A method is suggested to express the effective bulk modulus of the solid frame of a poroelastic material as a function of the saturated bulk modulus. This method enables effective Biot theory to be described through the use of seismic dispersion measurements or other models developed for the effective saturated bulk modulus. The effective Biot theory is generalized to a poroviscoelastic model of which the moduli are represented by the relaxation functions of the generalized fractional Zener model. The latter covers the general Zener and the Cole-Cole models as special cases. A global search method is described to determine the parameters of the relaxation functions, and a simple deterministic method is also developed to find the defining parameters of the single Cole-Cole model. These methods enable poroviscoelastic models to be constructed, which are based on measured seismic attenuation functions, and ensure that the model dispersion characteristics match the observations.

To simulate seismic wave propagation in natural porous media, the second-order Biot wave equations in the frequency domain are numerically solved with the (2-D) mixed-staggered grid nine-point stencil finite-difference method using both homogeneous and heterogeneous media approaches. A comparison of numerical results on three synthetic model examples shows that a significant discrepancy may be caused with the homogeneous formulation that ignores the spatial gradients of the medium properties. Such an approach, whilst previously applied, is not suitable to use for reflected/refracted wave investigations. A new and easier method is introduced to obtain the spatial derivative terms in the governing partial differential equation through a rotated coordinate system without using the derivative chain rule. This method is to directly rotate the vectors of the wave equations in the rotated coordinate system to the vectors in the classic coordinate system to constitute the wave equation in the mixed grid system. By using this new method, the exact same results are obtained as those with the classic method using the derivative chain rule. The calculated wavefields of solid particle velocity and fluid flux from the second-order Biot wave equation are used to calculate the pore (fluid) pressure through Biot's first-order constitutive equation. The fluid pressure wavefields are almost entirely devoid of all identified shear waves. Attention is also drawn to the fact that the current finite-difference, frequency-domain modelling methods, designed for the highest frequency and the slowest shear wave velocity in the model, could result in undersampling of the Biot slow P wave.

Many rocks and layered/fractured sequences have a clearly expressed electrical anisotropy although it is rare in practice to incorporate anisotropy into resistivity inversion. In this contribution, we present a series of 2.5-D synthetic inversion experiments for various electrode configurations and 2-D anisotropic models. We examine and compare the image reconstructions obtained using the correct anisotropic inversion code with those obtained using the false but widely used isotropic assumption. Superior reconstruction in terms of reduced data misfit, true anomaly shape and position, and anisotropic background parameters were obtained when the correct anisotropic assumption was employed for medium to high coefficients of anisotropy. However, for low coefficient values the isotropic assumption produced better-quality results. When an erroneous isotropic inversion is performed on medium to high level anisotropic data, the images are dominated by patterns of banded artefacts and high data misfits. Various pole–pole, pole–dipole and dipole–dipole data sets were investigated and evaluated for the accuracy of the inversion result. The eigenvalue spectra of the pseudo-Hessian matrix and the formal resolution matrix were also computed to determine the information content and goodness of the results. We also present a data selection strategy based on high sensitivity measurements which drastically reduces the number of data to be inverted but still produces comparable results to that of the comprehensive data set. Inversion was carried out using transversely isotropic model parameters described in two different co-ordinate frames for the conductivity tensor, namely Cartesian versus natural or eigenframe. The Cartesian frame provided a more stable inversion product. This can be simply explained from inspection of the eigenspectra of the pseudo-Hessian matrix for the two model descriptions.

Abstract We present a set of 2.5D synthetic inversion experiments for a model comprising an isotropic block embedded within an anisotropic background. We examine and compare the image reconstructions obtained using the correct anisotropic code and those obtained using code based on the inappropriate but widely adopted isotropic assumption. Superior reconstruction in terms of reduced data misfit, true anomaly shape and position, and anisotropic background parameters were obtained when the correct anisotropic code was employed for media characterized by moderate to high coefficients of anisotropy. However, for low coefficient values, the isotropic inversion produced slightly better results because there are fewer parameters to determine. When an erroneous isotropic inversion is performed on medium to high level anisotropic data, the images are dominated by patterns of banded artefacts and high data misfits.

The computational efficiency of 2.5-D seismic wave modelling in the frequency domain depends largely on the wavenumber sampling strategy used. This involves determining the wavenumber range and the number of the sampling points, and overcoming the singularities in the wavenumber spectrum when taking the inverse Fourier transform to yield the frequency-domain wave solution. In this paper, we employ our newly developed Gaussian quadrature grid numerical modelling method and extensively investigate the wavenumber sampling strategies for 2.5-D frequency-domain seismic wave modelling in heterogeneous, anisotropic media. We show analytically and numerically that the various components of the Green's function tensor wavenumber-domain solutions have symmetric or antisymmetric properties and other characteristics, all of which can be fully used to construct effective and efficient sampling strategies for the inverse Fourier transform. We demonstrate two sampling schemescalled irregular and regular sampling strategies for the 2.5-D frequency-domain seismic wave modelling technique. The numerical results, which involve calibrations against analytic solutions, comparison of the different wavenumber sampling strategies and validation by means of 3-D numerical solutions, show that the two sampling strategies are both suitable for efficiently computing the 3-D frequency-domain wavefield in 2-D heterogeneous, anisotropic media. These strategies depend on the given frequency, elastic model parameters and maximum wavelength and the offset distance from the source.

Elastic anisotropy of core samples

It has been shown that the spectral method (Trefethen 2000) and the spectral element method (Komatitsch and Tromp 1999) have more attractive features than the traditional finite difference (FD) and the finite element (FE) numerical methods used in resistivity modelling. The main advantages lie in the capability to simulate complex physical models and the exponential power convergence. They have been successfully applied to fluid flow dynamic modelling (Boyd 1989), seismic wave simulations (Komatitsch and Tromp 1999) and electromagnetic computations (Martinec 1999). The spectral method uses some global series of orthogonal functions to represent the unknown solution at the irregular collocation points, subject to boundary conditions. The resulting linear system matrix is full. The spectral element method combines the spectral method and the finite element method, and it possesses the main advantages of each. This includes the capability to handle various model shapes, the sparse matrix format of the FEM and the exponential power convergence of the spectral method. In a recent paper (Zhou et al., 2008) we presented the theory for a new resistivity modelling method based on Gaussian Quadrature Grids (GQG). It readily enables calculation of the electric potential in 2.5-D / 3-D heterogeneous, anisotropic models having arbitrary surface topography. The method co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it transforms the 2.5-D / 3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.

We present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the 'Gussian quadrature grid' (GQG) method, cooperatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method but does not employ the constant elements or require the mesh generator to match the Earth's surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. We show it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system that can be solved by an iterative or matrix inversion method.

In this paper we develop analytic solutions for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ l , dG/dσ t , dG/dθ0) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real-time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth.

We have developed explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and it is developed from a formal perturbation analysis and by means of a numerical (finite-element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry. The Fréchet derivative expressions were derived for the 2.5D and 3D problems using constant-point and constant-block model parameterizations. Special cases such as an isotropic earth and tilted transversely isotropic (TTI) media emerge from the general solutions. Numerical examples were investigated for various sensitivities as functions of dip angle and strike of the plane of stratification in uniform TTI media.

................................................................................................III Statement...............................................................................................VI Acknowledgements................................................................................VII Chapter

Introduction In this paper we present and analyse DC resistivity sensitivity patterns for uniform anisotropic media. The sensitivity functions (or Fréchet derivatives) give the responsive change in measured electric potential for a perturbation in a model parameter at a particular point in the subsurface for a specific electrode configuration. The anisotropic model investigated is the common tilted transversely isotropic medium (TTI) which is defined by four model parameters. We examine the changes in the Fréchet derivatives of the Green’s functions with respect to both the longitudinal and transverse conductivity and the dip and azimuth angle of the symmetry axis (dG/dσ1, dG/dσt, dG/dθ0, dG/dϕ0) for varying model parameters. Secondly, we wish to illustrate the differences that exist between the sensitivities calculated using an isotropic assumption and those computed with the correct anisotropic formulation. It is shown that in certain cases gross errors in inversion may occur if isotropic Fréchet derivatives are mistakenly used. Here we work with two special forms of the derivatives -one taken with respect to the mean conductivity and the other with respect to the magnitude of the anisotropy- and investigate a range of possible cases.

We present a new numerical scheme for 2D/3D direct current resistivity modelling. This method co-operatively combines the solution of the variational principle of the partial differential equation, Gaussian global quadrature abscissae and local cardinal functions so that it has the main advantages of the finite element method and the spectral method. The formulation shows that the method is close to the spectral element method, but it does not require the element mesh or the element integrations, and it makes it much easier to deal with geological models having a 2D/3D complex topography than the traditional numerical methods. It can achieve a similar convergence rate to the spectral element method. We show it transforms the 2D/3D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.