Laminar Flow of Drilling Mud Due to Axial Pressure Gradient and External Torque

Abstract

Published in Petroleum Transactions, AIME, Volume 210, 1957, pages 310–317. Abstract Using three-dimensional, stress-deformation rate equations for a Bingham plastic, an approximate solution for the laminar flow of drilling mud between the drill pipe and casing is given for the case when the velocity gradients due to axial flow are large compared to velocity gradients due to rotation. Introduction The fact that drilling mud can be treated, to a first approximation, as a Bingham plastic has been established by several investigators. A Bingham plastic is a material that is rigid until a certain combination of stresses exceeds a critical value. When the stresses are sufficiently high to initiate flow, the reduced stresses have a constant ratio to the corresponding deformation rates at each point. Prediction of pressure and torque requirements to produce the desired flow of the material have been made by analytical solutions due to Bingham, Reiner, and Buckingham. One part of the present means of calculation that requires clarification is the flow between the drill pipe and casing. Present methods treat the flow resulting from the rotation of the drill pipe separate from the flow produced by the axial pressure gradient. Since the non-linear behavior of a Bingham plastic implies that solutions cannot be superposed, the present analytical investigation was undertaken to obtain a solution to the problem of combined axial and tangential flow. Equations have been proposed for the three-dimensional flow of a Bingham plastic and it has been shown that the proposed equations lead to the usual solutions from the one-dimensional theory. Applying these equations to the flow between two concentric cylinders the problem of axial flow only and tangential flow only have been solved. In addition to the separate problems of axial and tangential flow between concentric cylinders, the authors have determined the relationship between external torque on the inside cylinder and axial pressure gradient necessary to initiate flow. The solution given in Ref. 14 points out the complexity of the present problem.