Disclosed is a flow-rate measuring system for drilling muds and for multiphase mixtures (water/oil/gas) also with transport of solids or sand and in the presence of heavy oil. The system includes a flow-rate measuring system for drilling muds and/or for multiphase mixtures, wherein the measuring ports are equipped with an appropriate pre-chamber that enables elimination of the risk of failure of the measuring membranes in the case of particularly erosive mixtures and in the case of presence of solid by accumulation or transport. Furthermore, the particular vertical installation enables a compactness of the system in terms of horizontal encumbrance, enables installation where there is little horizontal space, and makes possible an undisturbed flow necessary for obtaining a higher measuring accuracy. The vertical installation enables installation, on the descending stretch, of a sensor for measuring the density that renders the measuring system autonomous (normally, the density is entered as external input).