John Walker, thanks for taking an interest. I know of 2 quarries between Wouldham and Burham.
Margarets Pit has been a private landfill site, used by the the paper mills for as long as I remember, so no public access in 2017. (Now landscaped and no longer a landfill)
Peter's Pit has been a building site for some years, including 2017, so not there either.
I am not aware of any other quarry that meets your description.
My place is in North Kent rather than mid Kent.