arise vs well up

arise and well up both are verbs.

Word	Noun	Adjective	Verb	Adverb
arise	No	No	Yes	No
well up	No	No	Yes	No

As verbs, well up is a hyponym of arise; that is, well up is a word with a more specific, narrower meaning than arise:

Other hyponyms of arise include resurge, come forth, emerge, come, follow, swell, head.

arise (verb)	well up (verb)
get up and out of bed	come up (as of feelings and thoughts, or other ephemeral things)
result or issue
rise to one's feet
move upward
take part in a rebellion; renounce a former allegiance
come into existence; take on form or shape
originate or come into being