The entry for the language at Языки мира: Кавказские языки. - М., 1999 has the following description:
The "проницающий" (apparently, the one under question, due to the identity of affixes described) case marks indirect objects or adverbials of verbs denoting the direct contact, penetration, several other verbs. It is marked by -x: ас хене-х тIотI кхацIдинас 'I grabbed the tree with a hand'; о алзне-х дахь валъẽ 'He crossed to the other bank of Alazani'; окхус хене-х дикI дишдиẽ 'He struck a tree with an axe'; о со-х гIази-х ва 'He is better than me'.
Yet the whole set of cases described in the article is somewhat different from one on Wikipedia, so some deviations might be expected.