This paper presents the results of a numerical study of the optical mode confinement in whispering gallery mode disk nanocavities with hyperbolic dispersion using nanolayered hyperbolic metamaterial with different Al fill fractions. The fundamental properties of the optical modes and resonance frequencies for the disk nanocavities are studied using the numerical finite-element method. Numerical simulations show that light can be well confined in a disk nanocavity with a radius of up to an order of magnitude smaller than free-space resonant wavelength. This paper will also focus on how Purcell factor and quality factor of the disk nanocavities are affected by the fill fraction of the aluminum in the nanolayered metamaterial. Potential future applications for disk nanocavities with hyperbolic dispersion include silicon photonics optical communications networks, ultrafast LEDs, and biological nanoparticles sensing.