I just graduated with an MS Geoscience and so my past projects are focused on soil identification, reservoir modelling, well and water testing etc. using a lot of Geology equipment and software.

I am trying to apply for the role of "Project Consultant" but it is not a Geoscience company. One of the requirements is an MS in (any) Natural Science or Engineering or Economics etc etc... They don't require any previous consulting experience.

The point is, my actual academic training doesn't seem to be relevant. They want people with good communication and organization skills and so on.

I completed A LOT of projects in undergrad and graduate school but my work experience is limited to teaching undergraduate class and part time clerical work to make ends meet while at University.

Some of these projects I was a team lead, and for others I was doing it independently (meaning I was the only researcher/investigator and I had an academic adviser or committee).

How do I list Geological projects and make it more general? The company is NOT going to care about the Geological equipment or software etc.

An example of one project I did was I used a radiometric dating technique to find the age of a lava flow in Canada. I did ALL the lab work got all the results, wrote a paper and presented it at the host University although I allegedly had an academic adviser.

How do I translate that into something that catches HR's interest?

Any helpful ideas or feedback would be appreciated!

Demonstrate major achievements - ideally what you think the company would look at think "we would like that here". Try and show how you worked very hard, led research etc
If your academic training is not relevant yet they are not looking for someone with previous consulting experience then your MSc ticks one of the boxes they want. Without knowing the role it's very tough to help further.
I can give you my opinion from an IT consultancy / hiring perspective - it is the norm for a an academic level to be included on a job but very rare that the hiring manager pays much attention to it. I do actually like to focus on it in an interview to see that the candidate enjoyed their research, what motivated them to do it, what they gained from it etc it is my way of gauging if a candidate has the attitude I am looking for.