GeologyRocks

It depends on the driller and drilling site geology, but a lot of times drilling mud is made up of various materials that are not indigenous to the specific site. They use different clays for different texture, viscosity, etc. You could probably test for those.

Injecting some sort of dye might be a bad idea since it may take a lot of dye to show up (but then again it might not if the hole and volcano are close and there is a clear fracture between the two).

But I also wonder if these tests would be definitive since I think a case could be made that any fractures between the hole and volcano could be the result of the volcano material's pressure as it comes up and out. I'm sure that fractures surrounding rock.  Is there the same mud coming out of the well hole?  If not, then it would seem that there is no fracture connecting the two, so the two are unrelated.

How long after the drilling of the hole did the volcano start? I remember seeing pictures and short stories but don't remember much.

edit: just read that it's about 600 feet from the well and started soon after the punched through to the formation they were going for at something like 9000 feet down. But I don't know. It's hard to fault the drilling company for something they didn't know would happen, and really, how do you know it's even a possibility. You could say they were at fault for drilling into the unknown, but that's what all oil and even water well drillers do. They also said they had drilled something like 15-20 wells in the nearby area with no indication that this sleeping giant was down there. I don't know how you find fault with anybody except mother nature unless you can find specific evidence that the drillers knew that this reservoir of pressurized mud was down there and drilled anyway.

I suppose you're right in that the tests wouldn't be definitive, especially so long after the eruption began, but they'd still be useful to have. The well company still deny any physical link at all between their well and the eruption in the initial stages at least, so leakage of tracer in to the initial material would pretty much blow that out of the water. A total lack would absolutely absolve them. Perhaps it's too late for that though, unless some initial samples were preserved.

There is actually no mud at all coming out of the well as it was sealed off with a blowout preventer due to a well kick. A kick is not really out of the ordinary though so it's not possible to say for certain that this kick was the beginning of the mudflow, or indeed caused by the same fluid.

Perhaps they didn't know about the pressurised layers, but from what I have read, even though they are undetectable until drilled into, they are fairly commonplace and the very reason blowout preventers and well casings exist. The criticism of the well operators isn't that they hit these layers, but that they hadn't taken the nessecary precautions in case of such an event: their well casing wasn't deep enough and so the kick, or their attempts to kill it fractured the strata allowing the blowout. This should have been avoidable.

I've just realised this may be sounding very much like I believe they are to blame, so I'll just clarify that I am undecided but am discussing the scenarios proposed by others and echoing their sentiments. Personally I am slightly leaning towards a natural cause with the evidence that is available.

That's what I was unsure about, if maybe they neglected to do something they should have. They may have been lulled into thinking it wouldn't happen if their other wells in the area had no problems, but this is analogous to what causes malpractice lawsuits in medicine. They get to thinking "oh, the last 10 people that have come in here with these symptoms have had the flu, this guy surely just has the same so I'll just prescribe the same thing" instead of looking deeper or running all the tests, only to find out later that this 11th guy had something more seriously wrong.

However I still think you'd need something specifically linking any fractures caused by the blowout of the well to the mud flow. That may be difficult, as you have indicated. And even though the hole in the surface is 600 feet from the well hole, underground could be a whole different story. The source could be a foot from the well hole 9000 feet down, or it could be a mile away and just followed a number of convenient fractures, karst, bed planes, etc. to the output hole it has now.  Maybe all the holes they have drilled have contributed, releasing a little rock pressure with every hole or vibrations and other things associated with drilling causing unrest.  Then this final hole went over the tipping point.  Or... none of it may have anything to do with it.

Can they tell how deep the source of the mudflow is?  I'm sure that would be tough, but if they could conclude that the source is like 20,000 feet down, then I wouldn't think drilling to only 9,000 feet would have any affect.  Maybe taking temperature of the newly spouted mud and comparing it with geothermal gradient? 

The temperature was about 100 degrees c, which corresponds to about 1700m, corresponding to overpressured marine shales. The borehole was at nearly twice that depth when the event happened. Mud composition when compared with shale samples from the borehole suggests a depth of 1219-1828m. Analysing gas and water components suggests a mixed deep and shallow origin.