Discovered by (Austin,1962) and extensively promoted by (Searle, 1975), speech acts (SA) have been the object of extensive discussion in the philosophical and the linguistic literature, as well as in computational linguistics where the detection of SA have shown to be an important step in many down stream NLP applications. In this paper, we attempt to measure for the first time the role of SA on urgency detection in tweets, focusing on natural disasters. Indeed, SA are particularly relevant to identify intentions, desires, plans and preferences towards action, providing therefore actionable information that will help to set priorities for the human teams and decide appropriate rescue actions. To this end, we come up here with four main contributions: (1) A two-layer annotation scheme of SA both at the tweet and subtweet levels, (2) A new French dataset of 6,669 tweets annotated for both urgency and SA, (3) An in-depth analysis of the annotation campaign, highlighting the correlation between SA and urgency categories, and (4) A set of deep learning experiments to detect SA in a crisis corpus. Our results show that SA are correlated with urgency which is a first important step towards SA-aware NLP-based crisis management on social media.