Graphene devices for analog and radio frequency (RF) applications are prone to low frequency noise (LFN) due to its up conversion to undesired phase noise at higher frequencies. Such applications demand the use of short channel graphene transistors (GFETs) that operate at high electric fields in order to ensure a high speed. Electric field is inversely proportional to device length and proportional to channel potential, so it gets maximized as the drain voltage increases and the transistor's length shrinks. Under these conditions though, short channel effects like velocity saturation (VS) should be considered. The reduction of LFN data due to the VS effect at short channel GFETs operating at high drain potential is for the first time shown in the present work. Carrier number and mobility fluctuations have been proven to be the main sources that generate LFN in GFETs. While their contribution to the bias dependence of LFN in long channels has been thoroughly investigated, the way in which VS phenomenon affects LFN in short channel devices under high drain voltage conditions has not been well understood. In this paper we have proposed a physics-based analytical LFN model that works under both low and high electric field conditions. The implemented model is validated with experimental data from CVD grown back-gated single layer GFETs operating at gigahertz frequencies. The model accurately captures the reduction of LFN especially near the charge neutrality point because of the effect of the VS mechanism. Moreover, an analytical expression for the effect of contact resistance on LFN is derived. This contact resistance contribution is experimentally shown to be dominant at high gate voltages and is accurately described by the proposed model. The noise parameter related to LFN at contacts is found to have an exponential dependence with contact resistance, and to our knowledge, this is shown for the first time.