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\title{Finite element approximation of coupled seismic and electromagnetic waves in  gas hydrate-bearing sediments}
\author{Juan E. Santos$^\dag$ }
\institution{%
$^\dag$ CONICET, Departamento de Geof\'\i sica Aplicada,\\
Facultad de Ciencias Astron\'omicas y Geof\'\i sicas,
 Universidad Nacional de La Plata\\
and Department of Mathematics, Purdue University. \\ $ $ \\
{Joint work with Fabio I. Zyserman and Patricia M. Gauzellino}\\ $ $ \\
{\magenta 2009 SIAM Annual Meeting , Denver,  Colorado,  July  6-10 2009}}
%{Joint work with Fabio I. Zyserman and Patricia M. Gauzellino}
%\subtitle{{\magenta 2007 SEG Annual Meeting, San Antonio, Texas, September 
%23-28, 2007} }

\begin{document}
\maketitle





%----------------------------------------------- SLIDE -
%\begin{slide}{Electroseismic  Modeling. I }
%{\small
%\begin{itemize}

%Seismic waves propagating through
%near-surface layers of the Earth  may  induce 
%electromagnetic disturbances 
%that can be measured at the surface {\magenta { }(seismoelectric effect)}.
%\vskip1.cm
%Also,  recent tests 
%suggest that the reciprocal process, {\it i.e.}  surface measurable  
%acoustic disturbances induced by 
% electromagnetic fields {\magenta { }(electroseismic effect)},  is also possible.



%\end{itemize} %
%}
%\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{Electroseismic  Modeling. I}
\vskip.5cm
{\large
Within a fluid saturated porous medium there
exists a nanometer-scale separation of electric charge in which a bound
charge existing on the surface of the solid matrix (normally of negative sign)
is balanced by adsorbed positive ions of the surrounding fluid, setting an immobile {\magenta { } double  (Stern) layer}, having  
a  thickness   of about 10 nm. 
}
\end{slide}





%----------------------------------------------- SLIDE -
\begin{slide}{Electroseismic  Modeling. II}
\vskip-.5cm
{\large
Further from the surface of the solid matrix there exists a distribution 
of mobile counter ions,
forming the so called {\magenta { }diffuse layer}. When an electric field is applied to
this system, the ions in the diffuse layer move, dragging the pore fluid
along with it because of the {\magenta { }viscous traction}. This is known as {\magenta { }electro-osmosis} 
and is responsible for the  {\magenta { }electroseismic phenomena}.


%\end{itemize} 
}
\end{slide}


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

\begin{slide}{Electroseismic  Modeling. III}
\begin{center}
\vspace*{.1cm}
%\vspace*{-1.cm}
%\includegraphics[width=6cm,height=6cm]{figure4.ps}\\ 
\includegraphics[width=7cm]{double_layer.eps}\\ 
%\vspace*{-0.5cm}
{}
\end{center}
\end{slide}


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

\begin{slide}{2D Model and Sources. I}
\vskip.1cm
{\large 
Assume that the subsurface is an  horizontally layered fluid-saturated porus medium.
\vskip.3cm
Consider two different electromagnetic sources, both in the $y$-direction:
 \vskip.1cm 
1) an {\magenta { }infinite current line $J^{ext}_e$},
 
\vskip.1cm
2) an {\magenta { } infinite magnetic dipole (infinite solenoid) $J^{ext}_m$}.

}
\end{slide}
%-----------------------------------------Slide

\begin{slide}{2D Model and Sources. II}
\vskip.1cm
{\large 
{\magenta { }$J^{ext}_m$ } induces electromagnetic fields  $(E_x(x,z),E_z(x,z))$, 
and $H_y(x,z)$,  and  fluid and solid displacements  
$(u^s_x(x,z),u^s_z(x,z))$ and $(u^f_x(x,z), u^f_z(x,z))$, respectively.

\vskip.5cm 
 This is known as the  {\magenta { }PSVTM-mode},  in which  compressional and vertically polarized shear seismic  waves{\magenta { } (PSV-waves)} are present.

}
\end{slide}
%-----------------------------------------Slide

\begin{slide}{2D Model and Sources. III}
\vskip.1cm
{\large 
{\magenta { }$J^{ext}_e$}  induces  electromagnetic  fields 
 $(H_x(x,z),H_z(x,z))$  and  $E_y(x,z)$,  and 
fluid and solid displacements  $u^s_y(x,z)$  and $u^f_y(x,z)$, respectively.
\vskip.5cm
This is known as the  {\magenta { }SHTE-mode}, where only horizontally polarized 
seismic waves {\magenta { } (SH-waves)} are present. 

}
\end{slide}

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

\begin{slide}{2D Model and Sources. IV}
\begin{center}
\vspace*{-0.3cm}
%\vspace*{-1.cm}
%\includegraphics[width=6cm,height=6cm]{figure4.ps}\\ 
\includegraphics[width=10cm]{figdef.ps}\\ 
%\vspace*{.05cm}
{}
\end{center}
{3D layered subsurface, 2D model $\O = \O^a\cup \O^B$ and electromagnetic sources. }
\end{slide}






%----------------------------------------------- SLIDE -
\begin{slide}{\small {Formulation in Terms of Scattered Fields $E^s, H^s$}. PSVTM-mode. }
\vskip-.5cm
{\small
Let 
\begin{eqnarray*}\label{def_sigma}
\sigma(x,z) = \sigma^p(z) + \sigma^s(x,z),
\end{eqnarray*}
$\sigma^p(z)$: background conductivity, \ \  
$\sigma^s(x,z)$:  conductivity anomaly.
Let {\magenta { } $E^s = E - E^p$, $H^s = H - H^p$:  scattered fields.}

For the {\magenta { }PSVTM-mode}, solve for the primary fields  $E^p, H^p$ solution of: 
\begin{eqnarray*} 
&&\nabla\times E^p = -i \o \mu H^p + J_m^s, \quad\text{in}\quad  \O,
\label{max.p1}\\
&& \nabla\times H^p = \sigma^p E^p\label{max.p2}, \quad\text{in}\quad  \O.
\end{eqnarray*}
Analytical expressions for $E^p, H^p$ are known for the case that $\O$ is  the whole space $R^3$ and $\sigma_p$ is constant. 
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{Differential Model for the PSVTM-mode. I}
\vskip-.6cm
{\small
In terms of the scattered electromagnetic fields the equations for 
{\magenta { }PSVTM  electroseismic modeling } are  
  (Pride, 1994):
\vskip-.8cm
\begin{eqnarray*}
&&\sigma E^s - \text{curl} H^s_y =  -\sigma^s E^p, \quad   \O,\label{modf.2a}\\
&&\text{curl} E^s + i \o \mu H^s_y =0, \quad  \O, \label{modf.1}\\
&&-\omega^2 \rho_b u^s  - \omega^2  \rho_f u^f -
\nabla\cdot \tau(u) =0,  \quad \O^B,\label{modf.3}\\
&& -\omega^2 \rho_f u^s  + \eta (\kappa(\o))^{-1} i \o u^f  + \nabla p_f
\label{modf.4}\\
&&\qquad 
=  \eta (\kappa(\o))^{-1}{\magenta { } L(\o)} \left(E^p + E^s\right),    \,  \,\, \O^B,\nonumber\\ \\ 
&&\tau_{lm}(u)=2 N\,\varepsilon_{lm}(u^s)+\delta_{lm}
\left(\lambda_c \,\nabla\cdot u^s - \alpha K_{av}\, \xi\right),\label{mod.1e}\\
&& p_f(u) =  - \alpha K_{av} \, \nabla\cdot u^s + K_{av} \xi.\label{mod.1f}
\end{eqnarray*}
$\tau_{lm}(u)$: stress tensor, \,\,\,\, $p_f(u)$:  fluid pressure.
}
\end{slide}

%----------------------------------------------- SLIDE -
\begin{slide}{Differential Model for the PSVTM-mode. II}
\vskip-.4cm
{\small
{\magenta { }$L(\omega)$: complex coupling coefficient}, depending on  the effective  permeability, porosity, fluid permittivity and salinity among other parameters.
\vskip.3cm
{\magenta { } $\kappa(\o)$ : Dynamic permeability}:
\begin{eqnarray*}\label{def_perm}
\kappa(\o) = \kappa_0 \left[\left(1 + i \dfrac{\o}{\o_c}\dfrac{4}{m}\right)^{\frac12} + i\dfrac{\o}{\o_c}\right]^{-1} = \kappa_r(\o) - i \kappa_i(\o),
\end{eqnarray*}
\vskip.1cm
$\kappa_0$:        effective permeability
\vskip.2cm
 $m$: dimensionless parameter in the range  $ 4\le m \le 8$.
\vskip.2cm
The electroseismic equations were  solved employing absorbing boundary conditions at the artificial boundaries.
}
\end{slide}



%----------------------------------------------- SLIDE -
\begin{slide}{\small {Weak Formulation for the PSVTM-mode. I}}
%\vskip.5cm
{\tiny
Find $(E^s, H^s_y,u^s, u^f)
\in  H(\text{curl}, \O)\times L^2(\O)\times [H^1(\O_B)]^2\times
H(\text{div},\O_B)$ such that
\begin{eqnarray*}
&&(\sigma E^s,\psi) -(H^s_y, \text{curl} \psi) 
+ \left\langle a(1-i) E^s\cdot \chi, \psi\cdot\chi\right\rangle
 +(\text{curl} E^s, \varphi) + (i \o \mu H^s_y, \varphi)\\
&&\qquad   = -\left(\sigma^s E^p , \psi\right) ,\quad  
(\psi,\varphi)\in H(\text{curl}, \O)\times L^2(\O)\\ \\
&&- \o^2 \left(\rho_b  u^s, v^s\right)_{\O_B} 
- \o^2 \left(\rho_f  u^f, v^s\right)_{\O_B}- \o^2 \left(\rho_f  u^s, v^f\right)_{\O_B}
 + \left(  \eta (\kappa(\o))^{-1} i \o u^f, v^f\right)_{\O_B}\\
&&+ \sum_{l,m}\left(\tau_{lm}(u), \varepsilon_{lm}(v^s)\right)_{\O_B} - \left(p_f(u), \nabla\cdot v^f\right)_{\O_B}+ i \o \left\langle {\mathcal D} S_{\G_p}(u), S_{\G_P}(v)\right\rangle_{\G_p}\nonumber\\
&&\qquad = \left( {\magenta { } L(\o)}\eta (\kappa(\o))^{-1} \left(
E^p + E^s\right), v^f\right)_{\O_B},\nonumber\quad (v^s,v^f)\in [H^1(\O_B)]^2\times
H(\text{div},\O_B) .\nonumber\\
%\end{eqnarray*}
%\begin{eqnarray*}
&&S_{\G_p}(u)= \left(u^s\cdot\nu,u^s\cdot\chi,
 u^f\cdot\nu\right)
\end{eqnarray*} 
$\nu$: unit outer normal on $\G_B= \p \O_B$,\,\,\,\, 
$\chi$: a unit tangent on $\G_B$ oriented counterclockwise,
\vskip.2cm
${\mathcal D}$: a positive definite matrix,\,\,\,  $a = \left(\frac{\sigma}{2 \o \mu} \right)$  
%\end{eqnarray*}

}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {A  Finite Element Procedure for the  PSVTM-mode. I}}
\vskip-.5cm
{\small
${\mathcal T}^h(\O)$: nonoverlapping  
 partition of $\O=\O_a\cup\O_B$ into rectangles $\O_j$ 
of diameter bounded by $h$ 

\vskip.5cm
For  the  {\magenta { } electric vector   $E^s$ and the magnetic scalar $H^s_y$}   we use  
\begin{eqnarray*}
&&{\mathcal V}^{h}=\{\psi \in H(\text{curl},\O):\psi|\O_{j}\in {\mathcal V}^h_{j}
\equiv P_{0,1}(\O_j)\times P_{1,0}(\O_j)\},\\ \\
&&{\mathcal W}^{h}=\{\varphi \in L^2(\O): \varphi|\O_{j}\in 
{\mathcal W}^h_{j} \equiv  P_0(\O_j)\}.
\end{eqnarray*}
\vskip1cm
{\magenta { }DOF}: tangential componets of $E^s$ at the mid points of the sides of each 
$\O_j$ and  the value of $H^s_y$ at the center of $\O_j$.
 

}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {A  Finite Element Procedure for the PSVTM-mode. II}}
\vskip-.1cm
{\small
For  each component of the {\magenta { }solid displacement $u^s$}  we use the space  ${\mathcal NC}^{h}$: 
$$
\widehat R=[-1,1]^2, \qquad
\widehat {\mathcal NC}(\widehat R)=\mbox{Span}\{1,\hat x, \hat z,\widetilde  \alpha(\hat x)-
\widetilde \alpha(\hat z)\}
$$
$$
\widetilde \alpha(\hat x) =  \hat x^2 - \frac{5}{3}  \hat x^4.
$$
{\magenta { }DOF}: values of $u^s$   at the  midpoint of each edge of $\widehat R$.
\vskip-.2cm
\begin{eqnarray*}
\hskip-1cm&&{\mathcal NC}^{h} = \{ v \, :  v_j=  v|_{\O_j}\,  \in 
{\mathcal NC}^h(\O_j) , ~ 
v_j(\xi_{jk}) =v_k(\xi_{jk})\,\,  \forall (j,k)\} .
\end{eqnarray*}

The space ${\mathcal NC}^h$  yields  about half the numerical dispersion than that of standard bilinear elements.
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {A Finite Element Procedure for the  PSVTM-mode. III}}
\vskip1cm
{\small

For the {\magenta { }fluid displacement $u^f$ } vector we use:
\begin{eqnarray*}
&&{\mathcal M}^{h} = \{ v \in H(\text{div},\O_B)\, :  v_j=  v|_{\O_j}\,  \in 
 P_{1,0}\times P_{0,1}(\O_j)\}
\end{eqnarray*}

 
{\magenta { }DOF}:  value of $u^f\cdot \nu$ at the mid points of each side of $\O_j$.
}
\end{slide}
%----------------------------------------------- SLIDE -
\begin{slide}{\small {A Finite Element Procedure for the PSVTM-mode. IV}}
\vskip-.5cm
{\small
Find $(E^{s,h}, H^{s,h}_y,u^{s,h}, u^{f,h})
\in {\mathcal V}^h \times {\mathcal W}^h\times [{\mathcal NC}^h]^2 \times
{{\mathcal M}}^h$ such that
\begin{eqnarray*}
&&(\sigma E^{s,h},\psi) -(H^{s,h}_y, \text{curl} \psi) 
+ \left\langle a(1-i) E^{s,h}\cdot \chi, \psi\cdot\chi\right\rangle\\
&& +(\text{curl} E^{s,h}, \varphi) + (i \o \mu H^{s,h}_y, \varphi)\\
&&\qquad   = -\left(\sigma^s E^p , \psi\right) ,\quad  
(\psi,\varphi)\in  {\mathcal V}^h \times {\mathcal W}^h \\ \\
&&- \o^2 \left(\rho_b  u^{s,h}, v^s\right)_{\O_B} 
- \o^2 \left(\rho_f  u^{f,h}, v^s\right)_{\O_B}- \o^2 \left(\rho_f  u^{s,h}, v^f\right)_{\O_B}\\
&& + \left(  \eta (\kappa(\o))^{-1} i \o u^{f,h}, v^f\right)_{\O_B}
+ \sum_{l,m}\left(\tau_{lm}(u^h), \varepsilon_{lm}(v^s)\right)_{\O_B}\\
&& - \left(p_f(u^h), \nabla\cdot v^f\right)_{\O_B}+ i \o \left\langle {\mathcal D} S_{\G_p}(u^h), S_{\G_P}(v)\right\rangle_{\G_p}\nonumber\\
&& = \left({\magenta { }  L(\o)}\eta (\kappa(\o))^{-1} \left(
E^p + E^{s,h}\right), v^f\right)_{\O_B},\nonumber  (v^s,v^f)\in[{\mathcal NC}^h]^2 \times
{{\mathcal M}}^h.
 \end{eqnarray*} 

}
\end{slide}


%----------------------------------------------- SLIDE -
\begin{slide}{\small {A Priori Error Estimates for the PSVTM-mode.}}
\vskip-.5cm
{\small
{\it A priori}  error estimate:
\vskip.4cm 

{\magenta { }{\bf Theorem:}}

for $\o > 0$ and sufficiently small $h>0$,
\begin{eqnarray*}
&& \|E^s -E^{s,h}\|_0 + \|\text{curl}(E^s -E^{s,h})\|_0 +  \|H^s_y - H_y^{s,h}\|_0 
\\
&& \qquad+ \|u^s -u^{s,h}\|_{1,h,\O_B} + \|u^f -u^{f,h}\|_{0,\O_B} \\
&& \qquad + \|(E^s - E^{s,h})\cdot\chi\|_{0,\G} + \|u^s -u^{s,h}\|_{0,\G_B} \\
&&\qquad\qquad\le C(\o) \left[ h\left(\|E^s \|_1 +  \|\text{curl} ~E^s \|_1 + \|H_y^s \|_1\right)\right.\\
&& \qquad\qquad\quad  + \left. h^{1/2}\left(\|u^s \|_{2,\O_B} 
+ \|u^f \|_{1,\O_B}  +  \|\nabla\cdot u^f \|_{1,\O_b} \right)\right]. 
\nonumber
\end{eqnarray*}


}
\end{slide}

%----------------------------------------------Slide
\begin{slide}{\small{Numerical Electroseismic Modeling for the PSVTM-mode. I} }
\vskip.5cm
{\bf
{\small
{\magenta {} Gas hydrates (GH)} are crystalline molecular 
 complexes composed of water and 
{\magenta {} natural gas, mainly methane}, which form under certain   conditions of low temperature, 
high pressure and gas concentration. They are  found  in 
permafrost regions and seafloor
sediments along the continental margins. 


\vskip.8cm
\noi
{\magenta {} GH } are ice-like structures within the pore space that cause strong
changes in the {\magenta {} conductivity and phase velocities} of the seismic waves, 
making possible  to detect its presence  using electroseismic data.

}
}
\end{slide}

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

\begin{slide}{\small{Numerical Electroseismic Modeling for the PSVTM-mode. II}}
\begin{center}
\vspace*{-.7cm}
\includegraphics[width=12.cm]{psvtm_model.eps}\\
\vspace*{0.2cm}
\end{center}
\tiny{\bf  Brine-saturated sandstone subsurface model including a layer  
 containing  {\magenta {} BRINE + GH}  in the pore space. 
The black  dot at the top  indicates  the  infinite solenoid source in the $y-$ direction located at $(x=0, z=0)$.
The geophones are located near the earth surface, indicated  with inverted triangles. Distances   are given  in meters. The model was discretized with a $2241\times 1121$  mesh and solved for 100 frequencies with a main source frequency of 20 Hz. The reference mesh size was $h^* = 1.75$ m and  the diffusion length  at 20 Hz was $10 $ m.  The CPU time running  with 20 processors in the steele parallel computer at Purdue University was 7 hours.

}
\end{slide}

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

\begin{slide}{\tiny{z-component of solid acceleration (left) and fluid pressure (right)  at t=0.2 s.  GH-saturation $S_{gh}= 0.8$}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6cm,height=6.1cm]{snapshot-t2-soliZ.eps}\includegraphics[width=6cm,height=6.1cm]{snapshot-t2-press.eps} \\
%\vspace*{0.2cm}
\end{center}
%\vskip-.5cm
\tiny{\bf  The  snapshots show upward and downward travelling wavefronts generated by the presence of the  the GH-bearing layer. 
}
\end{slide}
%---------------------------------------SLIDE -----------------------

\begin{slide}{\tiny{z-component of solid acceleration(left) and fluid pressure (right) at  t=0.3 s. GH-saturation $S_{gh}= 0.8$}}
\begin{center}
%\vspace*{-.5cm}
\includegraphics[width=6cm,height=6.1cm]{snapshot-t3-soliZ.eps}\includegraphics[width=6cm,height=6.1cm]{snapshot-t3-press.eps} \\
%\vspace*{0.2cm}
\end{center}
%\vskip-.5cm
\tiny{\bf  The  snapshots show upward travelling wavefronts generated by the presence of the  the GH-bearing layer.
}
\end{slide}

% -----------------------------------------Slide -------------
\begin{slide}{\small{Traces of z-component of solid acceleration for GH-saturation  $S_{gh}$=0.8}}
\begin{center}
\vspace*{-.5cm}
\includegraphics[width=6.2cm]{z-traces-sgh08.eps}\\
%\vspace*{0.2cm}
\end{center}
\vskip-.5cm
\tiny{\bf  Notice that the amplitude of the arrivals increase  (smaller receiver numbers) as we move away from 
the location of the infinite solenoide source located at x=0. 
}
\end{slide}

% -----------------------------------------Slide -------------
\begin{slide}{\small{Traces of z-component of solid acceleration for different GH saturations}}
\tiny{\bf  The  electroseismic experiment  is able to discriminate between different GH saturations}
\begin{center}
\unitlength 1 mm
\begin{picture}(100,100)(0,0)
\put(-22,40){\resizebox{75mm}{!}{\includegraphics{GH-saturation.eps}}}
\put(45,40){\resizebox{75mm}{!}{\includegraphics{GH-saturation-a.eps}}}
\end{picture}
%\includegraphics[width=8cm]{trazas-capas.eps}\includegraphics[width=8cm]{trazas-capas1a.eps} \\
%\vspace*{0.2cm}
\end{center}
%\tiny{\bf  The model is able to discriminate between different GH saturations}
\end{slide}

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

\begin{slide}{\small{Traces of z-component of solid acceleration for different layer widths.}}
\begin{center}
%\vspace*{-.5cm}
\includegraphics[width=6.2cm]{trazas-capas.eps}\includegraphics[width=6.2cm]{trazas-capas1a.eps} \\
%\vspace*{0.2cm}
\end{center}
\vskip-.5cm
\tiny{\bf  The   arrival
 at  t $\approx .3$ sec. is associated with wavefronts  generated at  the top of the GH-saturated layer. The 10 and 50 m width layers are seen as  single reflectors. The second arrival at about .38 for the 200 m layer width case  corresponds to the wavefront generated at the bottom of the GH bearing layer. Notice that  a  
{\magenta {}seismic source } of the same frequency can not {\magenta {}see} the 10 m  width  layer. 
}
\end{slide}





\end{document}

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

\begin{slide}{\small{ Snapshot at time T4 of  $u_y^{s,h}$ for the SHTE-mode}}
\begin{center}
\vspace{-.3cm}
\includegraphics[width=8.cm]{shte_snapshot4.eps}\\
%\vspace*{0.2cm}
\end{center}
\tiny{\bf  Some of the  reflected waves generated at the top and bottom boundaries 
of  the solid 2- fluid 1 have already arrived to the earth surface.  The  wave front travelling downwards corresponds to a shear wave generated by the presence of the infinite electric line source normal to the plane and has already arrived at the solid 2- fluid 1  region, as can be seen in the change in curvature of the wavefront.

}
\end{slide}

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

\begin{slide}{\tiny{Time histories  of  $u_y^{s,h}$ for the SHTE-mode for receivers near the earth surface.}}
\vspace{-.5cm}
\begin{center}
%\vspace*{1cm}
\includegraphics[width=6.cm]{shte_trazas.eps}\\
\vspace*{-0.4cm}
\end{center}
\tiny{\bf  The reflections before and after .42 ms correspond to the shear waves 
reflected at the top and bottom boundaries of the solid 2-fluid 1 boundaries.

}
\end{slide}
















