An orthogonal-plano-cylindrically layered (OPCL) medium consists of materials stratified planarly and layered concentrically in the orthogonal directions. The numerical mode matching (NMM) method has previously been shown to be a fast and robust semi-analytical solver to investigate the propagation of electromagnetic (EM) waves in such a complex but isotropic or transversely isotropic medium. In this paper, several important improvements have been made to extend applications of this efficient solver to the anisotropic OCPL medium. The formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application. The perfectly matched layer (PML) is incorporated along the radial direction as an absorbing boundary condition (ABC) to make the NMM method more accurate and efficient for unbounded low conductivity media and applicable to lossless media. We manipulate the weak form of Maxwell's equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem. Finally, we also offer formulas for computing EM fields excited by a magnetic dipole located at any position with an arbitrary orientation. Numerical results have demonstrated the efficiency and accuracy of this method.Index Terms-Numerical mode matching, electromagnetic field, stratified planarly and layered concentrically medium, anisotropic and inhomogeneous medium, perfectly matched layer, cylindrical coordinate singularity , magnetic dipole.

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The goal of this project is to develop techniques for monitoring hydraulic fractures in reservoirs by injecting electrically conductive, dielectric, or magnetically permeable proppants. The contrasts between the properties of the proppants and the subsurface provided the basis for imaging using geophysical methods. The initial experiments focused on a series of small, shallow fractures; however, the goal of the project is to develop methods applicable to oil-field fractures.The project began by screening different proppant types using laboratory and numerical analyses that have been ongoing by researchers at the Advanced Energy Consortium (AEC). This work identified Loresco coke breeze and steel shot as materials that could create significant electrical or magnetic contrasts with most geological formations. These proppants were tested by creating hydraulic fractures in a shallow field setting consisting of highly weathered residual saprolite near Clemson University in western South Carolina. Six hydraulic fractures were created in highly monitored cells by injecting the contrasting proppants at a depth of approximately 1.5 m. This created sub-horizontal fractures filled with proppant approximately 10 mm thick and extending 3 to 5 m in maximum dimension.Each cell had a dense array of electrodes and magnetic sensors on the surface, as well as four shallow vertical electrode arrays that were used to obtain data before and after hydraulic fracturing. Net vertical displacement, cores and trenching were used to characterize the fracture geometries.Hydraulic fracture geometries were estimated by inverting pre-and post-injection geophysical data using various codes. Data from cores and excavation show that the hydraulic fractures formed a saucer-shape with a preferred propagation in the horizontal direction. The geophysical inversions generated images with remarkably similar form, size, and location to the ground truth from direct observation. Displacement and tilt data appear promising as a constraint on fracture geometry.