An Improved Complex Image Theory for Fast 3D Resistivity Modeling and Its Application to Geosteering in Unparallel Layers

Abstract

Abstract Azimuthal resistivity logging-while-drilling (LWD) tools are widely used in geosteering because of their azimuthal sensitivity and depth of investigation compared to those of other LWD tools such as nuclear, acoustic, or gamma ray measurements. Compared to conventional resistivity tools, azimuthal resistivity LWD measurements can provide additional information including distance to bed interface, relative dip, and resistivity anisotropy. Because of the computing efficiency requirement, modeling and inversion of azimuthal resistivity LWD measurements are usually based on the 1D parallel layer model: i.e., all bed interfaces are infinitely large and parallel to each other. Clearly this 1D model assumption does not apply to some realistic situations such as when the tool is navigating in an unparallel layered formation, or approaching a fault. 3D full wave simulations such as finite difference or finite element methods can handle the complex cases, but they are generally too slow for real jobs, not to mention the inversion based on forward modeling. An approximation method called "complex image theory" was proposed for geophysical prospecting and recently introduced to well logging. This theory approximates electromagnetic wave reflection by an interface between local and adjacent beds as signal radiated from a virtual source with a complex distance from the observation point. The complex image theory can be several orders of magnitude more efficient than 3D simulations. However, it also has several limitations: This method only works in resistive beds with conductive shoulders, and measurement cannot be too close to bed interfaces. Those shortcomings greatly limit this method to more extensive applications. An improved complex image theory is proposed here to tackle the aforementioned difficulties. This improved theory can handle finite-sized interfaces, any distance from the tool to a bed interface, and the scenarios where the source is in a conductive bed instead of a resistive one. An efficient and robust inversion scheme is also implemented based on the improved complex image theory. This proposed method can greatly facilitate formation evaluation and decision making of geosteering in complex scenarios. Synthetic data, laboratory experiments, and field jobs are shown here to demonstrate the effectiveness of this improved complex image theory.