Analysis of Drillstrings in Curved Boreholes
Authors: Fischer F. Joseph 
Published: December 2007

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.
This paper was prepared for the 49th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Texas, Oct. 6–9, 1974. Permission to copy is restricted to an abstract of not more than 300 words. illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
An analysis of the static deformation of drillstrings in two-dimensional curved boreholes is presented. A drillstring is imagined to consist of a finite number of constant-property beam-columns. The borehole axis lies in a single vertical plane and is assumed to be a smooth curve comprised of circular arcs and straight-line segments. The borehole diameter can vary along the borehole axis.
The associated mathematical model is embodied in a computer program, SCHADS (Static Curved Hole Analysis of Drill Strings). In addition to the deformation of the drillstring, the program also yields the locations of contact between the drillstring elements and the borehole walls, the contact forces, the forces and moment within the drillstring, and the resultant force which the bit exerts on the formation being drilled.
The program is intended as a "tool" for the drilling engineer with which he may assess the relative merits of many candidate drillstring hookups, weights-on-bit, etc., which might be considered as reasonable choices to accomplish a given drilling objective. In particular, remedies for unwanted hole deviation and casing wear arising from large string - borehole contact forces can be obtained by inexpensive numerical simulation of actual drilling conditions.

Introduction
Boreholes that possess curvature along their extents are by no means uncommon. This curvature may be either unexpected and thus usually undesirable, or else expected and thus the result of a concerted effort on the part of the operator. In either event, experience indicates that an improved understanding of the behavior of a drilling assembly within a curved borehole would be highly desirable. This understanding may pertain to locations and magnitudes of assembly-borehole contact forces or perhaps to the directions of the bit assembly and the resultant bit force relative to the borehole axis. This paper discusses a mathematical model and associated computer program which provide this and other information for general drilling assemblies within curved boreholes.
In the present mathematical model, a drillstring is considered to be an inextensible, elastic beam-colum. The geometric and material properties of the drillstring are assumed to be properties of the drillstring are assumed to be piecewise constant. piecewise constant.