Analyzing the verb “to head”

This is both a usage question and a grammatical analysis question. I am familiar with complex transitive verbs, such as "to place", where one has to have at least one complement, besides the direct object, to make the phrase grammatical.

There is no such thing, apparently, as a complex intransitive verb. But then I find that I cannot analyze a sentence like "She headed up the drive to the service entrance." The verb "to head" (in this sense) clearly needs at least one adverbial phrase as complement, but it cannot take a direct object. So my first question would be: how do you call this kind of verb?

I also note that this sentence has two adverbial phrases after the verb, and I’m not sure how to analyze them. I suspect "to the service entrance" is a complement to "the drive" rather than to the verb phrase, mainly because I’ve never seen "head" with a preposition other than "for" in this context. Does this analysis make sense to you?