%\documentclass[pdf,autumn,slideColor,colorBG]{prosper} % 2 %\documentclass[pdf,contemporain,slideColor,colorBG]{prosper} %\documentclass[pdf,lignesbleues,slideColor,colorBG]{prosper} % este lignesbleues es el mejor %\documentclass[pdf,azure,slideColor,colorBG]{prosper} %este azure es el mejor %\documentclass[pdf,blends,slideColor,colorBG]{prosper} %\documentclass[pdf,capsules,slideColor,colorBG]{prosper} %\documentclass[pdf,nuancegris,slideColor,colorBG]{prosper} \documentclass[pdf,serpaggi,slideColor,colorBG]{prosper} % 1 \usepackage[latin1]{inputenc} \usepackage{amsmath} %\usepackage{psfig,epsfig} \usepackage{epsfig} \def\noi{\noindent} \def\cl{\centerline} \def\bs{\bigskip} \def\ms{\medskip} \def\sm{\smallskip} \bibliographystyle{plain} \def\nn{\nonumber} \renewcommand\a{\alpha} \newcommand{\ii}{{\rm i}} \def\b{\beta} \newcommand{\bc}{\rm I\!\!\!C} \newcommand{\bh}{\rm I\!H} \newcommand{\br}{\rm I\!R} \newcommand{\brn}{{\rm I\!R}\,^n} \def\ai{\left\langle\langle} \def\ad{\right\rangle\rangle} \def\bus{{\mathbf{u}}^s} \def\bun{{\mathbf{u}}^n} \def\buw{{\mathbf{u}}^w} \def\bnab{\mathbf \nabla} \def\d{\delta} \def\D{\Delta} \def\sD{{\mathcal{D}}} \def\dfrac{\displaystyle\frac} \def\diag{\mathrm{diag\,}} \def\dint{\displaystyle\int} \def\doint{\displaystyle\oint} \def\dist{\mathrm{dist\,}} \def\Div{\mathrm{div\,}} \def\ds{\displaystyle} \def\dsum{\displaystyle\sum} \def\e{\varepsilon} \def\g{\gamma} \def\G{\Gamma} \def\hp{\widehat p} \def\k{\kappa} \def\lam{\lambda} \def\Lam{\Lambda} \def\Lie{\mathrm{Lie}} \def\nab{\nabla} \renewcommand\o{\omega} \renewcommand\O{\Omega} \def\og{\overline{g}} \def\OS{\overline{S}} \def\p{\partial} \newcommand{\pp}[2]{\dfrac{\partial {#1}}{\partial {#2}}} \newcommand{\ppt}{\frac{\partial}{\partial t}} \def\rank{\mathrm{rank}} \def\lma{\left\langle} \def\rma{\right\rangle} \def\s{\sigma} \def\sC{{\rho}} \def\V{{\mathcal{V}}} \renewcommand\t{\theta} \def\tK{\widetilde K} \def\tp{\widetilde p} \newcommand\var{\varphi} \def\wse{\widetilde {E}} \def\wsa{\widetilde {A}} \def\wsc{\widetilde {C}} \def\z{\zeta} \def\wh{\widehat} \def\vx{{\bf x}} \newcommand{\dd}[2]{\frac{\mbox{d} {#1}}{\mbox{d} {#2}}} \def\du{\dot u} \def\dU{\dot U} \newcommand{\DD}[2]{\frac{\mbox{D} {#1}}{\mbox{D} {#2}}} \def\stn{\dot{R}_{N}^S(\o)} \def\ctn{\dot{R}_{N}^C(\o)} \def\stlam{\dot{R}_{\lam_c}^S(\o)} \def\ctlam{\dot{R}_{\lam_c}^C(\o)} \def\stbu{\dot{R}_{B_1}^S(\o)} \def\ctbu{\dot{R}_{B_1}^C(\o)} \def\stbd{\dot{R}_{B_2}^S(\o)} \def\ctbd{\dot{R}_{B_2}^C(\o)} \def\stmu{\dot{R}_{M_1}^S(\o)} \def\ctmu{\dot{R}_{M_1}^C(\o)} \def\stmd{\dot{R}_{M_2}^S(\o)} \def\ctmd{\dot{R}_{M_2}^C(\o)} \def\stmt{\dot{R}_{M_3}^S(\o)} \def\ctmt{\dot{R}_{M_3}^C(\o)} \def\deps{\dot\varepsilon} \def\dxi{\dot\xi} \def\ds#1#2{\dfrac {\partial #1}{\partial #2}} \title{Numerical Simulation in Applied Geophysics. From the Mesoscale to the Macroscale} \vskip-.5cm \author{Juan E. Santos ,$^\dag$ Patricia M. Gauzellino $^*$, Gabriela B. Savioli$^{**}$ and Robiel Martinez Corredor$^{\dag,\dag}$ } \institution{% $^\dag$ CONICET, Instituto del Gas y del Petr\'oleo, {\magenta Facultad de Ingenier\'\i a UBA}, UNLP and Department of Mathematics, {\magenta Purdue University}. \\ $^*$ Facultad de Cs. Astr. y Geof\'\i sicas, UNLP \\ $**$ Instituto del Gas y del Petr\'oleo, {\magenta Facultad de Ingenier\'\i a UBA \\ $^{\dag,\dag}$ Facultad de Ingenier\'\i a, UNLP } {\magenta Jornadas de Cloud Computing, LIDI, Fac. de Inform\'atica, UNLP, 18 Junio 2013}} \begin{document} \maketitle %----------------------------------------------- SLIDE - \begin{slide}{\small Introduction. I} {\bf {\small \begin{itemize} \item {\red {} Seismic wave propagation } is a common technique used in hydrocarbon exploration geophysics, mining and reservoir characterization and production. \item Local variations in the fluid and solid matrix properties, fine layering, fractures and craks at the mesoscale (on the order of centimeters) are common in the earth's crust and induce attenuation, dispersion and anisotropy of the seismic waves observed at the macroscale. \item These effects are caused by equilibration of wave-induced fluid pressure gradients via a {\magenta { } slow-wave diffusion} process. \end{itemize} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small Introduction. II} {\bf {\small \begin{itemize} \item Due to the {\magenta { }extremely fine meshes} needed to properly represent these type of mesoscopic-scale heterogeneities, numerical simulations are very expensive or even not feasible. \item {\magenta { }Suggested approach:} Use a {\magenta { } numerical upscaling procedure } to determine the complex and frequency dependent stiffness at the macroscale of an equivalent viscoelastic medium including the mesoscopic-scale effects. \item To determine the {\blue{} complex stiffness} coefficients of the {\magenta{}equivalent medium}, we solve a set of boundary value problems (BVP's) for the wave equation of motion in the frequency-domain using the finite-element method (FEM). \end{itemize} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small Introduction. III} {\bf {\small \begin{itemize} \item The BVP's represent {\blue{} harmonic tests} at a finite number of frequencies on a representative sample of the material. \item {\blue {}Numerical rock physics} offer an alternative to laboratory measurements. \item Numerical experiments are inexpensive and informative since the physical process of wave propagation can be inspected during the experiment. \item Moreover, they are repeatable, essentially free from experimental errors, and may easily be run using alternative models of the rock and fluid properties. \end{itemize} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small The Mesoscale. Anisotropic poroelasticity. I} {\bf {\small \begin{itemize} \item To model mesoscopic effects we need a suitable differential model. \item As example, we will describe the numerical experiments on a representative sample of fluid-saturated sample having a dense set of horizontal fractures modeled as very thin and compliant layers. \item A dense set of horizontal fractures in a fluid-saturated poroelastic medium behaves as a {\magenta { } viscoelastic transversely isotropic (VTI)} medium when the average fracture distance is much smaller than the predominant wavelength of the traveling waves. \item This leads to frequency and angular variations of velocity and attenuation of seismic waves. \end{itemize} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The Mesoscale. Anisotropic poroelasticity. II}} \vskip-.5cm {\small \begin{itemize} \item For fluid-saturated poroelastic media (Biot's media), White et al. (1975) were the first to introduce the mesoscopic-loss mechanism in the framework of Biot's theory. \item For fine layered poroelastic materials, the theories of Gelinsky and Shapiro (GPY, {\bf 62}, 1997) and Krzikalla and M\"uller (GPY, {\bf 76}, 2011) allow to obtain the {\blue { }five} complex and frequency-dependent stiffnesses of the {\magenta { }equivalent VTI medium}. \item To test the model and provide a more general modeling tool, we present a {\magenta { }numerical upscaling procedure} to obtain the complex stiffnesses of the effective VTI medium. \end{itemize} } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The Mesoscale. Anisotropic poroelasticity. III}} {\small \begin{itemize} \item We employ the Finite Element Method {\blue { } (FEM)} to solve {\magenta { } Biot's equation of motion} in the space-frequency domain with boundary conditions representing {\blue { } compressibility and shear harmonic} experiments. \item The methodology is applied to saturated isotropic poroelastic samples having a dense set of horizontal fractures. \item The samples contained mesoscopic-scale heterogeneities due to patchy brine-CO$_2$ saturation and fractal porosity and consequently, fractal permeability and frame properties. \end{itemize} } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf TIV media and fine layering. I}} \vskip-.5cm {\bf {\small Let us consider isotropic fluid-saturated poroelastic layers. {\blue { } $ {\mathbf u^s}({\mathbf x}), {\mathbf u^f}({\mathbf x})$ } : time Fourier transform of the displacement vector of the solid and fluid relative to the solid frame, respectively. \smallskip {\blue { }${\mathbf u} = ({\mathbf u^s}, {\mathbf u^f})$} \vskip.2cm {\magenta { }${\mathbf \sigma_{kl}(u)}, {\mathbf p_f(u)}$}: Fourier transform of the total stress and the fluid pressure, respectively \vskip.3cm On each plane layer $n$ in a sequence of $N$ layers, the {\magenta { }frequency-domain stress-strain relations } are \begin{eqnarray*}\label{st_1} &{\mathbf \sigma}_{kl}(u) = 2 \mu\,\varepsilon_{kl}(u^s)+\delta_{kl} \left( \lambda_{_G} \,\nabla\cdot u^s + \alpha M \nabla\cdot u^f \right),\\ &{\mathbf p_f(u) }= - \alpha M \nabla\cdot u^s - M \nabla\cdot u^f.\label{st_7} \end{eqnarray*} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small TIV media and fine layering. II} {\bf {\small {\magenta { } Biot's equations in the diffusive range:} \begin{eqnarray*}\label{motion1} &&\nabla\cdot\sigma(u) = 0, \\ && \ii \o \dfrac{\eta}{\kappa} u^f(x,\o) + \nabla p_f(u)=0,\label{motion2} \end{eqnarray*} \vskip.2cm $\o=2\pi f$: angular frequency \vskip.2cm $\eta$: fluid viscosity \hskip1cm $\kappa$: frame permeability } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf TIV media and fine layering. III}} \vskip-.5cm {\bf {\small ${\mathbf \tau}_{{\mathbf i} {\mathbf j}}$: stress tensor of the equivalent VTI medium For a {\blue { }closed system( $\nabla\cdot u^f = 0) $}, the corresponding {\magenta { }stress-strain relations}, stated in the space-frequency domain, are \begin{eqnarray*} &&\tau_{11}(u) = {p}_{11} ~\epsilon_{11}(u^s) + p_{12} ~\epsilon_{22}(u^s) + p_{13} ~\epsilon_{33}(u^s), \label{st_GL_1}\\ &&\tau_{22}(u) = p_{12} ~\epsilon_{11}(u^s) + p_{11} ~\epsilon_{22}(u^s) + p_{13} ~\epsilon_{33}(u^s), \label{st_GL_2}\\ &&\tau_{33}(u) = p_{13} ~\epsilon_{11}(u^s) + p_{13} ~\epsilon_{22}(u^s) + p_{33} ~\epsilon_{33}(u^s), \label{st_GL_3}\\ &&\tau_{23}(u) = 2~ p_{55} ~\epsilon_{23}(u^s), \label{st_GL_4}\\ &&\tau_{13}(u) = 2 ~ p_{55} ~\epsilon_{13}(u^s),\label{st_GL_5} \\ &&\tau_{12}(u) = 2 ~ p_{66} ~\epsilon_{12}(u^s). \label{st_GL_6} \end{eqnarray*} {\small This approach provides the complex velocities of the fast modes and takes into account {\magenta { }interlayer flow effects}.} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. I}} \vskip1cm {\bf {\small To determine the complex stiffness we solve Biot's equation in the 2D case on a reference square $\O=(0,L)^2$ with boundary $\G$ in the $(x_1, x_3)$-plane. Set {\blue { }$\G = \G^L\cup\G^B\cup\G^R\cup\G^T$}, where \begin{eqnarray} {\blue { }\G^L} = \{ (x_1, x_3)\in \G: {\blue { }x_1=0}\}, \quad {\blue { }\G^R} = \{ (x_1,x_3)\in \G: {\blue { }x_1=L}\}, \nonumber\\ \nonumber\\ {\blue { }\G^B} = \{ (x_1, x_3)\in \G: {\blue { } x_3=0}\}, \quad {\blue { }\G^T }= \{ (x_1, x_3)\in \G:{\blue { } x_3=L}\}.\nonumber \end{eqnarray} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. II}} \vskip-.5cm {\bf {\small The sample is subjected to {\blue { } harmonic compressibility and shear tests} described by the following sets of {\magenta { } boundary conditions}. \vskip .1cm {\blue { } $p_{33(\o)}$}: \begin{eqnarray*}\label{bc1a} &&\sigma(u) \nu \cdot \nu = -\Delta P, \quad (x_1, x_3)\in \G^T,\label{bc1b}\\ &&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G^T\cup\G^L\cup\G^R,\label{bc1c}\\ %&&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G^L\cup\G^R,\label{bc1d}\\ &&u^s\cdot\nu = 0, \quad (x_1, x_3)\in \G^L\cup\G^R,\label{bc1e}\\ &&u^s = 0, \quad (x_1, x_3)\in \G^B,\quad u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{bc1g}. \end{eqnarray*} {\magenta { } $\nu$}: the unit outer normal on $\G$ {\magenta { } $\chi$}: a unit tangent on $\G$ so that ${\blue { } \{\nu,\chi\}}$ is an orthonormal system on $\G$. Denote by $V$ the original volume of the sample and by $\Delta V(\omega)$ its (complex) oscillatory volume change. } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. III}} \vskip-.5cm {\bf {\small In the quasistatic case \begin{eqnarray*}\label{c33-dv} \frac{\Delta V(\omega)}{V}=-\frac{\Delta P}{{\magenta { } p_{33}(\omega)}}, \end{eqnarray*} Then after computing the average $u^{s,T}_3(\omega)$ of the vertical displacements on $\Gamma^T$, we approximate \[ \Delta V(\omega)\approx L u^{s,T}_3(\omega) \] which enable us to compute ${\magenta { } p_{33}(\omega)}$ \vskip.5cm To determine {\blue { }$p_{11}(\omega)$} we solve an identical boundary value problem than for {\magenta { }$p_{33}$} but for a 90$^{\rm o}$ rotated sample. } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. IV}} \vskip-.5cm {\bf {\small {\blue { }$p_{55}(\omega)$:} the boundary conditions are \begin{eqnarray*}\label{s_bc1a} &&-\sigma(u) \nu = g, \quad (x_1, x_3)\in \G^T\cup\G^L\cup\G^R,\label{s_bc1b}\\ &&u^s = 0, \quad (x_1, x_3)\in \G^B,\label{s_bc1f}\\ &&u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{s_bc1g}, \end{eqnarray*} where \begin{eqnarray*} g = \begin{cases} (0, \Delta G),\quad (x_1, x_3)\in \G^L,\\ (0, -\Delta G),\quad (x_1, x_3)\in \G^R,\\ (-\Delta G, 0), \quad (x_1, x_3)\in \G^T.\end{cases} \end{eqnarray*} } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. V}} \vskip-.5cm {\bf {\small The change in shape suffered by the sample is \begin{equation}\label{shear} \tan [ \theta(\omega) ] =\frac{\Delta G}{{\blue { } p_{55}(\omega)}}. \end{equation} $\theta(\omega)$: the angle between the original positions of the lateral boundaries and the location after applying the shear stresses. Since $\tan [\theta(\omega)] \approx u^{s,T}_1(\omega)/L$, where $u^{s,T}_1(\omega)$ is the average horizontal displacement at $\Gamma^T$, $p_{55}(\omega)$ can be determined from \vskip.2cm to determine {\magenta { }$p_{66}(\omega)$} (shear waves traveling in the $(x_1, x_2)$-plane), we rotate the layered sample 90$^{\rm o}$ and apply the shear test as indicated for {\blue { }$p_{55}(\omega)$}. } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The harmonic experiments to determine the stiffness coefficients. VI}} \vskip-.5cm {\bf {\small {\blue { }$p_{13}(\omega)$}: the boundary conditions are \begin{eqnarray*}\label{bc1a_13} &&\sigma(u) \nu \cdot \nu = -\Delta P, \quad (x_1, x_3)\in \G^R\cup \G^T,\label{bc1b_13}\\ &&\sigma(u) \nu \cdot \chi = 0, \quad (x_1, x_3)\in \G,\label{bc1c_13}\\ &&u^s\cdot\nu = 0, \quad (x_1, x_3)\in \G^L\cup\G^B,\quad u^f\cdot\nu=0, \quad (x_1, x_3)\in \G\label{bc1g_13}. \end{eqnarray*} In this experiment $\epsilon_{22}= \nabla\cdot u^f = 0$, so that \begin{eqnarray}\label{st_backus_2d} &&\tau_{11} = p_{11} \epsilon_{11} + {\magenta { } p_{13}} \epsilon_{33},\quad \tau_{33} = {\magenta { }p_{13} }\epsilon_{11} + p_{33} \epsilon_{33}, \end{eqnarray} $\epsilon_{11}, \epsilon_{33}$: the strain components at the right lateral side and top side of the sample, respectively. Then, \vskip-.2cm %Then, once {\green { }$p_{11}$} and {\green { }$p_{33}$} have been determined, %\begin{equation} \label{p13} %p_{13}(\o) = \frac {p_{11} %\epsilon_{11} - p_{33} \epsilon_{33}}{\epsilon_{11} - \epsilon_{33}}.\nonumber %\end{equation} \[ p_{13}(\o) = \left(p_{11} \epsilon_{11} - p_{33} \epsilon_{33}\right)/ \left(\epsilon_{11} - \epsilon_{33}\right). \] %\end{equation} } } \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{Schematic representation of the oscillatory compressibility and shear tests in $\O$ }} \begin{center} \vskip-.3cm %\vspace*{0.1cm} \includegraphics[width=7cm,height=6.5cm]{fig1.eps}\\ %\vspace*{-0.2cm} {\small a): $p_{33}$,\hskip.3cm b): $p_{11}$, \hskip.3cm c): $p_{55}$,\hskip.3cm d): $p_{13}$ \hskip.3cm e): $p_{66}$} \end{center} \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small{\bf Examples}. I } \vskip-.8cm {\bf { A set of numerical examples consider the following cases for a square poroelastic sample of 160 cm side length and 10 periods of 1 cm fracture, 15 cm background: \begin{itemize} \item Case 1: A brine-saturated sample with fractures. \item Case 2: A brine-CO$_2$ patchy saturated sample without fractures. \item Case 3: A brine-CO$_2$ patchy saturated sample with fractures. \item Case 4: A brine saturated sample with a fractal frame and fractures. \end{itemize} The discrete boundary value problems to determine the complex stiffnesses $p_{IJ}(\o)$ were solved for 30 frequencies using a public domain sparse matrix solver package. \vskip.01cm Using relations not included for brevity, the $p_{IJ}(\o)$'s determine in turn the energy velocities and dissipation coefficients shown in the next figures. } } \end{slide}%----------------------------SLIDE ------------------------------- %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Polar representation of the qP energy velocity vector at 50 Hz for cases 1, 2 and 3 }}} \begin{center} \vspace*{-.7cm} \includegraphics[width=6.cm,height=6.cm] {fig5_VqP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\ \vspace*{0.3cm} {\tiny Velocity anisotropy caused by the fractures in cases 1 and 3 is enhanced for the case of patchy saturation, with lower velocities when patches are present. The velocity behaves isotropically in case 2. } \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Dissipation factor of the qP waves at 50 Hz for cases 1, 2 and 3 }}} \begin{center} \vspace*{-.5cm} \includegraphics[width=6.cm,height=6.cm] {fig3_1000_over_qP_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\ \vspace*{0.3cm} {\tiny Fractures induce strong Q anisotropy for angles normal to the fracture plane, enhanced by patchy saturation.} \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Fluid pressure distribution at 50 and 300 Hz. Compressibility test for $p_{33}$ for case 3.}}} \vspace*{0.9cm} \begin{center} %\vspace*{.4cm} \includegraphics[width=5.9cm] {fig7_presion_fluido_p33_relation_1-15_10percent_sat_color_50Hz.eps} \includegraphics[width=5.9cm]{fig8_presion_fluido_p33_relation_1-15_10percent_sat_color_300Hz.eps} %\vspace*{0.3cm} %{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)} \end{center} {\small Compression is normal to the fracture plane. Attenuation is stronger at 300 Hz.} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Polar representation of the qSV energy velocity vector at 50 Hz for cases 1, 2 and 3 }}} \vspace*{-0.7cm} \begin{center} %\vspace*{.4cm} \includegraphics[width=6.2cm,height=6.2cm] {fig12_VqSV_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\ %\vspace*{0.3cm} {\tiny Velocity anisotropy is induced by fractures (cases 1 and 3 ). Patchy saturation does not affect the anisotropic behavior of the qSV velocities. Case 2 shows isotropic velocity, with higher velocity values than for the fractured cases} \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Dissipation factor of qSV waves at 50 Hz for cases 1, 2 and 3 }}} \vspace*{-0.5cm} \begin{center} %\vspace*{.4cm} \includegraphics[width=6.2cm,height=6.2cm] {fig10_1000_over_qsv_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\ %\vspace*{0.3cm} {\tiny qSV anisotropy is strong for angles between 30 and 60 degrees. The lossless case 2 is represented by a triangle at the origin } \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Polar representation of the qSH energy velocity vector at 50 Hz for cases 1, 2 and 3 }}} \vspace*{-0.7cm} \begin{center} %\vspace*{.4cm} \includegraphics[width=6.2cm,height=6.2cm] {fig13_VSH_polar_co2_50Hz_relation_15-1_10percent_sat_color.eps}\\ %\vspace*{0.3cm} {\tiny SH velocity anisotropy is observed to be induced by fractures. Cases 1 and 3 are almost indistinguishable. Velocity for case 2 is isotropic. SH waves are lossles, since $p_{55}$ and $p_{66}$ are real.} \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Lam\'e coefficient (GPa) for the brine-saturated fractal porosity-permeability sample of case 4.}}} \begin{center} %\vspace*{.4cm} \includegraphics[width=8cm] {fig20_lambda_fractal_relation_1-15_160x160_color.eps} %{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)} \end{center} {\tiny $\text{log} ~ \kappa(x,z) = \left\langle \text{log} ~\kappa \right\rangle + f(x,z)$, $f(x,z)$: fractal representing the spatial fluctuation of the permeability field $\kappa(x,z)$.} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Dissipation factor of qP waves at 50 Hz for cases 1 and 4. }}} \begin{center} %\vspace*{.4cm} \includegraphics[width=6.cm,height=6.cm] {fig21_1000_over_qP_relation_1-15_fractal_por_perm_50Hz_color.eps}\\ \vspace*{0.3cm} {\tiny Note the increase in Q anisotropy for qP waves for angles normal to the fracture plane.} \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {Dissipation factor of qSV waves at 50 Hz for cases 1 and 4.}}} \begin{center} %\vspace*{.4cm} \includegraphics[width=6.cm,height=6.cm] {fig22_1000_over_qsv_relation_1-15_fractal_por_perm_50Hz_color.eps}\\ \vspace*{0.3cm} {\tiny Note the increase in Q anisotropy for angles between 30 and 60 degrees} \end{center} \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf The Macroscale. I} } \vskip-.5cm {\bf {\small The model: $\O$ consists of an anisotropic rotated TIV layer between two isotropic half-spaces. The upper half space has P- and S-wave velocities equal to 1890 m/s and 592 m/s, respectively, while the lower half-space has P- and S-wave velocities of 2320 m/s and 730 m/s. The anisotropic layer, with a thickness of 300 m, is an viscoelastic medium with five complex and frequency dependent stiffness obtained using the upscaling procedure. The medium is rotated by 20 degrees , i.e., the angle between the symmetry axis and the vertical direction. The average density in $\O$ is 2300 kg/m3. The mesh consists of square cells having side length 3.7 m. The source is a dilatational perturbation of central frequency is 30 Hz, located 150 m below the first interface (an asterisk in the figures). } } \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\small{\bf {The Macroscale. II.}}} \begin{center} \vspace*{-.4cm} \includegraphics[width=7cm] {model.eps} %{\tiny Compression is normal to the fracture plane ($p_{33}$ experiment)} \end{center} \vskip.2cm {\small The Macroscopic TIV model.} \end{slide} %--------------------------------------------------------Slide \begin{slide}{\small{The Macroscale. III. Seismic modeling.}} {\bf {\small We solve the following boundary value problem at the macroscale (in $\O$): \begin{eqnarray*}\label{mod3} &&\hskip-.5cm-\o^2 \rho u - \nabla\cdot\tau(u) = F,\quad \O\\ &&-\tau(u) \nu = i \o {\mathcal D} u,\quad \partial \O,\text{(absorbing bounday condition, $D>0$)}\label{mod4} \end{eqnarray*} $u =(u_x, u_z)$: displacement vector, \hskip.2cm $ \rho$: average density. \vskip.1cm \noi $\tau(u)$: stress-tensor of our {\magenta { } equivalent viscoelastic material}, defined in terms of the $p_{IJ}'s$. \vskip.2cm Instead of solving the global problem associated with the above model, we obtained the solution using a parallel FE iterative hybridized domain decomposition procedure employing a nonconforming FE space. \vskip.1cm A run in santalo under MPI with 100 processors required 10 hours of CPU time for 1000x1000 mesh, 6.000.000 unknowns and 900 frequencies. } } \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{\bf {The Macroscale. IV. Snapshots of the vertical displacement at 200 ms (a and c) and 500 ms (b and d ).}}} \begin{center} \vspace*{-.4cm} \includegraphics[width=7.cm] {fv6.eps}\\ {\tiny Ideal isotropic case (left, a and b) and real (rotated TIV) case (right, c and d). } \end{center} \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small{The Macroscale. IV. CO$_2$ injection and seismic monitoring.}} {\small \begin{itemize} \item Storage of CO$_2$ in geological formations is a procedure employed to reduce the amount of greenhouse gases in the atmosphere to slow down global warming. \item Geologic sequestration involves {\magenta { } injecting CO$_2$ into a target geologic formation} at depths typically >1000 m where pressure and temperature are above the critical point for CO$_2$ (31.6C, 7.38 MPa). \item Example of injection into the Utsira Sand, a saline aquifer at the Sleipner field, North Sea. \item {\magenta { }Time-lapse seismic surveys} aim to monitor the migration and dispersal of the CO$_2$ plume after injection. \end{itemize} } \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\small {The Macroscale. V. Permeability map (Darcy units) of the formation.}} \begin{center} {\small %\vspace*{-.4cm} \includegraphics[width=12cm,height=6.5cm]{kx_xz_300x5x400.gnu.eps} %\vspace*{-0.2cm} {\tiny The thin lines represent mudstone layers within the formation. } } \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\small{The Macroscale. VI. Injection Modeling. Black-Oil simulator.}} %\vskip -.6cm \begin{center} \includegraphics[width=5.6cm,height=6.cm]{sat_CO2_600x400_frac_2a_10km_dt005_test3.gnu.eps} \includegraphics[width=5.6cm,height=6.cm]{sat_CO2_600x400_frac_6a_10km_dt005_test3.gnu.eps}\\ {\tiny CO$_2$ saturation distribution after 2 years (left) and 6 years(right) of CO$_2$ injection } \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{The Macroscale. VII. P-wave phase velocity $v_p(\o)$(m/s) and attenuation coefficient $Q_p(\o)$ at 50 Hz.}} \vskip -.5cm \begin{center} \includegraphics[width=5.9cm,height=5.2cm]{vp_map_2a_gnu_ve_con_base.eps} \includegraphics[width=5.9cm,height=5.2cm]{qp_map_2a_gnu_ve_con_base.eps}\\ {\tiny Observe a decrease in P- wave velocity (left) and a decrease in attenuation coefficient $Q_p$ (right) in zones of CO$_2$ accumulation. Mesoscopic attenuation adn dispersion effects are taking into account using a White model. } \end{center} \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{Traces before and after 2 years of CO$_2$ injection. Line Source. Both figures shown in the SAME SCALE. }} \vskip-.2cm {\bf {\small %\vspace*{-.4cm} \includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base_solo_brine.ps}\quad \includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base.ps}\\ %\vspace*{-0.2cm} \tiny{ Traces of the z-component of the particle velocity before (left) and after 2 years (right) CO$_2$ injection. The two arrivals seen on the left Figure come from the base and top of the Utsira. The strong later arrivals seen on the right Figure are due to P-waves reflected at CO$_2$ accumulations and the base of the Utsira.} } } \end{slide} %----------------------------SLIDE ------------------------------- \begin{slide}{\tiny{Traces before and after 2 and 6 years of CO$_2$ injection. Line Source.}} %\vskip-.7cm {\bf {\small %\vspace*{-.4cm} \includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_2a_con_base.ps}\quad \includegraphics[width=5.2 cm,height=5.2cm,angle=-90]{gather_vz_line_600x200_6a_con_base.ps}\\ %\vspace*{-0.2cm} \tiny{Traces of the z-component of the particle velocity after 2 years (left) and 6 years (right) of CO$_2$ injection. The earlier arrivals on the right Figure are due to waves reflected at the CO$_2$ accumulations below the shallow mudstone layers}. } } \end{slide} %----------------------------------------------- SLIDE - \begin{slide}{\small {\bf Conclusions} } \vskip-.5cm {\bf {\small \begin{itemize} \item Numerical upscaling procedures allow to represent mesoscopic scale heterogeneities in the solid frame and saturant fluids, fractures and craks affecting observations at the macroscale. \item THE FEM is a useful tool to solve local problems in the context of Numerical Rock Physics and global problems at the macroscale. \item The techniques presented here to model acoustics of porous media can be extended to other fields, like ultrasound testing of quality of foods, groundwater flow and contamination among others. \item Thanks for your attention !!!!!. \end{itemize} } } \end{slide} \end{document} \end{document}