A Systematic Approach to Well Surveying Calculations

Abstract The information used in well surveying calculations is studied and formalized. A general, natural, mathematical approach to the problem is presented and four special cases are developed: presented and four special cases are developed: (1) triangular or trapezoidal, (2) secant, (3) quadratic, and (4) minimum curvature. These methods are compared and their properties analyzed for reasonableness. Two of these methods are new and have appeared promising when applied to real and test data. Introduction Several new approaches have been proposed to the problem of performing well surveying calculations; and it has been pointed out that improvements in directional drilling tools and techniques justify a better treatment than the old tangential methods. However, a systematic analysis of the problem in a standard mathematical form has not been published. This paper will try to cover several of the more useful methods in a natural mathematical framework and to suggest several alternative approaches and directions for improvement. Formulation of the Problem In performing a directional survey of a well, a tool measures an inclination angle with respect to the vertical and a bearing angle with respect to North at a number of points, Pi, in the well. Thus, we know three things: (1) the location of the first point, Po, at the surface, (2) the distance between any two points, Pi - and Pi+1, along the arc length, and (3) the direction of the wellbore at each point, Pi (as given by the inclination and bearing). Pi (as given by the inclination and bearing).If one considers using a smooth approximation to the wellbore rather than straight-line segments joined at angles, the question of what degree of smoothness is to be used arises. For example, cubic spline approximations could be used to insure second-order smoothness. There are two difficulties that arise. In the first place, the authors know of no reason why a wellbore should have second-order smoothness. Second, the calculations become very difficult, partly because more than two points must be dealt with at a time. In this paper we shall follow the common practice of only analyzing the wellbore from one point, say P0, to the next point, P1, and assume that the process is repeated until P1, and assume that the process is repeated until the final point, Pn, is located. Referring to Fig. 1, we assume that the Z-axis is vertical and the X-axis is in the direction North. Thus the Y-axis will point East. The angle of inclination will be 0 and the angle from North, which is essentially the bearing, will be, where and . Hence, we can state our problem mathematically as follows.