Transient modeling of one-dimensional solid-liquid flow in conduits

We consider the transient modeling of liquid-solid flow in conduits. Current models available in the literature are typically based on a three layer - two phase representation which, for the general transient case, results in a six-field model, entailing significant complexity in analysis and implementation. In this paper, we propose a transient three layer model represented by three equations (a third order PDE) by assuming a static momentum balance and then exploiting certain properties of the continuity relations. We illustrate how the transient model can be completed by introducing deposition-erosion closure relations from the literature, and describe its implementation by an upwind numerical scheme. Finally, some of the transient properties of the model are illustrated in six simulations comparing the simulated trends with six cuttings transport experiments found in the literature, motivated by application to horizontal drilling.