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Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules, so the chase is not guaranteed to terminate. Several acyclicity conditions provide sufficient conditions for chase termination. In this paper, we present two novel such conditions—modelfaithful acyclicity (MFA) and model-summarising acyclicity (MSA)—that generalise many of the acyclicity conditions known so far in the literature. Materialisation provides the basis for several widely-used OWL 2 DL reasoners. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2 DL; furthermore, some systems go beyond OWL 2 RL, but they provide no termination guarantees. In this paper we investigate whether various acyclicity conditions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 DL ontologies are MSA, and that the facts obtained via materialisation are not too large. Thus, our results suggest that principled extensions to materialisationbased OWL 2 DL reasoners may be practically feasible.