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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. After determining the dry-rock stiffness tensor, the proposed methodology  enables to analyze  the effects caused by 
 fluid saturation and seismic source frequencies, ranging from laboratory experiments and well logging to field seismic studies. Numerical results exhibit excellent agreement with laboratory  phase velocities measured at 1 MHz, 
 with errors below 5 \%. The procedure  is then validated as an effective tool for evaluating critical phenomena 
 such as attenuation and dispersion caused by wave-induced fluid flow (WIFF).
Once the model parameters have been calibrated, the stiffness tensor can be extrapolated to other areas within the block using only mineralogical and petrophysical information (porosity and permeability). This significantly simplifies the geomechanical and seismic characterization of unconventional reservoirs.
\end{abstract}

\section{Introduction}
The Vaca Muerta formation  consists of  a  sequence of very thin layers  
which at long wavelengths compared 
with the average layer thickness. It  behaves as
%is characterized by
a transversely isotropic anisotropy with a vertical axis - VTI  \citep{sosa2018}, although it may change its elastic symmetry due to the presence of fractures.

These type of formations were studied using different approaches.

\cite{white75b} obtained the complex and frequency-dependent
P-wave stiffness associated with the symmetry axis of thin poroelas-
tic layers.

\cite{Backus1962} obtained the average elasticity
constants in the case when the single layers are transversely
isotropic with the symmetry axis perpendicular to the
layering plane.


\cite{Gelinsky1997} determined  the 
relaxed and unrelaxed 
stiffnesses of the equivalent  poro-viscoelastic medium to a periodic 
sequence of finely layered 
horizontally homogeneous material, while    
 \cite{KM_11} 
combined the two previous models to determine the five 
stiffnesses of the equivalent TIV medium.  

The objective of this study is to verify that by knowing the
mineralogy and applying these  theoretical models of rock physics it is feasible   to obtain accurate estimates the elastic 
tensor coefficients $p_{ij}$ and their  associate phase velocities. 


\section*{Theory for VTI media}

%\subsection{Mesoscopic-flow attenuation theory for anisotropic poroelastic media}

%The Vaca Muerta formation  consists of  a  sequence of very thin layers  
%which at long wavelengths compared with the average layer thickness behaves as
%is characterized by
%a transversely isotropic anisotropy with a vertical axis - VTI  \citep{sosa2018estratigrafia}, although it may change its elastic symmetry due to the presence of fractures. 
%Let us consider $x_1$ and $x_3$ as the horizontal and vertical coordinates, respectively. 
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 given by \cite{Tsvankin2001},
\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  $p_{12}=p_{11}-2p_{66}$.
 
\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.

\cite{Gelinsky1997} determined  the 
relaxed and unrelaxed 
stiffnesses of the equivalent  poro-viscoelastic medium to such periodic sequences, while    
\cite{KM_11} 
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, computed as shown in the Appendix (Gelinsky and Shapiro, 1997). 
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 (see \cite{carcione05}). 
 

\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}. 

The objective of this study is to verify that, by knowing the mineralogy and applying theoretical rock physics models,
we can obtain an accurate estimate of the elastic tensor coefficients $p_{ij}$.

The numerical experiments consider an alternating  periodic sequence of 1 mm  thickness. Each period consists  of eighty layers, seventy nine  layers of  Material 1 and one  layer of Material 2.

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, 2005), we obtained the values  $K_s$, $K_m$,  $\mu_m$ for the dry core as shown in Table 1. 
 We also computed  the average density of Material 1. 
Besides, the single  layer of Material  2 consists of dry  kerogen. Its properties can be seen in Table 1. Both materials have 6 \% porosity and 
2.75 10$^{18}$ m$^2$ permeability.

\begin{figure} 
%\vspace{0.5cm}
\includegraphics[width=8cm,angle=0]{fig1.png}
\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}
\[
%\begin{tabular}{cccccc}
\begin{tabular}{lccccc}
\hline
Property &  Material 1 (Composite) & Material 2 (kerogen)  \\
\hline 
$K_s$ (GPa)  & 34.34    & 7.0    \\
$K_m$ (GPa)  & 29.19    & 1.29   \\
$\mu_m$ (GPa) & 15.69    & 0.36  \\
$\rho$ ($kg/m^3$)  & 2487    &  1400  \\
\end{tabular}
\]
\end{center}

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

We compute the numerical stiffness coefficients $p_{ij}$ applying the methodology described in section "Mesoscopic-flow attenuation theory for VTI media" and the data of Table 1.  We consider a 1 mm thickness periodic sequence, each period having seventy nine layers of Material 1 and one layer of Material 2. 
The mesh size is 1.25 10$^{-5}$ mm, then the  sizes of layers 1 and 2 are  9.875  10$^{-4}$ mm  and 1.25 10$^{-5}$ mm, respectively. 
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 from the numerical experiments, 
the measured  ones and  error  percentage. Frequency is 1 MHz }
%\end{center}
\[
%\begin{tabular}{cccccc}
\begin{tabular}{lccccc}
\hline
Phase velocity (m/s)& Computed  & Measured&  error (\%) \\
\hline 
$vp_{11}$  & 4538.29    & 4331      &  4.79  \\
$vp_{33}$  & 4008.93    & 4217.47   &  4.96  \\
$vp_{55}$  & 2098.69    & 2193.61   &  4.33  \\
$vp_{66}$  & 2581.97    & 2464      &  4.79   \\
\end{tabular}
\]
\end{center}

In  all computed velocities the  error of less than 5\% was  obtained,  which is considered a very good match since it is    within the range of measurement errors. Hence we can then rely on the numerically estimated values of the stiffness coefficients $p_{ij}$  shown in Table 3.


\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}

Figure \ref{Fig2} shows the polar representation  of  phase  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]{fig2.eps}
\caption {Polar representation  of  phase  velocities of qP,
qSV  and SH waves at 1 MHz. The  medium consists of a periodic sequence 
of seventy 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



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

This subsection considers the same sample of the previous experiment, 
but with the two periodic layers saturated by 100 \% oil and 100 \% gas.  Figure \ref{Fig3}  
displays velocities of waves parallel (‘11’ waves) and normal (‘33’ waves)   to
the layering plane, as function of frequency. It may be observed that parallel velocities are higher than normal ones, but the kind of fluid affects more the normal velocities. 

The corresponding  attenuation factors 1000/Q11 and 1000/Q33 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{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.

\section{ACKNOWLEDGMENTS}
We would like to thank Pluspetrol S.A. and Solaer Ingenier\'\i a S.A. for providing us with the measured data. This work was partially funded by the Universidad de Buenos Aires (UBACyT 20020190100236BA).


\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=8cm,angle=0]{fig3.eps}
\caption {Wave phase velocity as function of frequency for '11' and '33' waves. 
The  medium consists of a  periodic sequence  of seventy nine 
layers of the composite porous solid including 23 \% of kerogen  as a mineral  and  one  kerogen layer. 
The pore space in both Materials is saturated with 100 \% oil and 100 \% gas.  
}
% \vspace{0.3cm}
\label{Fig3} 
\end{figure}
%\vskip20cm

\begin{figure} 
\vspace{0.5cm}
\includegraphics[width=8cm,angle=0]{fig4.eps}
\caption {Attenuation factor 1000/Q  as function of frequency for '11' and '33' waves.  
The  medium consists of a  periodic sequence  of seventy nine  layers of  a composite  porous solid including 23 \% of kerogen as a mineral and  one  kerogen layer. 
The pore space in both Materials is saturated with 100 \% oil and 100 \% gas.  
}
% \vspace{0.3cm}
\label{Fig4} 
\end{figure}
\vskip10cm

\append{}
Following \cite{Gelinsky1997}, the relaxed stiffnesses are given 
by the expressions
\begin{equation}\label{eff}
\begin{array}{l}
c_{66}^r =  \langle \mu \rangle , \\
c_{11}^r - 2 c_{66}^r = c_{12}^r  = 2 \left \langle \myfrac{\lambda_m \mu}{E_m} \right \rangle + 
\left \langle \myfrac{\lambda_m }{E_m} \right \rangle^2  \left \langle \myfrac{1}{E_m} \right \rangle^{-1} \\
\hspace{3.6cm} +  \myfrac{{B_6^\ast}^2}{B_8^\ast}, \\
c_{13}^r  = \left \langle \myfrac{\lambda_m}{E_m} \right \rangle   
\left \langle \myfrac{1}{E_m} \right \rangle^{-1} +  \myfrac{B_6^\ast B_7^\ast}{B_8^\ast}, \\
c_{33}^r = \left[ \left \langle \myfrac{1}{E_m} \right \rangle - \left \langle \myfrac{\alpha}{E_m} \right \rangle^2  
\left \langle \myfrac{E_G}{M E_m} \right \rangle^{-1} \right]^{-1} , \\
c_{55}^r = \langle \mu^{-1} \rangle^{-1} , \\
\ \ \ \ \ B_6^\ast =  - B_8^\ast \left( 
2 \left \langle \myfrac{\alpha \mu}{E_m} \right \rangle + \left \langle \myfrac{\alpha}{E_m} \right \rangle
\left \langle \myfrac{\lambda_m}{E_m} \right \rangle \left \langle \myfrac{1}{E_m} \right \rangle^{-1} \right)  , \\
\ \ \ \ \ B_7^\ast =  - B_8^\ast \left \langle \myfrac{\alpha}{E_m} \right \rangle \left \langle \myfrac{1}{E_m} \right \rangle^{-1}   , \\
\ \ \ \ \ B_8^\ast =
\left[ \left \langle \myfrac{1}{M} \right \rangle + \left \langle \myfrac{\alpha^2}{E_m} \right \rangle - 
\left \langle \myfrac{\alpha}{E_m} \right \rangle^2 \left \langle \myfrac{1}{E_m} \right \rangle^{-1} 
\right]^{-1} ,
\end{array}
\end{equation}
where $  \left \langle \cdot  \right \rangle$ means weighted average between layers.

When there is no interlayer flow (unrelaxed regime), the coefficients are,  
\begin{equation}\label{effnf}
\begin{array}{l}
c_{66} =  c_{66}^r , \\
c_{11} - 2 c_{66} = c_{12} = 2 \left \langle \myfrac{(E_G - 2 \mu)  \mu}{E_G} \right \rangle + 
\left \langle \myfrac{E_G - 2 \mu}{E_G} \right \rangle^2  \left \langle \myfrac{1}{E_G} \right \rangle^{-1}, \\
c_{13} = \left \langle \myfrac{E_G - 2 \mu}{E_G} \right \rangle   
\left \langle \myfrac{1}{E_G} \right \rangle^{-1} , \\
c_{33} = 
\left \langle \myfrac{1}{E_G} \right \rangle^{-1} , \quad 
c_{55} =  c_{55}^r .
\end{array}
\end{equation}

\bibliographystyle{seg}  % style file is seg.bst
\bibliography{image25}

\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}
