Once a mineral estate has been severed from the surface estate on a tract of land, that mineral estate becomes a fee simple estate in and of itself. This means that it's not subject to abandonment and also receives, as the authors of Texas Law of Oil and Gas Second Edition note, the benefit of the doctrine of accretion (the gradual buildup of land from water). Moreover, you can receive ad valorum taxes separate from what's imposed on the surface estate and contracts regarding the mineral estate must comply with the Statute of Frauds, et al.
Because it's a fee estate, the owner of the mineral rights has the same basic rights that any other fee owner has, including the right to use, convey, and dispose of the property, as well as creating lesser estates (such as a Life Estate).
Adverse Possession is the next section, but it takes up a decent amount of space, so I'll address that another time.