An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homoge We have developed a finite-difference solution for neous Dirichlet condition is employed along the subsur three-dimensional (3-D) transient electromagnetic face boundaries. Numerical dispersion is alleviated by problems. The solution steps Maxwell's equations in using an adaptive algorithm that uses a fourth-order dif time using a staggered-grid technique. The time-step ference method at early times and a second-order meth ping uses a modified version of the Du Fort-Frankel od at other times. Numerical checks against analytical, method which is explicit and always stable. Both integral-equation, and spectral differential-difference so conductivity and magnetic permeability can be func lutions show that the solution provides accurate results. tions of space, and the model geometry can be arbi Execution time for a typical model is about 3.5 trarily complicated. The solution provides both elec hours on an IBM 3090/600S computer for computing tric and magnetic field responses throughout the earth. the field to 10 ms. That model contains 100 x 100 x 50 Because it solves the coupled, first-order Maxwell's grid points representing about three million unknowns equations, the solution avoids approximating spatial and possesses one vertical plane of symmetry, with derivatives of physical properties, and thus overcomes the smallest grid spacing at 10 m and the highest many related numerical difficulties. Moreover, since resistivity at 100 n . m. The execution time indicates the divergence-free condition for the magnetic field is that the solution is computer intensive, but it is valu incorporated explicitly, the solution provides accurate able in providing much-needed insight about TEM results for the magnetic field at late times. responses in complicated 3-D situations.

In this paper, we present a nonsplitting perfectly matched layer (NPML) method for the finite‐difference simulation of elastic wave propagation. Compared to the conventional split‐field approach, the new formulation solves the same set of equations for the boundary and interior regions. The nonsplitting formulation simplifies the perfectly matched layer (PML) algorithm without sacrificing the accuracy of the PML. In addition, the NPML requires nearly the same amount of computer storage as does the split‐field approach. Using the NPML, we calculate dipole and quadrupole waveforms in a logging‐while‐drilling environment. We show that a dipole source produces a strong pipe flexural wave that distorts the formation arrivals of interest. A quadrupole source, however, produces clean formation arrivals. This result indicates that a quadrupole source is more advantageous over a dipole source for shear velocity measurement while drilling.

Electric anisotropy is considered an important property of hydrocarbon reservoirs. Its occurrence has great influence on estimation of formation water saturation and other properties derived from electromagnetic (EM) measurements. Conventional tools using coaxial coils often underestimate formation resistivity and thus overestimate water saturation. Multicomponent EM sensors provide the additional information needed for better resistivity‐based formation evaluation. We have developed a finite‐difference method to simulate multicomponent EM tools in a 3‐D anisotropic formation. The new method can model inhomogeneous media with arbitrary anisotropy. By using the coupled Maxwell’s equations, our method consumes about the same computational time to model an anisotropic formation as it would take to model an otherwise isotropic formation. We have verified the finite‐difference method using layered‐earth models that are typically encountered in hydrocarbon exploration and development. Our results show that the newly developed simulation algorithm produces accurate results for different borehole and formation conditions.

Inversion of three‐dimensional transient electromagnetic (TEM) data to obtain electrical conductivity and permeability can be done by a time‐domain algorithm that extends to diffusive electromagnetic (EM) fields the imaging methods originally developed for seismic wavefields (Claerbout, 1971; Tarantola, 1984). The algorithm uses a conjugate‐gradient search for the minimum of an error functional involving EM measurements governed by Maxwell's equations without displacement currents. The connection with wavefield imaging comes from showing that the gradient of the error functional can be computed by propagating the errors back into the model in reverse time and correlating the field generated by the backpropagation with the incident field at each point. These two steps (backpropagation and cross correlation) are the same ones used in seismic migration. The backpropagated TEM fields satisfy the adjoint Maxwell's equations, which are stable in reverse time. With magnetic field measurements the gradient of the error functional with respect to conductivity is the cross correlation of the backpropagated electric field with the incident electric field, whereas the gradient with respect to permeability is the cross correlation of the backpropagated magnetic field with the time derivative of the incident magnetic field. Tests on two‐dimensional models simulating crosswell TEM surveys produce good images of a conductive block scatterer, with both exact and noisy data, and of a dipping conductive layer. Convergence, however, is slow.

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