New Trajectory Control Directional MWD Accuracy Prediction and Wellbore Positioning Method

is the side cutting in Z-axis due to RQ in t. It is clear that the drilling direction would not be the same as that of the resultant force and the magnitudes of planned/actual path depends on many influential factors, such as rock properties, formation characteristics, types of bit, etc. Hole Deviation Mathematical Definition: The wellbore trajectory is defined as a series of surveyed points in 3D space. These points along the planned path are called the Measured Depth (MD), associated with MD is north (N), east (E), Total Vertical Depth (TVD), Inclination (I) and azimuth (A), respectively, planned values North, East, True Vertical Depth, Inclination and Azimuth. These points are jointed together to form a continuous trajectory with a geometric calculations method. Eight components collectively define hole deviation control; they are based on lineal and angular differences between the actual and planned well paths.


INTRODUCTION
In the rectangular coordinate system shown in Fig. l, the side forces R P and R Q are acting along X-axis and Y-axis respectively. The resultant force R is combined by three mutually perpendicular components; they are R P , R Q and the weight on bit P B . The superscript (n) in the definitions of each relative change is refer to the respective values during the prior computing of hole deviation; (n-1) refers to values at planned hole drilled between the two foregoing hole deviation computations. The superscript ( * ) defines the measured data and the subscript (b) refers to current well bore total depth. Thus L is ( ) ) ( * n MD which is preferably somewhat short. Performing two successive coordinate axis rotations derive the equations for (V) and (H) the first rotation is by the deviation angle * about the TVD axis. The aforementioned vector is orthogonal to the planned path at MD * , then the required TVD" equals zero; i.e. Respective to hole deviation, a preferable method by which to mathematically represent the entire planned drill path is to parametrically define each Cartesian coordinate and hole inclination and azimuth, in terms of measured depth. That is the planned path is designed and then represented as follows: N MD = P 1(MD) ; E MD = P 2(MD); TVD MD = P 3(MD) ; MD = P 4(MD) ; MD = P 5(MD) The rate of change in lineal relationship between the planned and actual well paths is assumed to remain the same over small distances; this assumption is often completely valid. As the hole is drilled, it is necessary to determine where on the plan one would prefer the wellbore to exist. The linear distance between the current bottom hole location and a point on the planned path is computed with the 3D distance formulas. This is generally represented by Eq.1 Let MD * represent the measured depth along the planned path, whose respective Cartesian coordinates minimize the distance computed with Eq.1. Therefore, MD * found by taking the derivative of Eq.1 with respect to MD and setting the result equal zero.
The measured depth that sets the right hand side of Eq. 2 equal zero is MD * ; therefore, the denominator may be ignored and MD * is found by solving Eq. 2.
Well Bore Position Uncertainty: In 3D, the confidence region is most often depicted as ellipsoid because ellipsoids are the constant value contours of the 3D Guassian 2 probability density function. The technique used is based on the generalized linear regression model: where: ε C is the covariance matrix for the random vector ε . Maximization of Eq.4 with respect to β yields the following estimate βˆ and its covariance β C .
X T is the transpose of X. Assume we have (k) measurement can be written in the following form: in which each I j is a (3*3) identity matrix and 1 j k. The covariance matrix, ε C , can be written as: Guassion distribution with a probability density function in the following form: and because the covariance matrices, C ii , are diagonal, the probability density function reduces to: where, (x) is the element of the position covariance of matrix in the x-coordinates. (y) is the element of the position covariance of matrix in the y-coordinates. (z) is the element of the position covariance of matrix in the z-coordinates. The constant value contours of Eq.5 are family of ellipsoids defined by the equation of the quadratic expression in the exponent to a constant. For each ellipsoid, the length of the north, east and down semi-major axes are: where, (s) is the normalized length of the semi major principal axes of the confidence region ellipsoid. The mathematical basis of the HDC technique can be summarized by restating the basic formula in the following format: The covariance matrix of the HDC is given as: and making use of the inverse relations: where the superscript (sur) indicates the uncertainty is defined at a survey station. Error vectors due to bias error are given by: Defining the superscript (dep) to indicate uncertainty at an assigned depth, it may be shown that:  Survey bias at an assigned depth is calculated by: When calculating the uncertainty in the relative position between two surveys stations (K A , K B ), the uncertainty is given by: The uncertainty in this position error is expressed in the form of a covariance matrix:

RESULTS AND CONCLUSION
The error models for basic interference-correction MWD have been applied to the standard well profiles to generate position uncertainties in each well. The results of several combinations are tabulated in Table 1 and 2.

D, bias
Uncertainty at assigned depth, selected errors symmetric modeled as bias. Uncertainties at the tie line (MD=0) is zero; stations interpolated at whole multiples of station interval using minimum curvature and minimum distance methods; well plan way points included as additional stations; instrument tool face = borehole tool face Example 1 and 2 ( Table 2) compare the basic and interference in well Unity#30. Being a high inclination well running an approximately, the interference correction actually degrades the accuracy. The results are plotted in Fig. 5. Example 3 to 6 all represent the basic MWD error model applied to well RenMen#95.
They differ in that each uses a different permutation of the survey station/assigned depth and symmetric error/survey bias calculation options. The variation of lateral uncertainty and ellipsoid semi-major axis, characteristics is shown in Fig. 6. Example 7 breaks well Quan#95 into three depths intervals, with the basic and interference-correction models being applied alternately. This example is included as a test of error propagation ( Fig. 7 and 8).