Seismic wavefield modeling and inversion requires large computational resources. For applications that involve modeling wavefields through many closely related models (i.e. time-lapse FWI, iterative RTM imaging, etc.) it is of interest to compute the response locally within a region where the model changes instead of computing the response of the entire model. By using the method of multiple point sources, the wavefields inside a local domain can be simulated to mimic the wavefields that propagate in the full domain. The localized wavefields are exact within numerical precision for both forward and adjoint wavefields. FWI gradients for velocity and density model parameters can then be built from the local wavefields to update the model parameters efficiently inside the local domain for a more accurate representation of the subsurface. The wavefields and gradient at each iteration are computed locally at a cost orders of magnitude lower than in the full domain, thus enabling more computationally efficient inversions. Using a synthetic example, we show that inversion inside a local domain is comparable to inversion in the full domain, but is obtainable at a significantly lower cost.
In oil and gas production environments, controlled-source electromagnetics can be used to aid brownfield exploration, development, and reservoir monitoring efforts. However, such environments typically have many highly conductive steel-cased wells in the area of interest. We have developed a modeling algorithm using a method of moments (MoM) approach to calculate the electromagnetic response of multiple 3D steel-cased wells of arbitrary geometry in a layered earth conductivity model. This approach involves dividing each casing into a collection of segments, each carrying a uniform current density. A matrix is computed that describes how the casing segments interact with each other electromagnetically. Then, we solve a linear system for the current within each casing segment, given a transmitter of arbitrary frequency and location. From these currents we are able to solve for the secondary electromagnetic fields produced solely by the casings at any point in our layered model, and we add these to the primary fields produced by the transmitter. To validate the algorithm, we compared results with a pseudoanalytic MoM algorithm for a single vertical casing in a half-space. We also compare results with a finite-element solution using Comsol Multiphysics for vertical and single tilted wells buried in various layered earth models. Our results indicate a good match between these different approaches, with tilted casings in a layered model requiring further study. Finally, we applied our algorithm to a realistic synthetic model with three casings (one vertical and two deviated) extending into a layered earth model containing the classic thin resistive layer. This example illustrates how the algorithm can be used to compute the electromagnetic response of multiple steel casings. The example also illustrates how the electromagnetic field changes due to the presence of the casings and how the casings may be used to inject the signal at depth.
A B S T R A C TThe electromagnetic response of a horizontal electric dipole transmitter in the presence of a conductive, layered earth is important in a number of geophysical applications, ranging from controlled-source audio-frequency magnetotellurics to borehole geophysics to marine electromagnetics. The problem has been thoroughly studied for more than a century, starting from a dipole resting on the surface of a halfspace and subsequently advancing all the way to a transmitter buried within a stack of anisotropic layers. The solution is still relevant today. For example, it is useful for one-dimensional modelling and interpretation, as well as to provide background fields for two-and three-dimensional modelling methods such as integral equation or primary-secondary field formulations. This tutorial borrows elements from the many texts and papers on the topic and combines them into what we believe is a helpful guide to performing layered earth electromagnetic field calculations. It is not intended to replace any of the existing work on the subject. However, we have found that this combination of elements is particularly effective in teaching electromagnetic theory and providing a basis for algorithmic development. Readers will be able to calculate electric and magnetic fields at any point in or above the earth, produced by a transmitter at any location. As an illustrative example, we calculate the fields of a dipole buried in a multi-layered anisotropic earth to demonstrate how the theory that developed in this tutorial can be implemented in practice; we then use the example to examine the diffusion of volume charge density within anisotropic media-a rarely visualised process. The algorithm is internally validated by comparing the response of many thin layers with alternating high and low conductivity values to the theoretically equivalent (yet algorithmically simpler) anisotropic solution, as well as externally validated against an independent algorithm.
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