Vertical eigenstate method for simulation of induction and MWD resistivity sensors

A method for computing responses of induction or MWD (measurement-while-drilling) resistivity sensors in layered formations with invasion zones is presented. The geometry is assumed to have an axial symmetry resulting in a two-dimensional problem. The solution of the wave equation is expressed by a summation of eigenstates in the vertical direction in each cylindrical region. The eigenmodes in each region are found numerically, and the variation of fields in the horizontal direction is expressed in terms of analytical functions. Computed results are used to simulate logging tool responses in invaded formations. It is found that the apparent resistivity reading is a complicated function of the resistivities of the invaded zone, that of the uninvaded zone, bed thickness, and invasion diameter. Since no simple rule can be used to describe the characteristics of the investigation depth of the tool, the use of a computer simulation software, such as the one developed in this work, would be very useful to a log analyst in log interpretation