Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time. The results show that the volume of the "hot" zone when the emplacement of a constant volume of hot sills is all done will vary as a matter of two things: the overall rate at which the magma is emplaced (duh) and the thickness of the sills of magma (they are all wafer- or cylinder-shaped). You can see from the attached figure that the volume shows a maximum and then decreases from that maximum as sills get shorter and taller. My intuition suggests this is because du/dx and du/dy shrink with the thinner sills until du/dz is the only heat transfer of note (at the peak, the ratio of radius to height is ~300:1). That means heat is only escaping in effectively in one direction and thus more heat is retained in the zone. I guess as the sills get "too" thin, they simply lose heat to quickly between sill-emplacement events to keep the heat in the zone. I need to develop this for a journal paper I am working on, so is this sounding like the plausible cause of the curve shapes in my figure, or might there be an effect I am missing or not taking into account?