A numerical analysis of the effects of coatings and gaps upon relative dielectric permittivity measurement with time domain reflectometry

Fluid-filled gaps or dielectric coatings around parallel-wire transmission lines affect the ability of time domain reflectometry (TDR) to measure the water content of soils and other porous materials. We use a steady state, two-dimensional, finite element numerical solution of Laplace's equation to analyze these effects. We prove that the numerically determined electrostatic potential distribution and boundary fluxes can be used to calculate the equivalent relative dielectric permittivity measured by TDR by comparing the results of the numerical model with those obtained using existing analytical solutions for special cases. We then analyze the effects of fluid-filled concentric gaps that completely or partially surround TDR rods. The results show that an analytical solution due to Annan [1977b] for nonconcentric gaps can be used as a good approximation to predict the effect of concentric gaps or coatings that completely surround the rods. Coatings or gaps filled with low relative dielectric permittivity materials have a greater impact on the measured relative dielectric permittivity than those filled with high dielectric media. An increase in the thickness of the gap or coating for given rod diameters and separations increases the impact of the coating. To a lesser degree, the impact of a coating of a given thickness decreases with an increase in the ratio of the rod diameter to the rod separation. A gap or coating of a given thickness and relative dielectric permittivity will have a greater impact on the response of a three-rod probe than on that of a two-rod probe with the same rod diameter and separation of the outermost rods. Partial air gaps surrounding less than 30 degrees of the rod circumference are not likely to affect the probe response significantly.