U.S. patent application number 14/370913 was filed with the patent office on 2015-01-15 for wellbore positioning system and method.
This patent application is currently assigned to BP EXPLORATION OPERATING COMPANY LIMITED. The applicant listed for this patent is BP Exploration Operating Company Limited. Invention is credited to Steven James Sawaryn.
Application Number | 20150019134 14/370913 |
Document ID | / |
Family ID | 47666096 |
Filed Date | 2015-01-15 |
United States Patent
Application |
20150019134 |
Kind Code |
A1 |
Sawaryn; Steven James |
January 15, 2015 |
Wellbore Positioning System and Method
Abstract
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.
Inventors: |
Sawaryn; Steven James;
(Aberdeen, GB) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
BP Exploration Operating Company Limited |
Middlesex |
|
GB |
|
|
Assignee: |
BP EXPLORATION OPERATING COMPANY
LIMITED
Middlesex
US
|
Family ID: |
47666096 |
Appl. No.: |
14/370913 |
Filed: |
January 17, 2013 |
PCT Filed: |
January 17, 2013 |
PCT NO: |
PCT/EP2013/050863 |
371 Date: |
July 7, 2014 |
Current U.S.
Class: |
702/6 ; 175/24;
175/40 |
Current CPC
Class: |
E21B 7/04 20130101; E21B
47/022 20130101; E21B 41/0092 20130101 |
Class at
Publication: |
702/6 ; 175/40;
175/24 |
International
Class: |
E21B 41/00 20060101
E21B041/00; E21B 7/04 20060101 E21B007/04; E21B 47/022 20060101
E21B047/022 |
Foreign Application Data
Date |
Code |
Application Number |
Jan 27, 2012 |
EP |
12250013.5 |
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.
Description
TECHNICAL FIELD
[0001] 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.
BACKGROUND
[0002] As 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.
[0003] Referring to FIG. 1, the positional uncertainty 1 about a
point representing the calculated position of the centre of a
wellbore is commonly represented as an ellipsoid with its principal
axes aligned with the high-side, right-side and along-hole
directions. In this context, high-side is the direction normal to
the wellbore in the vertical plane and right-side is the direction
normal to both the wellbore direction and the high-side and so lies
in the horizontal plane. The ellipsoid usually also accounts for
the dimensions of the casing or open hole of the wellbore. The size
of the ellipsoid varies according to the wellbore trajectory shape,
survey instruments used in calculating the position, and selected
confidence limits.
[0004] 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 FIG. 1. Therefore, if a new
measurement is taken at a subsequent point along the trajectory of
the wellbore, the positional uncertainty decreases, then increases
as the distance from the measurement point increases.
[0005] 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.
[0006] 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.
[0007] If x and y are orthogonal coordinates in the normal plane
then the separation .delta. 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.
.DELTA.x.sub.0=x.sub.0,2-x.sub.0,1 (1)
.DELTA.y.sub.0=y.sub.0,2-y.sub.0,1 (2)
.delta.=(.DELTA.x.sub.0.sup.2+.DELTA.y.sub.0.sup.2).sup.1/2 (3)
[0008] In practical terms the minimum approach distance
.delta..sub.min between the wellbores must be greater than the sum
of the open hole and casing radii,
.delta..sub.min>(d.sub.h+d.sub.c)/2, where d.sub.h is the hole
diameter and d.sub.c is the casing diameter. This criterion
automatically satisfies the mathematical constraint
.delta..noteq.0.
[0009] Currently, the relationship between two adjacent ellipses is
approximated as a "separation factor", k.sub.s. 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).
k s = .delta. s 1 + s 2 ( 4 ) ##EQU00001##
[0010] 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.
[0011] For example, FIG. 2 shows how the currently used "centre
vector method" (CVM) is used to calculate a separation factor
between two wellbores. In the CVM the characteristic lengths
s.sub.1 and s.sub.2 are determined from the point of intersection
of each ellipse, marked A and B in FIG. 2, with the line .delta.
joining their centres. The separation factor k.sub.CVM is
calculated using Eq. 4. In this case the ellipses extend beyond
their points of intersection and will touch before the separation
factor reaches unity. Therefore, separation factors calculated
using this method may be too optimistic. Such overly optimistic
calculations of the separation factor can lead to safety issues
when planning and drilling wells based on computer simulations.
[0012] FIG. 3 shows an alternative method of calculating a
separation factor between two wellbores, the "pedal curve method"
(PCM). In the PCM, the characteristic lengths s.sub.1 and s.sub.2
are determined from the line that is both tangent to the ellipse
and is orthogonal to the line .delta. joining their centres. The
first step is to determine the points of tangency, marked A and B
in FIG. 3. In this case the tangent lines meet and the separation
factor k.sub.PCM reaches unity before the ellipses touch. Therefore
the separation factors calculated using this method may be too
conservative, leading to unnecessary shut-in of wells or missed
opportunities.
[0013] 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
FIGS. 4a and 4b, calculating the separation factor in terms of an
"expansion factor", k, by the simultaneous and equal expansion
(k>1) or contraction (k<1) of both ellipses until they touch,
is neither too optimistic nor too pessimistic. This expansion
factor calculation can increase the allowable proximity between two
adjacent wells whilst satisfying the geometrical and probabilistic
constraints. Although iterative methods can be used starting from
the elliptical conditions, there is no guarantee that such
iterative schemes converge towards the correct expansion factor
solution.
SUMMARY
[0014] In embodiments of the invention, there is provided a
computer-implemented method and a system according to the appended
claims.
[0015] 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:
[0016] 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;
[0017] 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
[0018] determining, based on the calculated expansion factor,
position data indicative of the relative positions of the wellbore
and the object.
[0019] 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.
[0020] 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.
[0021] 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.
[0022] 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.
BRIEF DESCRIPTION OF THE DRAWINGS
[0023] FIG. 1 shows a three-dimensional representation of a cone
which represents the positional uncertainty of a wellbore;
[0024] FIG. 2 shows a "centre vector method" for estimating the
separation between two ellipses;
[0025] FIG. 3 shows a "pedal curve method" for estimating the
separation between two ellipses;
[0026] FIG. 4a shows the determination of an "expansion factor" by
the simultaneous and equal expansion (k>1) of two ellipses;
[0027] FIG. 4b shows the determination of an "expansion factor" by
the simultaneous and equal contraction (k<1) of two
ellipses;
[0028] FIG. 5a shows the steps involved in a first method of
calculating an expansion factor;
[0029] FIGS. 5b-5e show an expansion of the ellipses carried out in
the first method of calculating the expansion factor;
[0030] FIG. 6a shows the steps involved in a second method of
calculating an expansion factor;
[0031] FIG. 6b shows an expansion of the ellipses carried out in
the second method of calculating the expansion factor;
[0032] FIGS. 7a-9b show dual and single sided expansion of various
configurations of ellipses;
[0033] FIG. 10 shows a wellbore positioning system according to the
present invention;
[0034] FIG. 11 shows an example of a planned wellbore in simplified
collision avoidance plot;
[0035] FIG. 12 shows the steps taken in determining the relative
position of a wellbore according to the present invention;
[0036] FIG. 13 shows a schematic diagram of a wellbore being
drilled into a formation.
DETAILED DESCRIPTION
[0037] In directional work an ellipse is generally defined by its
centre (x.sub.0, y.sub.0), the lengths of its semi-major and
semi-minor axes a and b and the orientation .theta. of the major
axis direction a relative to some reference direction.
[0038] 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.
[0039] 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)=Ax.sup.2+2Bxy+Cy.sup.2+2Dx+2Fy+H-a.sup.2b.sup.2k.sup.2=0
(5)
Where
A=b.sup.2 cos.sup.2.theta.+a.sup.2 sin.sup.2.theta. (6)
B=(b.sup.2-a.sup.2)sin .theta. cos .theta. (7)
C=a.sup.2 cos.sup.2.theta.+b.sup.2 sin.sup.2.theta. (8)
D=-y.sub.0B-x.sub.0A (9)
F=-x.sub.0B-y.sub.0C (10)
H=x.sub.0.sup.2A+2x.sub.0y.sub.0B+y.sub.0.sup.2C (11)
[0040] The quadratic may be represented in matrix form, as shown by
Eq. 12, where E is the 3.times.3 symmetric matrix. It is noted that
the expansion factor appears in only one of the matrix elements as
its square k.sup.2.
E ( x , y , k ) = [ x y 1 ] [ A B D B C F D F H - a 2 b 2 k 2 ] [ x
y 1 ] ( 12 ) ##EQU00002##
[0041] 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 Factor
[0042] The 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.
[0043] Referring to FIG. 5a, in step S501 the elliptical parameters
a.sub.1, b.sub.1, .theta..sub.1, x.sub.0,1, y.sub.0,1 of ellipse
E.sub.1 and a.sub.2, b.sub.2, .theta..sub.2, x.sub.0,2, y.sub.0,2
of ellipse E.sub.2 are input into a wellbore positioning system, as
described below with respect to FIG. 10. In step S502, a
determination is made as to whether the centres of the ellipses are
separated by a distance greater than .delta..sub.min, as explained
above in relation to Eq. 3. If the separation is not greater than
.delta..sub.min, it is determined in step S503 that the wellbores
physically interfere and the calculation is stopped in step S504.
Alternatively, if the ellipse centres are separated by more than
.delta..sub.min, the distance of closest approach .delta..sub.cr is
calculated in step S505, as explained below.
[0044] The distance of closest approach .delta..sub.cr 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 FIG.
5b, the ellipse E.sub.2 is translated towards E.sub.1 in the
direction joining their centres until it reaches the position
E.sub.2* when the ellipses touch externally. The orientations of
the two ellipses are maintained throughout. The ellipse E.sub.1 is
then transformed into a circle C.sub.1 and the same mathematical
transformation used to obtain the circle is applied to the ellipse
E.sub.2 (FIG. 5c). Note that the circle C.sub.1 and the ellipse
E.sub.2** remain connected at a respective tangent after the
transformation. The closest approach between the circle and the
ellipse is then found analytically, recovering the closest approach
between the ellipses E.sub.1 and E.sub.2* by applying the inverse
of the transformation used to obtain the circle (FIG. 5d). Details
of the relevant calculations are explained in the paper by Zheng
and Palffy-Muhoray, together with the solution of the resulting
quartic equation; the quartic equation is given in Appendix B.
[0045] The problem's symmetry is then used to determine the
expansion factor (FIG. 5e). It is noted that the translation of
E.sub.2 to E.sub.2* followed by a magnification of magnitude k, of
both E.sub.1 and E.sub.2* together (whilst maintaining their
relative position) about the centre of E.sub.1 is equivalent to the
magnification of magnitude k of each of E.sub.1 and E.sub.2 about
their respective centres, (Snapper, E., Troyer, R. J.: "Metric
Affine Geometry", 1971, Academic Press, London, 1, 36-55).
Therefore, the dual sided expansion factor k can be calculated from
the distance of closest approach using the scaling factor
k=.delta./.delta..sub.cr, and the expansion factor k is output
(step S507).
[0046] 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)|<.epsilon.,
where .epsilon. is some acceptable tolerance.
YKC Expansion Factor
[0047] A 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.
[0048] Referring to FIG. 6a, in step S601 the elliptical parameters
a.sub.1, b.sub.1, .theta..sub.1, x.sub.0,1, y.sub.0,1 of ellipse
E.sub.1 and a.sub.2, b.sub.2, .theta..sub.2, x.sub.0,2, y.sub.0,2
of ellipse E.sub.2 are input into a wellbore positioning system, as
described below with respect to FIG. 10. In step S602, a
determination is made as to whether dual sided or single sided
expansion is preferred. Single sided expansion may be preferred in
some cases because of the greater area of space obtained about the
expanded wellbore.
Single Sided Expansion
[0049] 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 FIG. 6a, in this case
the size of the first ellipse E.sub.1 is fixed and for a solution
to exist, the centre (x.sub.0,2, y.sub.0,2) of the second ellipse
E.sub.2 must lie outside its boundary (step S603). Mathematically
this requires the condition that E.sub.1(x.sub.0,2, y.sub.0,2,
1)>0. In step S604, if the centre of E.sub.2 does not lie
outside of E.sub.1, the system determines that no solution is
possible, and the calculation is stopped at step S605.
[0050] A characteristic cubic polynomial
P(.lamda.)=det(.lamda.E.sub.1-E.sub.2)=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
E.sub.1 and E.sub.2 are two ellipses with the characteristic
polynomial P(.lamda.) (where .lamda. is a multiplier) then they are
separated if and only if P(.lamda.) has two distinct negative roots
and they touch each other externally if and only if P(.lamda.) has
a double negative root. The ellipses are overlapping if P(.lamda.)
has no negative root.
[0051] For the purpose of calculating the expansion factor, the
expansion factor can be incorporated in the characteristic
polynomial giving
P(.lamda.)=det[.lamda.E.sub.1(k.sub.1)-E.sub.2(k.sub.2)]=0. For a
single sided expansion the first ellipse E.sub.1 is fixed so set
k.sub.1=1 and k.sub.2=k. The characteristic, cubic polynomial
becomes P(.lamda.)=det[.lamda.E.sub.1-E.sub.2(k)]=0 (step S606).
Using Choi's condition, the cubic's discriminant vanishes when the
ellipses touch, leaving a quartic equation in k.sup.2, as shown by
Eq.13. Taking the square root gives the expansion factor k.
.gamma..sub.4k.sup.8+.gamma..sub.3k.sup.6+.gamma..sub.2k.sup.4+.gamma..s-
ub.1k.sup.2+.gamma..sub.0=0 (13)
[0052] 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.
.gamma. 4 = a 2 4 b 2 4 ( z 12 2 - 4 a 1 2 a 2 2 b 1 2 b 2 2 ) ( 14
) .gamma. 3 = 2 a 2 2 b 2 2 [ 6 a 1 2 a 2 2 b 1 2 b 2 2 ( a 2 2 p 1
+ b 2 2 q 1 ) + 9 a 1 2 a 2 2 b 1 2 b 2 2 z 12 a 2 2 b 2 2 r 1 z 12
- z 12 2 ( a 2 2 p 1 + b 2 2 q 1 ) - 2 z 12 3 ] ( 15 ) .gamma. 2 =
- 27 a 1 4 b 1 4 a 2 4 b 2 4 + a 2 4 b 2 4 r 1 2 - 6 a 1 2 b 1 2 a
2 2 b 2 2 .times. 2 ( a 2 2 p 1 + b 2 2 q 1 ) 2 + 3 a 2 2 b 2 2 r 1
- 2 a 2 2 b 2 2 z 12 ( a 2 2 p 1 + b 2 2 q 1 ) ( 9 a 1 2 b 1 2 - 2
r 1 ) + z 12 2 ( a 2 2 p 1 + b 2 2 q 1 ) 2 + 12 a 2 2 b 2 2 r 1 [ (
a 2 2 b 1 2 + a 1 2 b 2 2 ) .times. cos 2 ( .theta. 1 - .theta. 2 )
+ ( a 1 2 a 2 2 + b 1 2 b 2 2 ) sin 2 ( .theta. 1 - .theta. 2 ) ] 2
( 16 ) .gamma. 1 = 2 ( [ a 2 2 p 1 + b 2 2 q 1 ] { - a 2 2 b 2 2 r
1 2 + a 1 2 b 1 2 [ 2 ( a 2 2 p 1 + b 2 2 q 1 ) 2 + 9 a 2 2 b 2 2 r
1 ] } - r 1 z 12 ( a 2 2 p 1 + b 2 2 q 1 ) 2 + 6 a 2 2 b 2 2 r 1 )
( 17 ) .gamma. 0 = r 1 2 4 a 2 2 b 2 2 r 1 + ( a 2 2 p 1 + b 2 2 q
1 ) 2 Where ( 18 ) .PHI. 1 = .DELTA. y cos .theta. 1 - .DELTA. x
sin .theta. 1 ( 19 ) 1 = .DELTA. x cos .theta. 1 + .DELTA. y sin
.theta. 1 ( 20 ) .PHI. 2 = .DELTA. y cos .theta. 2 - .DELTA. x sin
.theta. 2 ( 21 ) 2 = .DELTA. x cos .theta. 2 + .DELTA. y sin
.theta. 2 ( 22 ) p 1 = .PHI. 2 2 - a 1 2 sin 2 ( .theta. 1 -
.theta. 2 ) - b 1 2 cos 2 ( .theta. 1 - .theta. 2 ) ( 23 ) q 1 = 2
2 - a 1 2 cos 2 ( .theta. 1 - .theta. 2 ) - b 1 2 sin 2 ( .theta. 1
- .theta. 2 ) ( 24 ) r 1 = b 1 2 1 2 + a 1 2 ( .PHI. 1 2 - b 1 2 )
( 25 ) z 12 = a 1 2 [ a 2 2 sin 2 ( .theta. 1 - .theta. 2 ) + b 2 2
cos 2 ( .theta. 1 - .theta. 2 ) ] + b 1 2 a 2 2 cos 2 ( .theta. 1 -
.theta. 2 ) + b 2 2 sin 2 ( .theta. 1 - .theta. 2 ) ( 26 )
##EQU00003##
[0053] 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 FIG. 6b. This distance is equal to the expansion factor k
(step S607). Once calculated, the expansion factor is output at
step S608.
Dual Sided Expansion
[0054] 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 .delta..sub.min, as explained above in
relation to Eq. 3. If the separation is not greater than
.delta..sub.min, 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
.delta..sub.min, dual sided expansion can proceed; both ellipses
are expanded equally so set k.sub.1=k.sub.2=k. The characteristic
polynomial becomes P(.lamda.)=det[.lamda.E.sub.1(k)-E.sub.2(k)]=0
(step S612). This results in another quartic equation in k.sup.2;
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).
EXAMPLES
[0055] Some examples of elliptical configurations are shown in
FIGS. 7a to 9b. The configurations of two ellipses on FIGS. 7a, 8a
and 9a correspond to the configurations in FIGS. 7b, 8b and 9b,
respectively. In FIGS. 7a, 8a and 9a, the dashed ellipses represent
the expanded, osculating ellipses when a dual expansion method is
used. FIGS. 7b, 8b and 9b the dashed ellipse represents the
expansion of one of the ellipses in a single sided expansion.
[0056] 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:
[0057] FIG. 7a--the dual sided expansion for the ellipses
E.sub.1(3, 2, 0.degree., 0, 0) and E.sub.2(4, 2, 90.degree., 8, 0)
gives the expansion factor k=1.6;
[0058] FIG. 7b--the single sided expansion for the ellipses
E.sub.1(3, 2, 0.degree., 0, 0) and E.sub.2(4, 2, 90.degree., 8, 0)
gives the expansion factor k=2.5;
[0059] FIG. 8a--the dual sided expansion for the ellipses
E.sub.1(3, 2, 30.degree., -3, -2) and E.sub.2(2, 1, 135.degree., 1,
1) gives the expansion factor k=1.25568;
[0060] FIG. 8b--the single sided expansion for the ellipses
E.sub.1(3, 2, 30.degree., -3, -2) and E.sub.2(2, 1, 135.degree., 1,
1) gives the expansion factor k=2.01033;
[0061] FIG. 9a--the dual sided expansion for the ellipses
E.sub.1(7, 1, 135.degree., -2, 0) and E.sub.2(5, 1, 150.degree., 1,
0) gives the expansion factor k=0.814767; and
[0062] FIG. 9b--the single sided expansion for the ellipses
E.sub.1(7, 1, 135.degree., -2, 0) and E.sub.2(5, 1, 150.degree., 1,
0) gives the expansion factor k=0.695637.
[0063] Referring to the configurations of FIGS. 7a, 8a and 9a,
Table 1 shows a comparison of the CVM and PCM separation factors
(k.sub.CVM and k.sub.PCM, respectively) with the dual sided
expansion factor (k.sub.ZPM) for the three elliptical
configurations.
TABLE-US-00001 Factor Ellipses k.sub.CVM k.sub.PCM k.sub.ZPM
E.sub.1 E.sub.2 [-] [-] [-] 3, 2, 0.degree., 0, 0 4, 2, 90.degree.,
8, 0 1.60000 1.60000 1.60000 3, 2, 30.degree., -3, -2 2, 1,
135.degree., 1, 1 1.25593 1.24452 1.25568 7, 1, 135.degree., -2, 0
5, 1, 150.degree., 1, 0 0.91190 0.32055 0.81477
[0064] 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 FIG. 7a; calculations agree only in a
special case where the major or minor axes of the ellipses are
collinear. The differences in the calculated factors in any
particular case may be much more pronounced as eccentricities
increase.
System
[0065] As shown in FIG. 13, wellbore drilling systems generally
comprise drilling equipment 4 arranged to drill a wellbore 5 into
the one or more hydrocarbon-bearing reservoirs in a formation 6.
The drilling system typically comprises a controller 7 arranged to
control the drilling equipment. An existing wellbore 8 is also
shown.
[0066] 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 FIG. 10. A
reservoir model 121 may be employed. As known in the art, a
reservoir model is a conceptual 3-dimensional construction of a
reservoir that is constructed from incomplete data with much of the
inter-well space estimated from data obtained from nearby wells or
from seismic data. In conjunction with this, a trajectory model
123, that is, a computer model that constructs 2D and/or 3D
representations of the geographical locations and/or trajectories
of wellbores may be employed. The trajectory model may comprise or
make use of a collision avoidance plot, also known as a travelling
cylinder or normal plane diagram. An expansion factor calculation
tool 111, as described further below, can calculate the expansion
factor as explained above. Using the reservoir model 121, the
trajectory model 123 can use information such as the volume and
shape of the reservoir 3 (including the arrangement of overlying
rock formations and the locations of any faults or fractures in the
rock formations and sub-surface hazards), the porosity of the
oil-bearing rock formations, the location of existing production
well(s) and injection well(s), in combination with the results of
the expansion factor calculation tool 111, to provide an indication
as to the possible trajectory of a planned wellbore.
[0067] In one arrangement, referring to FIG. 10, the expansion
factor calculation tool 111 and optionally the reservoir model 121,
the trajectory model 123 and an optimisation tool 125 are executed
by the wellbore positioning system 100. The wellbore positioning
system 100, which is for example a control system on a platform,
can comprise conventional operating system and storage components
such as a system bus connecting a central processing unit (CPU)
105, a hard disk 103, a random access memory (RAM) 101, and I/O and
network adaptors 107 facilitating connection to user input/output
devices and interconnection with other devices on a network N1. The
Random Access Memory (RAM) 101 contains operating system software
131 which controls, in a known manner, low-level operation of the
wellbore positioning system 100. The server RAM 101 contains the
software tools and models 111, 121, 123 and 125 during execution
thereof. Each item of software is configurable with measurement
and/or predetermined data stored in a database or other storage
component which is operatively coupled or connected to the wellbore
positioning system 100; in the system of FIG. 2, storage component
DB1 stores all such data relating to the expansion factor
calculation tool 111 and is accessible thereby, while storage
component DB2 stores all other data for use by the other components
of the system 100.
[0068] 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.
[0069] 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.
[0070] 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-a.sup.2b.sup.2k.sup.2--see Appendix A), H can be
made. The determinants can then be expanded and the equations
simplified using this program. Additionally, Mathematica.TM. 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.TM. within an EXCEL.TM.
spreadsheet.
[0071] 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 FIG. 11.
[0072] FIG. 11 shows a simplified collision avoidance plot which
may be produced by the trajectory model 123 upon calculation of the
expansion factor; the x and y axes represent length in metres. In
FIG. 11 the dashed ellipse represents the tolerable errors,
including an acceptable operational margin, for a planned wellbore
at some point in space. The solid ellipses represent the tolerable
errors surrounding three adjacent, drilled wellbores. By
inspection, at this position in the wellbore the planned wellbore
is heavily constrained and its position at this point cannot be
moved within the collision avoidance plot without infringing the
space in which the other wellbores may lie. The centre vector
method is generally excluded in such a scenario as it is overly
optimistic. Using the pedal curve method, a well planner would
conclude that the planned wellbore could not be threaded through
this point. A well planner using the expansion factor calculation
method, which honours the geometry, would conclude that the well
could, with care, pass through this point. This is confirmed by the
common sense approach that, visually, the dashed ellipse fits
comfortably within the available space.
[0073] 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 FIG. 11, then the wellbore would have to be planned around the
existing wellbores. Such activities add to the tortuosity of the
wellbore's trajectory, which increases torque and drag forces,
and/or may be difficult to achieve with the available tools. In
some cases the detour may not be possible. In subsurface terms, the
detour may make it difficult to achieve optimum alignment to a
target. If so, oil and gas reserves and production may be adversely
affected.
[0074] 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.
Method
[0075] Referring to FIG. 12, the steps involved in a first
embodiment of a computer-implemented method for determining one or
more operating modes for the wellbore drilling system are
shown.
[0076] In step S1201, the input data is received by the wellbore
positioning system 100.
[0077] 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 FIGS. 5a-5e, 6a and 6b. The expansion factor
calculation tool is then run in step S1203, and generates, at step
S1204, position data indicative of a relative position or proximity
of the planned wellbore to the existing or simulated wellbore or
object. This data may be output in various forms, for example, as
coordinates of a 2D or 3D simulation of a reservoir, or as a
collision avoidance plot.
[0078] 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.
[0079] 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.
[0080] 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.
[0081] 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.
[0082] 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.
[0083] 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 Representations
[0084] The 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 Form
[0085] The ellipse E.sub.0(x,
y)=b.sup.2x.sup.2+a.sup.2y.sup.2-a.sup.2b.sup.2=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.
E 0 ( x , y ) = [ x y 1 ] [ b 2 0 0 0 a 2 0 0 0 - a 2 b 2 ] [ x y 1
] ( A - 1 ) ##EQU00004##
[0086] Writing the ellipse E.sub.0(x, y)=xE.sub.0x.sup.T gives
E _ 0 = [ b 2 0 0 0 a 2 0 0 0 - a 2 b 2 ] ( A - 2 )
##EQU00005##
[0087] The matrix T translates a point on the ellipse by an amount
x.sub.0 in the x direction and y.sub.0 in the y direction. The
rotation matrix R rotates a point by an amount .theta. clockwise
about the origin. The scaling matrix S scales a point by a factor k
relative to the origin.
T _ = [ 1 0 0 0 1 0 - x 0 - y 0 1 ] ( A - 3 ) R _ = [ cos .theta. -
sin .theta. 0 sin .theta. cos .theta. 0 0 0 1 ] ( A - 4 ) S _ = [ k
- 1 0 0 0 k - 1 0 0 0 1 ] ( A - 5 ) ##EQU00006##
[0088] 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=xTRSE.sub.0S.sup.TR.sup.TT.sup.Tx.sup.T (A-6)
Inverse Transform
[0089] 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-a.sup.2b.sup.2k.sup.2.
x 0 = CD - BF B 2 - AC ( A - 7 ) y 0 = AF - BD B 2 - AC ( A - 8 ) a
= 2 ( AF 2 + CD 2 + GB 2 - 2 BDF - ACG ) ( B 2 - AC ) [ ( A - C ) 2
+ 4 B 2 - ( A + C ) ] ( A - 9 ) b = 2 ( AF 2 + CD 2 + GB 2 - 2 BDF
- ACG ) ( B 2 - AC ) [ - ( A - C ) 2 + 4 B 2 - ( A + C ) ] ( A - 10
) .theta. = { 0 B = 0 , A < C .pi. / 2 B = 0 , A > C 1 2 tan
- 1 ( 2 B A - C ) B .noteq. 0 , A < C .pi. 2 + 1 2 tan - 1 ( 2 B
A - C ) B .noteq. 0 , A > C ( A - 11 ) ##EQU00007##
Second Quadratic Form
[0090] 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 .theta.=[sin .theta., cos .theta.]. In Zheng and
Palffy-Muhoray's paper these authors assume k=1 throughout.
[ x - x 0 , y - y 0 ] [ m 11 m 12 m 21 m 22 ] [ x - x 0 y - y 0 ] =
k 2 Where ( A - 12 ) [ m 11 m 12 m 21 m 22 ] = 1 b 2 [ I _ + ( b 2
a 2 - 1 ) .theta. _ T .theta. _ ] ( A - 13 ) ##EQU00008##
APPENDIX B
Expansion Factors
YKC Expansion Factor (Single Sided)
[0091] For a single sided expansion the characteristic polynomial
becomes P(.lamda.)=det[.lamda.E.sub.1-E.sub.2(k)]=0, Eq. B-1. For
conciseness substitute .chi.=k.sup.2. 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.
det { .lamda. [ A 1 B 1 D 1 B 1 C 1 F 1 D 1 F 1 H 1 - a 1 2 b 1 2 ]
- [ A 2 B 2 D 2 B 2 C 2 F 2 D 2 F 2 H 2 - a 2 2 b 2 2 .chi. ] } = 0
( B - 1 ) w 3 ( .chi. ) .lamda. 3 + w 2 ( .chi. ) .lamda. 2 + w 1 (
.chi. ) .lamda. + w 0 ( .chi. ) = 0 ( B - 2 ) ##EQU00009##
[0092] Choi, 2008 showed that the cubic discriminant equals zero
when the ellipses touch externally, Eq. B-3.
w.sub.2.sup.2w.sub.1.sup.2-4w.sub.2.sup.3w.sub.0+18w.sub.3w.sub.2w.sub.1-
w.sub.0-4w.sub.3w.sub.1.sup.3-27w.sub.3.sup.2w.sub.0.sup.2=0
(B-3)
[0093] By inspection, the coefficients w.sub.j are of the form
w.sub.j=u.sub.j+v.sub.j.chi.
[0094] Making this substitution, Eq. B-3 then becomes a quartic
equation in .chi., Eq. B-4.
( v 1 2 v 2 2 - 4 v 1 3 v 3 ) .chi. 4 ( 2 u 2 v 1 2 v 2 + 2 u 1 v 1
v 2 2 - 4 u 0 v 2 3 - 12 u 1 v 1 2 v 3 + 18 u 0 v 1 v 2 v 3 ) .chi.
3 + ( u 2 2 v 1 2 + 4 u 1 u 2 v 1 v 2 + u 1 2 v 2 2 - 12 u 0 u 2 v
2 2 - 12 u 1 2 v 1 v 3 + 18 u 0 u 2 v 1 v 3 + 18 u 0 u 1 v 2 v 3 -
27 u 0 2 v 3 2 ) .chi. 2 + ( 2 u 1 u 2 2 v 1 + 2 u 1 2 u 2 v 2 - 12
u 0 u 2 2 v 2 - 4 u 1 3 v 3 + 18 u 0 u 1 v 3 ) .chi. + u 1 2 u 2 2
- 4 u 0 u 2 3 = 0 ( B - 4 ) ##EQU00010##
[0095] Note that both u.sub.3=0 and v.sub.0=0 and
u 0 = - a 1 4 b 1 4 ( B - 5 ) u 1 = a 1 2 b 1 2 { - a 2 2 b 1 2 cos
2 ( .theta. 2 - .theta. 1 ) - a 1 2 a 2 2 sin 2 ( .theta. 2 -
.theta. 1 ) + b 2 2 cos 2 ( .theta. 2 - .theta. 1 ) + b 2 2 - b 1 2
sin 2 ( .theta. 2 - .theta. 1 ) + ( .DELTA. x 0 cos .theta. 2 +
.DELTA. y 0 sin .theta. 2 ) 2 ] + a 2 2 ( .DELTA. y 0 cos .theta. 2
- .DELTA. x 0 sin .theta. 2 ) 2 } ( B - 6 ) u 2 = a 2 2 b 2 2 [ - a
1 2 b 1 2 + b 1 2 ( .DELTA. x 0 cos .theta. 1 + .DELTA. y 0 sin
.theta. 1 ) 2 + a 1 2 ( .DELTA. y 0 cos .theta. 1 - .DELTA. x 0 sin
.theta. 1 ) 2 ] ( B - 7 ) v 1 = - a 1 2 b 1 2 a 2 2 b 2 2 ( B - 8 )
v 2 = - a 2 2 b 2 2 { a 1 2 a 2 2 sin 2 ( .theta. 1 - .theta. 2 ) +
b 2 2 cos 2 ( .theta. 1 - .theta. 2 ) + b 1 2 a 2 2 cos 2 ( .theta.
1 - .theta. 2 ) + b 2 2 sin 2 ( .theta. 1 - .theta. 2 ) } ( B - 9 )
v 3 = - a 2 2 b 2 2 ( B - 10 ) ##EQU00011##
[0096] Then substitute Eqs. B-5 to B-10 into Eq. B-4.
Simplification gives the coefficients .gamma..sub.i of the quartic
equation, Eq. 13 given in the specific description.
YKC Expansion Factor (Dual Sided)
[0097] The derivation of the dual sided expansion with the
characteristic polynomial
P(.lamda.)=deq[.lamda.E.sub.1(k)-E.sub.2(k)]=0 proceeds in the same
way. Again for conciseness substitute .chi.=k.sup.2.
det { .lamda. [ A 1 B 1 D 1 B 1 C 1 F 1 D 1 F 1 H 1 - a 1 2 b 1 2
.chi. ] - [ A 2 B 2 D 2 B 2 C 2 F 2 D 2 F 2 H 2 - a 2 2 b 2 2 .chi.
] } = 0 ( B - 11 ) ##EQU00012##
[0098] The coefficients .gamma..sub.i of the quartic equation may
be calculated as Eq. B-12 to B-24.
.PHI. 1 = .DELTA. y cos .theta. 1 - .DELTA. x sin .theta. 1 ( B -
12 ) 1 = .DELTA. x cos .theta. 1 + .DELTA. y sin .theta. 1 ( B - 13
) .PHI. 2 = .DELTA. y cos .theta. 2 - .DELTA. x sin .theta. 2 ( B -
14 ) 2 = .DELTA. x cos .theta. 2 + .DELTA. y sin .theta. 2 ( B - 15
) r 1 = .PHI. 1 2 a 1 2 + 1 2 b 1 2 ( B - 16 ) r 2 = .PHI. 2 2 a 2
2 + 2 2 b 2 2 ( B - 17 ) p 1 = a 1 2 a 2 2 sin 2 ( .theta. 1 -
.theta. 2 ) + b 1 2 + b 2 2 cos 2 ( .theta. 1 - .theta. 2 ) + b 1 2
a 2 2 cos 2 ( .theta. 1 - .theta. 2 ) + b 2 2 sin 2 ( .theta. 1 -
.theta. 2 ) ( B - 18 ) p 2 = a 2 2 a 1 2 sin 2 ( .theta. 1 -
.theta. 2 ) + b 1 2 cos 2 ( .theta. 1 - .theta. 2 ) + b 2 2 a 1 2
cos 2 ( .theta. 1 - .theta. 2 ) + a 2 2 + b 1 2 sin 2 ( .theta. 1 -
.theta. 2 ) ( B - 19 ) .gamma. 4 = 1 2 { b 1 2 b 2 2 sin 2 (
.theta. 1 - .theta. 2 ) + a 2 2 [ b 1 2 cos 2 ( .theta. 1 - .theta.
2 ) - b 2 2 ] + a 1 2 [ a 2 2 sin 2 ( .theta. 1 - .theta. 2 ) - b 1
2 + b 2 2 cos 2 ( .theta. 1 - .theta. 2 ) ] } 2 .times. ( 2 a 1 4 [
a 2 2 sin 2 ( .theta. 1 - .theta. 2 ) + b 2 2 cos 2 ( .theta. 1 -
.theta. 2 ) ] 2 + 2 b 1 4 [ a 2 2 cos 2 ( .theta. 1 - .theta. 2 ) +
b 2 2 sin 2 ( .theta. 1 - .theta. 2 ) ] 2 + a 1 2 b 1 2 { a 2 2 b 2
2 [ - 5 + cos 4 ( .theta. 1 - .theta. 2 ) ] + 4 ( a 2 4 + b 2 4 )
sin 2 ( .theta. 1 - .theta. 2 ) cos 2 ( .theta. 1 - .theta. 2 ) } )
( B - 20 ) .gamma. 3 = 2 ( - 9 a 1 2 b 1 2 a 2 2 b 2 2 r 1 p 2 + 6
a 1 2 b 1 2 r 2 p 2 2 - 9 a 1 2 b 1 2 a 2 2 b 2 2 r 2 p 1 - r 1 p 2
2 p 1 + 6 a 2 2 b 2 2 r 1 p 1 2 - r 2 p 2 p 1 2 ) ( B - 21 )
.gamma. 2 = 18 a 1 2 b 1 2 a 2 2 b 2 2 r 1 r 2 - 12 a 1 2 b 1 2 r 2
2 p 2 + r 1 2 p 2 2 - 12 a 2 2 b 2 2 r 1 2 p 1 + 4 r 1 r 2 p 1 p 2
+ r 2 2 p 1 2 ( B - 22 ) .gamma. 1 = 2 ( 2 a 2 2 b 2 2 r 1 3 + 2 a
1 2 b 1 2 r 2 3 - r 1 2 r 2 p 2 - r 2 2 r 1 p 1 ) ( B - 23 )
.gamma. 0 = r 1 2 r 2 2 ( B - 24 ) ##EQU00013##
ZPM Expansion Factor
[0099] Zheng and Palffy-Muhoray approach leads to a quartic
equation in the variable Q, Eq. (B-25).
tan 2 .phi. ( .zeta. + 1 - Q 2 ) ( Q b 2 ' + 1 ) 2 = ( Q 2 - 1 ) (
Q b 2 ' + 1 + .zeta. ) 2 ( B - 25 ) ##EQU00014##
[0100] This can be written in the standard form, Eq. (B-26)
.psi..sub.4Q.sup.4+.psi..sub.3Q.sup.3+.psi..sub.2Q.sup.2+.psi..sub.1Q+.p-
si..sub.0=0 (B-26)
[0101] Where the quartic coefficients .psi..sub.i are
.psi. 4 = - 1 b 2 '2 ( 1 + tan 2 .phi. ) ( B - 27 ) .psi. 3 = - 2 b
2 '2 ( 1 + .zeta. + tan 2 .phi. ) ( B - 28 ) .psi. 2 = - tan 2
.phi. - ( 1 + .zeta. ) 2 + 1 b 2 '2 [ 1 + ( 1 + .zeta. ) tan 2
.phi. ] ( B - 29 ) .psi. 1 = 2 b 2 ' ( 1 + tan 2 .phi. ) ( 1 +
.zeta. ) ( B - 30 ) .psi. 0 = ( 1 + .zeta. + tan 2 .phi. ) ( 1 +
.zeta. ) ( B - 31 ) ##EQU00015##
To avoid a clash of symbols, note that the nomenclature used here
differs from that used by Zheng and Palffy-Muhoray. The variables
.zeta. and .phi. used here are defined in their paper.
APPENDIX C
Nomenclature
[0102] a=Ellipse semi-major axis length, L, ft
[0103] a=Unit vector in the major axis direction
[0104] b=Ellipse semi-minor axis length, L, ft
[0105] d=Diameter, L, ft
[0106] k=Expansion scale factor, dimensionless
[0107] m=Element of a transformation matrix
[0108] p=Substituted variable
[0109] r=Substituted variable
[0110] s=Characteristic length for a separation factor, L, ft
[0111] u=Substituted variable
[0112] v=Substituted variable
[0113] w=Coefficients of the YKC cubic equation
[0114] x=Ordinate in the normal plane, L, ft
[0115] y=Ordinate in the normal plane, L, ft
[0116] z=Substituted variable
[0117] A=First ellipse quadratic form coefficient
[0118] B=Second ellipse quadratic form coefficient
[0119] C=Third ellipse quadratic form coefficient
[0120] D=Fourth ellipse quadratic form coefficient
[0121] E=Ellipse
[0122] F=Fifth ellipse quadratic form coefficient
[0123] G=Sixth ellipse quadratic form coefficient
[0124] H=Modified sixth ellipse quadratic form coefficient
[0125] E=Ellipse matrix representation
[0126] R=Rotation matrix
[0127] S=Scaling matrix
[0128] T=Translation matrix
[0129] I=Unit matrix
[0130] P=Polynomial
[0131] Q=Independent variable of the ZPM quartic equation
Greek Symbols
[0132] .delta.=Centre to centre distance between ellipses, L,
ft
[0133] .gamma.=Coefficients of the YKC quartic equation
[0134] .phi.=Substituted or temporary variable
[0135] .differential.=Substituted or temporary variable
[0136] .psi.=Coefficients of the ZPM quartic equation
[0137] .chi.=Square of the expansion factor, dimensionless
[0138] .theta.=Ellipse orientation angle to major axis, radians
[0139] .phi.=First variable defined by ZPM=
[0140] .zeta.=Second variable defined by ZPM
[0141] .lamda.=Multiplier
[0142] .DELTA.=A difference in a parameter
Subscripts and Superscripts
[0143] 0=Condition at an origin
[0144] '=Transformed condition
[0145] 1,2,3,4=First, second etc.
[0146] c=Casing
[0147] cr=Critical, or closest approach
[0148] h=Hole
[0149] i=Index
[0150] j=Index
[0151] s=Separation factor
* * * * *