Referring to the ongoing Petroleum Engineering standards, the present paper aims to examine the operating reality of the mud pumping systems acting at the well site mainly while drilling. Particularly referring to the floating offshore rigs and installations (and to their constraints) some case histories are summarized as a start point, in order to remark both the fundamental energetic and strategic importance of the rig "pumping heart" while drilling and/or working over the well, and the possible implications upon the out coming risk standards and uncertainty degrees. According to nowadays machinery, devices and different regulations, risk matrix approach will be furthermore performed in order to check and to carefully point out what has still to be improved and the influence degree proper of each of the components of the global system.
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