Patricia M. Gauzellino

SIMULATION OF WAVES IN PORO-VISCOELASTIC ROCKS SATURATED BY IMMISCIBLE FLUIDS. NUMERICAL EVIDENCE OF A SECOND SLOW WAVE

We present an iterative algorithm formulated in the space-frequency domain to simulate the propagation of waves in a bounded poro-viscoelastic rock saturated by a two-phase fluid. The Biot-type model takes into account capillary forces and viscous and mass coupling coefficients between the fluid phases under variable saturation and pore fluid pressure conditions. The model predicts the existence of three compressional waves or Type-I, Type-II and Type-III waves and one shear or S-wave. The Type-III mode is a new mode not present in the classical Biot theory for single-phase fluids. Our differential and numerical models are stated in the space-frequency domain instead of the classical integrodifferential formulation in the space-time domain. For each temporal frequency, this formulation leads to a Helmholtz-type boundary value problem which is then solved independently of the other frequency problems, and the time-domain solution is obtained by an approximate inverse Fourier transform. The numerical procedure, which is first-order correct in the spatial discretization, is an iterative nonoverlapping domain decomposition method that employs an absorbing boundary condition in order to minimize spurious reflections from the artificial boundaries. The numerical experiments showing the propagation of waves in a sample of Nivelsteiner sandstone indicate that under certain conditions the Type-III wave can be observed at ultrasonic frequencies.

Reflection and transmission coefficients of a single layer in poroelastic media

Wave propagation in poroelastic media is a subject that finds applications in many fields of research, from geophysics of the solid Earth to material science. In geophysics, seismic methods are based on the reflection and transmission of waves at interfaces or layers. It is a relevant canonical problem, which has not been solved in explicit form, i.e., the wave response of a single layer, involving three dissimilar media, where the properties of the media are described by Biots theory. The displacement fields are recast in terms of potentials and the boundary conditions at the two interfaces impose continuity of the solid and fluid displacements, normal and shear stresses, and fluid pressure. The existence of critical angles is discussed. The results are verified by taking proper limits-zero and 100% porosity-by comparison to the canonical solutions corresponding to single-phase solid (elastic) media and fluid media, respectively, and the case where the layer thickness is zero, representing an interface separating two poroelastic half-spaces. As examples, it was calculated the reflection and transmission coefficients for plane wave incident at a highly permeable and compliant fluid-saturated porous layer, and the case where the media are saturated with the same fluid.

Numerical electroseismic modeling: A finite element approach

Abstract Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nedelec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart–Thomas–Nedelec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.

NONCONFORMING FINITE ELEMENT METHODS FOR THE THREE-DIMENSIONAL HELMHOLTZ EQUATION: ITERATIVE DOMAIN DECOMPOSITION OR GLOBAL SOLUTION?

Iterative domain decomposition (DD) nonconforming finite element methods for the Helmholtz equation attempt to solve two problems. First, there exists no efficient algorithms able to solve the large sparse linear system arising from the discretization of the equation via the standard finite elements method. Secondly, even when DD methods generally yield small matrices, standard conforming elements, such as Q1 elements, force the transmission of a relatively large amount of data among subdomains. In this paper, we compared performance of global methods and a set of DD techniques to solve the Helmholtz equation in a three-dimensional domain. The efficiency of the algorithms is measured in terms of CPU time usage and memory requirements. The role of domain size and the linear solver type used to solve each local problem within each subdomain was also analyzed. The numerical results show that iterative DD methods perform far better than global methods. In addition, iterative DD methods involving small subdomains work better than those with subdomains involving a large number of elements. Properties of the iterative DD algorithms such as scalability, robustness, and parallel performance are also analyzed.

Waves in a fluid-saturated poroelastic matrix composed of two weakly coupled solids

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.

Waves in poroelastic solid saturated by a single-phase fluid

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.

Wave propagation in partially frozen porous media

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.

The macro-scale. Wave propagation in transversely isotropic media

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.

Solution of differential equations using the finite element method

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.

A poroelastic solid saturated by two immiscible fluids

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.