Patent Publication Number: US-8532942-B2

Title: Monitoring system for well casing

Description:
PRIORITY CLAIM 
     The present application claims priority from PCT/US2009/036646, filed 10 Mar. 2009, which claims priority from U.S. provisional application 61/035,822, filed 12 Mar. 2008, which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This invention relates generally to systems and methods for detecting deformation of a casing of a well in a formation and, more specifically, to a system that includes strings of interconnected strain sensors. 
     BACKGROUND 
     Electromagnetic investigation tools are often used to take measurements at points along the length of a borehole in an earth formation. Wells in formations are commonly reinforced with casings that prevent the wells from collapsing. However, forces applied by the formation may cause the casing to bend, buckle, or otherwise deform. Where the deformation results in a significant misalignment of the well axis, the production that can be gained from the well can may be partially or completely lost. In either case, additional time and expense is necessary to repair or replace the well. 
     The ability to detect an early stage of deformation would allow for changes in production practices and remedial action. 
     SUMMARY 
     The present disclosure provides a system and method for detecting deformation of a casing in a formation. The system includes non intersecting strings of interconnected sensors such that the risk of damage is reduced. The strings are arranged to facilitate qualitative and/or quantitative analysis of data from the interconnected sensors. 
     According to an exemplary embodiment, a system for monitoring deformation of a substantially cylindrical casing includes at least two strings of interconnected sensors. The strings are wrapped around the casing so as not to intersect one another. The strings include a first string that includes a first series of at least two segments. The first series of at least two segments includes a first segment arranged at a first angle with respect the casing axis and a second segment arranged at a second angle with respect to the casing axis. 
     In certain embodiments, first series of at least two segments further includes a third segment arranged at a third angle. In certain embodiments, the strings include a second string that is arranged at a substantially constant third angle. In certain embodiments, the segments extend for arc distances that are at least half of the circumference of the casing. 
     Grooves are formed in the casing and the strings are at least partially recessed in the grooves. 
     The system further includes a data acquisition unit and a computing unit for collecting and processing data measured at the sensors. In certain embodiments, at least one of the interconnected sensors measures strain. In certain embodiments, at least one of the interconnected sensors measures temperature. 
     According to one aspect of the disclosure, the strings include a second string that includes a second series of at least two segments. The second series of at least two segments includes a third segment that is arranged at a third angle with respect the casing axis and a fourth segment arranged at a fourth angle with respect to the casing axis. 
     In certain embodiments, the first series of at least two segments is substantially the same as the second series of at least two segments. According to an exemplary embodiment, axial distances of the segments are substantially equal to one another. In such embodiments, the first string and the second string can be positioned relative to one another such that segments that have different wrap angles are represented within distance intervals along the axial length of the casing. According to another exemplary embodiment, arc distances of the segments are substantially equal to one another. In such embodiments, the first string and the second string can be positioned relative to one another such that segments that have the same wrap angle are represented within distance intervals along the axial length of the casing. 
     According to another aspect of the disclosure, the strings include optical fibers and the sensors include periodically written wavelength reflectors. In certain embodiments, the wavelength reflectors are reflective gratings such as fiber Bragg gratings. 
     In such embodiments, strings provide a wavelength response that includes reflected wavelengths corresponding to sensors. Each reflected wavelength is substantially equal to the sum of a Bragg wavelength and a change in wavelength. The change in wavelength corresponds to a strain measurement. 
     Strings can be arranged such that subsets of the wavelength responses can be grouped according to wrap angle and such that at least one of the grouped subsets includes substantially continuous measurements along the longitudinal axis of the casing. Strings can also be arranged such that subsets of the wavelength responses can be grouped according to wrap angle and such that at least one of the grouped subsets includes substantially continuous measurements along the circumference of the casing. 
     According to another aspect of the disclosure, a method of imaging deformation of a cylindrical casing includes measuring an amount of strain at a plurality of positions on a casing, determining the deformation of the casing from the strain measurements, and projecting an image of the deformed casing. The strain is measured by receiving signals from at least two strings of interconnected sensors that are wrapped around the casing so as not to intersect one another. At least one of the strings includes a series of at least two segments. The series of at least two segments includes a first segment arranged at a first angle with respect the casing axis and a second angle arranged at a second angle with respect to the casing axis. A memory or computer readable medium includes computer executable instructions for execution of the method. 
     According to another aspect of the invention, a cylindrical casing includes at least two grooves for receiving at least two strings of interconnected sensors. The grooves are arranged so as not to intersect one another. At least one of the grooves includes a series of at least two segments. The series of at least two segments includes a first segment arranged at a first angle with respect the casing axis and a second segment arranged at a second angle with respect to the casing axis. 
     The foregoing has broadly outlined some of the aspects and features of the present invention, which should be construed to be merely illustrative of various potential applications of the invention. Other beneficial results can be obtained by applying the disclosed information in a different manner or by combining various aspects of the disclosed embodiments. Accordingly, other aspects and a more comprehensive understanding of the invention may be obtained by referring to the detailed description of the exemplary embodiments taken in conjunction with the accompanying drawings, in addition to the scope of the invention defined by the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a partial cross-sectional view of a well reinforced with a casing. 
         FIG. 2  is a partial side view of the casing of  FIG. 1  and a system for measuring deformation of the casing. 
         FIGS. 3-7  illustrate exemplary arrangements of strings of the system of  FIG. 2 . 
         FIG. 8  is a graph illustrating signals relating to the arrangement of strings shown in  FIG. 3 . 
         FIG. 9  is a graph illustrating signals relating to the arrangement of strings shown in  FIG. 4 . 
         FIG. 10  is a graph illustrating signals relating to the arrangement of strings shown in  FIG. 5 . 
         FIG. 11  is a graph illustrating signals relating to the arrangement of strings shown in  FIG. 6 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As required, detailed embodiments of the present invention are disclosed herein. It must be understood that the disclosed embodiments are merely exemplary of the invention that may be embodied in various and alternative forms, and combinations thereof. As used herein, the word “exemplary” is used expansively to refer to embodiments that serve as illustrations, specimens, models, or patterns. The figures are not necessarily to scale and some features may be exaggerated or minimized to show details of particular components. In other instances, well-known components, systems, materials, or methods have not been described in detail in order to avoid obscuring the present invention. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present invention. 
     Systems and methods are described herein in the context of determining deformation of a well casing. However, the present disclosure is also applicable to other cylindrical objects in a borehole where the systems and methods are used to detect and monitor deformation of the object during production or other non-production operations such as completion, gravel packing, frac packing, production, stimulation, and the like. The cylindrical objects may be in the form of a well bore tubular, a drill pipe, a production tube, a casing tube, a tubular screen, a sand screen, and the like. 
     The teachings of the present disclosure may also be used in other environments where pipes expand, contract, or bend. Examples of such environments include refineries, gas plants, and pipelines. 
     Herein, a suffix (a, b, c, etc.) or subscript (1, 2, 3, etc.) is affixed to an element numeral that references like-elements in a general manner so as to differentiate a specific one of the like-elements. For example, groove  30   a  is a specific one of grooves  30 . 
     Casing 
     Referring to  FIG. 1 , a well  10  is drilled in a formation  12 . To prevent well  10  from collapsing or to otherwise line or reinforce the well  10 , a casing  14  is formed in well  10 . In the exemplary embodiment, casing  14  is formed from steel tubes that are inserted into well  10 . Cement is poured between casing  14  and formation  12  to provide a bonded cement sheath  16 . However, in alternative embodiments, casing  14  may be formed from other materials and according to alternative methods. 
     For purposes of teaching, coordinate systems are now described. A Cartesian coordinate system can be used that includes an x axis, a y axis, and a z axis that are orthogonal to one another. The z axis corresponds to the longitudinal axis of casing  14  and any position on casing  14  can be established according to an axial position z and a position in the x-y plane, which is perpendicular to the z axis. In the illustrated embodiment, casing  14  is cylindrical and any position on casing  14  can be established using a Cylindrical coordinate system. Here, the z axis is the same as that of the Cartesian coordinate system and a position lying in the x-y plane is represented by a radius r and a position angle α and referred to as a radial position rα. Radius r defines a distance of the radial position rα from the z axis and extends in a direction determined by position angle α to the radial position rα. Here, position angle α is measured from the x axis. 
     A bending direction represents the direction of a bending moment on casing  14 . The bending direction is represented by a bending angle β that is measured relative to the x axis. A reference angle φ is measured between bending angle β and position angle α. 
     Deformation 
     Casing  14  may be subject to forces, such as shear forces and compaction forces exerted, for example, by formation  12  or by the inflow of fluid between formation  12  and casing  14 . These forces can cause casing  14  to deform. An example of a force F causing deformation of casing  14  is illustrated in  FIG. 2 . 
     System 
     Continuing with  FIG. 2 , casing  14  includes a system  20  for detecting deformation. System  20  includes strings  22  of interconnected strain sensors  24  that are wrapped around casing  14  such that sensors  24  are positioned along the axial length and circumference of casing  14 . 
     System  20  further includes a data acquisition unit  38  and a computing unit  40 . Data acquisition unit  38  collects the response at the sensors  24  of each of the strings  22 . The response and/or data representative thereof are provided to computing unit  40  to be processed. Computing unit  40  includes computer components including a data acquisition unit interface  42 , an operator interface  44 , a processor unit  46 , a memory  48  for storing information, and a bus  50  that couples various system components including memory  48  to processor unit  46 . 
     Strings of Interconnected Sensors 
     There are many different suitable types of strings  22  of interconnected sensors  24  that can be associated with system  20 . For example, strings  22  can be waveguides such as optical fibers and sensors  24  can be wavelength-specific reflectors such as periodically written fiber Bragg gratings (FBG). An advantage of optical fibers with periodically written fiber Bragg gratings is that fiber Bragg gratings are less sensitive to vibration or heat and consequently are far more reliable. In alternative embodiments, strain sensors  24  can be other types of gratings, semiconductor strain gages, piezoresistors, foil gages, mechanical strain gages, combinations thereof, and the like. 
     Sensors  24  are not limited to strain sensors. Rather, in certain applications, sensors  24  are temperature sensors. 
     According to a first exemplary embodiment described herein, strings  22  are optical fibers and sensors  24  are fiber Bragg gratings. 
     A wavelength response λ n  of a string  22  is data representing reflected wavelengths λ r  at sensors  24 . The reflected wavelengths λ r  each represent a fiber strain ε f  measurement at a sensor  24 . 
     Generally described, reflected wavelength λ r  is substantially equal to a Bragg wavelength λ b  plus a change in wavelength Δλ. Specifically, reflected wavelength λ r  is equal to Bragg wavelength λ b  when fiber strain ε f  measurement is substantially zero. When fiber strain ε f  measurement is non-zero, reflected wavelength λ r  differs from Bragg wavelength λ b . The difference is change in wavelength Δλ and thus change in wavelength Δλ is the part of reflected wavelength λ r  that is associated with fiber strain ε f . Bragg wavelength λ b  provides a reference from which change in wavelength Δλ is measured at each of sensors  24 . The relationship between change in wavelength Δλ and fiber strain ε f  is described in further detail below. 
     Multiple Strings and Multiple Wrap Angles 
     In the illustrated embodiments, system  20  includes a plurality of strings  22  and each string  22  winds substantially helically at least partially along the length of casing  14 . Certain of strings  20  include a series of segments S that are arranged at different inclinations, hereinafter referred to as wrap angles θ. Typically, the series is at least partially repeated. 
     In general, wrapping strings  22  at an angle is beneficial in that strings  22  only experience a fraction of the strain experienced by casing  14 . Each wrap angle θ is effective for a range of strain. Accordingly, the use of multiple strings  22  with different wrap angles θ expands the overall range of strain that system  20  can measure. For example, a string with a wrap angle of 20° may fail at one level of strain while the same string with a wrap angle of 30° or more may not fail at the same level of strain or at a slightly higher level of strain. 
     The use of multiple strings  22  with different wrap angles θ facilitates determining Poisson&#39;s ratio v, as described in further detail below. Poisson&#39;s ratio v may be an undetermined parameter where casing  14  nonelastically deforms or yields under higher strains. For example, where casing  14  is steel, Poisson&#39;s ratio v may be near 0.3 while deformation is elastic, but trends toward 0.5 after deformation becomes non-elastic and the material yields. 
     Another advantage of wrapping casing  14  with multiple strings  22  is that there is added redundancy in case of failure of one of strings  22 . The additional data collected with multiple strings  22  makes recovery of a 3-D image an overdetermined problem thereby improving the quality of the image. 
     Non Intersecting Method of Wrapping Multiple Strings at Multiple Angles 
     In the illustrated embodiments, strings  22  are arranged so as not to intersect one another. Referring to  FIG. 2 , grooves  30  are formed in casing  14  and strings  22  are at least partially recessed in grooves  30 . As strings  22  are arranged so as to not intersect one another, the depth of grooves  30  is minimized and, accordingly, the effect of grooves  30  on the integrity of casing  14  is minimized. Conversely, were strings  22  to be arranged to intersect, at least part of the depth of grooves  30  would have to be increased at regions of intersection so that strings  22  would not protrude out of grooves  30 . However, the increased depth of grooves  30  would have a greater effect on the integrity of casing  14 . Alternatively, if the depth of grooves  30  is not increased, overlapping strings  22  would protrude outside grooves  30  thereby increasing the risk of being damaged. 
     Exemplary arrangements of strings  22  are now described. In general, the description of an arrangement of strings  22  is applicable to an arrangement of grooves  30  as strings  22  are received in grooves  30 . In other words, a string  22  and a corresponding groove  30  follow substantially the same path. 
     Referring to  FIGS. 3-7 , casings  14  are shown in an unrolled or flattened condition to illustrate arrangements of strings  22 . In other words, axial position z is plotted on the vertical axis and radial position rα is plotted on the horizontal axis. 
     Generally described, each casing  14  is wrapped with strings  22  that wind substantially helically at least partially along the axial length of casing  14 . At least one of strings  22  includes a series of segments S that are arranged at different inclinations or wrap angles θ. The illustrated wrap angles θ are measured with respect to x-y planes that are represented by notional dotted lines although equivalent alternative formulations can be achieved by changing the reference plane used to measure wrap angles θ. 
     Segments S are arranged at wrap angles θ such that, as segments S are wrapped around casing  14 , segments S longitudinally ascend an axial distance L along the axial length of casing  14  and transversely extend an arc distance C around the circumference of casing  14 . 
     As mentioned above, wrap angle θ can be selected according to a range of strains that system  20  is likely to encounter or designed to measure. The lengths of segments S may then be selected in any manner so long as strings  22  do not intersect and overlap one another. Exemplary methods for selecting the lengths of segments S are now described. As described in further detail below, the selection of the lengths of segments S facilitates qualitative and quantitative analysis of wavelength responses λ n . 
     Arrangements of Strings 
     Referring to  FIGS. 3 and 4 , the illustrated arrangements include first string  22   a  and second string  22   b  where first string  22   a  has a repeating series of segments S 1 , S 2  and string  22   b  has a repeating series of segments S 3 , S 4 . Strings  22   a ,  22   b  are substantially similar to one another as segments S 1 , S 3  have substantially the same length and wrap angle θ and segments S 2 , S 4  have substantially the same length and wrap angle θ. Specifically, wrap angle θ 1  is substantially equal to wrap angle θ 3  and wrap angle θ 2  is substantially equal to wrap angle θ 4 . 
     The difference in position of strings  22   a ,  22   b  relative to one another, generally referred to herein as phase, is selected such that the strings do not intersect. Phase can be indicated by the distance and direction between reference points p on strings  22 . Reference points p may be selected where a series of segments S begins or ends or at a meeting point of segments S. For example, referring to  FIG. 3 , a reference point p on first string  22   a  and a corresponding reference point p on second string  22   b  have different axial positions z and radial positions rα. Referring to  FIG. 4 , a reference point p on first string  22   a  and a corresponding reference point p on second string  22   b  have the same axial position z and different radial positions rα. 
     For clarity, in  FIGS. 3 and 4 , first string  22   a  is illustrated as a relatively thicker line, second string  22   b  is illustrated as a relatively thinner line, segments S 1 , S 3  are illustrated as solid lines, and segments S 2 , S 4  are illustrated as dashed lines. 
     Referring to  FIG. 3 , lengths of segments S 1 , S 2 , S 3 , S 4  are selected such that axial distances L 1 , L 2 , L 3 , L 4  are substantially the same and equal to constant distance intervals N measured along the axial length of casing  14 . Phase is selected such that segments S 1 , S 4  are represented within every other distance interval N and segments S 2 , S 3  are represented within other distance intervals N. 
     As described in further detail below, when wavelength responses λ n  of both strings  22   a ,  22   b  are plotted on the same graph with respect to axial position z, subsets u of wavelength responses λ n  can be grouped according to wrap angle θ such that a group of subsets u represents a substantially continuous series of measurements along the axial length of casing  14  for one wrap angle θ value. Referring momentarily to  FIG. 8 , for the arrangement of  FIG. 3 , subsets u of wavelength responses λ n1 , λ n2  that correspond to wrap angles θ 1 , θ 3  can be combined and subsets of wavelength responses λ n1 , λ n2  that correspond to wrap angles θ 2 , θ 4  can be combined. 
     Referring to  FIG. 4 , lengths of segments S 1 , S 2 , S 3 , S 4  are selected such that arc distances C 1 , C 2 , C 3 , C 4  are substantially the same. Specifically, each arc distance C 1 , C 2 , C 3 , C 4  is substantially half of the circumference of casing  14 . 
     Phase is selected such that segments S 1 , S 3  are represented within every other distance interval N and segments S 2 , S 4  are represented within other intervals N. Here, distance intervals N change in length in an alternating manner according to different axial distances L 1 , L 2 , L 3 , L 4 . 
     Referring momentarily to  FIG. 9 , for the arrangement of  FIG. 4 , subsets u of wavelength responses λ n1 , λ n2  that correspond to wrap angles θ 1 , θ 3  can be combined and subsets u of wavelength responses λ n1 , λ n2  that correspond to wrap angles θ 2 , θ 4  can be combined. Here, the groups of subsets u are interrupted and only partially represented along the axial length of casing  14  but effectively measure around the entire circumference of casing  14 . 
     In alternative embodiments, the lengths of segments S 1 , S 2 , S 3 , S 4  can be constrained so as to be substantially equivalent. 
     The teachings of the present disclosure are not limited to a system having two strings  22  where each string  22  is arranged to include two wrap angles θ. Referring to  FIGS. 5-7 , embodiments of system  20  are described that include at least two strings  22  where at least one of strings  22  is arranged to include inclinations of at least two wrap angles θ. 
     Referring to  FIG. 5 , system  20  includes first string  22   a  and second string  22   b . Here, string  22   a  has a repeating series of segments S 1 , S 2  with different wrap angles θ 1 , θ 2 . String  22   b  has a substantially constant wrap angle θ 3  although, for purposes of teaching, string  22   b  is described as a series of segments S 3  that have the same wrap angle θ 3 . 
     Segments S 1 , S 2  extend axial distances L 1 , L 2  and arc distances C 1 , C 2  that are determined by the lengths of segments S 1 , S 2  wrap angles θ 1 , θ 2 . String  22   b  has wrap angle θ 3  where notional segments S 3  extend an axial distance L 3  that is substantially equal to the sum of axial distances L 1 , L 2  and extend an arc distance C 3  that is substantially equal to the sum of arc distances C 1 , C 2 . String  22   a  effectively varies about a constant angle of inclination along the length thereof and the constant angle of inclination is substantially equal to wrap angle θ 3 . Strings  22   a ,  22   b  are therefore approximately parallel to one another although phased such that variations of string  22   a  from a parallel path do not cause strings  22   a ,  22   b  to intersect one another. 
     Referring to  FIG. 6 , system  20  includes three strings  22   a ,  22   b ,  22   c  and each string  22   a ,  22   b ,  22   c  includes the same series of segments S 1 , S 2 , S 3  although, for simplicity, only string  22   a  is labeled. Similar to the arrangement of  FIG. 3 , lengths of segments S 1 , S 2 , S 3  such that axial lengths L 1 , L 2 , L 3  are substantially the same. Strings  22   a ,  22   b ,  22   c  are phased such that reference points p on strings  22   a ,  22   b ,  22   c  have different axial positions z and the same radial position rα. 
     The previously described arrangements of  FIGS. 3-5  can include additional strings  22  arranged at one or more wrap angles. For example, referring to  FIG. 7 , string  22   c  is added to the arrangement of  FIG. 3 . Here, string  22   c  has a substantially constant wrap angle θ 5  that can be determined as described for wrap angle θ 3  for the arrangement of  FIG. 5 . 
     Relationship Between Change in Wavelength and Strain 
     An equation that may be used to relate change in wavelength Δλ and fiber strain εf imposed on sensors  24  is given by Δλ=λ b (1−Pe)Kε f . As an example, Bragg wavelength λ b  may be approximately 1560 nanometers. The term (1−P e ) is a fiber response which, fox example, may be 0.8. Bonding coefficient K represents the bond of sensor  24  to casing  14  and, for example, may be 0.9 or greater. 
     The fiber strain ε f  may be associated with strain at a sensor  24  position on casing  14  according to 
     
       
         
           
             
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     Fiber strain ε f  measured by sensor  24  at a position on casing  14  is a function of axial strain ε a  at the position, radius of curvature R at the position, Poisson&#39;s ratio v, wrap angle θ of segment S on which sensor  24  is located, and radial position which is represented in the equation by radius r and reference angle φ. Fiber strain ε f  is measured, wrap angle θ is known, and radius r is known. Poisson&#39;s ratio v is typically known for elastic deformation of casing  14  and unknown for non-elastic deformation of casing  14 . Radius of curvature R, reference angle φ, and axial strain ε a  are typically unknown and are determined through analysis of wavelength response λ n . Similarly, Poisson&#39;s ratio v can be determined through analysis of wavelength response λ n  where Poisson&#39;s ratio v is unknown. 
     Analysis of Wavelength Response 
     Referring to  FIGS. 8-11 , wavelength responses λ n  of strings  22  are plotted on the same graph. These measurements represent fiber strain ε f  measurements made at each sensor  24  by system  20 . Here, wavelength responses λ n  are plotted with respect to axial positions z of sensors  24  or along the longitudinal axis of casing  14 . 
     As mentioned above, each reflected wavelength λ r  of wavelength response λ n  is substantially equal to Bragg wavelength λ b  plus change in wavelength Δλ. As change in wavelength Δλ is dependent on wrap angle θ, a shift in wavelength response λ n  (tracked by dotted lines) is observed where sensors  24  in series are on segments S that are arranged at different wrap angles θ. For example, a shift is observed approximately at axial positions z where segments S interface. As previously mentioned, subsets u of wavelength responsesλ n  that correspond to one wrap angle θ can be grouped together to effectively provide information that would be provided by a string  22  wrapped at a single wrap angle θ. 
     Generally described, in response to axial strain ε a  on casing  14 , wavelength response λ n  is typically observed as a constant (DC) shift from Bragg wavelength λ b . In response to bending of casing  14  that corresponds to a radius of curvature R, wavelength response λ n  is typically observed as a sinusoid (AC). A change in Poisson&#39;s ratio v modifies both the amplitude of the axial strain ε a  shift and the amplitude of the sinusoids. In any case, signal processing can be used to determine axial strain ε a , radius of curvature R, reference angle φ, and Poisson&#39;s ratio v at sensor  24  positions. Examples of applicable signal processing techniques include inversion where a misfit is minimized and turbo boosting. The signal processing method can include formulating wavelength response λ n  for one wrap angle as the superposition of a constant shift and a sinusoid. 
       FIG. 8  represents exemplary wavelength responses λ n1 , λ n2  measured by system  20  where strings  22   a ,  22   b  are arranged as shown in  FIG. 3 . Here, wavelength responses λ 1 , λ 2  are unique for axial strain ε a , radius of curvature R, and Poisson&#39;s ratio v. 
       FIG. 9  represents exemplary wavelength responses λ n1 , λ n2  measured by system  20  where strings  22   a ,  22   b  are arranged as shown in  FIG. 4 . Here, wavelength responses λ n1 , λ n2  are unique for radius of curvature R. Specifically, subsets u within one of distance intervals N spread apart with decreasing radius of curvature R. 
       FIG. 10  represents exemplary wavelength responses λ n1 , λ n2  measured by system  20  where strings  22   a ,  22   b  are arranged as shown in  FIG. 5 . As wrap angle θ of string  22   b  is substantially constant, there is no shift due to change in wrap angle θ. 
       FIG. 11  represents exemplary wavelength responses λ n1 , λ n2 , λ n3  measured by system  20  where strings  22   a ,  22   b ,  22   c  are arranged as shown in  FIG. 6 . The result is similar to that of  FIG. 8  however this arrangement provides three groups of subsets u corresponding to three different wrap angle θ values. 
     The law does not require and it is economically prohibitive to illustrate and teach every possible embodiment of the present disclosure. Hence, the above-described embodiments are merely exemplary illustrations of implementations set forth for a clear understanding of the principles of the invention. Variations, modifications, and combinations may be made to the above-described embodiments without departing from the scope of the claims. All such variations, modifications, and combinations are included herein by the scope of this disclosure and the following claims.