Method for measuring lateral displacements of conduit trajectories

Abstract

A method for measuring lateral displacements of a conduit trajectory relative to a reference trajectory uses a measurement tool configured to measure an eccentricity of a selected measurement point on the conduit trajectory relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory. The chord line moves with the tool as the tool is moved between selected measurement points. An incremental eccentricity is calculated for each measurement point, and a first system of equations is assembled relating lateral displacements of the conduit trajectory to the calculated incremental eccentricities. Boundary conditions are applied to the first system of equations to create a second system of equations. Error suppression is applied to the second system of equations to create a third system of equations, which may be solved for the lateral displacements of the trajectory.

Description

[0001] METHOD FOR MEASURING LATERAL DISPLACEMENTS OF CONDUIT TRAJECTORIES FIELD

[0002] The present disclosure relates to methods for measuring lateral displacements of the trajectory of a conduit (such as a tubular conduit disposed in a wellbore) relative to a reference trajectory, based on multiple local measurements of the conduit made by a logging tool moving along the conduit.

[0003] BACKGROUND

[0004] Conduits of various types are commonly used for conveying fluids and protecting equipment like electric wires or mechanical cables. As used in the present disclosure, the term “conduit” may be broadly understood as referring to any type of elongate (but not necessarily linear) structure that has or defines an internal bore. Non-limiting examples of conduits in accordance with this meaning include generally cylindrical elements defining a substantially circular bore, such as buried or above-ground steel pipelines, and steel casing strings installed in subterranean wells. However, uncased boreholes drilled into the earth, as well as subterranean tunnels (such as roadway and railway tunnels), which are not necessarily circular in cross-section, also may be conduits in the context of the present disclosure.

[0005] In many applications, conduits require significant structural capacity to perform their functions (for example, pipelines and casing strings in petroleum wells). Some applications also require that internal access within a conduit be maintained to facilitate integrity assessments, or to permit passage of tools required for remote maintenance of equipment at the far end of a conduit (for example, downhole pumps in water and petroleum wells).

[0006] Subterranean conduits may be subject to various influences that can alter their physical shape. Mechanical loading, such as lateral loads resulting from geomechanical movements or buckling deformations resulting from axial loads, can change a subterranean conduit’s path (alternatively referred to herein as a “conduit trajectory”). Changes in a conduit’s cross-section may also be associated with mechanical loading (for example: cross-section ovalization will result from bending of a tubular conduit; and changes in a tubular conduit’s diameter can result from high axial loads and / or differential pressure acting on the conduit). A comprehensive assessmentof conduit deformation considers changes in both conduit cross-section and conduit trajectory, and the foregoing examples demonstrate how knowing the modes of deformation can help in identifying loading conditions that contribute to conduit deformations.

[0007] Measurement of conduit deformations is useful in assessing both the functional capacity and the internal accessibility of a conduit. Additionally, conduits may deform due to change in the surrounding environment; for example, well casings cemented in geological formations that move in response to hydraulic fracturing, and pipelines traversing regions experiencing subsidence caused by permafrost thaw. In such situations, a measurement of conduit trajectory deformation can be used as a measurement of the surrounding environment (e.g., as a measurement of the geological movements and the thaw-induced subsidence in the above examples).

[0008] Internal measurements are commonly taken in many types of conduits (a process that may be called “logging”) using tools that are centralized in a conduit and that take measurements at selected spatial intervals as the tools are moved within and along the length of the bore of the conduit. Examples of prior art tools for taking cross-section geometry measurements of a conduit bore, including mechanical caliper and ultrasonic tools, are disclosed in:

[0009] • U. S. Patent No. 4,186,494 (Peruchon et al.);

[0010] • U. S. Patent No. 4,559,709 (Beseme et al.);

[0011] • U. S. Patent No. 4,876,672 (Petermann et al.);

[0012] • U. S. Patent No. 10,605,582 (Ohmer et al.); and

[0013] • U. S. Patent No. 12,146,402 (Manders et al.).

[0014] A measurement tool (alternatively referred to herein as “logging tool”) may be conveyed within and along a selected length of a conduit bore during logging by a wireline cable, a coiled tubing string, or a tractor device. The position of the measurement tool along the length of the conduit may be determined based on the length of wireline or coiled tubing deployed in the conduit relative to a reference position, which is commonly one end of the conduit where the measurement tool enters and exits the conduit. The position of the measurement tool along the length of the conduit may alternatively be determined based on the rotation of a measurement wheel mounted to the measurement tool that rolls against the conduit bore.Various methodologies for interpretation of conduit cross-section deformations are also known in the prior art, but the prior art contains comparatively few examples of methodologies for conduit trajectory measurement and deformation interpretation.

[0015] U. S. Patent No. 9,605,938 (Helmore) describes a method of determining the geometry (trajectory) of a deformed conduit comprising the step of selecting a deformation model to constrain the processing of measurement data. However, trajectory deformations are often complex and unreliably represented by user-selected deformation models, which introduces bias error in the interpretation of the conduit trajectory. For example, a buckled tubing string in a petroleum well may have an oscillating trajectory that varies in both amplitude and wavelength. If the user-selected deformation model assumes a constant amplitude and wavelength, then it cannot accurately represent the real shape of the tubing string trajectory.

[0016] U. S. Patent No. 11,150,374 (Capoglu et al.) describes methods for identifying a (pipe bend) deformation comprising the steps of identifying a zone with a deformation based on a point-wise eccentricity between a casing string and a pipe string (disposed within the bore of the casing string), and computing a match between an eccentricity profile and a database of patterns. However, computing a (best-fit) match to a pattern in a database similarly introduces bias error in the measurement of the conduit trajectory, because conduit trajectory deformations will typically differ from the limited number of patterns in the database.

[0017] BRIEF SUMMARY

[0018] In general terms, the present disclosure discloses methods, systems, and apparatus for measuring lateral displacements of conduit bore trajectories and determining conduit bore shapes with improved accuracy over prior art methods.

[0019] In a first aspect, the present disclosure discloses a method for measuring lateral displacements of a trajectory of a conduit along a selected length of the conduit, relative to a selected reference trajectory, where the conduit has a conduit bore, and the method includes the steps of:• receiving measurements of the conduit trajectory taken by a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:

[0020] o positions of the first and second datum points relative to the measurement reference point are defined by the geometry of the measurement tool;

[0021] o the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory; and

[0022] o the measurements of the conduit trajectory comprise an eccentricity of each of the selected measurement points along the selected length of the conduit, relative to the chord line;

[0023] • calculating, by means of one or more processors, an incremental eccentricity of each selected measurement point, wherein the incremental eccentricity is the difference between the measured eccentricity of the conduit trajectory at the measurement point and an eccentricity of the reference trajectory at the measurement point;

[0024] • assembling, by means of the one or more processors, a first system of equations relating lateral displacements of the conduit trajectory to the incremental eccentricities of the selected measurement points;

[0025] • creating, by means of the one or more processors, a second system of equations comprising a geometry matrix by applying, to the first system of equations, selected boundary conditions to make the number of unknowns associated with the first system of equations equal the number of equations in the first system of equations;

[0026] • creating, by means of the one or more processors, a third system of equations by applying selected error suppression to the second system of equations; and

[0027] • solving, by means of the one or more processors, the third system of equations for the lateral displacements of the conduit trajectory.In one embodiment of the method, the measurement tool used to take the measurements of the conduit trajectory comprises a rigid tool body centralized relative to the conduit at locations corresponding to the first and second datum points. In a variant of this embodiment, the length of the chord line may be the same for all of the selected measurement points.

[0028] In another embodiment of the method, the measurement tool comprises:

[0029] • a first rigid segment having an outer end and an inner end, with the outer end of the first rigid segment corresponding to the first datum point; and

[0030] • a second rigid segment having an outer end and an inner end, with the outer end of the second rigid segment corresponding to the second datum point, and with the inner end of the second rigid segment being joined to the inner end of the first rigid segment by means of an articulating connection.

[0031] In a further embodiment of the method, the measurements of the conduit trajectory were taken with the measurement tool not centralized relative to the conduit, and the measurement tool was further configured to measure the locations of the first and second datum points relative to the measurement tool.

[0032] In variant embodiments of the method, the measurement tool that took the conduit trajectory methods may have comprised:

[0033] • a rigid tool body configured with three pluralities of caliper fingers spaced apart at selected intervals along the length of the tool body; or

[0034] • a rigid tool body configured with an upper centralizer, a lower centralizer, and an ultrasonic measurement head.

[0035] In some embodiments of the method, the movement of the measurement tool along the selected length of the conduit may be within the conduit bore, while in other embodiments the movement of the measurement tool along the selected length of the conduit may be external to the conduit.

[0036] The selected boundary conditions applied to the first system of equations may include the boundary condition that the lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory is zero.Alternatively, the selected boundary conditions applied to the first system of equations may include the boundary condition that the lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory is equal to that of a nearest point of the plurality of selected measurement points.

[0037] As a further alternative, the selected boundary conditions applied to the first system of equations may include the boundary condition that the lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory decreases linearly to zero from that of a nearest one of the plurality of selected measurement points to a conduit trajectory point associated with a chord line furthest from the selected length of the conduit trajectory.

[0038] As a yet further alternative, the selected boundary conditions applied to the first system of equations may include the boundary condition that the lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory along a chord line at a selected end of the selected length of the conduit trajectory is zero.

[0039] In one embodiment of the method, the selected error suppression applied to the second system of equations may comprise one or more signal conditioning techniques to filter the incremental eccentricities, which may be selected from the group consisting of

[0040] • applying a constant offset correction;

[0041] • applying a variable offset correction based on the measurement tool inclination and rotation;

[0042] • applying a long- wavelength filter; and

[0043] • applying a short- wavelength filter.

[0044] In an alternative embodiment, the second system of equations may be pre-multiplied by a transpose of the geometry matrix and thereby forming a mathematically-equivalent system of equations comprising a geometric stiffness matrix, and the selected error suppression applied to the second system of equations may include matrix conditioning of the mathematically-equivalent system of equations (instead of or in addition to signal conditioning). The matrix conditioning may include addition of a foundation conditioning matrix to the geometric stiffness matrix.

[0045] In a variant embodiment, the matrix conditioning may include addition of a flexural conditioning matrix to the geometric stiffness matrix.By way of non-limiting example, the third system of equations may be solved using a direct solution algorithm or by using an iterative solution algorithm.

[0046] The lateral displacements of the conduit trajectory measured using methods in accordance with the present disclosure may be used to produce a three-dimensional representation of the shape of the conduit bore, which in turn may be used to determine a drift diameter of a conduit bore (wherein the conduit is a pipe string disposed in a petroleum well). In such a case, the conduit bore shape determined in accordance with the method may serve as a reference for determining where along its length the conduit wall thickness can be reduced (such as by milling) to increase the drift diameter of the conduit bore without reducing the conduit’s structural strength and integrity below safe levels.

[0047] In a second aspect, the present disclosure discloses a system for measuring lateral displacements of a trajectory of a conduit at selected measurement points along a selected length of the conduit, relative to a selected reference trajectory, wherein the conduit has a conduit bore, and the system comprises one or more processors configured to perform the steps of:

[0048] • receiving measurements of the conduit trajectory taken by a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:

[0049] o positions of the first and second datum points relative to the measurement reference points are defined by the geometry of the measurement tool;

[0050] o the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory; and

[0051] o the measurements of the conduit traj ectory taken by the measurement tool comprise eccentricities of each of the selected measurement points along the selected length of the conduit, relative to the chord line;• calculating an incremental eccentricity of each selected measurement point, wherein the incremental eccentricity is the difference between the measured eccentricity of the conduit trajectory at the measurement point and an eccentricity of the reference trajectory at the measurement point;

[0052] • assembling a first system of equations relating lateral displacements of the conduit trajectory to the incremental eccentricities of the selected measurement points;

[0053] • creating a second system of equations comprising a geometry matrix by applying, to the first system of equations, selected boundary conditions to make the number of unknowns associated with the first system of equations equal the number of equations in the first system of equations;

[0054] • creating a third system of equations by applying selected error suppression to the second system of equations; and

[0055] • solving the third system of equations for the lateral displacements of the conduit traj ectory.

[0056] The one or more processors may be further configured to use the measured lateral displacements of the conduit trajectory to produce a three-dimensional representation of the shape of the conduit bore, and / or to determine a drift diameter of the conduit bore based on the three-dimensional representation of the shape of the conduit bore.

[0057] In one variant embodiment, the system for measuring lateral displacements of a trajectory of a conduit may comprise a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:

[0058] • positions of the first and second datum points relative to the measurement reference point are defined by the geometry of the measurement tool; and

[0059] • the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory.

[0060] In this variant embodiment, the measurement tool may comprise a rigid tool body centralized relative to the conduit at locations corresponding to the first and second datum points. Alternatively, the measurement tool in this variant embodiment may comprise:• a first rigid segment having an outer end and an inner end, said outer end of the first rigid segment corresponding to the first datum point; and

[0061] • a second rigid segment having an outer end and an inner end, said outer end of the second rigid segment corresponding to the second datum point, and said inner end of the second rigid segment is joined to the inner end of the first rigid segment by means of an articulating connection.

[0062] It is not essential for the measurement tool to be centralized relative to a conduit with respect to which the tool is to take conduit trajectory measurements, and for use in cases where the measurement tool is not intended to be centralized relative to the conduit, the measurement tool may be further configured to measure the locations of the first and second datum points on the conduit trajectory relative to the measurement tool.

[0063] In a third aspect, the present disclosure discloses a measurement tool (alternatively referred to a “logging tool”) comprising:

[0064] • a first rigid segment having an outer end and an inner end;

[0065] • a second rigid segment having an outer end and an inner end, with the inner end of the second rigid segment being joined to the inner end of the first rigid segment by means of an articulating connection;

[0066] • a first centralizer, for centralizing the outer end of the first rigid segment within a bore of a conduit in which the logging tool may be disposed;

[0067] • a second centralizer, for centralizing the outer end of the second rigid segment within the conduit bore;

[0068] • a third centralizer, for centralizing the articulating connection within the conduit bore; and • a sensor for measuring an articulation angle between the first and second rigid segments.

[0069] The present disclosure further discloses a logging tool comprising:

[0070] • a rigid tool body;

[0071] • a first measurement means configured to measure an orthogonal eccentricity of a first selected measurement point on the conduit trajectory relative to a first measurement reference point on the rigid tool body;• a second measurement means configured to measure an orthogonal eccentricity of a second selected measurement point on the conduit trajectory relative to a second measurement reference point on the rigid tool body; and

[0072] • a third measurement means configured to measure an orthogonal eccentricity of a third selected measurement point on the conduit trajectory relative to a third measurement reference point on the rigid tool body;

[0073] wherein the first measurement point, the second measurement point, and the third measurement point are colinear. Each of the first, second, and third measurement means may be selected from the group consisting of a plurality of mechanical caliper fingers and an ultrasonic measurement head, such that the logging tool optionally could incorporate both caliper fingers and one or more ultrasonic measurement heads.

[0074] BRIEF DESCRIPTION OF DRAWINGS

[0075] Embodiments in accordance with the present disclosure will now be described with reference to the accompanying Figures, in which numerical references denote like parts, and in which:

[0076] FIGURE 1 is a simplified drawing of a typically-configured caliper-type logging tool performing measurements in a petroleum well.

[0077] FIGURE 2 is a schematic representation of cross-sectional measurements of a conduit bore taken using a caliper-type logging tool as in FIG. 1.

[0078] FIGURE 3 is a schematic representation of an eccentricity measurement being taken using a typically-configured caliper-type logging tool at one selected location within the bore of a conduit that follows a trajectory with a constant radius.

[0079] FIGURE 4 is a schematic representation of eccentricity measurements being taken using a typically-configured caliper-type logging tool at three different selected locations along a conduit that follows a trajectory with varying curvature.

[0080] FIGURE 5 is a flow chart showing the general steps of a non-limiting embodiment of a method in accordance with the present disclosure.FIGURE 6 is a schematic representation of eccentricity measurements of a 2D trajectory being taken using a typically-configured caliper-type logging tool, relative to a local tool coordinate system.

[0081] FIGURE 7 is a simplified drawing of an ultrasonic-type logging tool performing measurements in a petroleum well.

[0082] FIGURE 8 is a schematic representation of eccentricity measurements being taken at multiple locations along a section of a conduit trajectory.

[0083] FIGURE 9 is a simplified drawing of an articulating logging tool performing measurements in a petroleum well.

[0084] FIGURE 10 is a schematic representation of eccentricity measurement of a 2D trajectory being taken using an articulating logging tool.

[0085] FIGURE 11 is a simplified drawing of a logging tool, capable of simultaneously measuring eccentricity at three different locations along a conduit trajectory, performing measurements in a petroleum well.

[0086] FIGURE 12 is a schematic representation of a logging tool capable of simultaneously measuring eccentricity at three different locations along a conduit trajectory.

[0087] DETAILED DESCRIPTION

[0088] It is common for cross-section measurement tools to be centralized within the bore of a conduit at axial locations remote from the measurement transducers of the tool. Most caliper-type logging tools for subterranean wells are examples of this type of configuration. These tools typically have a slender, generally cylindrical body on the order of two meters long with a centralizer located at each end. A plurality of measurement transducers (commonly called “caliper arms” or “caliper fingers”) measure radial displacement from the tool body to the inside surface of the conduit bore, and are located along the length of the tool body between the centralizers. In some configurations, the caliper fingers are located close to one centralizer to minimize the eccentricity of the measured cross-section relative to the tool body resulting from well curvature. In other configurations, the caliper fingers are located closer to the midpoint between the centralizers, and analysis of the cross-section measurement includes calculations to account foreccentricity. Alternatively, some measurement tools may have the measurement transducers not located between the centralizers, but on an adjacent rigid extension of the tool (cantilevered). Logging is performed by moving a tool along a length of a conduit and obtaining measurement readings at a plurality of locations along the conduit.

[0089] FIG. 1 is a simplified drawing of a non-limiting example of atypically-configured calipertype logging tool 20 taking measurements within a bore 12 of a steel pipe conduit 10 installed in a petroleum well, with bore 12 having an inside surface 14. Conduit 10 follows a path characterized by a conduit trajectory 11. Tool 20 comprises an elongate rigid tool body 21, a plurality of caliper fingers 22 distributed circumferentially around tool body 21, an upper rollerstyle centralizer 23 and a lower roller-style centralizer 24. In this example, conduit trajectory 11 is defined as the curve following the cross-sectional center of bore 12 of conduit 10. Upper centralizer 23 and lower centralizer 24 centralize tool body 21 within bore 12 at two locations as tool 20 moves along conduit 10. Tool 20 may be conveyed along the length of conduit 10 by any functionally suitable means, such as (by way of non-limiting example) a wireline cable, a coiled tubing string, or a tractor device known to persons of ordinary skill in the art.

[0090] Caliper fingers 22 measure radial displacements from a measurement reference point m, which is located along the length of tool body 21, to inside surface 14 of bore 12 based on a polar coordinate system. Tool 20 is designed to measure the cross-sectional shape of conduit 10 in a plane (alternatively referred to as “measurement plane”) that is perpendicularly transverse to tool body 21 at measurement reference point m. FIG. 2 is a schematic representation of caliper readings C1to €)■ of the surface of bore 12 taken with tool 20. In this non-limiting example, tool 20 has j = 24 caliper fingers. Measurement reference point m is a reference datum for measurements of radial displacement in a polar coordinate system. The cross-section shape of bore 12 may be inferred by interpolation of caliper readings to Cj (e.g. the dashed curve in FIG. 2 joining all of the caliper readings). Tool 20 takes a set of caliper readings to Cj at each selected measurement location along a selected length of conduit 10.

[0091] Similar to many cross-section measurement tools, tool 20 comprises orientation sensors that measure:

[0092] • inclination Qincof tool 20 relative to a vertical axis defined by Earth gravity; and• rotation 0rotof tool 20 relative to the top (or “high side”) when the inclination of tool 20 is non-zero ( / .<., tool 20 is not vertical and the measurement plane of caliper fingers 22 is not horizontal).

[0093] At each measurement location along a selected length of conduit 10, a selected measurement point c on trajectory 100 (representing a cross-sectional center of bore 12 of conduit 10) may be defined from caliper readings to Cj. Any one of multiple mathematical methods known to persons of ordinary skill in the art for finding a geometric center of bore 12 from caliper readings to Cj may be used, such as (by way of non-limiting example), one of:

[0094] • an average of caliper readings C-t to Cj;

[0095] • a best-fit circle to caliper readings C-t to Cj;

[0096] • a best-fit ellipse to caliper readings Ci to Cj; or

[0097] • a Fourier decomposition of caliper readings C-t to Cj.

[0098] For the purposes of this disclosure, an eccentricity E is defined as the displacement in the measurement plane from measurement reference point m to selected measurement point c on trajectory 100.

[0099] Fundamental principles relating to measurement of lateral displacements of a conduit trajectory may be understood with reference to FIG. 3, which is a schematic representation of a typically-configured caliper-type logging tool 101 movably disposed within a conduit that follows a conduit trajectory 100 with a constant radius r (i.e., constant curvature of 1 / r). Tool 101 comprises two centralizers that locate tool 101 on trajectory 100 at two datum points a and b.

[0100] Chord line ab schematically represents tool 101 and the length L of tool 101 is defined as the distance between datum points a and b. Chord line ab is normal to the measurement plane of caliper fingers 22. The measurement plane is located a selected distance Lmfrom datum point a along chord line ab at measurement reference point m. As a matter of convention, datum point b may be selected as the front of tool 101 for taking measurements along a selected length of the conduit.

[0101] In petroleum wells, the curvature of an as-installed conduit (e.g., a casing string) is comparatively small (i.e., the radius of curvature r is comparatively large), such that the measured eccentricity E of the as-installed conduit, which depends on the curvature of the conduit trajectory,is small relative to the length of measurement tools. For example, consider a subterranean wellbore with a constant trajectory curvature expressed as a dogleg severity of 10 degrees over 30 meters, and a measurement tool with a chord line length L of 2 meters and a measurement reference point located at the chord line mid-point ( / .<., Lmof 1 meter). The associated measured eccentricity would be only 2.91 millimeters. Therefore, typical measurements of eccentricities of petroleum well conduits have small signal-to-noise ratios, and accurate determination of conduit trajectory shape from the eccentricity measurements is difficult.

[0102] For illustrative purposes, the conduit trajectories in FIGS. 2, 3, 5, 9 and 10 are shown with large curvatures compared to trajectories typical of petroleum wells.

[0103] More generally, the curvature of a conduit’s trajectory may vary continuously over the conduit’ s length and within any conduit length interval spanned by the tool as the tool moves along the conduit. For illustration, FIG. 4 is a schematic representation of a typically-configured calipertype logging tool 201 at three different locations (indicated by subscripts 1, 2, and 3) along a conduit with a trajectory 200 that varies in curvature along its length. The measured eccentricity also varies as tool 201 moves along the length of trajectory 200. Tool 201 may be conveyed along the conduit by (for example) a wireline cable, a coiled tubing string, or a tractor device known to persons of ordinary skill in the art.

[0104] If the centralizers perfectly centralize the measurement tool on a conduit trajectory at two locations, and if the exact distances between the measurement location and the centralizers are known, an accurate measurement of eccentricity would accurately reflect the geometry of the conduit trajectory. Furthermore, any variation in the measured eccentricity relative to that associated with the tool geometry in the original (as-installed) conduit trajectory would be associated with deformation of the conduit trajectory. However, because the eccentricity associated with conduit curvature over typical tool lengths is small, measurement uncertainties and analysis simplifications may result in significant error in the conduit trajectory interpretation. Such possible measurement uncertainties include:

[0105] • Centralization accuracy. Measurement tools often use centralizers that provide only approximate centralization, which may be sufficient for conduit cross-section shape analysis but insufficient for accurate eccentricity calculation.• Centralization location ambiguity. A centralizer may have multiple sets of arms at different axial locations, a logging tool assembly may be configured with additional (z.e., more than two) centralizers to provide more centralizing force, and the axial location of centralization of a centralizer may vary when the conduit cross-section shape changes. These factors can cause ambiguity with respect to the distance between the axial location of centralization and the axial location of the cross-section measurement.

[0106] • Tool lateral stiffness'. Real physical measurement tools are not perfectly rigid and can bend and sag, depending on tool inclination and centralizer capacity, potentially leading to significant error in the eccentricity measurement.

[0107] • Calibration errors'. These can include calibration errors due to wear or damage of components in mechanical tools, variation of fluid acoustic properties with ultrasonic tools, and other factors that can introduce offset or gain errors in eccentricity measurements.

[0108] • Measurement tool location uncertainty. Measurement tools are typically moved along a conduit by a wireline cable (or coiled tubing string), and the position of the tool along the conduit is measured by the amount of cable deployed into one end of the conduit. The cable inside the conduit may expand and contract with temperature changes, and may stretch and shrink with changes in the cable tension, creating uncertainty with respect to the measurement tool’s location in the conduit. Some tools record measurement location using a built-in odometer wheel, which can slip, resulting in measurement error.

[0109] A general method for measuring conduit trajectory deformation ideally will accommodate these uncertainties associated with measurement devices and additional uncertainties associated with internal conduit conditions, such as scale deposits or corrosion.

[0110] Additionally, a general method for measuring conduit trajectory deformation ideally will incorporate data from all the measurement locations. The measurement tool length is typically much longer than the measurement interval (z.e., many measurements are performed over any span along the conduit equal to the length of the tool), and the measured conduit length is typically much longer than the measurement tool length. For example, a measurement interval of about five millimeters is common for caliper-type logging tools, and a typical caliper tool is about two meterslong, spanning 500 measurement locations. A conduit length of 2,000 meters (or more) may be measured in a single logging run of a caliper tool, so that could produce data for 400,000 (or more) measurement locations.

[0111] Mathematical Basis

[0112] FIG. 5 is a flow chart indicating steps generally corresponding to an exemplary and nonlimiting embodiment of a method in accordance with the present disclosure for measuring lateral displacement of the trajectory of a conduit relative to a reference trajectory.

[0113] FIGS. 3 and 4 each illustrate the eccentricity of a point along a conduit trajectory from a chord line between two other points (i.e., two datum points) along the conduit trajectory. Accordingly, measurement of the eccentricity can locate a point in the trajectory relative to two datum points on the traj ectory. Considering a deformed conduit traj ectory (i. e., one that may differ from a selected reference trajectory), the eccentricity of the deformed conduit trajectory at a measurement location will depend on the deformed conduit trajectory at the datum points (i.e., the endpoints of the chord line where the chord line intersects the deformed conduit trajectory). Accordingly, the interpretation of conduit trajectory deformation at any point on the trajectory is a function of the trajectory deformation at other points on the trajectory. This implies that an interpretation of conduit traj ectory deformation from measurements of eccentricity might be found by solving a system of equations where the eccentricity measurement at each conduit trajectory location contributes to the solution for trajectory deformation at other conduit trajectory locations.

[0114] For further explanation, FIG. 6 schematically illustrates a conduit trajectory 300 located in a two-dimensional (2D) planar space that is measured with a logging tool 301. Tool 301 is centralized with trajectory 300 at datum points a and b, which define a chord line ab of length L.

[0115] A measurement reference point m is located on chord line ab at a distance Lmfrom datum point a. Tool 301 measures an eccentricity Emorthogonal to chord line ab from measurement reference point m to an associated selected measurement point c on trajectory 300. For each measurement location along the length of trajectory 300, a local cartesian coordinate system (z, x) can be defined with the z-axis parallel to a chord line ab, and with the x-axis orthogonal to the chord line ab. A position of trajectory 300 (or of a point on trajectory 300) in the direction of the x-axis will be herein referred to as a “lateral position”. A displacement i.e., change in position) of trajectory300 (or of a point on trajectory 300) relative to a selected reference trajectory in the direction of the x-axis will be herein referred to as a “lateral displacement”.

[0116] The measured eccentricity Emat a selected measurement point c on conduit trajectory 300 is the difference in lateral position between selected measurement point c and the associated measurement reference point m (on chord line ab), and the lateral position of measurement reference point m may be calculated by linear interpolation between the lateral positions of the datum points a and b:

[0117] Equation 1: Em= xc— xm

[0118] Equation 2: xm= xa+ R(xb- xa~) = [(1 - / ?) xa+ R xb]

[0119] where x variables represent the lateral positions of the points denoted by subscripts a, b, c, and m. The parameter R is referred to herein as a measurement length ratio, and is mathematically defined as follows:

[0120] Equation 3: R

[0121]

[0122] = ^m /

[0123] where Lmis the distance along chord line ab between datum point a and measurement reference point m, and L is the length of chord line ab (i.e., the distance between datum points a and b).

[0124] For many conduit logging tools, the value of measurement length ratio R is between zero and 1 i.e., measurement reference point m is located between datum points a and b). With such tools, the eccentricity measurement is most sensitive to conduit trajectory curvature when measurement length ratio R is 0.5 i.e., measurement reference point m is centered between datum points a and b). However, measurement length ratio R is not limited to this range, and may be less than zero or greater than 1 i.e., measurement reference point m may be located on a secant line colinear with chord line ab and not between datum points a and b). For example, FIG. 7 is a simplified drawing of a non-limiting example of an ultrasonic-type logging tool 30 performing measurements within bore 12 of steel pipe conduit 10 installed in a petroleum well. Ultrasonic-type logging tool 30 comprises an elongate rigid tool body 31, an ultrasonic measurement head 32, an upper rollerstyle centralizer 33, and a lower roller-style centralizer 34. Upper centralizer 33 and lower centralizer 34 centralize tool body 31 within bore 12 at two locations as logging tool 30 movesalong conduit 10. The ultrasonic measurement head 32 measures radial distances from tool body 31 to inside surface 14 of bore 12, and logging tool 30 performs multiple measurements along a selected length of conduit 10. Because measurement reference point m is below both upper centralizer 33 and lower centralizer 34, measurement length ratio R for logging tool 30 is greater than 1.

[0125] Equation 1 and Equation 2 can be combined and then rearranged as follows:

[0126] Equation 4: Em= xc— [(1 — / ?) xa+ R xb]

[0127] Equation 5: —(1 — / ?) xa+ xc— R xb= Em

[0128] A relationship similar to Equation 5 exists between a measured incremental eccentricity Em(i.e., change in eccentricity) and lateral displacements at points a, b, and c on the conduit trajectory:

[0129] Equation 6: — (1 — / ?) ua+ uc— R ub= Em= Em— Er

[0130] where u variables represent the lateral displacements of the conduit trajectory points denoted by subscripts a, b, and c referenced in Equation 5. The measured incremental eccentricity Emis the difference between the measured eccentricity Emon the conduit trajectory and a reference eccentricity Erat the same measurement point on the selected reference trajectory. The reference eccentricity Eris determined using a chord line ab having the same length L and same measurement length ratio R as the measured eccentricity Em. Typically, the selected reference trajectory will be a planned trajectory of the conduit or a trajectory for an earlier time in the life of the conduit, such as when the conduit was originally installed. Factors contributing to the difference between the measured eccentricity and the reference eccentricity include:

[0131] • deviations of the conduit trajectory from the selected reference trajectory, which may include:

[0132] o inaccuracy (error) in the selected reference trajectory; and

[0133] o conduit deformation from loads applied to the conduit; and

[0134] • errors in eccentricity measurement.The selected reference trajectory and the resulting reference eccentricities may be measured, calculated, or otherwise estimated based on available knowledge of the conduit. As a first example, when the conduit is a vertical well, the reference trajectory may be selected to be a straight vertical line, and reference eccentricities along the length of the conduit would all be zero. As a second example, when the conduit is a deviated or horizontal well, the reference trajectory may be selected to be a mathematical spline passing through the data points of a well survey, and the reference eccentricities calculated from the reference trajectory based on the geometry of the tool used to measure eccentricity.

[0135] The direction of the measured conduit trajectory 300 and the reference trajectory can change substantially and abruptly along the length of the conduit. Accordingly, chord line ab joining datum points a and b can also change direction as tool 301 moves along the conduit trajectory. However, the length of tool 301 (z.e., length of chord line ab) is usually large compared to the variations in lateral displacement within any length of 2 X L along trajectory 300 (i.e., within one chord line length L to either side of any point along trajectory 300). Thus, even when trajectory 300 changes direction abruptly, the change in direction of chord line ab is small and the distance between datum points a and b along the trajectory is approximately equal to chord line length L.

[0136] Equation 6 relates the lateral displacement ucat a measurement point c on conduit trajectory 300 to the lateral displacements uaand ubof trajectory 300 at datum points a and ft, and the measured incremental eccentricity AEm(which is the difference between the measured eccentricity Emat measurement point c and the reference eccentricity Erof the selected reference trajectory for the same tool geometry and location in the conduit). Equation 6 may be made more concise by introducing two substitutions:

[0137] Equation 7: gaua+ uc+ gbub= Em

[0138] where ga= — (1 — R) and gb= — R. A first system of equations (which is a system of linear equations) can be assembled by applying this relationship at every measurement location along a conduit trajectory.

[0139] FIG. 8 schematically illustrates a non-limiting example of a measurement tool located at a series of measurement locations numbered 1 to n along a conduit trajectory (shown as a straighttrajectory for illustrative simplicity). Most conduit logs containing data used to determine eccentricity have measurements taken at small, uniformly-spaced measurement intervals (AMDin FIG. 8) along the conduit trajectory, and the distance from the measurement reference point to either datum points is large compared to the measurement interval. However, for illustration simplicity, the chord line length L in FIG. 8 is only five measurement intervals, and the distance Lmfrom datum point a to the measurement reference point m is only two measurement intervals. (This measurement tool has a measurement length ratio R of 0.4.) Applying the relationship of Equation 7 at each measurement location along the conduit trajectory in the non-limiting example of FIG. 8 produces a first system of equations:

[0140] 9a + Itl + CJ / j Em l

[0141]

[0142] 9a U0 + u2+ 9b u$ = Em>29a Ul + u3+ 9b u6= Em>3Equation 8a:

[0143] 9 a Un-4 + Un-2 " I" 9b Un+1 £±Em n-2 9a Un-3 + Un-1 " I" 9b Un+2 ^Emin-1 9

[0144]

[0145] a Un-2 + Un " I" 9b Un+3 ^Em>n

[0146] The first system of equations may also be expressed in matrix (and vector) notation:

[0147] Equation 8b: [^]{u = {AEm}

[0148] where the vector {nJ contains the unknown lateral displacements, and the vector {AEm} contains the measured incremental eccentricities. The matrix [g] is rectangular with the number of rows equal to the number of equations in the first system of equations and the number of columns equal to the number of unknown lateral displacements.

[0149] The first system of equations is a system of linear algebraic equations. The first system of equations is mathematically underdetermined with the number of unknowns (alternatively expressed as the “degrees of freedom” associated with the system of equations) being greater than the number of equations. The extra degrees of freedom result from the conduit trajectory points associated with the centralizers that lie beyond the conduit length spanned by the measurement reference point of the tool: i.e., the conduit trajectory points utfor i < 1 and i > n. To solve the first system of equations, suitable and sufficient boundary conditions must be selected and appliedto the first system of equations to make the number of unknowns to equal the number of equations. Non-limiting examples of boundary conditions that may be selected include:

[0150] • no lateral displacement of the conduit trajectory beyond the conduit length spanned by the measurements: i.e., ut= 0 for t < 1 and t > n,

[0151] • lateral displacement of the conduit trajectory beyond the conduit length spanned by the measurements is equal to the lateral displacement at the boundaries of the conduit length spanned by the measurements: i.e., ut= u for i < 1 and ut= unfor i > n, • lateral displacement of the conduit trajectory beyond the conduit length spanned by the measurements decreases linearly from the lateral displacements at the measurement boundaries u and unto zero at the conduit trajectory points associated with the centralizers furthest from the boundaries of the conduit length spanned by the measurements i.e., u_ta= 0 and utb= 0 where iais the number of measurement intervals nearest in length to Lm, which is the distance between datum point a and measurement reference point m, and ibis the number of measurement intervals nearest in length to (L — Lm~), which is the distance between measurement reference point m and datum point ft); and

[0152] • no lateral displacement of the conduit traj ectory for one measurement tool length at one end of the conduit: i.e., either ut= 0 for — ta< t < tb, or ut= 0 for (n — ia) < i < (n + ih).

[0153] A second system of equations is created by applying selected boundary conditions to the first system of equations to make the degrees of freedom (alternatively expressed as the “number of unknowns” associated with the equations) equal to the number of equations. The second system of equations may be expressed in matrix (and vector) notation as follows:

[0154] Equation 9: [G] {it} = {AEm}

[0155] The vector {u} contains the remaining unknown lateral displacements after the application of boundary conditions to the first system of equations, and the vector { Em} contains the measured incremental eccentricities. Vector {u} is equal in size to vector {AEm}. The matrix [G] is square, and is referred to herein as a geometry matrix.Deformation interpretation of a conduit trajectory in three-dimensional (3D) space requires an expanded first system of equations that includes components of measured incremental eccentricities and components of lateral displacements along two axes (AHand ARin FIG. 2) that are orthogonal to chord line ab and orthogonal to each other. It is common for measurement tools to rotate as they are moved along a conduit. An orientation sensor may be used to measure rotation 6rotof rotate the section measurement data so that the orthogonal components of the measured incremental eccentricities and lateral displacements in the local coordinate system can be related to the global coordinate system.

[0156] Often suitable and sufficient boundary conditions may be no displacement of the conduit trajectory beyond the conduit length spanned by the measurements (“no displacement boundary conditions”): / .<., ut= 0 for i < 1 and i > n. This leads to a geometry matrix [G] with a banded structure having unity values on the main diagonal to couple eccentricity measurement locations, and off-diagonal terms of ga= — (1 — / ?) and gb= — R to couple datum points of each chord line. For the illustrative non-limiting example of FIG. 8, the geometry matrix [G] after application of the “no displacement boundary conditions” is:

[0157] 1 0 0 gb0

[0158] 0 1 0 0 gb

[0159] gao i o o

[0160] o gao i o

[0161]

[0162] o o ga0 1

[0163] Generally, tool geometry lengths L and Lmare not exact integer multiples of the conduit trajectory measurement intervalMD Therefore, datum points a and b for every chord line may not exactly coincide with one of the conduit trajectory measurement locations. The simplest approach in constructing the first system of equations is to couple each datum point to the nearest the conduit trajectory measurement location, as done in this non-limiting example. More sophisticated approaches that may improve accuracy include:• using interpolation factors to couple each datum point to the nearest two conduit trajectory measurement locations; and

[0164] • using a set of polynomial regression parameters to couple multiple conduit trajectory measurement locations to each datum point.

[0165] It is common for the conduit trajectory measurement intervalMDand the tool geometry lengths L and Lmto be constant for the entire measurement set, in which case the matrix structure repeats diagonally. However, if the measurement interval size varies and can be quantified, or if the tool geometry changes (for example, the distance from either datum points a and b to measurement reference point m may change when the logging tool centralizers are collapsed through a restriction of the conduit bore), then the geometry matrix [G] can be adjusted at each row to reflect the changing geometric relationships.

[0166] The second system of equations describes a relationship between conduit trajectory lateral displacements {u} and measured incremental eccentricities {AEm} for a measurement system geometry described by geometry matrix [G], It is analogous to systems of equations that are foundational to finite element analysis methods for solving structural engineering problems: for example, the tensor equations that relate stress and strain through constitutive relationships, and the strain-displacement matrix function that relates the strain distribution in an element to nodal displacements.

[0167] While it is usually mathematically possible to solve the second system of equations for the conduit trajectory lateral displacements {u}, the resulting solution will frequently be highly inaccurate. Measurement systems are not perfectly accurate, and eccentricity measurement errors (alternatively referred to as “measurement noise”) are usually not negligible compared to the small eccentricities associated with many conduit trajectories, as described earlier regarding conduits in petroleum wells. For example, a small offset in the eccentricity measurement signal would be interpreted as a constant curvature of large radius, which would accumulate over the conduit length into a large lateral offset in the solution that is not actually present in the conduit trajectory. Additionally, there are simplifying analysis assumptions related to the possible measurement uncertainties described earlier that introduce variances in solution results that are small but not negligible. Other possible issues include random measurement noise in the eccentricity data, and noise from the tool orientation sensor that produces oscillations in the solution that are not actuallypresent in the conduit trajectory. The first system of equations can also become decoupled into multiple subsystems when the value of measurement length ratio R is a member of the set { n<n =2, 3,4,...}, and then eccentricity measurement errors can produce additional artefacts in the solution that are not actually present in the conduit trajectory. If a solution method has no way to filter out (i.e., suppress) errors, the errors can accumulate to make the calculated solution meaningless.

[0168] Solution Error Suppression

[0169] Prior art methods for conduit trajectory measurement require assumptions to be made regarding the deformation shape (e.g., selection of a deformation model), and use the measurements to find a “best-fit” solution for the assumed deformation shape. The accuracy of the solution depends on the accuracy of the assumed deformation shape or model, which biases the trajectory deformation interpretation. The shape of conduit deformations is generally unknown and complex, making accurate assumptions difficult and prone to introducing additional error in the trajectory interpretation.

[0170] Embodiments of conduit trajectory measurement methods in accordance with the present disclosure include a step to suppress errors. The step to suppress errors does not require assumptions regarding the shape of conduit trajectory deformation, although such assumptions may be used without departing from the scope of the present disclosure. If the conditions to define datum points for eccentricity measurement are satisfied, arbitrary conduit trajectory deformations can be resolved. Assumptions are made regarding the character of measurement errors and errors associated with analysis assumptions, and these errors are usually more easily identified and characterized than the shape of deformations. Suitable error suppression techniques may then be applied to obtain reliable solutions for conduit trajectory lateral displacements {u}.

[0171] Filtering Eccentricity Data

[0172] In some embodiments of methods in accordance with the present disclosure, the eccentricity data may be filtered to suppress measurement errors (“measurement noise”) or other sources of error. A third system of equations may be created from a second system of equations by filtering the measured incremental eccentricities {AEm} to suppress errors:Equation 10: [G]{u) = {AEm}z

[0173]

[0174] where is the filtered measured incremental eccentricities.

[0175] A variety of signal conditioning techniques for time-series (“temporal”) data known to persons of ordinary skill in the art may be adapted to filter the spatial eccentricity data. Spatial signals with long- wavelength characteristics are analogous to temporal signals with low-frequency characteristics, and spatial signals with short- wavelength characteristics are analogous to temporal signals with high-frequency characteristics. Non-limiting examples of signal conditioning techniques are described in following paragraphs, and one or more of these techniques may be applied to spatial eccentricity data.

[0176] A constant offset bias correction: Eccentricity measurement errors may be constant in magnitude and direction relative to the measurement tool frame of reference. For example, such error may occur when a caliper-type measurement tool body is bent. If such error is found and quantified, the eccentricity measurement may be corrected with a corresponding subtraction of the error.

[0177] A variable offset correction based on measurement tool inclination and rotation:

[0178] Many conduit measurement tools are long and slender, and may sag under the force of gravity and thus introduce an eccentricity measurement error. At each measurement location, the magnitude of the eccentricity error will depend on the inclination of the tool, and the direction of the eccentricity error in the measurement tool frame of reference will depend on the rotational orientation of the measurement tool relative to vertical ( / .<., relative to gravity). If such error is found and the error dependencies on tool inclination and rotation are quantified, the eccentricity measurement may be corrected with a corresponding subtraction of the error.

[0179] A long-wavelength filter (which is analogous to a low-frequency filter for time-domain data) may be applied to the eccentricity data to suppress long-wavelength lateral displacements. Filter parameters may be selected based on the observed measurement noise characteristics of the eccentricity data. For example, in subterranean wells, long- wavelength lateral displacements of large magnitude may be caused by measurement noise if there are no known mechanisms for such trajectory movement. Additionally, long- wavelength lateral displacements often produce little strain and stress in the conduit, and do not limit the movement of equipment within the conduitbore, such that the impact of filtering out real long-wavelength lateral displacements from the measurement is negligible.

[0180] A short-wavelength filter (which is analogous to high-frequency filter for time-domain data) may be applied to the eccentricity data to suppress short- wavelength lateral displacements. Filter parameters may be selected based on the observed measurement noise characteristics of the eccentricity data. For example, in subterranean wells, short- wavelength lateral displacements that alternate direction along the conduit (i.e., oscillations) may be caused by measurement noise if there are no known mechanisms for such trajectory oscillations.

[0181] Matrix Conditioning

[0182] In addition (or as an alternative) to filtering eccentricity data, some embodiments of methods in accordance with the present disclosure may suppress measurement error by conditioning a second system of equations (herein referred to as “matrix conditioning”), analogous to the conditioning employed in finite element and finite difference numerical methods for structural analysis. The matrix conditioning technique is based on the principle of virtual work, in which a virtual incremental eccentricity is used to produce virtual work in the system. (The virtual incremental eccentricity is analogous to a virtual displacement field in finite element numerical methods.) Minimizing the virtual work leads to a solution for the conduit trajectory lateral displacements {u} as will be described in the following paragraphs.

[0183] There will be differences between the incremental eccentricities associated with a real physical conduit trajectory and the incremental eccentricities measured by the conduit logging system because measurement errors are introduced by the measurement system. Incremental eccentricities {AE} associated with “best estimate” conduit trajectory lateral displacements {u} may be defined as the incremental eccentricities from measurement { Em} plus eccentricity corrections {Af}. Equation 9 may then be recast as:

[0184] Equation 11: [G]{u] = {AE} = {AEm+ Ae}

[0185] A virtual work w£associated with the eccentricity corrections {Af} may be formulated by ascribing a “virtual stiffness” K£that relates a virtual force {F£} to the eccentricity correction {Af}:

[0186] Equation 12: {F£} = fAe}Equation 13: w£= {A£}T{F£} = {Ae}TfAe} = K£{ E}T{Ae}

[0187] A variation in the virtual work SwEmay be defined as:

[0188] Equation 14: Sw£= 6£{5A£}T{A£}

[0189] where {5Af} represents variations in the eccentricity corrections and may be related to variations in the conduit trajectory lateral displacements {5u} through the geometry matrix [G] (in the same way that the incremental eccentricities {AE} are related to the conduit trajectory lateral displacements {u} in Equation 9):

[0190] Equation 15a: [G]{5u} = {5A£}

[0191] Equation 15b: {5A£}T= {5it}T[G]T

[0192] The eccentricity corrections {Af} can be expressed in terms of the measured incremental eccentricities {AEm}, the geometry matrix [G] and the conduit trajectory lateral displacements {u} by rearranging Equation 11:

[0193] Equation 16: {Af} = [G] {it} — {AEm}

[0194] Substituting Equation 15b and Equation 16 into Equation 14 gives:

[0195] Equation 17: 5u / £= K£{8u}T[G]T[[G] {u} - {AEm}]

[0196] For a solution for the lateral displacements {u} that minimizes the virtual work w£, the variation in virtual work 6wemust be zero, and the following system of equations (derived from Equation 17) must be satisfied for a non-trivial solution for the eccentricity corrections {Af}:

[0197] Equation 18a:

[0198]

[0199] = [G]T{AEm}

[0200] Equation 18b: |G]{it}={^G)

[0201] where [G] = [G]T[G] is referred to herein as a geometric stiffness matrix, and {FG}=[G]T{AEm} is referred to herein as a geometric forcing vector. Prior to the application of matrix conditioningto suppress errors, the system of equations expressed by Equation 18 is mathematically equivalent to the second system of equations expressed by Equation 9:

[0202] • both sides of Equation 9 are pre-multiplied by the constant matrix [G]r(which is a transpose of the geometry matrix) to produce Equation 18, and

[0203] • Equation 18 and Equation 9 produce the same solution for conduit trajectory lateral displacements {u}.

[0204] One or more matrix conditioning techniques may be applied to the second system of equations expressed by Equation 18 based on the character of errors to be suppressed and to reflect physical constraints on the conduit structure. Finite element and finite difference numerical analysis methods known to persons of ordinary skill in the art employ systems of equations of similar structure, and may employ similar matrix conditioning techniques. Such techniques (by way of non-limiting example) may include:

[0205] • a foundation stiffness conditioning may be used to suppress long- wavelength errors, which may be identified when large-scale lateral movement of the conduit trajectory from the reference trajectory is prevented by a conduit’s physical support structure; and • flexural stiffness conditioning and shear stiffness conditioning may be used to suppress short- wavelength errors, which in the solution are often associated with noise in eccentricity measurements and imperfect analysis assumption, and can be identified because there are no plausible mechanisms for such short-wavelength oscillations in the conduit trajectory.

[0206] If the measurement data is of high quality, the amount of conditioning required to suppress errors is small. A trajectory analysis objective should be to reduce the conditioning as much as possible to minimize suppression of any real physical conduit trajectory lateral displacement in the calculated solution.

[0207] In general, matrix conditioning reflects a constraint on the lateral displacements {u} and may be incorporated into the second system of equations as an addition to the geometric stiffness matrix [G] to create a third system of equations:

[0208] Equation 19a: [[Kc] + [ (“l = cwhere [c] is a selected conditioning matrix, which reflects a stiffness that resists trajectory deformation from the reference trajectory. Keeping the terms in the conditioning matrix [c] small allows the geometric stiffness matrix [G] (measurement geometry terms) in the third system of equations to dominate the lateral displacement solution. When multiple matrix conditioning techniques are applied, multiple conditioning matrices are added to the geometric stiffness matrix:

[0209] Equation 19b: [

[0210]

[0211] IXG] + [KC1] + [KG2] + ••• ]{u} = {FG}

[0212] By way of non-limiting example, two matrix conditioning techniques that may be applied are a foundation conditioning matrix [Kfa] to suppress large-scale modes of error, and a flexural conditioning matrix |7Z] to suppress small-scale modes of error. The foundation conditioning matrix [Kd] may be defined by the following equation:

[0213] Equation 20: [Kfd] = kfd[I]

[0214] where kdis a selected foundation stiffness parameter, and [ / ] is an identity matrix of the same size as The flexural conditioning matrix [fz] may be constructed according to the following equation:

[0215] Equation 21: [Kfl] = (kfl) [Gf z[Gf z]

[0216] where kyZis a selected scalar flexural stiffness parameter, and [GyZ] is a. geometric flexural matrix defined as a matrix of the same size as the geometric stiffness matrix [G] with a structure consisting of: entries of 2 along the main diagonal; entries of -1 along both diagonals immediately adjacent to the main diagonal; and entries of zero elsewhere, as exemplified by the following:

[0217] - 2 -1 0. 0 - -1 2 -1 X X X ■

[0218] 0 -1 2 X X X:

[0219] Equation 22: [t^z] =

[0220] i X X X 2 -1 0

[0221] : X X X -1 2 -1

[0222]

[0223] - 0. 0 -1 2 -After the third system of equations (Equation 19) is solved to obtain a solution for lateral displacements {u}, Equation 16 may be used to calculate the eccentricity corrections {Af}, which indicate how much the matrix conditioning contributes to the solution.

[0224] In some embodiments of methods in accordance with the present disclosure, a third system of equations is created using both eccentricity signal conditioning and matrix conditioning:

[0225] Equation 23: [[KG] + [Kc]] (u) = [G]T{AEm} / = {FG f

[0226] where {FG}f is a geometric forcing vector based on filtered measured incremental eccentricities

[0227] Solving the System of Equations

[0228] The third system of equations is a system of linear algebraic equations, and may be solved for the lateral displacements {u} of the conduit trajectory using any suitable linear algebra solution algorithm known to persons of ordinary skill in the art. Suitable solution algorithms include direct solution algorithms and iterative solution algorithms. Direct solution algorithms include, but are not limited to, Gauss elimination, Gauss-Jordan elimination, LU decomposition, and Cholesky decomposition. Iterative solution algorithms include, but are not limited to, Jacobi iteration, Gauss-Seidel iteration, and relaxation methods.

[0229] The lateral displacements {u} of the conduit trajectory may be used to determine lateral positions {%} of the conduit trajectory relative to lateral positions {xr} of the selected reference trajectory:

[0230] Equation 24: {%} = {xr} + {it}

[0231] The lateral displacements {u} and lateral positions {%} are values expressed in the local cartesian coordinate system (z, x) of each measurement location along the selected length of the conduit trajectory. A coordinate system transformation may be performed at each measurement location to determine the conduit trajectory lateral displacements and lateral positions in the global coordinate system.Alternative Eccentricity Measurement

[0232] Conduit measurement tools as described in the present disclosure may be used as an alternative to prior art caliper-type logging tools to generate eccentricity data for the conduit trajectory interpretation method. Such alternative measurement tools may offer advantages in terms of additional measurement capabilities or may be configured to pass through more severe conduit bends compared to conventional caliper-type logging tools of similar size.

[0233] FIG. 9 is a simplified drawing of a non-limiting embodiment 40 of an articulating logging tool in accordance with the present disclosure performing measurements within bore 12 of steel pipe conduit 10 installed in a petroleum well. Articulating logging tool 40 comprises:

[0234] • a first rigid segment 41;

[0235] • a second rigid segment 42;

[0236] • an articulating connection 43 between first rigid segment 41 and second rigid segment 42;

[0237] • an upper centralizer 44;

[0238] • a middle centralizer 45; and

[0239] • a lower centralizer 46.

[0240] Upper centralizer 44 centralizes a location a of first rigid segment 41 within bore 12. Middle centralizer 45 centralizes a location c of articulating connection 43 within bore 12. Lower centralizer 46 centralizes a location b of second rigid segment 42 within bore 12.

[0241] FIG. 10 schematically represents eccentricity measurement with an articulating logging tool 401, similar to articulating logging tool 40, comprising:

[0242] • a first rigid segment 410 having an inner end 411, an outer end 412, and a length Lac, • a second rigid segment 420 having an inner end 421, an outer end 422, and a length Lftc; and

[0243] • an articulating connection between inner end 411 of first rigid segment 410 and inner end 421 of second rigid segment 420.

[0244] Articulating logging tool 401 has centralizers located at datum points a and b and at the articulating connection to centralize the tool at these points on a conduit trajectory 400, with the articulating connection being centralized at a selected measurement point c on conduit trajectory400. Articulating logging tool 401 has a sensor (of any suitable type known to persons of ordinary skill in the art) to measure an articulation angle 0acbbetween segments 410 and 420. The length L of a chord line aft joining datum points a and ft may be determined from articulation angle 0acbusing trigonometry (law of cosines):

[0245] Equation 25: L = 7 L2ac+ L2bc- 2LacLbccos 6acb

[0246] The orthogonal eccentricity Emof measurement point c from an associated measurement reference point m on chord line aft may also be determined from articulation angle 0acbusing trigonometry (including the law of sines):

[0247] Lbc L

[0248] Equation 27: Em= Lacsin 6bac=

[0249]

[0250] Angle 0bacbetween rigid segment 410 and chord line aft in FIG. 10 may be calculated using Equation 26. A distance Lmfrom datum point a to measurement reference point m may then be calculated from angle 0bac

[0251] Equation 28: Lm= Laccos 6bac

[0252] In some applications, the maximum eccentricity is small relative to length Lacand a simplifying approximation of Lm= Lacmay be used because 0bac~

[0253] FIG. 11 is a simplified drawing of a non-limiting embodiment 50 of a logging tool in accordance with the present disclosure performing measurements within bore 12 of steel pipe conduit 10 installed in a petroleum well. In contrast to prior art caliper-type logging tools that have a single set of caliper fingers, tool 50 has three sets of caliper fingers and is capable of simultaneously measuring eccentricities at three different locations along a conduit trajectory. Tool 50 comprises:

[0254] • an elongate rigid tool body 51;

[0255] • a first plurality of caliper fingers 52 distributed around tool body 51 at a location maalong the length of tool body 51;• a second plurality of caliper fingers 53 distributed around tool body 51 at a location mcalong the length of tool body 51; and

[0256] • a third plurality of caliper fingers 54 distributed around tool body 51 at a location mbalong the length of tool body 51.

[0257] A variant embodiment of a logging tool in accordance with the present disclosure comprises an elongate rigid tool body and three ultrasonic measurement heads. Each ultrasonic measurement head is located at a different location along the length of the tool body.

[0258] A further variant embodiment of a logging tool in accordance with the present disclosure comprises:

[0259] • an elongate rigid tool body;

[0260] • a first plurality of caliper fingers distributed around the tool body at a first location along the length of the tool body;

[0261] • a second plurality of caliper fingers distributed around the tool body at a second location along the length of the tool body; and

[0262] • an ultrasonic measurement head located at a third location along the length of the tool body.

[0263] FIG. 12 schematically represents an exemplary embodiment of an elongate logging tool 501, similar to tool 50, following a conduit trajectory 500. Tool 501 measures eccentricities Ea, Eft, and Ecof points a, b, and c on trajectory 500 at three points ma, mb, and mc, respectively, along the length of logging tool 501. As a non-limiting example, tool 501 is configured with a first plurality of caliper fingers at location ma, a second plurality of caliper fingers at location mb, and a third plurality of caliper fingers at location mc. The locations of datum points a and b that define a chord line ab are measured relative to the body of tool 501, which is represented in FIG.

[0264] 12 by a heavy straight tool body line (similar to the representation of tool body 51 in FIG. 11). Points ma, mb, and mcare located on this tool body line. Tool 501 does not need to be physically centralized at datum points a and b, and eccentricity measurement errors caused by insufficient physical centralization of tool 501 are eliminated.

[0265] A measurement eccentricity Em(perpendicular to chord line ab) and a measurement length ratio R may be calculated using geometric relationships and principles known to persons of skillin the art. A length I, of a chord line ab defined by datum points a and b for each measurement point may be calculated from eccentricity measurements Eaand Eb

[0266] Equation 29: L

[0267]

[0268] = 7 (Ea- Ed)2+ L2mab

[0269] where Lmabis the distance between locations maand mbof tool 501. A point d on chord line ab in FIG. 11 is defined where the vector of eccentricity measurement Ecintersects chord line ab.

[0270] The distance Edbetween location mcon tool 501 and point d may be calculated from eccentricity measurements Eaand Ebusing linear interpolation:

[0271] Equation 30: Ea= Ea- (Ea- Eb)

[0272]

[0273] Lmab

[0274] where Lmacis the distance between locations maand mcof tool 501. The distance Ladbetween datum point a and point d, and the distance Lbdbetween datum point b and point d may be calculated from the proportional relationship between chord line ab and the line in FIG. 11 representing tool 501:

[0275] Equation 31: Lad= L

[0276]

[0277] Lmab

[0278] Equation 32: Lbd=

[0279]

[0280] Lmab

[0281] where Lmbcis the distance between locations mband mcof tool 501. The measurement eccentricity Emmay then be calculated using the principle of similar triangles:

[0282] Equation 33: Ern= ^(E,:- Ecl)

[0283]

[0284] Lbd

[0285] The distance Ldmbetween point d and measurement reference point m is calculated also using the principle of similar triangles:

[0286] Equation 34: Lam=Em = (Ec>Ed)

[0287]

[0288] LmbcLbdThe distance Lmfrom datum point a to measurement reference point m on chord line ab is required to determine measurement length ratio R and may be calculated as follows:

[0289] Equation 35.

[0290]

[0291] Lad ^dm

[0292] In some applications, the angle between tool 501 and chord line ab is small ( / .<., \Ea~ Eb\ « Lmac) at all measurement locations such that simplifying approximations of L = Lmab and — ^mac may be used, and the eccentricity Emof measurement point c from a measurement reference point m on chord line ab may be calculated from eccentricity measurements Ea, Eb, and Ecas follows:

[0293] Equation 36: Em= Ec— [(1 — R)Ea+ REb]

[0294] Utility of Trajectory Lateral Displacement Measurement Results

[0295] The conduit trajectory lateral positions, calculated from the measured eccentricities, may be combined with the cross-section measurements of the logging tool to produce a three-dimensional shape measurement of the inside surface of the conduit bore (e.g., the inside surface of a petroleum well casing string).

[0296] The measured three-dimensional conduit bore shape can be used to determine drift diameters of the bore. The term “drift diameter” refers to the maximum diameter limit of a cylinder of a user-specified length that can be inserted into the bore. Methods for determining drift diameters of a conduit bore from the three-dimensional bore shape are found in the prior art, such as a method disclosed in US 9,605,938 (Helmore), and other methods known to persons of ordinary skill in the art. Petroleum well operators can select suitably-sized downhole equipment to insert into the conduit bore based on the calculated drift diameters. Examples of downhole equipment for petroleum wells include, but are not limited to, perforating guns, plugs, plug-drilling assemblies, and casing-milling assemblies.

[0297] The measured three-dimensional shape of a petroleum well casing bore may be used by the well operator to predict the amount of casing wall thickness removed by a milling tool in a milling operation, and then to set a safe operating pressure limit for the well after the milling operation.The conduit trajectory lateral positions and the measured conduit bore shape may also inform petroleum well operators about casing deformation and damage. Petroleum well operators may use this information in performing corrective actions, such as decommissioning a well, and in performing preventative actions, such as setting safe operating limits and changing the design of future wells to be more resistant to or more tolerant of casing deformation and damage.

[0298] The trajectory of a petroleum well casing string may be deformed by earth formation movement at a sedimentary geological plane. The conduit (i.e., casing) trajectory lateral displacements, calculated from the measured eccentricities using methods in accordance with the present disclosure, may be combined with the reference trajectory orientation (inclination and azimuth angles) and the geological plane orientation (strike and dip angles) where they intersect to estimate the magnitude and direction of the formation movement (using trigonometry). Petroleum well designers can use this information to design future well structures that are more resistant to the formation movement, or to design future well trajectories that will result in less casing deformation and damage at the geological plane.

[0299] When formation movement is induced by well operations, petroleum well operators can use the magnitude and direction of formation movement derived from the conduit trajectory lateral displacements to calibrate geomechanical earth models. The calibrated geomechanical earth models can then be used to determine appropriate well operation to limit future casing deformation and damage ( / .<., by taking preventative action).

[0300] # # # # #

[0301] It will be readily appreciated by persons of ordinary skill in the art that various modifications to embodiments in accordance with the present disclosure may be devised without departing from the scope of the present teachings, including modifications which may use equivalent mathematical functions.

[0302] It is to be especially understood that the scope of the present disclosure is not intended to be limited to described or illustrated embodiments and example data, and that the substitution of a variant of a claimed or illustrated element or feature, without any substantial resultant change in methodology or practical result, will not constitute a departure from the scope of the disclosure.In this patent document, any form of the word “comprise” is to be understood in its non-limiting sense to mean that any element or feature following such word is included, but elements or features not specifically mentioned are not excluded. A reference to an element or feature by the indefinite article "a" does not exclude the possibility that more than one such element or feature is present, unless the context clearly requires that there be one and only one such element or feature. Where an element or feature is referred to herein as being “selected”, this is to be understood as meaning “user-selected”, unless the context implies otherwise.

[0303] Relational and conformational terms such as (but not limited to) “axial”, “orthogonal”, “parallel”, “perpendicular”, “radial”, and “vertical” are not intended to denote or require absolute mathematical or geometrical precision. Accordingly, such terms are to be understood as denoting or requiring substantial precision only (e.g., “substantially parallel”) unless the context clearly requires otherwise.

[0304] Wherever used in this document, the terms “typical” and “typically” are to be interpreted in the sense of being representative of common usage or practice, and are not to be understood as implying essentiality or invariability.LIST OF DRAWING ELEMENTS AND MATHEMATICAL SYMBOLS

[0305] Label Description

[0306] 10 conduit (steel pipe of petroleum well)

[0307] 11 conduit trajectory

[0308] 12 conduit bore

[0309] 14 inside surface of conduit bore

[0310] 20 caliper-type logging tool

[0311] 21 tool body

[0312] 22 plurality of caliper fingers

[0313] 23 upper roller-style centralizer

[0314] 24 lower roller-style centralizer

[0315] 30 ultrasonic-type logging tool

[0316] 31 tool body

[0317] 32 ultrasonic measurement head

[0318] 33 upper roller-style centralizer

[0319] 34 lower roller-style centralizer

[0320] 40 articulating logging tool

[0321] 41 upper tool body

[0322] 42 lower tool body

[0323] 43 articulating connection

[0324] 44 upper centralizer

[0325] 45 middle centralizer

[0326] 46 lower centralizer

[0327] 50 caliper-type logging tool

[0328] 51 tool body

[0329] 52 first plurality of caliper fingers

[0330] 53 second plurality of caliper fingers

[0331] 54 third plurality of caliper fingers

[0332] 100 conduit trajectory with constant curvature

[0333] 101 caliper-type logging tool

[0334] 200 conduit trajectory with variable curvatureLabel Description

[0335] 201 caliper-type logging tool

[0336] 300 conduit trajectory with variable curvature

[0337] 301 caliper-type logging tool

[0338] 400 conduit trajectory with variable curvature

[0339] 401 articulating logging tool

[0340] 410 first rigid segment of 401

[0341] 411 inner end of first rigid segment 410

[0342] 412 outer end of first rigid segment 410

[0343] 420 second rigid segment of 401

[0344] 421 inner end of second rigid segment 420

[0345] 422 outer end of second rigid segment 420

[0346] 500 conduit trajectory with variable curvature

[0347] 501 logging tool with eccentricity measurement at three locations along tool body

[0348] a datum point on conduit trajectory defining a first end of chord line ab H axis towards top side of measurement tool in measurement plane of caliper fingers

[0349] AR axis towards right side of measurement tool measurement plane of caliper fingers

[0350] b datum point on conduit trajectory defining a second end of chord line ab c selected measurement point on conduit trajectory

[0351] d point of intersection of Econ chord line ab

[0352] caliper readings, radial distance measurement of conduit bore relative to tool body

[0353] Eameasured eccentricity of conduit trajectory at point ma

[0354] Ebmeasured eccentricity of conduit trajectory at point mb

[0355] Ec measured eccentricity of conduit trajectory at point mc

[0356] Ea. measured from point mcto point dLabel Description

[0357] Em measured eccentricity of conduit trajectory at measurement reference point m

[0358] Erreference eccentricity of reference trajectory at measurement reference point m

[0359] E “best estimate” incremental eccentricity of conduit trajectory

[0360] & Emmeasured incremental eccentricity of conduit trajectory

[0361] {FG} geometric forcing vector

[0362] F£virtual force

[0363] [6] geometry matrix

[0364] [Gfd geometry flexural matrix

[0365] i index of a measurement location along conduit trajectory

[0366] -a number of measurement intervals nearest in length to Lmib number of measurement intervals nearest in length to (L — Lm) j index of a caliper finger

[0367] [ ] identity matrix

[0368] [c] conditioning matrix

[0369] fed] foundation conditioning matrix

[0370] fed flexural conditioning matrix

[0371] [KG] geometric stiffness matrix

[0372] K£virtual stiffness

[0373] fd foundation stiffness parameter

[0374] fi flexural stiffness parameter

[0375] L length of chord line ab

[0376] Lac length of first rigid segment 410

[0377] Lbc length of second rigid segment 420

[0378] L-m distance between point a and point m

[0379] Lmab distance between point maand point mb

[0380] L-mac distance between point maand point mcLabel Description

[0381] m measurement reference point (on chord line ab) associated with a selected measurement point c on conduit trajectory

[0382] mameasurement point on logging tool

[0383] mbmeasurement point on logging tool

[0384] mcmeasurement point on logging tool

[0385] n number of measurement locations along conduit trajectory

[0386] r radius of curvature of a trajectory

[0387] R measurement length ratio

[0388] u lateral displacement of conduit trajectory

[0389] wEvirtual work

[0390] x lateral position of conduit trajectory in local cartesian coordinate system xrlateral position of reference traj ectory in local cartesian coordinate system z distance along z-axis in local cartesian coordinate system

[0391] MDmeasurement interval along conduit traj ectory

[0392] Af eccentricity correction

[0393] 8u variation in lateral displacement

[0394] 8wEvariation in virtual work

[0395] SAf variation in eccentricity correction

[0396] 0acbangle of articulation at measurement point c

[0397] 0bacangle between first rigid segment 410 and chord line ab

[0398] 0incinclination of measurement tool relative to vertical

[0399] 0rotrotation of measurement tool relative to the top (or “high side”) of the logging tool

Claims

WHAT IS CLAIMED IS:

1. A method for measuring lateral displacements of a trajectory of a conduit at selected measurement points along a selected length of the conduit, relative to a selected reference trajectory, said conduit having a conduit bore, wherein said method comprises the steps of:(a) receiving measurements of the conduit trajectory taken by a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:• positions of the first and second datum points relative to the measurement reference point are defined by the geometry of the measurement tool;• the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory; and• the measurements of the conduit trajectory comprise an eccentricity of each of the selected measurement points along the selected length of the conduit, relative to the chord line;(b) calculating, by means of one or more processors, an incremental eccentricity of each selected measurement point, wherein the incremental eccentricity is the difference between the measured eccentricity of the conduit trajectory at the measurement point and an eccentricity of the reference trajectory at the measurement point;(c) assembling, by means of the one or more processors, a first system of equations relating lateral displacements of the conduit trajectory to the incremental eccentricities of the selected measurement points;(d) creating, by means of the one or more processors, a second system of equations comprising a geometry matrix by applying, to the first system of equations, selected boundary conditions to make the number of unknowns associated withthe first system of equations equal the number of equations in the first system of equations;(e) creating, by means of the one or more processors, a third system of equations by applying selected error suppression to the second system of equations; and (f) solving, by means of the one or more processors, the third system of equations for the lateral displacements of the conduit trajectory.

2. The method as in Claim 1 wherein the measurement tool comprises a rigid tool body centralized relative to the conduit at locations corresponding to the first and second datum points.

3. The method as in Claim 2 wherein the measurements of the conduit trajectory were taken with the length of the chord line being the same for all of the selected measurement points.

4. The method as in Claim 1 wherein the measurement tool comprises:(a) a first rigid segment having an outer end and an inner end, said outer end of the first rigid segment corresponding to the first datum point; and(b) a second rigid segment having an outer end and an inner end, said outer end of the second rigid segment corresponding to the second datum point, and said inner end of the second rigid segment is joined to the inner end of the first rigid segment by means of an articulating connection;wherein the measurement tool is centralized relative to the conduit at the first datum point, the second datum point, and the articulating connection.

5. The method as in Claim 1 wherein the measurements of the conduit trajectory were taken with the measurement tool not centralized relative to the conduit, and with the measurement tool further configured to measure the locations of the first and second datum points on the conduit trajectory points relative to the measurement tool.

6. The method as in Claim 5 wherein the measurement tool comprises a rigid tool body configured with three pluralities of caliper fingers spaced apart at selected intervals along the length of the tool body.

7. The method as in Claim 1 or Claim 2 wherein the measurement tool comprises a rigid tool body configured with an upper centralizer, a lower centralizer, and an ultrasonic measurement head.

8. The method as in any one of Claims 1-7 wherein the measurements of the conduit trajectory were taken with the measurement tool being located within the conduit bore.

9. The method as in any one of Claims 1-8 wherein the selected boundary conditions applied to the first system of equations include the boundary condition that lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory is zero.

10. The method as in any one of Claims 1-8 wherein the selected boundary conditions applied to the first system of equations include the boundary condition that lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory is equal to that of a nearest point of the plurality of selected measurement points.

11. The method as in any one of Claims 1-8 wherein the selected boundary conditions applied to the first system of equations include the boundary condition that the lateral displacement of the conduit trajectory outside the selected length of the conduit trajectory decreases linearly to zero from that of a nearest one of the plurality of selected measurement points to a conduit trajectory point associated with a chord line furthest from the selected length of the conduit trajectory.

12. The method as in any one of Claims 1-8 wherein the selected boundary conditions applied to the first system of equations include the boundary condition that the lateral displacement of the conduit trajectory at a selected end of the selected length of the conduit trajectory is zero along a sufficiently long interval of the conduit trajectory to make the number of unknowns associated with the first system of equations equal the number of equations.

13. The method as in any one of Claims 1-12 wherein the selected error suppression applied to the second system of equations comprises one or more signal conditioning techniques to filter the incremental eccentricities.

14. The method as in Claim 13 wherein the one or more signal conditioning techniques to filter the incremental eccentricities are selected from the group consisting of(a) applying a constant offset correction;(b) applying a variable offset correction based on the measurement tool inclination and rotation;(c) applying a long- wavelength filter; and(d) applying a short- wavelength filter.

15. The method as in any one of Claims 1-14 wherein:(a) the second system of equations is pre-multiplied by a transpose of the geometry matrix forming a mathematically-equivalent system of equations comprising a geometric stiffness matrix; and(b) the selected error suppression applied to the second system of equations comprises matrix conditioning of the mathematically-equivalent system of equations.

16. The method as in Claim 15 wherein the matrix conditioning comprises addition of a foundation conditioning matrix to the geometric stiffness matrix.

17. The method as in Claim 15 or Claim 16 wherein the matrix conditioning comprises addition of a flexural conditioning matrix to the geometric stiffness matrix.

18. The method as in any one of Claims 1-17 wherein the step of solving the third system of equations is performed using a direct solution algorithm.

19. The method as in any one of Claims 1-17 wherein the step of solving the third system of equations is performed using an iterative solution algorithm.

20. The method as in any one of Claims 1-19 further comprising the step of using the measured lateral displacements of the conduit trajectory to produce a three-dimensional representation of the shape of the conduit bore.

21. The method as in Claim 20 further comprising the step of determining a drift diameter of the conduit bore based on the three-dimensional representation of the shape of the conduit bore.

22. The method as in Claim 21 wherein the conduit is a pipe string disposed in a petroleum well, and wherein the method further comprises the step of operating a milling tool within the conduit bore to reduce the conduit wall thickness in selected areas and thereby to increase the drift diameter of the conduit bore.

23. A system for measuring lateral displacements of a trajectory of a conduit at selected measurement points along a selected length of the conduit, relative to a selected reference trajectory, said conduit having a conduit bore, wherein said system comprises one or more processors configured to perform the steps of:(a) receiving measurements of the conduit trajectory taken by a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:• positions of the first and second datum points relative to the measurement reference point are defined by the geometry of the measurement tool;• the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory; and• the measurements of the conduit trajectory comprise an eccentricity of each of the selected measurement points along the selected length of the conduit, relative to the chord line;(b) calculating an incremental eccentricity of each selected measurement point, wherein the incremental eccentricity is the difference between the measured eccentricity of the conduit trajectory at the measurement point and an eccentricity of the reference trajectory at the measurement point;(c) assembling a first system of equations relating lateral displacements of the conduit trajectory to the incremental eccentricities of the selected measurement points;(d) creating a second system of equations comprising a geometry matrix by applying, to the first system of equations, selected boundary conditions to make the number of unknowns associated with the first system of equations equal the number of equations in the first system of equations;(e) creating a third system of equations by applying selected error suppression to the second system of equations; and(f) solving the third system of equations for the lateral displacements of the conduit trajectory.

24. The system as in Claim 23 wherein the one or more processors are further configured to use the measured lateral displacements of the conduit trajectory to produce a three-dimensional representation of the shape of the conduit bore.

25. The system as in Claim 24 wherein the one or more processors are further configured to determine a drift diameter of the conduit bore based on the three-dimensional representation of the shape of the conduit bore.

26. The system as in any one of Claims 23-25 further comprising a measurement tool configured to measure an orthogonal eccentricity of a selected measurement point on the conduit trajectory, relative to an associated measurement reference point on a chord line joining selected first and second datum points on the conduit trajectory, wherein:(a) positions of the first and second datum points relative to the measurement reference point are defined by the geometry of the measurement tool; and(b) the chord line will move with the measurement tool relative to the conduit trajectory, such that the locations of the first and second datum points on the conduit trajectory will change when the measurement tool is moved along the conduit trajectory.

27. The system as in Claim 26 wherein the measurement tool comprises a rigid tool body centralized relative to the conduit at locations corresponding to the first and second datum points.

28. The system as in Claim 26 wherein the measurement tool comprises:(a) a first rigid segment having an outer end and an inner end, said outer end of the first rigid segment corresponding to the first datum point; and(b) a second rigid segment having an outer end and an inner end, said outer end of the second rigid segment corresponding to the second datum point, and said inner end of the second rigid segment is joined to the inner end of the first rigid segment by means of an articulating connection.

29. The system as in Claim 26 wherein the measurement tool is not centralized relative to the conduit, and the measurement tool is further configured to measure the locations of the first and second datum points on the conduit trajectory relative to the measurement tool.

30. A logging tool comprising:(a) a first rigid segment having an outer end and an inner end;(b) a second rigid segment having an outer end and an inner end, with the inner end of the second rigid segment being joined to the inner end of the first rigid segment by means of an articulating connection;(c) a first centralizer, for centralizing the outer end of the first rigid segment within a bore of a conduit in which the logging tool may be disposed;(d) a second centralizer, for centralizing the outer end of the second rigid segment within the conduit bore;(e) a third centralizer, for centralizing the articulating connection within the conduit bore; and(f) a sensor for measuring an articulation angle between the first and second rigid31. A logging tool comprising:(a) a rigid tool body;(b) a first measurement means configured to measure an orthogonal eccentricity of a first selected measurement point on the conduit trajectory relative to a first measurement reference point on the rigid tool body;(c) a second measurement means configured to measure an orthogonal eccentricity of a second selected measurement point on the conduit trajectory relative to a second measurement reference point on the rigid tool body;(d) a third measurement means configured to measure an orthogonal eccentricity of a third selected measurement point on the conduit trajectory relative to a third measurement reference point on the rigid tool body; and(e) the first measurement point, the second measurement point, and the third measurement point are colinear.

32. The logging tool as in Claim 31 wherein each of the first, second, and third measurement means is selected from the group consisting of(a) a plurality of mechanical caliper fingers; and(b) an ultrasonic measurement head.

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