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2016 | Buch

Waves in poroelastic solid saturated by a single-phase fluid Kapitel lesen Erstes Kapitel lesen

Numerical Simulation in Applied Geophysics

verfasst von: Juan Enrique Santos, Patricia Mercedes Gauzellino

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Geosystems Mathematics and Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents the theory of waves propagation in a fluid-saturated porous medium (a Biot medium) and its application in Applied Geophysics. In particular, a derivation of absorbing boundary conditions in viscoelastic and poroelastic media is presented, which later is employed in the applications.

The partial differential equations describing the propagation of waves in Biot media are solved using the Finite Element Method (FEM).

Waves propagating in a Biot medium suffer attenuation and dispersion effects. In particular the fast compressional and shear waves are converted to slow diffusion-type waves at mesoscopic-scale heterogeneities (on the order of centimeters), effect usually occurring in the seismic range of frequencies.

In some cases, a Biot medium presents a dense set of fractures oriented in preference directions. When the average distance between fractures is much smaller than the wavelengths of the travelling fast compressional and shear waves, the medium behaves as an effective viscoelastic and anisotropic medium at the macroscale.

The book presents a procedure determine the coefficients of the effective medium employing a collection of time-harmonic compressibility and shear experiments, in the context of Numerical Rock Physics. Each experiment is associated with a boundary value problem, that is solved using the FEM.

This approach offers an alternative to laboratory observations with the advantages that they are inexpensive, repeatable and essentially free from experimental errors.

The different topics are followed by illustrative examples of application in Geophysical Exploration. In particular, the effects caused by mesoscopic-scale heterogeneities or the presence of aligned fractures are taking into account in the seismic wave propagation models at the macroscale.

The numerical simulations of wave propagation are presented with sufficient detail as to be easily implemented assuming the knowledge of scientific programming techniques.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Waves in poroelastic solid saturated by a single-phase fluid
Abstract
This chapter contains the derivation of Biot’s theory describing the propagation of waves in a porous elastic solid saturated by a single-phase fluid. After deriving the constitutive relations and the form of the potential and kinetic energy densities and the dissipation function, the lagrangian formulation of the equations of motion is given. Next, a plane wave analysis is performed showing the existence of two compressional waves and one shear wave. An example showing the behaviour of all waves as function of frequency for a sample of Nivelsteiner sandstone saturated by water, oil and gas is included.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 2. A poroelastic solid saturated by two immiscible fluids
Abstract
The derivation of Biot’s theory presented in Chapter 1 assumed a singlephase fluid. The case of a porous solid saturated by a two-phase fluid requires a generalized argument due to the presence of capillary pressure forces. Here capillary forces are included in the wave propagation model using a Lagrange multiplier in the virtual complementary work principle, leading to the derivation of the constitutive relations. Following the ideas given in Chapter 1, the potential and kinetic energy and dissipation functions are derived to obtain the lagrangian formulation of the equations of motion. In particular, the dissipation function is determined considering two-phase fluids and two-phase Darcy’s law. A plane wave analysis shows the existence of three compressional waves, denoted as P1, P2 and P3, and one shear wave. A numerical example is given showing the behaviour of all waves as function of saturation and frequency for a sample of Nivelsteiner sandstone saturated by either oil-water or gas-water, water being the wetting phase.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 3. A poroelastic solid saturated by a three-phase fluid
Abstract
A theory to describe the propagation of waves in a poroelastic solid saturated by a three-phase fluid is presented and anlyzed. The model takes into account the existence of two capillary and three relative permeability functions, assumed to depend on the non-wetting fluid saturations. A plane wave analysis predicts the existence of four compressional waves, denoted as P1, P2, P3 and P4, and one shear wave, all of them suffering attenuation and dispersion effects. A numerical example showing the behaviour of all waves as function of saturation and frequency in a sample of Nivelsteiner sandstone saturated by water, oil and gas is presented.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 4. Waves in a fluid-saturated poroelastic matrix composed of two weakly coupled solids
Abstract
This chapter presents a theory to describe wave propagation in a porous medium composed of two weakly coupled solids saturated by a single-phase fluid. The model is useful in realistic situations such as seismic wave propagation in permafrost or shaley sandstones. The constitutive relations are derived from the virtual work principle, allowing to determine the generalized forces of the system and the constitutive relations, given here for the isotropic case. The coefficients in the constitutive relations are defined for the cases of shaley sandstones and permafrost. After stating the equations of motion, including dissipative effects, a plane wave analysis reveals the existence of three compressional waves and two shear waves. The theory is applied to determine phase velocities of the fast compressional and shear waves in a sample of shaley sandstone, which are shown to fit experimental data.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 5. Absorbing boundary conditions in elastic and poroelastic media
Abstract
In applied geophysics, it is usually needed to simulate the propagation of waves in a bounded region of the subsurface. This requires the use of absorbing boundary conditions at the artificial boundaries of the subsurface model so that spurius reflections be mimimized. This Chapter presents the derivation of this type of absorbing boundary conditions in elastic and poroelastic media. They will be used later in the Chapters dedicated to simulate wave propagation phenomena.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 6. Solution of differential equations using the finite element method
Abstract
The finite element method (FEM) is a useful tool to solve boundary value problems of interest in applied geophysics. 1-D finite element spaces are first defined and analyzed. The concept of continuous and discrete weak solutions is introduced and a priori error estimates are stated. The FEM is used to solve wave propagation problems and to characterize fine layered media in the 1-D case. Next, 2-D and 3-D conforming and non-conforming finite element spaces and defined over partitions of a bounded domain into triangular or rectangular elements in 2-D and tetrahedral or 3-rectangular elements in 3-D. These finite element spaces are used in the following Chapters to represent solid or fluid vector displacements in the boundary value problems to be formulated and solved using the FEM.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 7. Modeling Biot media at the meso-scale using a finite element approach
Abstract
Fast compressional or shear waves travelling through a fluid-saturated porous material (a Biot medium) containing heterogeneities on the order of centimeters (mesoscopic scale) suffer attenuation and dispersion observed in seismic data. The mesoscopic loss effect occurs because different regions of the medium may undergo different strains and fluid pressures. This in turn induces fluid flow and Biot slow waves causing energy losses and velocity dispersion. Numerical modeling of wave propagation using Biot’s equations of motion takes into account this phenomenon but is computationally very expensive. In the context of Numerical Rock Physics, this chapter presents compressibility and shear time-harmonic experiments to determine an equivalent viscoelastic medium having in the average the same attenuation and velocity dispersion than the original highly heterogeneous Biot medium. Each experiment is associated with a boundary-value problem that is solved using the finite element method.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 8. The meso-scale. Fractures as thin layers in Biot media and induced anisotropy
Abstract
A fluid-saturated poroelastic isotropic medium with aligned fractures behaves as a transversely isotropic and viscoelastic (TIV) medium when the predominant wavelength is much larger than the average distance between fractures. Here fractures are modeled as extremely thin and compliant porous layers. P-waves travelling in this type of medium generate fluid flow and slow (diffusion) Biot waves, causing attenuation and dispersion of the fast modes (mesoscopic loss). This chapter presents a set of compressibility and shear time-harmonic experiments on highly heterogeneous fractured poroelastic samples to determine the five complex and frequency dependent stiffnesses characterizing the equivalent TIV medium. These experiments are identified with boundary-value problems which solutions are computed using the finite element procedures.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 9. Fractures modeled as boundary conditions in Biot media and induced anisotropy
Abstract
The acoustic response of fractures within a Biot medium is modeled using boundary conditions imposing relations among displacements, stresses and fluid pressure along fractures. Besides, a Biot medium with a dense set of aligned fractures behaves as a transversely isotropic and viscoelastic (TIV) medium when the average fracture distance is much smaller than the predominant wavelength of the travelling waves. The stiffness coefficients of an equivalent TIV medium to a Biot medium with aligned fractures are determined using a set of boundary-value problems formulated in the space-frequency domain at the continuous level, which are solved using the finite element method.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 10. The macro-scale. Seismic monitoring of CO2 sequestration
Abstract
Storage of carbon dioxide (CO2) in deep saline aquifers and aging oil reservoirs is a valid alternative approach for reducing the amount of greenhouse gases in the atmosphere. This procedure can be modeled by the combined use of multi-phase fluid flow and wave propagation. The flow simulator is used to model the CO2 injection, and seismic monitoring is applied to determine the spatiotemporal distribution of CO2 after several years of injection. The simultaneous flow of brine and CO2 is modeled with the Black-Oil formulation for two-phase flow in porous media, while wave propagation is formulated using an isotropic viscoelastic model. The CO2 saturation and pressure data computed with the flow simulator is used in a petrophysical model to determine the complex bulk and shear moduli of the formation. Wave propagation is performed using a finite element domain decomposition procedure. The simulation is used to model CO2 injection and flow and compute time-lapse seismograms corresponding to the Utsira aquifer at Sleipner field with the objective of identifying the spatio-temporal distribution of CO2 after injection.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 11. Wave propagation in partially frozen porous media
Abstract
The propagation of waves in a fluid-saturated poroelastic medium which matrix is composed of two weakly coupled solids is simulated using an iterative finite element domain decomposition algorithm. The equations of motion are formulated in the space-frequency domain including dissipation in the solid matrix and frequency correction factors in the mass and viscous coupling coefficients. First order absorbing boundary conditions are employed at the artificial boundaries of the computational domain. The algorithm is applied to simulate wave propagation in a sample of partially frozen Berea sandstone at ultrasonic frequencies.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Chapter 12. The macro-scale. Wave propagation in transversely isotropic media
Abstract
The propagation of seismic waves in a fluid-saturated poroelastic medium containing a dense set of aligned fractures is simulated using a non-conforming finite element (FE) domain decomposition procedure. The macroscopic properties of this fractured medium are determined using the set of time-harmonic up-scaling experiments developed in Chapter 8, with the fractures modeled as extremely thin, highly permeable and compliant porous layers. This approach yields a complex and frequency dependent stiffness matrix defining an equivalent transversely isotropic viscoelastic (TIV) medium at the macro-scale. The FE procedure to simulate wave propagation in TIV media is completely analogous to the one explained in Chapter 10 for isotropic viscoelastic media. Fracture induced anisotropy and the influence of different fluids filling the fractures are analyzed in the numerical examples.
Juan Enrique Santos, Patricia Mercedes Gauzellino
Backmatter
Metadaten
Titel
Numerical Simulation in Applied Geophysics
verfasst von
Juan Enrique Santos
Patricia Mercedes Gauzellino
Copyright-Jahr
2016
Electronic ISBN
978-3-319-48457-0
Print ISBN
978-3-319-48456-3
DOI
https://doi.org/10.1007/978-3-319-48457-0

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