Two-dimensional simulations of ambipolar or double-injection field-effect transistors (FETs) based on organic materials as active layer are presented in this article. These organic FETs are of interest because of the direct recombination accompanied by light emission. When modeling such FETs, one problem is that even advanced existing models based on the Pao-Sah description neglect both the actual contact properties and the recombination process. We present numerical simulations taking account these peculiarities. Current-voltage characteristics are used as reference data, recently obtained on an organic heterostructure ambipolar FET with a bottom Au contact and a Mg top contact [Rost et al., J. Appl. Phys. 95 5872 (2004)]. To investigate the basic ambipolar effects without the additional influences of the special structure, a single-layer model structure is considered in this simulation study. Simulated current-voltage characteristics indeed feature all specific characteristics arising from the ambipolar operation. Their origin becomes clear by inspecting the internal field and concentration profiles. The simulations reveal a complex interplay between the properties of the source and gate contacts, interface charges, the ratio of the electron and hole mobilities, doping, and recombination. Thereby, pronounced ambipolar operation depends sensitively on the combination of the barrier of the electron-injecting contact and the ratio of hole to electron mobilities. Possibilities and limitations of direct parameter extraction based on simple analytical expressions are demonstrated.