What is the nature of the sediment? If it has a big density difference from the liquid ultrasounds should work pretty well. If it is too similar to the liquid phase, you won't get much reflection. Laser range finding should also work with any wavelength for which the liquid is transparent. IR scatters less from fine particles, so if your liquid is slightly cloudy that might be better than visible, but you say the liquid is "clear", so I assume you can see the bottom with your eyes, in which case a red laser should be just fine.
Both ultrasound and lasers will have a large reflection from the top surface. You will need to exclude that signal. For ultrasound, the easiest way to do this is by time gating -- just ignore any signal at the receiver until after the initial reflection is detected. You will have to watch out for multiple bounces as well, but with a little trial-and-error this should work OK. This means you may not be able to use a plug-and-play module, since they usually sample only the first reflection.
One potential problem with ultrasound is that the speed of sound is temperature dependent. You may have to temperature correct your readings to get good enough accuracy.
With laser range finding, excluding the first reflection is potentially easier: just mount the sensor at a slight angle so that the top surface reflection misses the receiver, but this depends on the surface being flat enough for specular reflection. If you shine a laser pointer at the liquid and you see a lot of scatter from the liquid surface, you are going to have a problem. If you primarily see the spot on the sediment, an off-the-shelf laser range finder might work fine. In principle you can do time gating on a laser system too, but it is a lot more difficult since the round trim time is so small (a few ns).
One other approach to consider is to use a camera and do this with software image processing. If you can point a camera at the bottom corner of the barrel, you can just watch the sediment accumulate. This trades hardware simplicity for a lot more software work and calibration. If you could paint indicators on the inside wall of the barrel that would make this a lot easier, but I am assuming you can't do that. The problem here is that without reference marks, you are going to have a hard time compensating for the refraction of the liquid.