Zhiru Yu

A Mixed-Order Stabilized Bi-Conjugate Gradient FFT Method for Magnetodielectric Objects

Magnetodielectric materials have many emerging applications, but their scattering problems are challenging for volume integral equation solvers. In this paper, a mixed-order stabilized bi-conjugate gradient FFT (mixed-order BCGS-FFT) method for solving the scattering problem of magnetodielectric objects is presented. This method uses the volumetric roof-top basis functions as testing functions for the coupled field volume integral equation (CFVIE) and basis functions for the electric and magnetic flux densities (D,B). However, unlike the conventional weak form BCGS-FFT methods for dielectric objects, the mixed-order BCGS-FFT method uses the second-order curl conforming basis functions for the electric and magnetic vector potentials (F,A). The curl conforming basis functions preserve the continuity of the tangential components for F and A, while the use of second-order basis functions avoids the zero terms caused by the divergence operations on the vector potentials. Numerical examples show excellent performance of the mixed-order BCGS-FFT method over the currently available BCGS-FFT method as well as high accuracy of the mixed-order BCGS-FFT method on large contrast magnetodielectric scatterers.

The Mixed-Order BCGS-FFT Method for the Scattering of Three-Dimensional Inhomogeneous Anisotropic Magnetodielectric Objects

This paper presents the first FFT-based fast volume integral equation solver for inhomogeneous anisotropic magnetodielectric objects. The volume integral equations are formulated by employing mixed-order basis functions that expand the flux densities and vector potentials in the coupled field integral equations in terms of different sets of basis functions with different orders. Volumetric roof-top basis functions are used for flux densities whereas second-order curl conforming basis functions are used for vector potentials. A fast volume integral equations solver namely the BCGS-FFT method is then applied to accelerate the solution of this mixed-order weak-form formulation. Examples show that the mixed-order BCGS-FFT method has high accuracy compared to both analytical and numerical solutions. Examples also show the mixed-order BCGS-FFT method has high computational efficiency compared to commercial software.

Through-Casing Hydraulic Fracture Evaluation by Induction Logging II: The Inversion Algorithm and Experimental Validations

Evaluation of hydraulic fractures has been under intensive study since the last decade. Among published works, only a few have included casing effects. This paper focuses on the through-casing electromagnetic (EM) induction imaging method that evaluates hydraulic fractures with enhanced contrasts. An experimental system and an inverse scattering algorithm are presented. The laboratory scaled experimental system is built for the feasibility study of the EM induction imaging method for the through-casing hydraulic fracture evaluation. To develop the inverse scattering algorithm, fractures outside boreholes with metallic casing are modeled by a novel hybrid approximation method. This method combines the distorted Born approximation and the mixed order stabilized bi-conjugate gradient fast Fourier transform method to solve the forward scattering problem. The variational Born iterative method is applied to solve the nonlinear inverse problem iteratively. Experimental results show that the inverse scattering algorithm is effective for EM contrast enhanced through-casing hydraulic fracture evaluation.

Through-Casing Hydraulic Fracture Evaluation by Induction Logging I: An Efficient EM Solver for Fracture Detection

Hydraulic fracturing is an essential way to improve the production of unconventional shale oil and gas. It is important to characterize the produced fractures using either acoustic or electromagnetic (EM) methods. Conventional EM solvers in the low-frequency range face significant challenges by such multiscale problems where the fracture width is orders of magnitude smaller than its diameters. Furthermore, the cased borehole environment is extremely difficult to simulate with conventional EM solvers due to meshing difficulties and the multiscale nature of the problem. In this paper, we develop a hybrid distorted Born approximation and 3-D mixed ordered stabilized biconjugate gradient fast Fourier transform (DBA-BCGS-FFT) method to simulate the very challenging 2-D, 2-D-axisymmetric, and 3-D hydraulic fracture models under both open and cased borehole environments. Numerical examples show that this method has orders of magnitude higher efficiency than the finite element method. The capabilities of the DBA-BCGS-FFT method for the induction tool fracture mapping are demonstrated by comparing with laboratory experimental results and other reference results.

Application of mixed order BCGS-FFT on contrast enhanced oil reservoir imaging

Summary form only given. With the rapid advancements in material science and manufacturing technology, applications of materials with special properties have been widely expanded. Some specially made nanoparticles can be used as contrast agents to change the contrast modalities (ε, μ, and σ) of the unknown objects in electromagnetic imaging problem to improve the imaging quality. Due to the potentials of the enhanced contrast imaging, many attempts have recently been made to apply this technique to various fields, for example oil reservoir imaging. In the application of the contrast enhanced oil reservoir imaging, nanoparticles with large permittivity, magnetic permeability and/or electrical conductivity are injected to the reservoir through the injection well. As the nanoparticles propagate into the reservoir, the three contrast modalities of the oil can be significantly changed depending on the fluid saturation rate. The enhanced contrasts of the oil can result in a stronger scattered signal which will lead to an enhancement in imaging result. However, the low operating frequency and large contrast modalities of the nanoparticles can be challenging for currently available electromagnetic field solvers for scattering problems.

Multiphysics Coupling of Dynamic Fluid Flow and Electromagnetic Fields for Subsurface Sensing

Summary form only given. Electromagnetic (EM) measurement has been extensively applied in subsurface sensing while fluid flow modeling is capable of characterizing subsurface fluid flow behavior. The multiphysics coupling of the EM measurement and dynamic fluid flow analysis has significant potential to improve electromagnetic geophysical exploration with injecting electromagnetic contrast agents.

Application of BCGS-FFT and distorted born approximation for hydraulic fracturing detection and imaging

With the ever increasing number of research on hydraulic fracture aiming at improved oil production, forward and inverse solvers based on electromagnetic method to detect and reveal properties of hydraulic fracture have become an important subject of research. Most of existing forward and inverse methods are developed to simulate the well logging model, such as Method of Moments (MoM) and Born Approximation. Those methods have the advantages to reconstruct the geometrical and electromagnetic information of formation. However, they are not fast enough and the memory cost are large. Moreover, when those methods are used to simulate hydraulic fractures, they are not able to obtain the accurate result.

Mixed order integral equation formulation for the scattering from large inhomogeneous anisotropic magnetodielectric objects

We propose a mixed order formulation of the combined field volume integral equation (CFVIE) for inhomogeneous anisotropic magnetodielectric objects. This method uses the Rao-Wilton-Glisson (RWG) basis functions to expand flux densities while second order curl conforming basis functions are used to expand vector potentials. In this case, both dielectric and magnetic contrasts of the objects can be accounted for in the CFVIE formulation. Moreover, correct boundary conditions are imposed on both flux densities and vector potentials. Finally, the stabilized bi-conjugate gradient fast Fourier transform (BCGS-FFT) method is employed to solve the integral equations. Numerical examples show that this method can handle large size objects with inhomogeneous or anisotropic magnetodielectric materials. The scattered fields calculated by this method have excellent agreement with both MIE solution and the results from commercial software.

Three-dimensional MR reconstruction of high-contrast magnetic susceptibility by the variational born iterative method based on the magnetic field volume integral equation

To provide high‐quality and high‐contrast magnetic susceptibility mapping, a 3D MR reconstruction method for magnetic susceptibility based on the magnetic field volume integral equation with the variational Born iterative method (VBIM) is developed.

Fast Simulation of Scattering Problem for Magnetodielectric Materials With General Anisotropy in Layered Media

In this paper, the mixed-order stabilized biconjugate gradient fast Fourier transform (mixed-order BCGS-FFT) method is presented to solve the scattering problem of magnetodielectric materials with general anisotropy in layered media. While the volumetric roof-top functions are used as the testing functions for the coupled field volume integral equation and the basis functions for flux densities, the second-order curl conforming basis functions are applied to expand the vector potentials with the aim of both preserving the continuity of their tangential components and avoiding the zero terms that might otherwise be caused by the divergence operator. The layered medium Greens function (LMGF) is efficiently evaluated through the recursive matrix method along with an interpolation technique. Several numerical experiments are presented to demonstrate the high accuracy and efficiency of the method. Different from the previously published works that aim to solve the similar problem, the new contribution of this work is to extend the mixed-order BCGS-FFT method to accommodate the layered background medium. Therefore, the 3-D FFT acceleration for integral kernels associated with the LMGF as well as the interpolation technique has been implemented and combined with the mixed-order BCGS-FFT method.