Quantification of fluid losses at the topside is beneficial for early kick‐loss detection and automation of the drilling operation. A model‐based estimator is a useful tool for this purpose. The real‐time estimation of the amount of fluid losses with the cuttings removal could significantly help in this regard, especially for kick detection and automation. However, to the authors’ knowledge, there is no published literature on such attempts. Therefore, a simple dynamic mathematical model of the complete closed‐loop oil‐well drilling system is developed in this study for estimation of the fluid losses during the removal of drill cuttings at the topside, as well as for monitoring the flow of return fluid during drilling. Furthermore, this model could provide information about the topside fluid‐flow rates and fluid losses to other monitoring systems, such as kick‐ and loss‐detection systems and automation systems. The model is used to estimate both the mud‐pit level and the fluid losses during the removal of the drill cuttings through the solids‐removal equipment. The model‐order reduction of the flowline model using orthogonal collocation allows the model to be used in real‐time estimations and/or with control systems. It is simple, easy to implement, and, more importantly, shows the necessary dynamic behavior of both the bottomside and topside of a drilling operation simultaneously. The topside model can be used together with bottomside models of varying complexity to estimate both the bottomhole pressure and the fluid losses through the solids‐removal system.