Introduction to rigorous theory of drilling
Existing theories of drilling are based on certain empirical relationships, giving the ground for concluding and forming equations to describe drilling process. Such theories are more descriptive and cannot be attributed to the rigorous theory. Nowadays there is no valuable rigorous theory of drilling with the results based on the solutions of differential equations of the model adopted. The article proposes a variant for developing such a rigorous theory of drilling. The basis is the theory of a rotating body immersion in the ground, created by the author, and a new dynamic model of the drill string. It is assumed that the upper section of the column is twisted at a constant angular velocity, and rotation of the cutter tool (drill bit) and the columns may alternate immersion with short stop (jamming). For the model proposed, in addition to the strength properties of rocks at the bottom, some points have been taken into account for the first time: changes in the torsional stiffness of the column with increasing its length in the process of deepening of the well; current bit wear; resistance force from the wash fluid and cuttings; the tension of the rope block and tackle system. Values, such as the mechanical speed and drilling depth, the angular velocity of the bit, the force and the moment of resistance on the part of the face are not determined by empirical correlations, but, quite rightly, as a result of solutions of differential equations for the model adopted. Reliability of the results obtained is determined by the choice of the dynamic model of the drill string and by the precision of solutions of differential equations found. There is a possibility of subsequent refinements of the theory introduced by means of studying more accurate dynamic models of the drill string. Numerical solution of differential equations of the model allow, in particular, to represent graphically the change in the mechanical drilling speed from the axial pressure on the bit and the frequency of rotation is based on existing fundamental theories of drilling, which is currently determined only empirically. The approach is proposed to determine the optimal pressure on the drill bit and the dimensions of the layout bottom of the drill string.