The automatization of the drilling process opens the opportunity to faster reactions in case of unexpected drilling conditions, therefore reducing the risk that a drilling incident escalates to a serious situation. It also allows to push the drilling performance to be as close as possible to the limits of drillability as a function of the varying drilling conditions. But to achieve high level of drilling process automation, it is necessary to have access to accurate mathematical models of the complex physical system that is composed of the drilling rig, the drill-string, the drilling fluid and the borehole itself.

As the development of accurate heat transfer, mechanical and hydraulic models and their utilization in full scale drilling applications is a huge and complex task, it is tempting to recreate drilling automatization problems in a laboratory scale setup.

Because of sudden variations of the downhole drilling conditions, like when transitioning from soft to hard rock or when the bit is subject to large torque variations induced by interbedded rock layers, the boundary conditions at the bit change suddenly and require quick response from the automatic top-drive and hoisting system controllers. At a small laboratory scale, the necessary reaction times are of the order of milliseconds and therefore exclude any manual intervention. It is therefore crucial that the automatic control methods utilize precise mathematical models of the physical system to accurately estimate the limits by which the drilling process can be managed under safe conditions. For that reason, a general purpose mathematical model of small-scale drilling rigs has been developed.

First, the Rayleigh-Ritz method is used to determine the deflection of the drill-string and to estimate the side forces at the contact points along the drill-string and BHA (Bottom Hole Assembly). Then the dynamic response of the power transmission system is modelled for both variable frequency drive controlled tri-phase motors and for stepper motors, including friction effects at the contact points. Friction is modelled using Stribeck theory rather than the classical Coulomb laws of friction. Finally, the expected response of 3D accelerometers, that could be placed on the outside of a BHA component, is modelled to retrieve possible inclination variations and potential vibration modes such as torsional oscillations, forward or backward whirl.

The generality of the model is such that it can be used for many small-scale drilling rig designs.

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