An Improved Complex Image Theory for Fast Resistivity Modeling and Its Application to Geosteering

Summary azimuthal resistivity logging-while-drilling (LWD) tools with tilted antennas (Bittar 2002; Li et al. 2005) are widely used in geosteering because of their azimuthal sensitivity and the relatively large depth of investigation compared with other LWD tools such as nuclear, acoustic, or gamma ray measurements. Compared with conventional resistivity tools, azimuthal-resistivity LWD measurements can provide additional information including distance-to-boundary, relative dip angle, and resistivity anisotropy (Li et al. 2014). Because of the computing efficiency requirement, modeling and inversion of azimuthal-resistivity LWD measurements are usually based on a 1D parallel-layer model in practice (Zhong et al. 2008). Clearly, this 1D model assumption is not applicable to some realistic situations such as when the tool is navigating in unparallel-layer formation, or approaching a fault. The 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 that is based on iterative calls of forward modeling. An approximation method called complex image theory was proposed for geophysical prospecting (Wait 1969; Bannister 1986), and recently was introduced to well logging (Dong and Wang 2011; Wang and Liu 2014). 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 the 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 a very short distance from the tool to a bed interface as well as the scenarios in which the source is in a conductive bed instead of a resistive one. One can implement robust inversion schemes on the basis of this method. The effectiveness and efficiency of this method are verified by several numerical examples as well as laboratory tests and field jobs.