Complex Image Theory and Applications in Boundary Detection in Geo-Steering Using Data from a Directional Resistivity LWD Tool

Geo-steering is the process of controlling and adjusting the direction of the drilling bit in a horizontal or deviated well, in real time to keep the drilling in the desired zone. One of the most challenging steps of Geo-steering is the boundary detection, which is to calculate the distance from the bit to upper or lower boundary based on the measured data from an LWD tool. Generally, calculating the distance from bit to boundary is an inversion problem. To speed up the inversion process, a fast forward modeling algorithm is critical. In this study, the Complex Image Theory applied in finite conductivity layered media is derived to speed up the forward modeling of the geo-steering system. Two approximation results are shown in detail in dealing with the two general cases of dipole radiation. The first one is when the dipole is placed in the relative high resistive layer. The second one is when the dipole is placed in the relative high conductive layer. The algorithm is tested in both two-layer and three-layer cases and in high deviated well. Compared with the results from the full solution (the result from INDTRI), the Complex Image Theory has satisfactory accuracy and when the number of logging points is 600,000, it is 160 times faster. Error only exists in area two ft. or three ft. away from boundary. By considering the power of real source, the possibility of real application is investigated. The tolerance in different frequencies, spacing and conductivity combinations is discussed too. The simulation results show that the Complex Image Theory works in most geo-steering situations. The proposed method reduces the simulation time and improves the real-time performance of the control system. The distance inversion is developed for two-layer formation. The inversion results show that the algorithm works well even at the position 10 ft. away from the boundary. The anti-noise capacity of the propose method is measured by further involving random white noise in the simulation scenario. The relative error of simulation is as low as 5% in the area six ft. away from the boundary. With higher conductivity contrast formation, the proposed method is even more robust.