CN110847893B - Method for constructing borehole trajectory error elliptic cylinder - Google Patents
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Abstract
The invention provides a method for constructing a borehole trajectory error elliptic cylinder, which comprises the steps of obtaining inclination measurement data of a borehole trajectory by using a measuring instrument according to an industrial specification, solving a covariance matrix at each measurement point under a wellhead coordinate system NEH based on an ISCSSA standard and a model, wherein the covariance matrix represents an error ellipsoid family, and the probabilities of different error ellipsoids in the error ellipsoid family containing borehole trajectory errors are different; selecting an amplification factor to uniquely determine an error ellipsoid according to the actual drilling engineering requirement; determining the size and attitude of the error ellipsoid based on the covariance matrix of the error ellipsoid; determining the cross section ellipse of the error elliptic cylinder based on the size and the posture of the error ellipsoid according to the tangent condition of the elliptic cylindrical surface and the ellipsoid; and connecting the cross-section ellipses in series along the well track to form an error elliptic cylinder of the well track, so as to characterize and evaluate the error and the change of the well track along the well depth.
Description
Technical Field
The invention relates to the field of oil and gas well engineering, in particular to a method for constructing a borehole trajectory error elliptic cylinder in the oil and gas well engineering.
Background
The basic goal of wellbore trajectory monitoring and control is to determine the spatial location of any point on the wellbore trajectory, i.e., the wellbore trajectory location. However, due to measurement, calculation, etc. errors, the borehole trajectory positioning may not be absolutely accurate. Although the positioning accuracy of the borehole trajectory can be improved by means of error correction and the like, the errors cannot be completely eliminated, so that the borehole trajectory still has an uncertainty problem.
An error ellipsoid of each measuring point can be obtained based on the borehole trajectory error model, and the ellipsoid is used for representing uncertainty of the spatial position of the borehole trajectory. During the drilling process, error ellipsoids are required to be connected in series along the borehole trajectory to form an error elliptic cylinder for describing the on-way error condition of the borehole trajectory. At present, only error ellipsoids and evaluation methods of horizontal section error ellipses thereof (Liu Gonghui, dong Benjing and Gao Deli. Error ellipsoids (circles) and well collision probability analysis [ J ]. Drilling and production process, 2000,23 (3): 5-12.) exist, and no construction method of well track error elliptic columns exists.
Therefore, it is necessary to establish a corresponding characterization and evaluation method to make up for the deficiencies of the prior art.
Disclosure of Invention
In order to solve the above problems, the present invention provides a method for constructing a borehole trajectory error elliptic cylinder, the method comprising:
acquiring inclination measurement data of a well track by using a measuring instrument according to industrial regulations, and solving a covariance matrix at each measurement point under a wellhead coordinate system NEH based on an ISCSSA standard and a model, wherein the covariance matrix represents an error ellipsoid group, and the probabilities of different error ellipsoids in the error ellipsoid group containing well track errors are different;
selecting an amplification factor to uniquely determine the error ellipsoid according to the probability required by actual drilling engineering;
determining the size and the attitude of the error ellipsoid based on a covariance matrix of the error ellipsoid, wherein the size and the attitude of the error ellipsoid are characterized by three principal axis radii and three attitude angles, respectively;
determining a cross section ellipse of the error elliptic cylinder based on the size and the posture of the error ellipsoid according to the tangent condition of the elliptic cylindrical surface and the ellipsoid;
and connecting the cross-section ellipses in series along the well track to form an error elliptic cylinder of the well track, so as to characterize and evaluate the error and the change of the well track along the well depth.
According to the method for constructing the borehole trajectory error elliptic cylinder, the inclination measurement data preferably comprise a well inclination angle, an azimuth angle, a well depth and a tool face angle, and the covariance matrix of the error elliptic cylinder is a 3 x 3 matrix [ C ] under a wellhead coordinate system NEH] NEH Wherein: n is north coordinate, m; e is the east coordinate, m; h is the vertical depth coordinate, m.
According to the method for constructing a borehole trajectory error ellipsoid of the present invention, it is preferable that in the step of uniquely determining the error ellipsoid according to the probability, an amplification factor is obtained by using a relational expression between the probability and the amplification factor as follows, and the error ellipsoid is further determined:
wherein: k is an amplification factor and has no dimension; p is the error probability, decimal, of the borehole trajectory.
According to the method for constructing the borehole trajectory error elliptic cylinder, the size of the error ellipsoid is preferably calculated according to the following steps after the error ellipsoid is determined:
taking the main axis of the ellipsoid close to the high side direction of the ellipsoid as a U axis, taking the main axis of the ellipsoid close to the plumb direction as a W axis, and determining a V axis according to a right hand rule to enable the U axis, the V axis and the W axis to form a right hand coordinate system;
the sizes of error ellipsoids are represented by the three main shaft radiuses of the error ellipsoids, and a covariance matrix [ C ] is obtained by adopting a Jacobian method and a linear transformation method] NEH Characteristic value (λ) of U ,λ V ,λ W ) Then, the radius of the principal axis of the error ellipsoid is calculated as follows:
in the formula: r is the radius of the main shaft of the error ellipsoid, m; λ is the eigenvalue of the covariance matrix, m 2 。
According to the method for constructing the borehole trajectory error elliptic cylinder, preferably, after the error ellipsoid is determined, the attitude of the error ellipsoid is represented by three attitude angles of a main axis of the error ellipsoid, and the attitude of the error ellipsoid is calculated according to the following steps:
determining a characteristic value (lambda) 1 ,λ 2 ,λ 3 ) Corresponding feature vector (p) 1 ,p 2 ,p 3 ) These feature vectors are represented in the well head coordinate system O-NEH
The attitude angle of the error ellipsoid is calculated according to the following formula
In the formula: i. j and k are unit coordinate vectors on an N axis, an E axis and an H axis respectively; (P) 1N ,P 1E ,P 1H )、(P 2N ,P 2E ,P 2H ) And (P) 3N ,P 3E ,P 3H ) Are respectively a feature vector p 1 、p 2 And p 3 A component of (a); alpha is alpha W 、φ W And theta W Respectively the well inclination angle, azimuth angle and attitude angle of the main axis W of the error ellipsoid (°).
According to the method for constructing the error elliptic cylinder of the borehole trajectory, the method preferably further comprises the following sub-steps in the step of constructing the error elliptic cylinder:
establishing a standard equation of the error ellipsoid based on the size and the posture of the error ellipsoid;
determining the cross section ellipse of the error elliptic cylinder according to the tangent condition of the elliptic cylinder and the ellipsoid;
and characterizing the error elliptic cylinder by using a cross section ellipse, and connecting the cross section ellipses in series along the borehole trajectory to form the error elliptic cylinder of the borehole trajectory.
According to the method for constructing the borehole trajectory error elliptic cylinder, preferably, in the step of establishing the error elliptic equation, the rotation transformation relation between the elliptic main axis coordinate system UVW and the wellhead coordinate system NEH is
Wherein
Under a principal axis coordinate system UVW of an error ellipsoid, the equation of the error ellipsoid is
According to the method for constructing the borehole trajectory error elliptic cylinder, preferably, an error elliptic cylinder equation under a Regulas eye coordinate system xyz is established first, then the cross section ellipse of the error elliptic cylinder is determined according to the tangent condition of the elliptic cylinder and the elliptic cylinder,
the rotation transformation relation between the borehole coordinate system xyz and the wellhead coordinate system NEH is
Wherein
Wherein: the x axis points to the well-increasing inclined direction, the y axis points to the well-increasing azimuth direction, and the z axis points to the advancing direction of the well track;
in Reguladamole coordinate system xyz, the error ellipsoid equation is
Wherein
[B]＝[H] T [A] T
B i ＝[B i1 ,B i2 ,B i3 ]
r＝[x,y,z]
Based on an error ellipsoid equation under an eye Regulare coordinate system xyz, obtaining a cross section ellipsoid equation of an error elliptic cylinder according to the tangent condition of an elliptic cylinder surface and an ellipsoid as
Wherein
D ij ＝B ij -B i3 C j (i＝1,2,3；j＝1,2)
Expressing the cross-sectional ellipse equation in a matrix form
Wherein
According to the method for constructing the borehole trajectory error elliptic cylinder, preferably, in the step of characterizing the error elliptic cylinder, the cross section ellipse of the error elliptic cylinder is characterized by two main shaft radiuses and an attitude angle, and the calculation formula is
In the formula: sigma is the radius of the main shaft of the cross section ellipse of the error elliptic cylinder, and m; beta is the attitude angle of the cross section ellipse of the error elliptic cylinder, (°); g is the inverse of the matrix F;
the well track is provided with a series of measuring points, each measuring point is provided with an error ellipsoid, and the enveloping surfaces of the error ellipsoids form an error elliptic cylinder;
and (3) characterizing the error elliptic cylinder by using the cross section ellipse of the error elliptic cylinder, and connecting a series of cross section ellipses at each measuring point in series along the borehole trajectory to form the error elliptic cylinder of the borehole trajectory.
The method for constructing the elliptical column for the borehole trajectory error can evaluate the error and the change condition of the borehole trajectory along the well depth, can be used for preventing collision of adjacent wells, is widely applied to engineering design and construction of various wells with complex structures, and has wide application prospect.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 shows a flow diagram of a method of constructing a borehole trajectory error ellipsoid according to one embodiment of the present invention;
FIG. 2 shows a schematic diagram of an error ellipsoid characterization method according to an embodiment of the present invention; and
FIG. 3 shows a schematic diagram of a method for constructing an error elliptic cylinder according to one embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention are described in further detail below with reference to the accompanying drawings.
Aiming at the defects in the prior art, the invention provides the method for constructing the elliptic cylinder of the borehole trajectory errors, which can evaluate the errors of the borehole trajectory along the well depth and the change conditions of the errors and is suitable for wells with various complex structures. The general idea is as follows: identifying various error sources, and establishing a covariance matrix of the well track according to error propagation and accumulation principles; selecting an amplification factor according to the error probability, and determining the size and the posture of an error ellipsoid; and establishing an error ellipsoid and an error elliptic cylinder cross section elliptic equation to construct an error elliptic cylinder.
Referring to FIG. 1, a flow diagram of a method for constructing a borehole trajectory error ellipsoid is shown in accordance with one embodiment of the present invention.
In step S1 shown in FIG. 1, a covariance matrix of the borehole trajectory first needs to be established. Specifically, according to the error source identification, characterization, propagation and accumulation rules, a covariance matrix of the borehole trajectory is established in a wellhead coordinate system NEH to describe the uncertainty of the borehole trajectory.
In the process of establishing the covariance matrix, acquiring inclination measurement data of a well track by using a measuring instrument according to industrial regulations, and solving a covariance matrix at each measurement point under a wellhead coordinate system NEH based on an ISCSSA standard and a model, wherein the covariance matrix represents an error ellipsoid group, and the probabilities that different error ellipsoids in the error ellipsoid group contain well track errors are different.
The method specifically comprises the following substeps:
in sub-step S11, the measurement is carried out according to industry regulations. The method comprises the following steps: (1) strict and regular instrument checking; (2) the inclinometry distance is not more than 30m; (3) carrying out field tests such as gravity field, geomagnetic field, magnetic inclination angle, gyro drift and the like; (4) determining the nonmagnetic interval of the MWD according to the industry standard; (5) the use of MWD requires that the casing and adjacent well be kept remote.
In sub-step S12, various error sources are identified and characterized. Including the magnitude and weight functions of the various error sources.
In sub-step S13, an error propagation equation is established, expressed as follows:
in the formula：e i Errors of north coordinates N, east coordinates E and vertical depth coordinates H (3 × 1 vector) caused by error source i; epsilon i Is the ith error source; sigma i An error magnitude (scalar) that is an error source i;
is a weight function, and represents the influence of an error source i on the well depth, the well inclination angle and the azimuth angle (a 3 multiplied by 1 vector);
The effect of well depth, well offset angle and azimuth measurement errors on the N, E and H coordinates (3 x 3 matrix) is shown.
In sub-step S14, error accumulation is performed in the propagation mode.
Considering the correlation among various error sources, respectively carrying out error accumulation according to error propagation modes such as random error, system error, single well and global error and the like to obtain a 3 multiplied by 3 covariance matrix [ C ] at each measuring point] NEH 。
Then in step S2, a borehole trajectory error ellipsoid is determined.
Selecting an amplification coefficient of an error ellipsoid according to the requirement of the error probability of the borehole trajectory; and (5) solving an eigenvalue and an eigenvector of the covariance matrix, and determining the size and the posture of the error ellipsoid. Specifically comprising the following substeps:
s21: and selecting an amplification factor according to the error probability.
The covariance matrix at each survey point characterizes a family of ellipsoids, with different ellipsoids containing different probabilities of borehole trajectory error.
Specifically, an amplification factor is obtained by using a relation between the probability and the amplification factor as follows, and the error ellipsoid is determined:
wherein: k is an amplification factor and has no dimension; p is the error probability, decimal, of the borehole trajectory.
According to the specific requirement of the error probability of the well track, an error ellipsoid can be uniquely determined by selecting the amplification factor k. Generally, the amplification factor has a value ranging from k =1 to 3, and the borehole trajectory error is usually evaluated with k = 2. For wells with high requirements on borehole trajectory monitoring and control accuracy, a large coefficient k = 2.8 is preferably taken and placed, and the probability that the borehole trajectory is located in an error ellipsoid at the moment is more than 95%.
When the amplification factor k =1.0 to 4.0 and the step size is 0.5, the probabilities P are 19.87%, 47.78%, 73.85%, 89.99%, 97.07%, 99.34%, and 99.89%, respectively.
Next, in sub-step S22, an error ellipsoid size is calculated.
The error ellipsoid of the borehole trajectory has three principal axes. And taking the main axis of the ellipsoid close to the high side direction of the ellipsoid as a U axis, taking the main axis of the ellipsoid close to the plumb direction as a W axis, and determining a V axis according to a right-hand rule so that the U axis, the V axis and the W axis form a right-hand coordinate system. As shown in fig. 2.
The three principal axis radii of the error ellipsoid are used to characterize the error ellipsoid dimensions. Firstly, the covariance matrix [ C ] is obtained by the methods of Jacobian method and linear transformation] NEH Characteristic value (λ) of U ,λ V ,λ W ) Then, the radius of the principal axis of the error ellipsoid is calculated as follows:
in the formula: r is the radius of the main shaft of the error ellipsoid, m; λ is the eigenvalue of the covariance matrix, m2.
In sub-step S23 the error ellipsoid pose is calculated.
Three attitude angles of the main axis of the error ellipsoid are used for representing the attitude of the error ellipsoid.
First, a characteristic value (λ) is obtained 1 ,λ 2 ,λ 3 ) The corresponding feature vector (p 1, p2, p 3). Under the wellhead coordinate system O-NEH, these feature vectors can be represented as
Then, the attitude angle of the error ellipsoid is calculated as follows
In the formula: i. j and k are unit coordinate vectors on an N axis, an E axis and an H axis respectively; (P) 1N ,P 1E ,P 1H )、(P 2N ,P 2E ,P 2H ) And (P) 3N ,P 3E ,P 3H ) Are respectively a feature vector p 1 、p 2 And p 3 A component of (a); alpha (alpha) ("alpha") W 、φ W And theta W Respectively the well inclination angle, azimuth angle and attitude angle of the main axis W of the error ellipsoid (°).
Next in step S3, an error elliptic cylinder is constructed. Determining a cross section ellipse of the error elliptic cylinder based on the size and the posture of the error ellipsoid according to the tangent condition of the elliptic cylindrical surface and the ellipsoid; and connecting the cross-section ellipses in series along the well track to form an error elliptic cylinder of the well track, so as to characterize and evaluate the error and the change of the well track along the well depth.
Specifically, the method comprises the following substeps:
and a substep S31 of establishing an error ellipsoid equation.
Based on the result of step S2, the rotation transformation relation between the ellipsoid principal axis coordinate system UVW and the wellhead coordinate system NEH is
Wherein
Then, under the principal axis coordinate system UVW of the error ellipsoid, the error ellipsoid equation is
In sub-step S32, the error ellipsoid equation is converted to the eye Regulation coordinate system.
In one embodiment, the borehole coordinate system xyz is specifically established. And establishing a borehole coordinate system xyz by taking the high side of the borehole as an x axis, taking a borehole direction line, namely a borehole orbit tangent line, as a z axis, taking the y axis to be vertical to the x axis and the z axis, and forming a right-hand system by the x axis, the y axis and the z axis. Thus, the x-axis points in the increasing well deviation direction, the y-axis points in the increasing azimuth direction, and the z-axis points in the advancing direction of the borehole trajectory.
According to the differential geometry principle, the borehole coordinate system xyz and the wellhead coordinate system NEH are in rotational transformation relation
Wherein
Then, in the Reguladamole coordinate system xyz, the error ellipsoid equation is
Wherein
[B]＝[H] T [A] T
B i ＝[B i1 ,B i2 ,B i3 ]
r＝[x,y,z]
Next, in sub-step S33, a cross-sectional ellipse equation of the error elliptic cylinder is established.
There are a series of survey points along the borehole trajectory, each survey point having an error ellipsoid. The error ellipsoids are connected in series along the track of the borehole, and the enveloping surface is an elliptic cylindrical surface. As shown in fig. 3. At a measuring point, the elliptic cylindrical surface and the ellipsoid should be tangent, and the tangent point forms a closed space curve, which is the cross-section boundary curve of the error elliptic cylinder at the measuring point.
According to the tangent condition of the elliptic cylindrical surface and the ellipsoid, the cross section boundary curve of the error elliptic cylinder is
Wherein
D ij ＝B ij -B i3 C j (i＝1,2,3；j＝1,2)
Further, the formula (8) can also be written as
Wherein
Equations (9) and (10) are the cross-sectional ellipse equations of the error elliptic cylinder.
In sub-step S34, an error elliptic cylinder is characterized.
The cross-sectional ellipse of the error elliptic cylinder can be characterized by two principal axis radii and one attitude angle. Is calculated by the formula
In the formula: sigma is the radius of the main shaft of the cross section ellipse of the error elliptic cylinder, and m; beta is the attitude angle of the cross section ellipse of the error elliptic cylinder, (°); g is the inverse of the matrix F.
Thus, the cross-sectional ellipses at each survey point are connected in series along the borehole trajectory to form an error elliptic cylinder.
The method for constructing the elliptical column for the borehole trajectory error can evaluate the error and the change condition of the borehole trajectory along the well depth, can be used for preventing collision of adjacent wells, is widely applied to engineering design and construction of various wells with complex structures, and has wide application prospect.
The invention is further described below with reference to specific examples. The scope of the invention is not limited by the embodiments, but is set forth in the claims.
And (3) measuring and calculating the well track of a certain horizontal well according to the industrial regulations to obtain a calculation result taking true north as a reference datum, and the calculation result is shown in table 1. The geomagnetic field intensity of the well is 56356.51nT, the magnetic declination angle is-10.60 degrees, the magnetic dip angle is 64.72 degrees, and the meridian convergence angle is 0.876 degrees. According to the conventional practice in the industry, the large coefficient k =2 is taken and placed, and the borehole trajectory error elliptic cylinder construction method is adopted to obtain the evaluation results shown in the table 2. Table 1 and Table 2 list only a portion of the data, to be limited by space.
TABLE 1 borehole trajectory calculation results of the examples
| Well depth (m) | Oblique angle (degree) | Azimuth (°) | North coordinate (m) | East coordinate (m) | Vertical depth (m) |
| 0.00 | 0.00 | Is absent from | 0.00 | 0.00 | 0.00 |
| 900.00 | 0.00 | Is absent from | 0.00 | 0.00 | 900.00 |
| 1800.00 | 0.00 | Is absent from | 0.00 | 0.00 | 1800.00 |
| 2700.00 | 0.00 | Is absent from | 0.00 | 0.00 | 2700.00 |
| 3600.00 | 0.00 | Is absent from | 0.00 | 0.00 | 3600.00 |
| 4500.00 | 0.00 | Is absent from | 0.00 | 0.00 | 4500.00 |
| 5400.00 | 0.00 | Is absent from | 0.00 | 0.00 | 5400.00 |
| 6300.00 | 0.00 | Is absent from | 0.00 | 0.00 | 6300.00 |
| 7200.00 | 0.00 | Is absent from | 0.00 | 0.00 | 7200.00 |
| 7660.00 | 0.00 | (29.71) | 0.00 | 0.00 | 7660.00 |
| 7831.77 | 45.81 | 41.16 | 51.74 | 39.36 | 7814.05 |
| 8028.73 | 45.81 | 51.01 | 149.57 | 140.96 | 7951.35 |
| 8161.32 | 90.00 | 55.43 | 221.10 | 237.05 | 8000.00 |
| 8400.00 | 90.00 | 57.02 | 353.79 | 435.44 | 8000.00 |
| 8700.00 | 90.00 | 59.02 | 512.67 | 689.90 | 8000.00 |
| 9000.00 | 90.00 | 61.02 | 662.57 | 949.75 | 8000.00 |
| 9300.00 | 90.00 | 63.02 | 803.32 | 1214.67 | 8000.00 |
| 9600.00 | 90.00 | 65.02 | 934.73 | 1484.34 | 8000.00 |
| 9900.00 | 90.00 | 67.02 | 1056.65 | 1758.43 | 8000.00 |
| 10161.32 | 90.00 | 68.76 | 1155.00 | 2000.52 | 8000.00 |
TABLE 2 results of ellipsometry evaluation of borehole trajectory errors for the invention
In the embodiment, the borehole trajectory error elliptic cylinder is obtained by adopting the borehole trajectory error elliptic cylinder construction method, the size and the posture of the cross section ellipse of the error elliptic cylinder are calculated, and the error and the change condition of the borehole trajectory along the well depth can be evaluated.
It is to be understood that the disclosed embodiments of the invention are not limited to the particular structures, process steps, or materials disclosed herein but are extended to equivalents thereof as would be understood by those ordinarily skilled in the relevant arts. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting.
Reference in the specification to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. Thus, the appearances of the phrase "one embodiment" or "an embodiment" in various places throughout this specification are not necessarily all referring to the same embodiment.
Although the embodiments of the present invention have been described above, the above description is only for the convenience of understanding the present invention, and is not intended to limit the present invention. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (7)
1. A method of constructing a borehole trajectory error elliptic cylinder, the method comprising:
acquiring inclination measurement data of a well track by using a measuring instrument according to industrial regulations, and solving a covariance matrix at each measurement point under a wellhead coordinate system NEH based on an ISCSSA standard and a model, wherein the covariance matrix represents an error ellipsoid group, and the probabilities of different error ellipsoids in the error ellipsoid group containing well track errors are different;
selecting an amplification factor to uniquely determine the error ellipsoid according to the probability required by the actual drilling engineering;
determining the size and the attitude of the error ellipsoid based on a covariance matrix of the error ellipsoid, wherein the size and the attitude of the error ellipsoid are characterized by three principal axis radii and three attitude angles, respectively;
determining the cross section ellipse of the error elliptic cylinder based on the size and the posture of the error ellipsoid according to the tangent condition of the elliptic cylindrical surface and the ellipsoid;
connecting the cross-section ellipses in series along the well track to form an error elliptic cylinder of the well track, so as to characterize and evaluate the error and the change of the well track along the well depth;
the inclination measurement data comprises a well inclination angle, an azimuth angle, a well depth and a tool face angle, and the covariance matrix of the error ellipsoid is a 3 multiplied by 3 matrix [ C ] under a well head coordinate system NEH] NEH Wherein: n is north coordinate, m; e is the east coordinate, m; h is vertical depth coordinate m;
in the step of uniquely determining the error ellipsoid according to the probability, an amplification factor is obtained by using a relational expression between the probability and the amplification factor as follows, and the error ellipsoid is determined:
wherein: k is an amplification factor and has no dimension; p is the error probability, decimal, of the well track; r is the error ellipsoid radius.
2. The method of claim 1, wherein after the error ellipsoid is determined, the size of the error ellipsoid is calculated according to the following steps:
taking the main axis of the ellipsoid close to the direction of the high side of the ellipsoid as a U axis, taking the main axis of the ellipsoid close to the direction of the plumb as a W axis, and determining a V axis according to a right-hand rule to enable the U axis, the V axis and the W axis to form a right-hand coordinate system;
the sizes of error ellipsoids are represented by the three main shaft radiuses of the error ellipsoids, and a covariance matrix [ C ] is obtained by adopting a Jacobian method and a linear transformation method] NEH Characteristic value (λ) of U ,λ V ,λ W ) Then, the radius of the principal axis of the error ellipsoid is calculated as follows:
in the formula: r is the radius of the main shaft of the error ellipsoid, m; λ is the eigenvalue of the covariance matrix, m 2 。
3. The method of claim 2, wherein after the error ellipsoid is determined, the attitude of the error ellipsoid is characterized by three attitude angles of the principal axis of the error ellipsoid, and the attitude of the error ellipsoid is calculated by:
determining a characteristic value (lambda) 1 ,λ 2 ,λ 3 ) Corresponding feature vector (p) 1 ,p 2 ,p 3 ) These feature vectors are represented in the well head coordinate system O-NEH
The attitude angle of the error ellipsoid is calculated according to the following formula
In the formula: i. j and k are unit coordinate vectors on an N axis, an E axis and an H axis respectively; (P) 1N ,P 1E ,P 1H )、(P 2N ,P 2E ,P 2H ) And (P) 3N ,P 3E ,P 3H ) Are respectively a feature vector p 1 、p 2 And p 3 A component of (a); alpha (alpha) ("alpha") W 、φ W And theta W The inclination angle, azimuth angle and attitude angle of the main axis W of the error ellipsoid are (deg.) respectively.
4. The method of constructing a borehole trajectory error elliptical cylinder according to claim 3 further comprising, in the step of constructing the error elliptical cylinder, the sub-steps of:
establishing a standard equation of the error ellipsoid based on the size and the posture of the error ellipsoid;
determining the cross section ellipse of the error elliptic cylinder according to the tangent condition of the elliptic cylinder and the ellipsoid;
and characterizing the error elliptic cylinder by using a cross section ellipse, and connecting the cross section ellipses in series along the borehole trajectory to form the error elliptic cylinder of the borehole trajectory.
5. The method of claim 4, wherein in the step of establishing the error ellipsoid equation, the rotational transformation relationship between the ellipsoid principal axis coordinate system UVW and the wellhead coordinate system NEH is
Wherein
Under a principal axis coordinate system UVW of an error ellipsoid, the equation of the error ellipsoid is
In the formula: r 1 、R 2 、R 3 The three main shaft radiuses of an error ellipsoid under a main shaft coordinate system UVW.
6. The method of claim 5, wherein the error ellipsoid equation under the borehole coordinate system xyz is established, and the cross-sectional ellipse of the error ellipsoid is determined according to the tangent condition of the elliptic cylinder and the ellipsoid,
the rotation transformation relation between the borehole coordinate system xyz and the wellhead coordinate system NEH is
Wherein
Wherein: the x axis points to the well-increasing inclined direction, the y axis points to the well-increasing azimuth direction, and the z axis points to the advancing direction of the well track;
under the borehole coordinate system xyz, the error ellipsoid equation is
Wherein
[B]＝[H] T [A] T
B i ＝[B i1 ,B i2 ,B i3 ]
r＝[x,y,z]
Based on an error ellipsoid equation under a borehole coordinate system xyz, obtaining a cross section ellipsoid equation of an error ellipsoid according to the tangent condition of an elliptic cylindrical surface and an ellipsoid as
Wherein
D ij ＝B ij -B i3 C j (i＝1,2,3；j＝1,2)
Expressing the cross-sectional ellipse equation in a matrix form
Wherein
In the formula: r is i Is the radius of an error ellipsoid of the ith coordinate axis in the borehole coordinate system xyz.
7. The method of claim 6, wherein in the step of characterizing the error elliptic cylinder, the cross section ellipse of the error elliptic cylinder is characterized by two major axis radii and an attitude angle, and the calculation formula is
In the formula: sigma is the radius of the main axis of the cross section ellipse of the error elliptic cylinder, m; beta is the attitude angle of the cross section ellipse of the error elliptic cylinder, (°); g is the inverse of the matrix F, matrix
Matrix->
The well track is provided with a series of measuring points, each measuring point is provided with an error ellipsoid, and the enveloping surfaces of the error ellipsoids form an error elliptic cylinder;
and (3) characterizing the error elliptic cylinder by using the cross section ellipse of the error elliptic cylinder, and connecting a series of cross section ellipses at each measuring point in series along the borehole trajectory to form the error elliptic cylinder of the borehole trajectory.
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