# Browsing by Subject "Domain Decomposition Method"

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Item Embargo Efficient Simulations of Electromagnetic Induction Tool in a Deviated Borehole for Resistivity Inversions(2022) Zhong, YangFor the petroleum industry, layered medium subsurface detection plays an important role in discovering reservoirs and drilling wells. In geophysics, resistivity is an essential property for distinguishing formation layers or even small fractures. Well logging with electromagnetic induction tools can measure the subsurface resistivity. This measurement includes two steps: 1) directly measure the low-frequency response signals using the tool and 2) determine the subsurface geometric model and resistivity. The problem is that no formula can directly calculate the resistivity from the measured tool responses. A systematic solution is to combine forward electromagnetic simulations and inversion of the subsurface model. In this dissertation, two categories of inversion are investigated: Determine the proper subsurface model by 1) optimizing the objective function, such as data misfit, and 2) training a surrogate model for the inverse mapping. Many forward simulations are demanded for either estimating the data misfit of new candidate models or collecting data for training. Therefore, efficient electromagnetic simulation is critical for resistivity logging. From complex to simple, three types of simulation are discussed: 1) borehole simulation with real tool configuration, 2) borehole simulation with point sources as the virtual tool, and 3) simplified layered medium simulation with virtual tool. Three optimal methods are implemented, respectively: the domain decomposition method, the finite element boundary integral method, and the analytical method. The tool calibration and the borehole effects are studied in the comparison of these simulations. Ideally, the simplest forward simulation should be used in the inversion, and the additional effects can be extracted as correction terms. The optimization-based inversion of the formation model uses simulations of a virtual tool in the layered medium. The Occam inversion or Monte Carlo Markov chain can minimize the data misfit. Another special simulation for small fractures using the thin dielectric sheet approximate method collects the dataset of fracture models. Fracture parameters such as resistivity, extension, and tilt angle are accurately determined by machine learning methods. The surrogate model also tends to predict fracture properties correctly, even for the complete simulation result.

Item Open Access Numerical Solution of Multiscale Electromagnetic Systems(2013) TOBON, LUIS E.The Discontinuous Galerkin time domain (DGTD) method is promising in modeling of realistic multiscale electromagnetic systems. This method defines the basic concept for implementing the communication between multiple domains with different scales.

Constructing a DGTD system consists of several careful choices: (a) governing equations; (b) element shape and corresponding basis functions for the spatial discretization of each subdomain; (c) numerical fluxes onto interfaces to bond all subdomains together; and (d) time stepping scheme based on properties of a discretized

system. This work present the advances in each one of these steps.

First, a unified framework based on the theory of differential forms and the finite element method is used to analyze the discretization of the Maxwell's equations. Based on this study, field intensities (E and H) are associated to 1-forms and curl-conforming basis functions; flux densities (D and B) are associated to 2-forms and divergence-conforming basis functions; and the constitutive relations are defined by Hodge operators.

A different approach is the study of numerical dispersion. Semidiscrete analysis is the traditional method, but for high order elements modal analysis is prefered. From these analyses, we conclude that a correct discretization of fields belonging to different p-form (e.g., E and B) uses basis functions with same order of interpolation; however, different order of interpolation must be used if two fields belong to the same p-form (e.g., E and H). An alternative method to evaluate numerical dispersion based on evaluation of dispersive Hodge operators is also presented. Both dispersion analyses are equivalent and reveal same fundamental results. Eigenvalues, eigenvector and transient results are studied to verify accuracy and computational costs of different schemes.

Two different approaches are used for implementing the DG Method. The first is based on E and H fields, which use curl-conforming basis functions with different order of interpolation. In this case, the Riemman solver shows the best performance to treat interfaces between subdomains. A new spectral prismatic element, useful for modeling of layer structures, is also implemented for this approach. Furthermore, a new efficient and very accurate time integration method for sequential subdomains is implemented.

The second approach for solving multidomain cases is based on E and B fields, which use curl- and divergence-conforming basis functions, respectively, with same order of interpolation. In this way, higher accuracy and lower memory consumption are obtained with respect to the first approach based on E and H fields. The centered flux is used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration.

Numerical examples and realistic cases are presented to verify that the proposed methods are non-spurious and efficient DGTD schemes.