We have studied interface characteristics in a continuous fluid invasion model, first introduced by Cieplak and Robbins [Phys. Rev. Lett. 60, 2042 (1988)]. In this model, the interface grows as a response to an applied quasistatic pressure, which induces various types of instabilities. We suggest a variant of the model, which differs from the original model by the order of instabilities treatment. This order represents the relative importance of the physical mechanisms involved in the system. This variant predicts the existence of a third, intermediate regime, in the behavior of the roughness exponent as a function of the wetting properties of the system. The gradual increase of the roughness exponent in this third regime can explain the scattered experimental data for the roughness exponent in the literature. The growth exponent in this model was found to be around zero, due to the initial rough interface.