Analysis of 2D time-domain seismoelectric modeling

This work analyzes the equations for 2D seismoelectric modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations, coupled via an electrokinetc coefficient and employs absorbing boundary conditions at the artificial boundaries. Results on existence and uniqueness of the solution of the differential problem are presented.Facultad de Ciencias Astronómicas y Geofísica

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí

Evaluation of the stiffness tensor of a fractured medium with harmonic experiments

A fractured medium behaves as an anisotropic medium when the wavelength is much larger than the distance between fractures. These are modeled as boundary discontinuities in the displacement and particle velocity. When the set of fractures is plane, the theory predicts that the equivalent medium is transversely isotropic and viscoelastic (TIV). We present a novel procedure to determine the complex and frequency-dependent stiffness components. The methodology amounts to perform numerical compressibility and shear harmonic tests on a representative sample of the medium. These tests are described by a collection of elliptic boundary-value problems formulated in the space-frequency domain, which are solved with a Galerkin finite-element procedure. The examples illustrate the implementation of the tests to determine the set of stiffnesses and the associated phase velocities and quality factors.Fil: Santos, Juan Enrique. Universidad de Buenos Aires. Facultad de Ingeniería. Instituto del Gas y del Petróleo; Argentina. Universidad Nacional de La Plata; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Benedettini, Stefano. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale; ItaliaFil: Carcione, José M.. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale; Itali

Numerical experiments of fracture-induced velocity and attenuation anisotropy

Fractures are common in the Earth's crust due to different factors, for instance, tectonic stresses and natural or artificial hydraulic fracturing caused by a pressurized fluid. A dense set of fractures behaves as an effective long-wavelength anisotropic medium, leading to azimuthally varying velocity and attenuation of seismic waves. Effective in this case means that the predominant wavelength is much longer than the fracture spacing. Here, fractures are represented by surface discontinuities in the displacement u and particle velocity v as [κ · u + η · v], where the brackets denote the discontinuity across the surface, κ is a fracture stiffness and η is a fracture viscosity. We consider an isotropic background medium, where a set of fractures are embedded. There exists an analytical solution-with five stiffness components-for equispaced plane fractures and an homogeneous background medium. The theory predicts that the equivalent medium is transversely isotropic and viscoelastic. We then perform harmonic numerical experiments to compute the stiffness components as a function of frequency, by using a Galerkin finite-element procedure, and obtain the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts) and quality factors. The algorithm is tested with the analytical solution and then used to obtain the stiffness components for general heterogeneous cases, where fractal variations of the fracture compliances and background stiffnesses are considered.Este documento tiene una corrección (ver documento relacionado).Facultad de Ciencias Astronómicas y Geofísica

Anisotropic poroelasticity and wave-induced fluid flow: Harmonic finite-element simulations

A dominant P-wave attenuation mechanism in reservoir rocks at seismic frequencies is due to wave-induced fluid flow (mesoscopic loss). The P-wave induces a fluid-pressure difference at mesoscopic-scale inhomogeneities (larger than the pore size but smaller than the wavelength), generating fluid flow and slow (diffusion) Biot waves. The theory has been developed in the 1970s for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a finely layered medium, and has recently been generalized to all propagation angles. The new theory states that the fluid-flow direction is perpendicular to the layering plane and it is independent of the loading direction. As a consequence, the relaxation behaviour can be described by a single relaxation function, since the medium consists of plane homogeneous layers. Besides P-wave losses, the coupling between the qP and qSV waves generates shear-wave anisotropic velocity dispersion and attenuation. In this work, we introduce a set of quasi-static numerical experiments to determine the equivalent viscoelastic TI medium to a finely layered poroelastic medium, which is validated using a recently developed analytical solution. The modelling technique is the finite-element (FE) method, where the equations of motion are solved in the space-frequency domain. Numerical rock physics may, in many circumstances, offer an alternative to laboratory measurements. Numerical experiments are inexpensive and informative since the physical process of wave propagation can be inspected during the experiment. Moreover, they are repeatable, essentially free from experimental errors, and may easily be run using alternative models of the rock and fluid properties. We apply the methodology to the Utsira aquifer of the North Sea, where carbon dioxide (CO2) has been injected during the last 15 years. The tests consider alternating layers of the same rock saturated with gas and brine and a sequence of gas-saturated sandstone and mudstone layers, which represent possible models of the reservoir and cap rock of the aquifer system. The numerical examples confirm the new theory and illustrate the implementation of the harmonic tests to determine the complex and frequency-dependent effective stiffnesses and the associated wave velocities and quality factors.Facultad de Ciencias Astronómicas y Geofísica

Numerical experiments of fracture-induced velocity and attenuation anisotropy

Fractures are common in the Earth's crust due to different factors, for instance, tectonic stresses and natural or artificial hydraulic fracturing caused by a pressurized fluid. A dense set of fractures behaves as an effective long-wavelength anisotropic medium, leading to azimuthally varying velocity and attenuation of seismic waves. Effective in this case means that the predominant wavelength is much longer than the fracture spacing. Here, fractures are represented by surface discontinuities in the displacement u and particle velocity v as [κ · u + η · v], where the brackets denote the discontinuity across the surface, κ is a fracture stiffness and η is a fracture viscosity. We consider an isotropic background medium, where a set of fractures are embedded. There exists an analytical solution-with five stiffness components-for equispaced plane fractures and an homogeneous background medium. The theory predicts that the equivalent medium is transversely isotropic and viscoelastic. We then perform harmonic numerical experiments to compute the stiffness components as a function of frequency, by using a Galerkin finite-element procedure, and obtain the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts) and quality factors. The algorithm is tested with the analytical solution and then used to obtain the stiffness components for general heterogeneous cases, where fractal variations of the fracture compliances and background stiffnesses are considered.Este documento tiene una corrección (ver documento relacionado).Facultad de Ciencias Astronómicas y Geofísica

Fracture-induced anisotropic attenuation

The triaxial nature of the tectonic stress in the earth's crust favors the appearance of vertical fractures. The resulting rheology is usually effective anisotropy with orthorhombic and monoclinic symmetries. In addition, the presence of fluids leads to azimuthally varying attenuation of seismic waves. A dense set of fractures embedded in a background medium enhances anisotropy and rock compliance. Fractures are modeled as boundary discontinuities in the displacement u and particle velocity v as [ κ · u + ν · v] where the brackets denote discontinuities across the fracture surface, j is a fracture stiffness, and g is a viscosity related to the energy loss. We consider a transversely isotropic background medium (e.g., thin horizontal plane layers), with sets of long vertical fractures. Schoenberg and Muir's theory combines the background medium and sets of vertical fractures to provide the 13 complex stiffnesses of the long-wavelength equivalent monoclinic and viscoelastic medium. Long-wavelength equivalent means that the dominant wavelength of the signal is much longer than the fracture spacing. The symmetry plane is the horizontal plane. The equations for orthorhombic and transversely isotropic media follow as particular cases. We compute the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts), and quality factors. The effective medium ranges from monoclinic symmetry to hexagonal (transversely isotropic) symmetry from the low-to the high-frequency limits in the case of a particle-velocity discontinuity (lossy case) and the attenuation shows typical Zener relaxation peaks as a function of frequency. The attenuation of the coupled waves may show important differences when computed versus the ray or phase angles, with triplication appearing in the Q factor of the qS wave. We have performed a full-wave simulation to compute the field corresponding to the coupled qP-qS waves in the symmetry plane of an effective monoclinic medium. The simulations agree with the predictions of the plane-wave analysis.Fil: Carcione, Jose M.. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale; ItaliaFil: Santos, Juan Enrique. Universidad de Buenos Aires. Facultad de Ingeniería. Instituto del Gas y del Petróleo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata; ArgentinaFil: Picotti, Stefano. Istituto Nazionale di Oceanografia e di Geofisica Sperimentale; Itali

On the static and dynamic behavior of fluid saturated composite porous solids: A homogenization approach

The macroscopic description of the dynamical behavior of a porous solid composed of two nonwelded solid phases saturated by a single-phase fluid is derived using two-space homogenization techniques for periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two solids are described by the linear elastic equations, and the fluid by the linearized Navier-Stokes equations, with appropriate boundary conditions at the solid-solid and solid-fluid interfaces. The nonwelded interface between the two solid phases is represented by displacement and/or velocity discontinuities proportional to the stresses across the interface, while the stresses are assumed to be continuous. After performing the homogenization procedure, constitutive relations, Darcy's and Biot's type dynamic equations for the saturated composite porous material are obtained.Facultad de Ciencias Astronómicas y Geofísica