Numerical approximation improves well survey calculation

It is now possible to improve the precision of well survey calculations by order of magnitude with numerical approximation.

Although the most precise method of simulating and calculating a wellbore trajectory generally requires more calculation than other, less-accurate methods, the wider use of computers in oil fields now eliminates this as an obstacle.

The results of various calculations show that there is a deviation of more than 10 m among the different methods of calculation for a directional well of 3,000 m.¹ Consequently, it is important to improve the precision and reliability of survey calculation-the fundamental, necessary work of quantitatively monitoring and controlling wellbore trajectories.

Inclinometers can only give parameters from separate survey stations and cannot give the real profile of wellbore trajectory. Therefore, every method of survey calculation is based on some assumptions, and most of them in each course are straight lines, polygonal lines, cylinder helixes, or circular arcs.^{2 3} These models are simple, but they are not precise enough for exceptional wells such as extended-reach wells and multiple-target wells.

Theoretical basis

Engineering areas such as mathematics, mechanics, and other disciplines have used spline function since I.J. Schoenberg proposed it in 1946. The spline, made of a slender batten or other elastic material, was originally a simple device used for drawing a curve. Users lay it on the plate, put some paperweights on it, and form it into the shape of the given curve. When the bending deflection of the spline is tiny, Equation 1 can express the equation of its shape (see equation box).

Engineers can regard the spline as an elastic, slender beam, and the elementary theory of beams shows that the curve Equation 1 expresses is the shape of the beam loaded with a series of concentrated forces. The whole curve consists of sectional, cubic curves.

The first and second derivatives on the whole curve are continuous, and the inflection points are at the paperweights. With regard to efficacy, the paperweights operate simply as support points.

Mathematical spline is the curve that a series of sectional cubic curves approach on the elastic curve of a drawing spline. Its continuity is through to the second derivative, and its third derivative skips being continuous at the inflection points.

For many important engineering applications, the mathematical model of the drawing spline is highly accurate and true to nature.

During the process of drilling, the deformation of the drillstring is limited to the wellbore wall, and the contact points can be simplified as a series of fulcrums.

These fulcrums equal the paperweights used in a curve drawing, and the whole drillstring makes up a kind of elastic slender beam. Furthermore, a wellbore trajectory is continuous and smooth. Consequently, the drillstring in a wellbore is an elastic spline, and in every way simulates a wellbore trajectory using the spline function.⁴

In order to discuss the behavior of a function relation that has no explicit expression-such as that on some experiments' datasheets, observations, or surveys-it is common first to make an approximate mathematical relation. Interpolation is effective for creating such an approximate function. There are many interpolation methods, but the interpolating function of cubic spline is the best and the most common choice.

Trajectory spline function

Depth, inclination, and azimuth are the three essential parameters for determining trajectory, and, based on them, the engineer can calculate other parameters to simulate a real trajectory. As a rule, depth is the independent variable among these parameters.

For the whole trajectory or a part of it, assuming that there is a series of measured results arranged in the order of survey depth:

If a(L) and f(L) are no more than cubic multinomials over subinterval [L_i, L_i + ₁], and they have the continuous first and second derivatives over [a,b], and they meet Equation 2, then a(L) and f(L) are termed the inclination spline function and the azimuth spline function, respectively, based on the nodes {L_i}.⁴According to the definitions of inclination spline function, Equation 3 can express a(L). Except for the measured parameters, the equation shows that a(L) only relates to the second derivatives, mi, which meet the relations shown as Equation 4. But, because there are only n - 2 equations available here for n unknowns, (m₁,m₂, ..., m_n), there is a need for two more equations for the ends of the simulated trajectory.

There are generally three ways to give these supplementary equations:⁴

• When the rate of inclination change expresses as the first derivative, if å₁ and å_n are known, then Equation 5 can determine the boundary equations.
• When the curvature of the inclination curve expresses as the second derivative, if ä₁ and ä_n are known, then Equation 6 can determine the boundary equations.
• Undoubtedly, however, the best way to give the boundary condition from the first and the last several survey stations is: If cubic interpolation multinomials are used, then Equation 7 can determine the boundary equations.

When one of these boundary conditions joins Equation 4, a linear equation group will emerge. It can be proven that the coefficient matrix of the equation group is not singular. Thus, the equation provides a unique solution for the determination of a(L).

Similarly, Equation 8 can determine the azimuth spline function.

Trajectory curvature, torsion

A wellbore trajectory generally is a curve in space that is both curved and tortuous. For many years, engineers have focused critical attention on the trajectory curvature, regarding it as an important index.^5-7

Drilling practice shows, however, that a tight trip sometimes occurs even if trajectory curvature is not high and weight on bit cannot be applied successfully during regular drilling, especially when bent subs or houses are in the drillstring.

The research results show that the value of trajectory torsion is usually higher than that of trajectory curvature and changes more sharply. Evidently, there are many factors that influence frictional resistance-bottom hole assembly, hole curvature, inclination, performance of drilling mud, hole condition, and a difference in the degree of influence between hole curvature and its torsion.

Research also shows, however, that frictional resistance greatly increases as the drilling string is twisted in the tortuous hole even if its deflection is not as considerable. Therefore, hole torsion is another important parameter for describing and calculating a wellbore trajectory.₈

According to the definitions, the parameters relating to the hole curvature and torsion can be calculated by Equations 9, 10, and 11.^9-11

Survey calculation

Based on the idea and theory discussed previously, there are two methods for describing and calculating a drilled wellbore trajectory-the curve structure method and the numerical integral method.

In the curve structure method, according to the theory of curve, the shape of a curve in a vicinal area depends on its structural elements. For example, an arc can be the curve in which k = constant and t = 0.

Evidently, the results are more precise when the hole curvature and torsion are synchronously involved over every course length. Thus, it is reasonable to simulate a wellbore trajectory using the curve structural theory.