Dielectric dispersion model including polarization response of a shape distribution

Formation evaluation based on the complex dielectric dispersion in the megahertz to gigahertz frequency range has become a useful method for obtaining formation properties with the development of several logging tools. Such an evaluation relies on fitting the measured dielectric signals to a dielectric dispersion model and inverting for model parameters directly associated with certain formation petrophysical properties: the water-filled porosity, the brine salinity, and a water-phase tortuosity exponent. Hence, a model that can faithfully describe the dielectric dispersion governed by those formation properties is of importance for such a practice. By incorporating the polarization response of grains with a distribution of shapes, we have improved upon a widely accepted dielectric dispersion model, the bimodal model. To efficiently approximate the overall polarization response of ensembles of grains with continuous shape distributions, we establish two approaches that use a finite number of grain shapes: a Padé approximation and samplings with a Gauss quadrature rule. The latter method provides greater flexibility in what type of shape distribution can be introduced and how they are introduced. By adopting a specific class of shape distributions sampled with a Gauss quadrature rule, a new dielectric dispersion model for brine-saturated rocks is developed using a differential effective medium approach. The new model has the same number of model parameters and should have similar use cases, clay-free clastic and carbonate formations without complex pore structures, as those of the bimodal model. By fitting our model to available core measurements, the new model better describes the dielectric dispersion of brine-saturated rocks over a broader frequency range in comparison to the bimodal model, while giving good inverted petrophysical parameters. Based on the scaling behavior of the Maxwell-Wagner polarization, the new model would be more adequate for describing the dielectric dispersion of rocks saturated with higher salinity brine.