Wellbore Positioning System and Method
A method and a system are provided for determining the relative positions of a wellbore and an object. The wellbore is represented by a first ellipse and the object is represented by a second ellipse. The first ellipse represents the positional uncertainty of the wellbore and the second ellipse represents the positional uncertainty of the object. The method includes receiving input data relating to a measured or estimated position of the wellbore and the object. In addition, the method includes calculating an expansion factor representing an amount by which one, or both, of the first ellipse and the second ellipse can be expanded with respect to one or both of respective first and second sets of elliptical parameters so that the first and second ellipses osculate. Further, the method includes determining, based on the calculated expansion factor, position data indicative of the relative positions of the wellbore and the object.
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The present invention relates to a computer-implemented method and a system for determining the relative positions of a wellbore and an object. Spatial relationships between two ellipses, each of which represents the positional uncertainty of a wellbore, are utilized to determine the conditions governing osculation between the two ellipses, expressing the determination as an expansion scale factor.
BACKGROUNDAs the drilling of a wellbore in hydrocarbon reservoir (for example, an oil or gas reservoir) proceeds, the positional uncertainty at any point in a well is dependent on a number of factors, including the positional uncertainty of the surface location, the well's geographical location and trajectory and the various instruments used to survey the well. By positional uncertainty is meant positional uncertainty of the well's geographical location, positional uncertainty of its trajectory etc. The expected behaviours of these instrument types are presented as instrument performance models. Application of these models quantifies the uncertainty of the true wellbore position for a stated confidence.
Referring to
Using this model, at any time and point in space, which is to say positions down a wellbore and its surrounding volume, the resulting positional uncertainty about a wellbore along its trajectory is the envelope of the ellipsoids; a curved, continuous cone 2 with a curved end. The interference between two adjacent wells can be visualised as the interference between the two cones. The positional uncertainty can change over time as data is re-processed or more data is acquired. It also changes when the wellbore is resurveyed using a more accurate instrument system, for example when a high accuracy gyroscope is run at a casing point 3. This then narrows the cone, as shown in
To a good approximation, at any given point along the wellbore the intersection of a plane that is normal to the along-hole direction of the wellbore with the cone can be represented as an ellipse. Therefore, the problem of calculating the interference between two wells can be reduced to that of calculating the distance between two such ellipses. This simple geometrical model has been adopted by various standards organisations to define minimum acceptable separation distances between two wellbores, for example the Norwegian “Norsk Sokkels Konkurranseposisjon” (NORSOK) D-10.
The separation between wellbores in 3D can be represented in 2D using a collision avoidance plot, also known as a travelling cylinder or normal plane diagram. In this representation the intersection of, for example, an existing and a planned well (or two planned wells) is displayed on a plane, constructed normal to the planned well. The planned well is kept at the centre of the plot and therefore the relative separation between the planned well and the adjacent well is indicated by the locus of points obtained at successive depths. At any point in the subject well the plane also intersects the curved cone and at low or modest angles of incidence between wells the intersections with the cones appears, to good approximation as two ellipses. During drilling the as-drilled and projected positions are shown on the same plot. The planned well is also referred to as the subject or reference well.
If x and y are orthogonal coordinates in the normal plane then the separation δ between the wells can be calculated using Eqs. 1 to 3, below. In the absence of bias this is also the separation between the error ellipses. Further adjustments can be made if required.
Δx0=x0,2−x0,1 (1)
Δy0=y0,2−y0,1 (2)
δ=(Δx02+Δy02)1/2 (3)
In practical terms the minimum approach distance δmin between the wellbores must be greater than the sum of the open hole and casing radii, δmin>(dh+dc)/2, where dh is the hole diameter and dc is the casing diameter. This criterion automatically satisfies the mathematical constraint δ≠0.
Currently, the relationship between two adjacent ellipses is approximated as a “separation factor”, ks. In this representation the ellipses are related only by the line passing through their centres. Because of this, the calculation of the characteristic length s for each ellipse may be performed independently of the other. Two common methods are the centre vector method (CVM) and pedal curve method (PCM).
Because of mathematical difficulties, existing methods for calculating separation factors are approximations and may be either too optimistic or too conservative, particularly for ellipses with high eccentricities.
For example,
Although the separation factors calculated by either the centre vector or pedal curve methods are relatively easy to calculate, neither method is a faithful representation of the geometrical relationship between the two ellipses. As shown in
In embodiments of the invention, there is provided a computer-implemented method and a system according to the appended claims.
According to an embodiment of the invention, there is provided a computer-implemented method for determining the relative positions of a wellbore and an object, the wellbore being represented by a first ellipse and the object being represented by a second ellipse, wherein the first ellipse represents the positional uncertainty of the wellbore and the second ellipse represents the positional uncertainty of the object, the method comprising the steps of:
receiving input data relating to a measured or estimated position of the wellbore and the object, the position of the wellbore having a first set of parameters defining the first ellipse, and the position of the object having a second set of parameters defining the second ellipse;
calculating an expansion factor representing an amount by which one, or both, of the first ellipse and the second ellipse can be expanded with respect to one or both of respective first and second sets of elliptical parameters so that the first and second ellipses osculate, wherein calculating the expansion factor involves determining and solving a quartic equation that is based on the geometry of the ellipses; and
determining, based on the calculated expansion factor, position data indicative of the relative positions of the wellbore and the object.
Embodiments of the present invention utilize spatial relationships between two ellipses for determining the conditions governing osculation between the two ellipses (where osculation is the case in which the ellipses touch), expressing the determination as an expansion scale factor. Each expansion factor calculation involves using the smallest positive root of the quartic equation. The explicit schemes of the present invention offer improvements in both calculation efficiency and reliability over known methods of calculating a separation factor and over iterative methods of calculating an expansion factor.
Typically, the wellbore is a first wellbore, and the object is a second wellbore. Alternatively, the object may be a sub-surface hazard that is to be avoided when drilling the wellbore.
Methods are presented for the expansion of either one, or both ellipses. The computer-implemented methods can be used to increase the allowable proximity of two adjacent wellbores whilst satisfying the necessary geometrical and probabilistic constraints. The calculation method is consistent with existing industry wellbore uncertainty models. Since the determination of the osculating condition is exact the calculation is neither too optimistic nor too conservative.
Further features and advantages of the invention will become apparent from the following description of preferred embodiments of the invention, given by way of example only, which is made with reference to the accompanying drawings.
In directional work an ellipse is generally defined by its centre (x0, y0), the lengths of its semi-major and semi-minor axes a and b and the orientation θ of the major axis direction a relative to some reference direction.
Since the dimensions of each ellipse represents some confidence interval that the wellbore lies within its boundary, the equal expansion or contraction of both ellipses until they touch (i.e. osculate) is a measure of a potential collision between the wells. Since the point at which two ellipses touch is a function of both their sizes and orientations, conceptually, the available space can also be calculated by expanding only one ellipse with the other one fixed. Therefore both dual sided and single sided expansion can be applied to calculate a relevant expansion factor k.
Mathematically, it may be more convenient to represent the ellipse as a quadratic form, as shown by Eq. 5, incorporating the above elliptical parameters and the expansion factor k within the quadratic form's coefficients, which are shown by Eqs. 6 to 11. Details of both the transform and inverse transform are given in Appendix A.
E(x,y,k)=Ax2+2Bxy+Cy2+2Dx+2Fy+H−a2b2k2=0 (5)
Where
A=b2 cos2θ+a2 sin2θ (6)
B=(b2−a2)sin θ cos θ (7)
C=a2 cos2θ+b2 sin2θ (8)
D=−y0B−x0A (9)
F=−x0B−y0C (10)
H=x02A+2x0y0B+y02C (11)
The quadratic may be represented in matrix form, as shown by Eq. 12, where E is the 3×3 symmetric matrix. It is noted that the expansion factor appears in only one of the matrix elements as its square k2.
It is also possible to represent the ellipse in matrix form so that the square symmetric matrix is independent of the ellipse's origin, (Zheng, X., Palffy-Muhoray, P.: “Distance of Closest Approach of Two Arbitrary Hard Ellipses in 2D”). A summary of the representation is included in Appendix A.
ZPM Expansion FactorThe following calculation, referred to hereinafter as the “ZPM” method, can be used to calculate the expansion factor using dual sided expansion, where each of two ellipses is expanded equally.
Referring to
The distance of closest approach δcr of two arbitrary hard ellipses in 2D can be determined using the method disclosed in Zheng, X., Palffy-Muhoray, P.: “Distance of Closest Approach of Two Arbitrary Hard Ellipses in 2D”. Referring to
The problem's symmetry is then used to determine the expansion factor (
Zheng and Palffy-Muhoray also describe a method for calculating the contact point and provided computer code for both of the closest approach and contact point calculations. Knowledge of the contact point may be used to verify the expansion factor results, checking for each ellipse that |E(x, y, k)|<ε, where ε is some acceptable tolerance.
YKC Expansion FactorA further method, referred to hereinafter as the “YKC” method, can be used to calculate an expansion factor using dual sided expansion, where each of the two ellipses is expanded equally, or single sided expansion, where only one ellipse is expanded while the other remains fixed. However, for dual expansion the ZPM approach is preferred. Tests show that it is more stable computationally, particularly for similarly sized ellipses with centres that are close together.
Referring to
For single sided expansion (output “Y” at step S602), the symmetry present in the dual sided expansion is broken and a different approach must be used. Referring to
A characteristic cubic polynomial P(λ)=det(λE1−E2)=0, which can be used to determine the separation conditions between two ellipses without explicitly calculating the contact point, was derived in Choi, Y. K.: “Collision Detection for Ellipsoids and Other Quadrics”, PhD Thesis, University of Hong Kong, March 2008. Choi showed that if E1 and E2 are two ellipses with the characteristic polynomial P(λ) (where λ is a multiplier) then they are separated if and only if P(λ) has two distinct negative roots and they touch each other externally if and only if P(λ) has a double negative root. The ellipses are overlapping if P(λ) has no negative root.
For the purpose of calculating the expansion factor, the expansion factor can be incorporated in the characteristic polynomial giving P(λ)=det[λE1(k1)−E2(k2)]=0. For a single sided expansion the first ellipse E1 is fixed so set k1=1 and k2=k. The characteristic, cubic polynomial becomes P(λ)=det[λE1−E2(k)]=0 (step S606). Using Choi's condition, the cubic's discriminant vanishes when the ellipses touch, leaving a quartic equation in k2, as shown by Eq.13. Taking the square root gives the expansion factor k.
γ4k8+γ3k6+γ2k4+γ1k2+γ0=0 (13)
After lengthy, computer assisted simplification, using a software program such as Mathematica, the coefficients of the quartic equation can be written as Eqs. 14 to 26. Further details are provided in Appendix B.
By inspection, this calculation of the closest distance of a point (which may represent an object) to an ellipse is equivalent to the single sided expansion of a unit circle (which is a special case of an ellipse) centred on the point against the ellipse, as shown in
For dual sided expansion, in step S609 a determination is made as to whether the centres of the ellipses are separated by a distance greater than δmin, as explained above in relation to Eq. 3. If the separation is not greater than δmin, it is determined in step S610 that the wellbores physically interfere and the calculation is stopped in step S611. Alternatively, if the ellipse centres are separated by more than δmin, dual sided expansion can proceed; both ellipses are expanded equally so set k1=k2=k. The characteristic polynomial becomes P(λ)=det[λE1(k)−E2(k)]=0 (step S612). This results in another quartic equation in k2; details of the coefficients are provided in Appendix B. When solved (step S613), the smallest positive root of the equation gives the expansion factor k, at which point the calculation stops (step S614).
EXAMPLESSome examples of elliptical configurations are shown in
In using the ZPM dual sided expansion method and the YKC single sided expansion method, the expansion factors for these configurations are calculated as follows:
Referring to the configurations of
From the separation and expansion factors of Table 1, it can be seen that the factors are calculated to be the same value (1.6) for the configuration of
As shown in
In order to determine optimum settings of the various components of the wellbore drilling system, the wellbore positioning system 100 comprises suitable computer-implemented models, software tools and hardware, as shown in
In one arrangement, referring to
Input data received by receiving means of the system 100 comprise the elliptical parameter values and are based on a measured position of an existing wellbore or an estimated (i.e. modelled or simulated) position of a planned wellbore. Such estimated input data can be modelled or estimated upon planning a wellbore, for example upon an initial assessment or appraisal of a reservoir when developing a new field. Alternatively, in the case where the position of a planned wellbore is being determined in order to avoid an object other than another wellbore, such as a sub-surface hazard, the input data includes measurement data relating to the position of the object.
The measurement data may comprise specific measured values as directly measured by suitably positioned measurement equipment such as survey instruments 12, or may comprise values derived from a number of separate positional measurements. Therefore, the raw measured data may, if necessary or preferred, be manipulated by appropriate software and executed by the CPU 105 of the system 100, in order to generate measurement or estimated position data that are suitable for inputting into the expansion factor calculation tool 111. Such manipulation may comprise using the reservoir and/or trajectory models to determine the parameter values of the two ellipses.
The expansion factor calculation tool 111 may comprise a software program such as Mathematica. This program can be used in a number of ways during the calculation of the expansion factor. Firstly by making use of its symbolic manipulation, the substitutions, for example, for A, B, C, D, F, G (which is equivalent to H−a2b2k2—see Appendix A), H can be made. The determinants can then be expanded and the equations simplified using this program. Additionally, Mathematica™ is preferably employed to program the resulting quartic coefficients and solve the quartic equation. Alternatively, the expressions can be programmed in, for example, Visual Basic™ within an EXCEL™ spreadsheet.
An optimisation tool 125 may be provided to assist in the planning and drilling of wellbores. The optimisation tool may be used in conjunction with the trajectory model 123 to compute an optimal position for the wellbore in 2D or an optimal trajectory in 3D, based on input data including the calculated expansion factor and the measured or estimated input data that relates to the position of one or more existing wellbores or objects. In the case where a number of positions or trajectories are possible, the optimisation tool 125 may be programmed with rules that take into account additional data representing, for example, threshold values representing practical limits to the degree of curvature of the wellbore trajectory. In this way, the optimisation tool 125 can determine an optimum alignment of the trajectory, as explained further below with reference to
The use of the expansion factor in the wellbore positioning method and system of the invention is advantageous in the planning and drilling of wellbores, as it provides more space in which to plan and optimise the trajectories of wellbores. However, if a planner concluded that it was not possible to drill through the gap of
The wellbore positioning system 100 is preferably operatively connected to a controller 133 of the wellbore drilling system, for example via the network N1. The controller 133 of the wellbore drilling system is automatically configured with the one or more operating modes determined by the system 100, the controller 133 being arranged to apply the one or more operating modes.
MethodReferring to
In step S1201, the input data is received by the wellbore positioning system 100.
At step S1202, the input data are input into the expansion factor software tool 111, the calculations of which are described above in relation to
At step S1205, the generated position data are used to determine one or more operating modes of the wellbore drilling system. The operating mode can represent an instruction or suggested setting for the drilling system, which can subsequently be applied to the drilling system. The determination can include the step of comparing, in accordance with a predetermined set of rules (which can be set using a collision avoidance plot implemented by the trajectory model 123), the calculated position data to predetermined known or threshold position data that is accessible from the database DB2. For example, the determination may be based on a known position of an existing wellbore or a sub-surface hazard.
Software executed by the CPU 105 of the system 100 determines, on the basis of the determined position data, the one or more operating modes of the wellbore drilling system. The expansion factor calculation tool 111, the reservoir model 121 and/or the trajectory model 123 may be configured to determine the operating mode(s) upon generation of the position data, or a separate software component may be provided. Additional technical and physical constraints determined by the reservoir model 121 or the trajectory model 123 may be taken into account in order to determine the operating mode, and can be stored and accessed from the databases DB1 and DB2 as necessary.
For example, the operating mode can comprise an instruction to go ahead with the drilling of a planned wellbore or not, this determination being based on a determination by the trajectory model 123 that the trajectory of the planned wellbore under consideration is drillable. Alternatively or additionally, the operating mode can comprise one or more specific configuration settings for the wellbore drilling system, such as a drilling speed or trajectory.
The software component used to determine the operating mode is configured to use a predetermined set of rules in conjunction with input data such as the calculated expansion factor, in order to determine the operating mode. These rules are stored in and accessible from the database DB1 and DB2 as necessary.
The computer-implemented method can further include an optional step, S1206, of applying or inputting the determined operating mode into a controller of the wellbore drilling system.
The above embodiments are to be understood as illustrative examples of the invention. It is to be understood that any feature described in relation to any one embodiment may be used alone, or in combination with other features described, and may also be used in combination with one or more features of any other of the embodiments, or any combination of any other of the embodiments. Furthermore, equivalents and modifications not described above may also be employed without departing from the scope of the invention, which is defined in the accompanying claims.
APPENDIX A Ellipse RepresentationsThe derivations using the YKC conditions depend on the ability to translate freely between the ellipse representations. The first and second quadratic forms are mathematically equivalent.
First Quadratic FormThe ellipse E0(x, y)=b2x2+a2y2−a2b2=0 with semi-major axis a and semi-minor axis b, aligned with the x and y axes and centred on the origin can be represented as a quadratic form, Eq. A-1.
Writing the ellipse E0(x, y)=xE0xT gives
The matrix T translates a point on the ellipse by an amount x0 in the x direction and y0 in the y direction. The rotation matrix R rotates a point by an amount θ clockwise about the origin. The scaling matrix S scales a point by a factor k relative to the origin.
Note that the translation, rotation and scaling operations do not commute and therefore the order in which these operations are performed is important. Combining these transformations as shown in Eq. A-6 gives the ellipse E(x, y, k) in the body of the paper, Eqs. 5 and 12.
E=xTRSE0STRTTTxT (A-6)
The coordinates of the ellipse's centre, semi-major and semi-minor axes and orientation may be recovered from the first quadratic form using the inverse transform, Eqs. A-7 to A-11. Although the inverse transform is not used in either the separation or expansion factor calculations it provides an effective means of testing the correctness of the transform. Note that the constant G is equivalent to H−a2b2k2.
The ellipse can also be represented so the symmetric matrix is independent of the ellipse's origin, Eq. A-12 and A-13, (Zheng and Palffy-Muhoray, 2010). Here I is the identity matrix and the vector θ=[sin θ, cos θ]. In Zheng and Palffy-Muhoray's paper these authors assume k=1 throughout.
For a single sided expansion the characteristic polynomial becomes P(λ)=det[λE1−E2(k)]=0, Eq. B-1. For conciseness substitute χ=k2. Expanding the determinant gives a cubic polynomial which coefficients are functions of the coefficients of the quadratic forms and the square of the expansion factor, Eq. B-2.
Choi, 2008 showed that the cubic discriminant equals zero when the ellipses touch externally, Eq. B-3.
w22w12−4w23w0+18w3w2w1w0−4w3w13−27w32w02=0 (B-3)
By inspection, the coefficients wj are of the form wj=uj+vjχ
Making this substitution, Eq. B-3 then becomes a quartic equation in χ, Eq. B-4.
Note that both u3=0 and v0=0 and
Then substitute Eqs. B-5 to B-10 into Eq. B-4. Simplification gives the coefficients γi of the quartic equation, Eq. 13 given in the specific description.
YKC Expansion Factor (Dual Sided)The derivation of the dual sided expansion with the characteristic polynomial P(λ)=deq[λE1(k)−E2(k)]=0 proceeds in the same way. Again for conciseness substitute χ=k2.
The coefficients γi of the quartic equation may be calculated as Eq. B-12 to B-24.
Zheng and Palffy-Muhoray approach leads to a quartic equation in the variable Q, Eq. (B-25).
This can be written in the standard form, Eq. (B-26)
ψ4Q4+ψ3Q3+ψ2Q2+ψ1Q+ψ0=0 (B-26)
Where the quartic coefficients ψi are
To avoid a clash of symbols, note that the nomenclature used here differs from that used by Zheng and Palffy-Muhoray. The variables ζ and φ used here are defined in their paper.
APPENDIX C Nomenclaturea=Ellipse semi-major axis length, L, ft
a=Unit vector in the major axis direction
b=Ellipse semi-minor axis length, L, ft
d=Diameter, L, ft
k=Expansion scale factor, dimensionless
m=Element of a transformation matrix
p=Substituted variable
r=Substituted variable
s=Characteristic length for a separation factor, L, ft
u=Substituted variable
v=Substituted variable
w=Coefficients of the YKC cubic equation
x=Ordinate in the normal plane, L, ft
y=Ordinate in the normal plane, L, ft
z=Substituted variable
A=First ellipse quadratic form coefficient
B=Second ellipse quadratic form coefficient
C=Third ellipse quadratic form coefficient
D=Fourth ellipse quadratic form coefficient
E=Ellipse
F=Fifth ellipse quadratic form coefficient
G=Sixth ellipse quadratic form coefficient
H=Modified sixth ellipse quadratic form coefficient
E=Ellipse matrix representation
R=Rotation matrix
S=Scaling matrix
T=Translation matrix
I=Unit matrix
P=Polynomial
Q=Independent variable of the ZPM quartic equation
Greek Symbolsδ=Centre to centre distance between ellipses, L, ft
γ=Coefficients of the YKC quartic equation
φ=Substituted or temporary variable
∂=Substituted or temporary variable
ψ=Coefficients of the ZPM quartic equation
χ=Square of the expansion factor, dimensionless
θ=Ellipse orientation angle to major axis, radians
φ=First variable defined by ZPM=
ζ=Second variable defined by ZPM
λ=Multiplier
Δ=A difference in a parameter
Subscripts and Superscripts0=Condition at an origin
′=Transformed condition
1,2,3,4=First, second etc.
c=Casing
cr=Critical, or closest approach
h=Hole
i=Index
j=Index
s=Separation factor
Claims
1. A computer-implemented method for determining the relative positions of a wellbore and an object, the wellbore being represented by a first ellipse and the object being represented by a second ellipse, wherein the first ellipse represents a positional uncertainty of the wellbore and the second ellipse represents a positional uncertainty of the object, the method comprising the steps of:
- receiving input data relating to a measured or estimated position of the wellbore and the object, the position of the wellbore having a first set of parameters defining the first ellipse, and the position of the object having a second set of parameters defining the second ellipse;
- calculating an expansion factor representing an amount by which one, or both, of the first ellipse and the second ellipse can be expanded with respect to one or both of respective first and second sets of elliptical parameters so that the first and second ellipses osculate, wherein calculating the expansion factor comprises determining and solving a quartic equation that is based on the geometry of the first and second ellipses; and
- determining, based on the calculated expansion factor, position data indicative of the relative positions of the wellbore and the object.
2. The method of claim 1, wherein the first and second ellipses are expanded equally, and wherein the calculation of the expansion factor further comprises:
- solving the quartic equation to determine a distance between a centre of the first ellipse and a centre of the second ellipse when the second ellipse is translated towards the first ellipse along a line joining the centres of the first and second ellipses so that the first and second ellipses osculate; and
- calculating the expansion factor based on the determined distance and a scale factor.
3. The method of claim 1, wherein either the first and second ellipses are expanded equally, or one of the first and second ellipses is expanded, so that the first and second ellipses osculate, and wherein the calculation of the expansion factor further comprises:
- applying the first and second sets of elliptical parameters to a polynomial equation, the solution of which represents a separation condition of the ellipses;
- determining the quartic equation from the polynomial equation; and
- solving the quartic equation to calculate the expansion factor.
4. The method of claim 3, wherein the ellipses osculate when the polynomial equation has a double root.
5. The method of claim 1, wherein the wellbore is a first planned or drilled wellbore, and the object is a second planned or drilled wellbore.
6. The method of claim 1, wherein the object is a sub-surface hazard.
7. The method of claim 1, wherein the first and second sets of elliptical parameters are derived from a measured or estimated position of the wellbore and the object.
8. The method of claim 1, further comprising the step of determining a trajectory of the wellbore in a three-dimensional simulation.
9. The method of claim 1, further comprising the step of optimising the position of the wellbore relative to the object.
10. A computer-implemented method for determining one or more operating modes of a wellbore drilling system, the wellbore drilling system being arranged to drill a wellbore in a rock formation, the method comprising the steps of:
- receiving position data determined according to the method of claim 1;
- inputting said position data into a wellbore trajectory model;
- operating the wellbore trajectory model so as to generate trajectory data indicative of a trajectory of the wellbore; and
- determining, on the basis of the trajectory data, said one or more operating modes of the wellbore drilling system.
11. The method of claim 10, further comprising the steps of:
- automatically configuring a controller of the wellbore drilling system with the one or more operating modes determined by the wellbore positioning system; and
- applying the one or more operating modes.
12. A wellbore positioning system arranged to determine the relative positions of a wellbore and an object, the wellbore being represented by a first ellipse and the object being represented by a second ellipse, wherein the first ellipse represents a positional uncertainty of the wellbore and the second ellipse represents a positional uncertainty of the object, the system comprising:
- data receiving means arranged to receive input data relating to a measured or estimated position of the wellbore and the object, the position of the wellbore having a first set of parameters defining the first ellipse, and the position of the object having a second set of parameters defining the second ellipse;
- expansion factor calculation means arranged to calculate an expansion factor representing an amount by which one, or both, of the first ellipse and the second ellipse can be expanded with respect to one or both of respective first and second sets of elliptical parameters so that the first and second ellipses osculate, wherein calculating the expansion factor comprises determining and solving a quartic equation that is based on the geometry of the first and second ellipses; and
- position determining means arranged to determine, based on the calculated expansion factor, position data indicative of the relative positions of the wellbore and the object.
13. The wellbore positioning system of claim 12, further comprising operating mode determining means arranged to determine, on the basis of the position data, one or more operating modes of a wellbore drilling system.
14. The wellbore positioning system of claim 12, the system being operatively connected to a controller of the wellbore drilling system such that the controller of the wellbore drilling system is automatically configured with the one or more operating modes determined by the wellbore positioning system, the controller being arranged to apply the one or more operating modes.
15. A computer program product comprising a set of instructions which, when executed by a computing device, is configured to cause the computing device to carry out the method according to claim 1.
16. The computer program product of claim 15, comprising a computer readable storage medium.
Type: Application
Filed: Jan 17, 2013
Publication Date: Jan 15, 2015
Applicant: BP EXPLORATION OPERATING COMPANY LIMITED (Middlesex)
Inventor: Steven James Sawaryn (Aberdeen)
Application Number: 14/370,913
International Classification: E21B 41/00 (20060101); E21B 7/04 (20060101); E21B 47/022 (20060101);