Explosive basaltic eruptions pose significant threats to local communities, regional infrastructures and international airspace. They produce tephra plumes that are often associated with a lava fountain, complicating their dynamics. Consequently, source parameters cannot be easily constrained using traditional formulations. Particularly, mass flow rates (MFRs) derived from height observations frequently differ from field deposit-derived MFRs. Here, we investigate this discrepancy using a novel integral plume model that explicitly accounts for a lava fountain, which is represented as a hot, coarse-grained inner plume co-flowing with a finer-grained outer plume. The new model shows that a plume associated with a lava fountain has higher variability in rise height than a standard plume for the same initial MFR depending on initial conditions. The initial grain-size distribution and the relative size of the lava fountain compared to the surrounding plume are primary controls on the final plume height as they determine the strength of coupling between the two plumes. We apply the new model to the August 29, 2011 paroxysmal eruption of Mount Etna, Italy. The modeled MFR profile indicates that the field-derived MFR does not correspond to that at the vent, but rather the MFR just above the lava fountain top. High fallout from the lava fountain results in much of the erupted solid material not reaching the top of the plume. This material deposits to form the proximal cone rather than dispersing in the atmosphere. With our novel model, discrepancies between the two types of observation-derived MFR can be investigated and understood.