This dissertation focuses on a FFT based forward EM solver and its application in inverse problems. The main contributions of this work are two folded. On the one hand, it presents the first scaled lab experiment system in the oil and gas industry for through casing hydraulic fracture evaluation. This system is established to validate the feasibility of contrasts enhanced fractures evaluation. On the other hand, this work proposes a FFT based VIE solver for hydraulic fracture evaluation. This efficient solver is needed for numerical analysis of such problem. The solver is then generalized to accommodate scattering simulations for anisotropic inhomogeneous magnetodielectric objects. The inverse problem on anisotropic objects are also studied.
Before going into details of specific applications, some background knowledge is presented. This dissertation starts with an introduction to inverse problems. Then algorithms for forward and inverse problems are discussed. The discussion on forward problem focuses on the VIE formulation and a frequency domain solver. Discussion on inverse problems focuses on iterative methods.
The rest of the dissertation is organized by the two categories of inverse problems, namely the inverse source problem and the inverse scattering problem.
The inverse source problem is studied via an application in microelectronics. In this application, a FFT based inverse source solver is applied to process near field data obtained by near field scanners. Examples show that, with the help of this inverse source solver, the resolution of unknown current source images on a device under test is greatly improved. Due to the improvement in resolution, more flexibility is given to the near field scan system.
Both the forward and inverse solver for inverse scattering problems are studied in detail. As a forward solver for inverse scattering problems, a fast FFT based method for solving VIE of magnetodielectric objects with large electromagnetic contrasts are presented due to the increasing interest in contrasts enhanced full wave EM imaging. This newly developed VIE solver assigns different basis functions of different orders to expand flux densities and vector potentials. Thus, it is called the mixed ordered BCGS-FFT method. The mixed order BCGS-FFT method maintains benefits of high order basis functions for VIE while keeping correct boundary conditions for flux densities and vector potentials. Examples show that this method has an excellent performance on both isotropic and anisotropic objects with high contrasts. Examples also verify that this method is valid in both high and low frequencies. Based on the mixed order BCGS-FFT method, an inverse scattering solver for anisotropic objects is studied. The inverse solver is formulated and solved by the variational born iterative method. An example given in this section shows a successful inversion on an anisotropic magnetodielectric object.
Finally, a lab scale hydraulic fractures evaluation system for oil/gas reservoir based on previous discussed inverse solver is presented. This system has been setup to verify the numerical results obtained from previously described inverse solvers. These scaled experiments verify the accuracy of the forward solver as well as the performance of the inverse solver. Examples show that the inverse scattering model is able to evaluate contrasts enhanced hydraulic fractures in a shale formation. Furthermore, this system, for the first time in the oil and gas industry, verifies that hydraulic fractures can be imaged through a metallic casing.
