ASEG Extended Abstracts - 3D Electromagnetics, 2003

This paper presents a 3-D finite-difference method (FDM) for analysis of electromagnetic measurement-while-drilling (MWD) tool response. The FDM is formulated with cylindrical coordinates, enabling accurate simulation of a cylindrical tool geometry. The algorithm allows the simulation of inhomogeneous media with arbitrary conductivity and magnetic permeability distributions. Material averaging is applied to both conductivity and magnetic permeability based on a mixture of harmonic and arithmetic averages. With the use of an irregular finite-difference grid, a complex MWD antenna with cavities and slots is accurately modeled. The system of linear equations is solved with the Lanczos decomposition method or the quasi-minimum-residual method, depending on the volume of output data. Our results show the 3-D FDM can accurately predict the tool response. It is shown that ferrite inserted in antenna slots increases the effective radiating length of the slots. The ferrite also helps reduce the eddy currents in the metal. However, it is not necessary to use ferrite with a relative magnetic permeability greater than 200 for the particular antenna geometry considered. For typical MWD tool spacings of 1 m or so, the field by a slotted antenna can be well approximated by that of a dipole antenna.

In an attempt to overcome long computer run times in direct calculation of resistivity parameter Jacobians in 3-D EM inversion, we combine forward calculations for the staggered grid finite difference (FD) method with Jacobians estimated from the forward results using the integral equations (IE) method. The second-order electric field formulation in the FD method yields the electric field everywhere in the earth efficiently. The IE formulation for the Jacobians utilizes the electric field just within the parameter of interest to define a Born term for the Jacobian, plus a depolarization term reflecting the incremental change in the electric field in all parameters. The amplitude of the depolarization decreases rapidly with distance from the parameter allowing a lumped cell approach to setting up the IE expression to be solved for the Jacobian. The lumped cells may grow geometrically from the parameter in accord with EM scaling, requiring only 100-200 cells to be considered for each parameter. Although this constitutes a sizeable pre-multiplier, the expression for computational cost of the total Jacobian set is only linearly proportional to the number of parameters.

We examine the relationship between the seven invariants of the complex MT tensor, which we previously proposed as a vehicle for testing the dimensionality of the regional conductivity structure prior to an analysis of MT data, and the three invariants of the real ‘phase tensor', recently introduced as an innovative aid in the treatment of MT data. It is found that the relevant invariants, and the necessary conditions on them for galvanically distorted data to be consistent with ID, 2D or 3D regional structures, agree in almost every detail for the two approaches. The new method does lead, however, to an improved normalisation of the eighth (dependent) invariant previously introduced. It is shown that the phase tensor can be expressed as a sum of three simple matrices, clearly associated with ID, 2D and 3D regional conductivity structures respectively. It is further shown that it can be depicted graphically as a single Mohr circle that retains the principal properties of the separate real and imaginary Mohr circles associated with the MT tensor. The simplicity and elegance of the phase tensor method is achieved by dispensing with the capability of distinguishing between galvanically distorted and undistorted data in ID and 2D regions, a distinction that is ultimately unimportant and unnecessary with real data. The paper concludes with a simple illustrative example of the theory applied to a real MT dataset from NE Australia. A shallow ID regional conductivity structure associated with a sedimentary basin is revealed, and a 2D anomaly with calculated strike angle is also identified.

In this paper, we present a Global Integral and Local Differential Equation (GILD) method for the magnetic field propagation in nanometer materials. A new 3D boundary integral equation for the magnetic field is proposed. We prove the equivalence between the new magnetic integral equation and the Galerkin equation for the magnetic field. We calculate the electromagnetic (EM) field in a nonlinear dielectric nanometer material. Simulations show that the GILD magnetic field modelling method in nanometer materials is accurate and fast. When extremely high frequency EM fields (Gamma ray) propagate through and interact with nanometer materials, we observe that there are EM numerical quanta in the GILD solution of EM scattering radiation field. The GILD magnetic field modelling is useful for EM engineering, geophysics and material sciences, and for investigating the physical and chemical properties and production in nanometer materials. The magnetic integral equation can be used to calculate the EM field in super conducting media.

We have developed a three-dimensional (3-D) resistivity inversion code for resistivity tomography, where resistivity data are measured using electrodes installed in several boreholes as well as at the earth surface. The algorithm is based on finite element approximations for forward modeling and an ACB (Active Constraint Balancing) algorithm is adopted to enhance the resolving power of the smoothness constrained least-squares inversion. Resolution analysis with numerical examples shows that 3-D resistivity tomography is a promising tool to get a high-resolution 3-D image of subsurface structures. We have also shown that the topographic effect should be incorporated in the inversion to get an accurate 3-D image without artefacts. In the application of the method to the field data set acquired at a granite quarry, we could successfully delineate the 3-D attitude of a fault or fracture at the site.

Moreover, we have shown that attributes of subsurface structures can be accurately defined by combining 3-D resistivity tomography and the borehole radar reflection method.

In regions where cross-beddings or faults in the earth are well developed, an isotropic earth might be an inadequate geophysical model. The parallel alignment of these structures results in the change of conductivity of the earth with the direction of the current flow (electrical anisotropy). Due to the huge memory and time requirements, general 3D isotropic modelling for these fine structures is practically impossible, so that some global parameters, e.g., the electrical anisotropy, can be very helpful in the interpretation of EM measurements over these structures.

In this paper, the EM field is represented by two scalar potentials, describing the poloidal and toroidal part of the magnetic field, for which I obtain two coupled ordinary differential equations in the vertical coordinate. To stabilize the numerical calculation, the wave number domain is divided into two parts. For small wave numbers, the EM field is continued in the anisotropic earth from layer to layer using the continuity conditions. For large wave numbers, the EM field is calculated by a Green’s function.

For seafloor EM exploration, where the transmitter and receiver (T-R) are usually positioned at the seafloor, the EM field is solved in the air half-space and in conductive seawater. At the bottom of the sea, they are connected to the field in the anisotropic earth. The apparent resistivities defined from the EM impedance are introduced to present the calculation results. Numerical experiments show that when the medium above the T-R is very conductive, as is the case of salt water, the earth anisotropy under the sea can only be explored at relatively high frequencies. At low frequency, the EM field concentrates in the conductive sea, so that the apparent resistivity reflects the true resistivity of the salt water. Polar plots of apparent resistivities at different frequencies can identify the anisotropic character of the earth, e.g., the principal anisotropic orientations.

Interpretation of niagnetotelluric data over inhomogeneous geological structures is still a challenging problem in geophysical exploration. We have developed a new 3-D MT inversion method and a computer code based on full nonlinear conjugate gradient inversion and quasi-analytical approximation for forward modeling solution. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. However, in order to control the accuracy of the inversion, our method allows application of the rigorous forward modeling in the intermediate steps of the inversion procedure and for the final inverse model. The 3-D niagnetotelluric inversion code QAINV3D based on QA approximation, has been tested on synthetic models and applied to the practical MT data collected in an area with complex geology. The inversion of 3-D MT data can be done within a few minutes on a PC to generate a 3-D image of subsurface formations on a large grid with tens of thousands of cells.