The Research of Ellipse Parameter Fitting Algorithm of Ultrasonic Imaging Logging in the Casing Hole

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1 Appled Mathematcs, 04, 5, 37-3 Publshed Onlne May 04 n ScRes. The Research of Ellpse Parameter Fttng Algorthm of Ultrasonc Imagng Loggng n the Casng Hole Jhu Tu, Bn Yang Electroncs & Informaton School of Yangtze Unversty, Jngzhou, Chna Electronc Informaton Engneerng School of HEBI College of Vocaton and Technology, Henan, Chna Emal: green666@6.com Receved 5 March 04; revsed 5 Aprl 04; accepted Aprl 04 Copyrght 04 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract In order to estmate ntensty and drecton of pressure acted on ol well n horzontal, ths paper proposes an algorthm of ellpse fttng based on least square method for sectons of ol well wth ultrasonc magng loggng data. Fve parameters about ellpse correspondng to the sectons of ol well are extracted and ther geometry deformatons and drecton are estmated. The tested results show that the fttng algorthm s robust and the estmatng error of parameter s small, whch have very mportant practcal sgnfcance on processng the well loggng. Keywords BHTV, Ellpse Fttng, Image Correcton, Tool Eccentrcty. Introducton The acoustc mage log s an acoustc televewer or ultrasonc borehole magng tool. The tool uses a rotatng acoustc transducer/recever operatng at ultrasonc frequences. The travel tme of the sound from the transducer to the wellbore wall and back to the recever s measured. Ths provdes a complete mage of wellbore dameter (a calper log). Black colour ndcates large dameter, and whte s near bt sze. Fractures and wellbore breakout are easly seen (black) [] []. In the casng hole, when the ultrasonc wave non-normal ncdence s caused by the eccentrc nstrument, the ampltude of ultrasonc reflecton wave from the nner wall wll be reduced; the echo tme wll be not accurate; and can often appear two peces of vertcal black strps n the casng well. Ths two dark bands cover up a lot of real nformaton wellbore casng. The log nterpretaton workers are unable to make the rght judgments. Owng to the eccentrcty of the nstruments durng the process of actual loggng of- How to cte ths paper: Tu, J.H. and Yang, B. (04) The Research of Ellpse Parameter Fttng Algorthm of Ultrasonc Imagng Loggng n the Casng Hole. Appled Mathematcs, 5,

2 ten appear, t s an mportant challenge to fnd a method of elmnatng eccentrc tme mage dark band [3] [4]. Durng the process of well loggng, ultrasonc transducer rotatng scannng a crcle gets the several tme callback data n a week. Accordng to these calper callback tme data can be calculated the well radus values, whch provde the estmate data for the ellpse fttng of well secton. Ths paper uses a least squares curve fttng method to calculate the center pont, the major axs, mnor axs and slope of the ellpse, and then judges the geometrc deformaton and drecton of the well cross secton usng these parameters of the ellpse, further to determne whether a downhole nstrument produce eccentrc, whch provdes a good theoretcal bass for the eccentrc nstrument correcton.. The Instrument Eccentrc Prncple In cased hole, when the downhole nstrument generates eccentrcty, acoustc transducer launch oblque ncdence, the eccentrc nstrument wll affect the ultrasonc echo ampltude and tme. The followng accounts for the eccentrc effect on magng from the prncple of ultrasonc magng. When the devce s the eccentrc, the cross secton of the well casng s shown n Fgure. Transducer n the y drecton s eccentrc and the eccentrcty dstance s D. The ncdent and echo wave of transducer s shown n Fgure, the nstrument eccentrc causes the non-normal ncdence and reflecton of the ultrasonc wave, even f the same condtons of reflecton coeffcent n the well week, the part reflecton wave devate the range accepted by transducer, the transducer can t receve the echo nformaton. At the same tme, the ncdent angle of the ultrasound wave s changed n the dfferent rotatonal poston because of the transducer eccentrcty, the values of the echo ampltude s also changed n the same smoothness degree of the well week. The ultrasonovson mage caused by the tool eccentrcty can often appear two peces of vertcal black strps n the casng well, the dark place s correspondng near the eccentrc vertcal drecton. Therefore, n the case of a downhole nstrument eccentrc ultrasonc televson mage can no longer completely correctly reflect the acoustc mpedance characterstcs of the wall of well, the echo wave tme cannot reflect the sze of the well radus, whch affects the nterpretaton and judgment of loggng mage. 3. Ellpse Fttng Algorthm Durng the process of ultrasonc televson magng loggng, ultrasonc transducer rotatng scannng a crcle gets the several tme callback data n a week, whch estmates the parameters for the ellpse fttng of well secton. The ellptcal curve fttng uses least squares fttng algorthm, the basc algorthm [5] s descrbed as follows: Fgure show that gets ponts echo tme T ( =,,, ), n () gets the dstance R ( =,,, ) from the devce center pont to the wall of well, V s the ultrasonc velocty n the mud. In () convert polar coordnates to rectangular coordnate system. Fgure. The secton dagram of the nstrument eccentrc n the casng hole; The ampltude dagram of Acoustc echo. 38

3 Fgure. The ellpse n the rectangular coordnate system. R = V T =,,, () x = R cosθ y = Rsnθ π θ = As we all known, arbtrary ellptc equatons on the plane can be expressed as: x Axy By Cx Dy E Usng pont data, constructng objectve functon s as follows: =,,, () = 0 (3) ( ABCDE,,,, ) ( x + Ax y + By ) F = Cx Dy E (4) Accordng to the prncple of least squares, the F s mnmze, there wll be: From above, the normal equatons are as follows: F F F F F = = = = = 0 A B C D E 3 3 xy xy xy xy xy xy xy y xy y y A x y B 3 xy xy x xy x C = x D 3 xy y xy y y E x y xy y x y x Solvng the lnear equatons of order 5 n (6), thus gets the general ellptc quadratc coeffcent A, B, C, D and E. Accordng to (7)-(), calculate the parameters of the ellpse as shown n Fgure 3: the center coordnates (x 0, y 0 ), the major axs a, the mnor axs b, the angle between the long axs and x-axs θ. (5) (6) 39

4 Fgure 3. The fttng result wthout the nose; The fttng result wth the nose. a = b = x BC AD = A 4B 0 y θ = tan D AD = A 4B 0 a bb ab b ( ACD BC D + 4BE A E) ( A 4B) B A + ( B) + ( ) ( ACD BC D + 4BE A E) ( A 4B) B+ A + ( B) + ( ) (7) (8) (9) (0) () 4. Testng and Results We performed experments on smulated data and real data to show the performance of our proposed model. All numercal experments were performed on 64-bt Wndows 7 on a desktop wth an Intel CPU at 3.0 GHz and 4 GB memory. ) The testng of smulated data In order to verfy the correctness of the algorthm, we produce a group data of ellpse accordng to the theory of ellptc equaton model, and add the whte Gaussan nose, then ft the ellpse curve accordng to the data wth nose and wthout nose respectvely. Fgure 3 shows the results of fttng. Smulaton results show that, wthout the nose the fttng parameters of ellptc equaton to draw the curve s completely covered by the orgnal ellptc curve, whch shows to concde; wth the nose the fttng parameters of ellptc equatons to draw the curve s also very well reflect the cross secton shape of the orgnal calper curve. Therefore, Ellpse fttng method based on the least squares fttng well estmates the ellptc curve parameters, even f superposton nose s very bg, the fttng algorthm s also very robust. ) The testng of real data In order to verfy the practcablty and valdty of the algorthm, we use the actual loggng data to test the algorthm. Fgure 4 shows the result of fttng ellpse for the real well data 6 before and after delete the outlers. 30

5 Fgure 4. The fttng result of real data wthout deletng outlers; The fttng result of real data wth deletng outlers. Fgure 4 s the result of fttng after the medan flter. Fgure 4 s the result of fttng after deletng the oulers. The red lne s the raw data, the blue pont lne s the well data of medan flter, the blue lnes s the well data after deletng the oulers, the black lnes s the curve accordng to the parameter of ellpse fttng. All the results show that the curve by ftted after deletng the oulers respects the secton shape of the well. 6. Concluson Ths paper fts ellpse for the ol well secton based on the method of least squares and extracts fve parameters of ellpse. From the testng results of theoretcal model data and real data, the algorthm proposed by ths paper s robust and the parameters extracted by ths algorthm are accurate. Accordng to those parameters of ellpse fttng, the geometrc deformaton and drecton of the well cross secton are estmated. So the nformaton about horzontal pressure and drecton can be obtaned, meanwhle those parameters provde a good theoretcal bass for the eccentrc nstrument correcton. References [] Pang, J.F., et al. (008) Loggng Prncple and Instrument. Scence Press, Bejng, [] Lu, F., Gao, H.J. and L, J. (009) A Hgh Performance Ultrasonc TV Imagng Loggng Tool. Well Loggng Technology, 33, [3] Yu, H.Q., Qu, W.L. and Huang, Z.L. (997) The Influences of Tool Eccentrcty n a Casng Well on Ultrasonc Echo. Journal of Huazhong Unversty of Scence and Technology, 5, [4] Yu, H.Q., Qu, W.L. and Huang, Z.L. (999) Influence of Tool Eccentrcty n a Well Casng on the Ultrasonc Televewer Image and Its Correcton Processng. Appled Acoustcs, 8, 6-0, 43. [5] Lu, S.G., L, P. and a, Y.L. (00) Evaluaton of the Form Error of Ellpse Based on Least Square Method. Acta Metrologca Snca, 3,