In emergency situations users of social networks convey all sorts of what have been called communicative intentions, well-known since the work of Austin (1962) and Searle (1969) as speech acts (SA). While speech acts have been the focus of close scrutiny in the philosophical and linguistic literature (see (Portner, 2018) for extended discussion), their role has been only rarely understood and exploited in processing social media content about crisis events, our focus here. Current work on communicative intentions in social media are topic-oriented, focusing on the correlation between SA and specific topics such as crisis (e.g., earthquakes) but also politics, celebrities, cooking, travel, etc. It has been observed that people globally tend to react to natural disasters with SA distinct from those used in other contexts (e.g., celebrities, which are essentially made up of comments). Here, we explore the further hypothesis of a correlation between different SA types and urgency and propose an in depth linguistic and computational analysis of communicative intentions in tweets from an urgency-oriented perspective. Indeed, SA are mostly relevant to identify intentions, desires, plans and preferences towards action and to ultimately produce a system intended to help rescue teams. Our contribution is four-fold and consists of: (1) A two-layer annotation scheme of speech acts both at the tweet and sub-tweet levels, (2) A new French dataset of about 13K tweets annotated for both urgency and SA, targeting both expected (e.g., storms) and unexpected or sudden (e.g., building collapse, explosion) events, (3) A thorough analysis of the annotations studying in particular the correlation between SA and the urgency of the message, SA and intentions to act categories (e.g., human damages), and SA and crisis types, finally, (4) A set of deep learning experiments to detect SA in crises related corpora. Our results show a strong correlation between SA and urgency annotations at both the tweet and sub-tweet levels with a particular salient correlation in the latter case, which constitutes a first important step towards SA-aware NLP-based crisis management on social media.