation.} 
% \vspace{0.3cm}
\label{Fig12} 
\end{figure}
\vskip20cm


\section{conclusions}

We have presented  a rock physics simulation-based methodology for estimating the stiffness tensor of a dry core sample from the Vaca Muerta formation in Neuqu\' en, Argentina,  using mineralogical characterization as input data. The proposed methodology  was  tested by comparing  the 
numerical phase velocities with those measured in laboratory at 1 MHz, yielding excellent agreement. Furthermore, the procedure  can be applied to study the influence of fluid saturation and different  source frequencies at laboratory and field scales.


 \begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure4.eps}
\caption {'33'  waves phase velocity as function of frequency.} 
% \vspace{0.3cm}
\label{Fig4} 
\end{figure}


\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=7cm,angle=0]{figure5.eps}
\caption {Attenuation factor 1000/Q  as function of frequency for  '33' waves.  
}
% \vspace{0.3cm}
\label{Fig5} 
\end{figure}
\vskip10cm



\end{document}

% \section{references}
% 
%   
% 
% %%%%%%%%%%%%%%  5
% \bibitem{CP_06}
% J. M. Carcione  and S. Picotti, P-wave seismic attenuation by slow-wave diffusion,
% Effects of inhomogeneous rock properties, Geophysics 71 (2006)  O1-O8. 
% 
% \bibitem{Carcione2005} 
% Jose M. Carcione, Hans B. Helle, Juan E. Santos and Claudia L. Ravazzoli, 
% A constitutive equation and generalized Gassmann modulus
% for multimineral porous media, Geophysics,70 (2);  doi: 10.1190/1.1897035
% 
% %%%%%%%%%%%%%%   6 
% \bibitem{GS_97}
% S. Gelinsky  and S. A. Shapiro,  Poroelastic Backus-averaging for
% anisotropic, layered fluid and gas saturated sediments, 
% Geophysics 62 (1997) 1867-1878.
% 
% %\bibitem[{Gonz{\'a}lez~Celis et~al.(2020)}]{gonzalez2020matrix}
% %Gonz{\'a}lez~Celis, R. et~al. (2020) Matriz energ{\'e}tica mundial y el %cambio
% %  clim{\'a}tico: Estado actual.
% %\newblock \textit{Master Thesis - Gestión Sostenible de la Energía}.
% 
% 
% \bibitem[{Grechka and Tsvankin(2003)}]{grechka2003feasibility}
% Grechka, V. and Tsvankin, I. (2003) Feasibility of seismic characterization of
%   multiple fracture sets.
% \newblock \textit{Geophysics}, \textbf{68}, 1399--1407.
% 
% %%%%%%%%%%%%%%%   7
% 
% \bibitem{KM_11} 
% F. Krzikalla  and T. M. M\"uller,  
% Anisotropic P-SV-wave dispersion and attenuation due to interlayer 
% flow in thinly layered porous rocks, Geophysics 76 W 135 (2011); doi:10.1190/1.3555077.
% 
% \bibitem{Santos_19} 
% J.E. Santos, G.B. Savioli, J. M. Carcione and J. Ba, Effect of capillarity and relative permeability on Q anisotropy of hydrocarbon source rocks, Geophys. J. Int. (2019) 218, 1199–1209, doi: 10.1093/gji/ggz217.
% 
% 
% \bibitem{white_75} 
% J. E. White, N. G. Mikhaylova  and F. M. Lyakhovitskiy,  
% Low-frequency seismic waves in fluid saturated layered rocks, 
% Physics of the Solid Earth  11 (1975) 654-659. 
% 
% 
% %\end{thebibliography}



\end{document}


Note that the macros for the \verb#\tensor# command has been changed
to force tensors to be bold uppercase, in compliance with current SEG
submission standards. This is so that documents typeset to the old
standards will print out according to the new ones: e.g., tensor
$\tensor{t}$ (note converted to uppercase).

\subsection*{Figures}
\renewcommand{\figdir}{Fig} % figure directory

Figure~\ref{fig:waves} shows what it is about.

\plot{waves}{width=\columnwidth}
{This figure is specified in the document by \texttt{
    $\backslash$plot\{waves\}\{width=$\backslash$columnwidth\}\{This caption.\}}.
}

\subsubsection{Multiplot} 

Sometimes it is convenient to put two or
more figures from different files in an array (see
Figure~\ref{fig:exph,exgr}). Individual plots are
Figures~\ref{fig:exph} and~\ref{fig:exgr}.

\multiplot{2}{exph,exgr}{width=0.4\textwidth}
{This figure is specified in the document by \texttt{
    $\backslash$multiplot\{2\}\{exph,exgr\}\{width=0.4$\backslash$textwidth\}\{This caption.\}}.
}

The first argument of the \texttt{multiplot} command specifies the
number of plots per row.

\subsection{Tables}

The discussion is summarized in Table~\ref{tbl:example}.

\tabl{example}{This table is specified in the document by \texttt{
    $\backslash$tabl\{example\}\{This caption.\}\{\ldots\}}.
}{
  \begin{center}
    \begin{tabular}{|c|c|c|}
      \hline
      \multicolumn{3}{|c|}{Table Example} \\
      \hline
      migration\rule[-2ex]{0ex}{5ex} & 
      $\omega \rightarrow k_z$ & 
      $k_y^2+k-z^2\cos^2 \psi=4\omega^2/v^2$ \\
      \hline
      \parbox{1in}{zero-offset\\diffraction}\rule[-4ex]{0ex}{8ex} &
      $k_z\rightarrow\omega_0$ &
      $k_y^2+k_z^2=4\omega_0^2/v^2$ \\
      \hline
      DMO+NMO\rule[-2ex]{0in}{5ex} & $\omega\rightarrow\omega_0$ & 
      $\frac{1}{4}
      v^2k_y^2\sin^2\psi+\omega_0^2\cos^2\psi=\omega^2$ \\
      \hline
      radial DMO\rule[-2ex]{0in}{5ex} & $\omega\rightarrow\omega_s$ &
      $\frac{1}{4}v^2k_y^2\sin^2\psi+\omega_s^2=\omega^2$\\
      \hline
      radial NMO\rule[-2ex]{0in}{5ex} & $\omega_s\rightarrow\omega_0$ &
      $\omega_0\cos\psi=\omega_s$\\
      \hline
    \end{tabular}
  \end{center}
}

\section{ACKNOWLEDGMENTS}

I wish to thank Ivan P\v{s}en\v{c}\'{\i}k and Fr\'ed\'eric Billette
for having names with non-English letters in them.  I wish to thank
\cite{Cerveny} for providing an example of how to make a bib file that
includes an author whose name begins with a non-English character and
\cite{forgues96} for providing both an example of referencing a Ph.D.
thesis and yet more non-English characters.

\append{Appendix example}

According to the new SEG standard, appendices come before references.

\begin{equation}
\frac{\partial U}{\partial z} = 
\left\{
  \sqrt{\frac{1}{v^2} - \left[\frac{\partial t}{\partial g}\right]^2} +
  \sqrt{\frac{1}{v^2} - \left[\frac{\partial t}{\partial s}\right]^2}
\right\}
\frac{\partial U}{\partial t}
\label{eqn:partial}
\end{equation}
It is important to get equation~\ref{eqn:partial} right.

\append{Another appendix}

\begin{equation}
\frac{\partial U}{\partial z} = 
\left\{
  \sqrt{\frac{1}{v^2} - \left[\frac{\partial t}{\partial g}\right]^2} +
  \sqrt{\frac{1}{v^2} - \left[\frac{\partial t}{\partial s}\right]^2}
\right\}
\frac{\partial U}{\partial t}
\label{eqn:partial2}
\end{equation}
Too lazy to type a different equation but note the numeration.

The error comparison is provided in Figure~\ref{fig:errgrp}.

\plot*{errgrp}{width=0.8\textwidth} {This figure is specified in the
  document by \texttt{
    $\backslash$plot*\{errgrp\}\{width=0.8$\backslash$textwidth\}\{This caption.\}}.  }

\onecolumn

\append{The source of the bibliography}
\verbatiminput{example.bib}

\twocolumn

\bibliographystyle{seg}  % style file is seg.bst
\bibliography{example}

\end{document}

\begin{center}
{\bf Table 3.  Numerical $p_{ij}$ values from the VTI experiments (Pa). Frequency is 1 MHz}
%\end{center}
%\label{table1}       
%---------
\[
%\begin{tabular}{cccccc}
\begin{tabular}{lccccc}
\hline
$p_{11}= (47881607223.002250, 34039.066160)$\\
\hline 
%(47881607223.002250, 34039.066160)\\
 $p_{33}$ = (37362983980.381652, 223305.323451) \\
\hline 
%(37362983980.381652, 223305.323451) \\

 $p_{55}$ = (10239564921.821903, 0.0) \\
\hline 
%(10239564921.821903, 0.0)\\

 $p_{66}$ = (10239564921.827581, 0.0) \\
\hline 
%(10239564921.827581, 0.0)

 $p_{13}$ = (13930827662.989664,        10086.232336) \\
\hline 
%(13930827662.989664,        10086.232336)
\end{tabular}
\]
\end{center}

\section*{Theory for VTI media}

A VTI medium is characterized by five independent elastic moduli $c_{11}, c_{13}, c_{33} ,c_{55}, c_{66}$. In this case, the stress-strain relationship in Voigt notation is 
\begin{equation}
%\[
\begin{bmatrix}
\sigma_{11} \\
\sigma_{22} \\
\sigma_{33} \\
\sigma_{23} \\
\sigma_{31} \\
\sigma_{12} \\
\end{bmatrix}
 =
\begin{bmatrix}
c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\
c_{12} & c_{11} & c_{13} & 0 & 0 & 0 \\
c_{13} & c_{13} & c_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & c_{55} & 0 & 0 \\
0 & 0 & 0 & 0 & c_{55} & 0 \\
0 & 0 & 0 & 0 & 0 & c_{66} \\
\end{bmatrix} \cdot
\begin{bmatrix}
\epsilon_{11} \\
\epsilon_{22} \\
\epsilon_{33} \\
2\epsilon_{23} \\
2\epsilon_{31} \\
2\epsilon_{12} \\
\end{bmatrix}
% =
% C \cdot
% \begin{bmatrix}
% \epsilon_{11} \\
% \epsilon_{22} \\
% \epsilon_{33} \\
% 2\epsilon_{23} \\
% 2\epsilon_{31} \\
% 2\epsilon_{12} \\
% \end{bmatrix}
%\]
\end{equation}
with the vectors $\bf{\sigma}$ and $\bf{\epsilon}$ being the stress and the strain, respectively, and $c_{12}=c_{11}-2c_{66}$.

 Attenuation and dispersion of the material can be represented
 %is considered 
 through an equivalent viscoelastic medium, characterized by complex and frequency dependent stiffness $p_{ij}$, with  the imaginary part representing 
 attenuation. Therefore, with  $\bf  {\tilde \sigma}$ and $\bf{ \tilde \epsilon}$ denote the stress and strain of the equivalent viscoelastic VTI medium we have the relations
 \begin{equation}\label{pij}
%\[
\begin{bmatrix}
\tilde \sigma_{11} \\
\tilde \sigma_{22} \\
\tilde \sigma_{33} \\
\tilde \sigma_{23} \\
\tilde \sigma_{31} \\
\tilde \sigma_{12} \\
\end{bmatrix}
 =
\begin{bmatrix}
p_{11} & p_{12} & p_{13} & 0 & 0 & 0 \\
p_{12} & p_{11} & p_{13} & 0 & 0 & 0 \\
p_{13} & p_{13} & p_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & p_{55} & 0 & 0 \\
0 & 0 & 0 & 0 & p_{55} & 0 \\
0 & 0 & 0 & 0 & 0 & p_{66} \\
\end{bmatrix}\cdot
\begin{bmatrix}
\tilde \epsilon_{11} \\
\tilde \epsilon_{22} \\
\tilde \epsilon_{33} \\
2 \tilde \epsilon_{23} \\
2 \tilde \epsilon_{31} \\
2 \tilde \epsilon_{12} \\
\end{bmatrix}
\end{equation}

where  $p_{12}=p_{11}-2p_{66}$.
 
White (1975) and later Carcione (2006)
%\cite{white75b} and \cite {carcione06} 
described the equivalent viscoelastic medium of a periodic sequence of 
alternating thin horizontally homogeneous  layers of thickness $d_1$ 
and $d_2$,  with symmetry axes normal  to the layering plane. Their 
theory gives the complex and frequency dependent stiffness $p_{33}(\omega)$, as: 
\begin{equation}\label{p33}
p_{33}(\omega)= \left[ \frac{1}{c_{33}} 
+ \frac{2 (r_2 - r_1)^2} {\ii \omega (d_1 + d_2) (I_1 + I_2)} \right]^{-1},
\end{equation}
where
\begin{equation}\label{acp33}
r = \frac{\alpha M}{E_G}, \\ I =  \frac{\eta}{\kappa a} {\rm coth} 
\left( \frac{a d}{2} \right), \\ a = \sqrt{\frac{ \ii \omega \eta E_G}{\kappa M E_m}}
\end{equation}
for each single layer. In Eq.\eqref{acp33}, $\alpha = 1-\myfrac{K_m}{K_s}$, $E_G = E_m + \alpha^2 M$,  $E_m = K_m + \myfrac{4}{3} \mu$, $M = \left [\myfrac{\alpha - \phi}{K_s}+\myfrac{\phi}{K_f} \right ]^{-1}$, with $K_s$, $K_m$  and $K_f$ the solid grains, dry matrix and saturant fluid bulk moduli; $\mu$ is the shear modulus, $\eta$ is the fluid viscosity, $\kappa$ is the permeability, $\phi$ the porosity,  $\omega$ the frequency and $\ii$ the imaginary unit.

Gelinsky and Shapiro (1997)      determined  the 
relaxed and unrelaxed 
stiffnesses of the equivalent  poro-viscoelastic medium to such periodic sequences, while    
 Krzikalla  and  Müller (2011)
combined the two previous models, i.e., 
they computed $p_{33}$ as in  Eq.\eqref{p33} and obtained the other five 
stiffnesses of the equivalent TIV medium as,
%White model 
%has been generalized by Krzikalla and M\"uller (2011)
%to anisotropic media, 
\begin{equation}\label{stre-stra}
p_{ij}( \omega )  = c_{ij} + \left( \frac{ c_{ij} - c_{ij}^r}{c_{33} - c_{33}^r} \right)  [p_{33} (\omega)  - c_{33}] ,
\end{equation}
where $c_{ij}^r$ are the relaxed stiffnesses.

The main assumption in Eq. \eqref{stre-stra} 
%in \cite{KM_11} 
is  that the fluid-flow direction is perpendicular  to the  layering 
so that the relaxation behavior is described by a single
 stiffness, $p_{33} (\omega)$. 
Thus the theory is not  valid when the layers are heterogeneous.

 Velocity and quality factor of the qP, qS and SH waves for the equivalent anisotropic medium  were computed from the complex velocities.
 
