Electric field and potential distribution in layered homogeneous borehole fluids and its logging responses of resistivity logging through casing

The resistivity logging through casing is an important logging instrument to carry out formation evaluation in the production wells. The transmission line equation (TLE) is a useful tool to solve the huge resistivity differences between metal casing and formation. Except for the casing, borehole fluids (e.g., mud, water, oil and gas) still could affect the logging responses of the resistivity logging through casing, which has not been studied well. Furthermore, the conventional TLE cannot be applied to investigate the current and potential distributions in borehole fluids. A suitable algorithm needs to be studied to calculate the influences of borehole fluids on logging responses. In this paper, we propose a modified TLE, which includes the formation information outside the casing and borehole fluids. Then, the analytical solution of the potential distribution on casing wall is derived using the modified TLE. For the infinite casing case, the current and potential distributions in borehole fluids are further derived based on the calculated potential distributions on casing wall and characteristic of the axial symmetry. In addition, the logging responses of resistivity logging through casing are calculated and compared. Through the numerical modeling, a new phenomenon is discovered that the voltage is low at the regions where casing contact with electrode and different borehole fluids. A microscopic qualitative explanation is given using the ion damping oscillation model of a reflective cavity to explain the existence of these low voltage regions. Furthermore, the contacts of borehole fluids are observed on the logging responses of resistivity through casing.