Calculation of Tilted Coil Voltage in Cylindrically Multilayered Medium for Well-Logging Applications

Abstract

In cylindrical multilayered medium, we develop two tool models for electromagnetic (EM) well logging: a metal mandrel winding with a co-axial (tilted) transmitter coil and a tilted (coaxial) receiver coil. The voltages on receivers in those two models are proven to be the same and only zero-order harmonic of EM field need to be considered for voltage calculation. To calculate the voltage, two pseudo-analytical formulae are presented by using the integral of electrical field in spatial and wavenumber domain, respectively. Those two alternative pseudo-analytical formulae can be used to verify calculation accuracy of each other in some scenarios when other numerical methods do not work well. Furthermore, the reflection coefficients of EM fields, consisting of the ratios of the cylindrical functions, are introduced to avoid overflow problems in the numerical integral. Numerical examples corroborate the correctness and stability of the proposed formulae. These formulae help advance the forward modeling and inversion of logging-while-drilling (LWD) azimuthal resistivity measurements and new extra-deep azimuthal resistivity tools (EDAR). INDEX TERMS Directional antennas, receiving antennas, electromagnetic propagation in absorbing media, geophysical measurement techniques, well logging.

Study of the Electromagnetic Propagation in Stratified Anisotropic Media via Semi-Analytical Methods

“…, with {s,z} = 0 r{s,z} + iσ {s,z} /ω, (3)(4)(5)(6)(7)(8) both represented in cylindrical coordinates, where the subscript s = {ρ, φ}. The terms µ 0 = 4π × 10 −7 H/m and 0 = 8.854 × 10 −12 F/m are the values of permeability and permittivity in vacuum.…”

“…From equation 3-2we have that 13) projecting the componentẑ of the equation (3-13) ontoẑ and substituting in the equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12), we can find…”

“…The equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and are the homogeneous Helmholtz wave equations for E z and H z , respectively. In order to solve these equations in cylindrical coordinates, the method of separation of variables can be used [55].…”

“…with k z given by equation (4)(5)(6)(7)(8)(9)(10)(11). Validating the conditions (4-5) and (4)(5)(6)(7)(8) or with a similar procedure to the previous one, we can find for a source radiating for z ≶ 0 in general, the local reflection and transmission coefficients for the TM mode as,…”

“…The EM wave propagation in complex geophysical media has been studied for many authors [2,3,4,5] due to its importance to many areas of extractive industry (including mining and the petroleum exploitation) as well as the precise agriculture. The oil industry is one of the most prominent in the demand for novel solutions [6,7]. In special, there is a continuous development of sensors for profiling the soil to favor cost-effective oil wells.…”