oh, you evil evil man.

you made me pull out my calculus book.
the pressure graph is roughly a line of equation: y= -(15/650)x + 15,
or y = -.023x + 15. or x = (15 - y)/.023
the flow rate for a 60ml/30sec double, is actually about 75ml/30sec (i measured it for the thermofilter design), or 150ml/min. if we assume that full brew pressure (whatever it might be) doesn't occur until 5 seconds after the pump is activated (the 'ramp up time'), then we have a much higher flow rate for those first 5 seconds. brew pressure is determined by the lesser value of puck resistance or relief valve setting.
total volume should equal low (ramp up) pressure volume plus high (brew) pressure volume.
Vt = Vr + Vb
flow rate at 9 bar is, as you noted, 250ml/min.
flow rate for ramp up (0 to 9 bar) is an average of 450ml/min (650 max, 250 min).
if ramp up is 5 seconds, then Vr is:
(450ml/min) / (60 sec/min) * 5 sec = 37.5ml, or half the total target volume.
the 9 bar flow rate with a 25 second flow period will pass 104.1ml, giving a total shot volume of 141.6ml. a very long double indeed.
now let's hold to that 75ml number as the target for total volume. given the figure for Vr above, then Vb would be the same.
(37.5ml) / (25 sec) * (60 sec/min) = 90ml/min
plug that into the line formula (or look it up on the graph) and this gives a brew pressure of about 13 bar.
i'm sure that someone with better math skills can work all that out into a function where we only have to plug in the variables to work out what the pressure and flow will be for any given volume/time relationship.
now, if the relief valve is set for 9 bar, then part of that copious flow will vent to drain (or reservoir). interesting, though, that half the shot volume is poured into/through the puck before the pump comes up to full pressure. i think, perhaps, preinfusion is 3 or 4 seconds, instead of the assumed 5. i hope.
--barry "possibly wrong"