First principle modeling of the lower hybrid (LH) current drive in tokamak plasmas is a longstanding activity, which is gradually gaining in accuracy thanks to quantitative comparisons with experimental observations. The ability to reproduce simulatenously the plasma current and the non-thermal bremsstrahlung radial profiles in the hard x-ray (HXR) photon energy range represents in this context a significant achievement. Though subject to limitations, ray tracing calculations are commonly used for describing wave propagation in conjunction with Fokker-Planck codes, as it can capture prominent features of the LH wave dynamics in a tokamak plasma-like toroidal refraction. This tool has been validated on several machines when the full absorption of the LH wave requires the transfer of a small fraction of power from the main lobes of the launched power spectrum to a tail at a higher parallel refractive index. Conversely, standard modeling based on toroidal refraction only becomes more challenging when the spectral gap is large, except if other physical mechanisms may dominate to bridge it, like parametric instabilities, as suggested for JET LH discharges (Cesario et al 2004 Phys. Rev. Lett. 92 175002), or fast fluctuations of the launched power spectrum or 'tail' LH model, as shown for Tore Supra (Decker et al 2014 Phys. Plasma 21 092504). The applicability of the heuristic 'tail' LH model is investigated for a broader range of plasma parameters as compared to the Tore Supra study and with different LH wave characteristics. Discrepancies and agreements between simulations and experiments depending upon the different models used are discussed. The existence of a 'tail' in the launched power spectrum significantly improves the agreement between modeling and experiments in plasma conditions for which the spectral gap is large in EAST and Alcator C-Mod tokamaks. For the Alcator C-Mod tokamak, the experimental evolution of the HXR profiles with density suggests that this model is valid up to a line-averaged density of (n) over bar (e) similar or equal to 1.0 x 10(+20) m(-3), a statement that is confirmed by simulations of the HXR scaling law with density. While simulations with GENRAY/CQL3D codes have ascribed the fast decrease of the HXR emission with density to parasitic absorption in the scrape-off layer by collisional damping, an alternative interpretetation based on an enhanced refraction as the LH wave propagates in the vicinity of the X-point is provided by C3PO/LUKE codes. The consequences for the predictions of LH current in ITER are discussed.