Cubic Spline Interpolation for. Petroleum Engineering Data

Transcription

1 Appled Mathematcal Scences, Vol. 8, 014, no. 10, HIKARI Ltd, Cubc Splne Interpolaton for Petroleum Engneerng Data * Samsul Arffn Abdul Karm Department of Fundamental and Appled Scences, Unverst Teknolog PETRONAS Bandar Ser Iskandar, Tronoh, Perak Darul Rdzuan, Malaysa * Correspondng author Muhammad Azuddn Mohd Rosl Petroleum Engneerng Department Unverst Teknolog PETRONAS Bandar Ser Iskandar, Tronoh, Perak Darul Rdzuan, Malaysa Muhammad Izzatullah Mohd Mustafa Reservor Engneer, Emerson Process Management Level 11, Menara Chan, 138 Jalan Ampang, 50450, Kuala Lumpur, Malaysa Copyrght 014 Samsul Arffn Abdul Karm et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Interpolaton and fttng the data to produce the nterpolatng curves or fttng curves are mportant n Ol and Gas (O&G) ndustry. Data nterpolaton s useful for scentfc vsualzaton for data nterpretaton. One of the effcent methods for data nterpolaton s cubc splne functon. In ths paper two types of cubc splne wll be used for data nterpolaton. The frst one s cubc splne nterpolaton wth C contnuty. Meanwhle the second splne s Pecewse Cubc Hermte 1 Splne (PCHIP) wth C contnuty. Numercal comparson between both cubc splnes and lnear splnes wll be dscussed n detals. From all numercal results, t was ndcated that cubc splne gves good results. Keywords: Petroleum Engneerng Data; Data nterpolaton; cubc splne; contnuty

2 5084 Samsul Arffn Abdul Karm et al. I. INTRODUCTION Data fttng, data approxmaton and data nterpolaton are mportant n vsualzaton of the data that may be obtaned from numercal experment or data collecton from well drllng, petroleum etc. Karm and Yahya [1] has dscussed n detals the used of cubc splne smoothng for ol and gas data nterpretaton. The Seabed Logng Data (SBL) s used for ther purposed. From the results, they concluded that cubc splnes smoothng gves qute mpressve results. The common method for data nterpolaton s a cubc splne functon. There exst many types of splne bass functon wth respectve degrees and ts respectve knots. But from lterature, the most sutable splne for many applcatons s cubc splnes nterpolaton. One of the man reasons why cubc splne s the most utlzng bass functon for data nterpolaton s that t s the lower degree splnes that can acheve the C contnuty [ - 9]. Besdes the used of cubc splne nterpolaton 1 there exst another cubc splne called as PCHIP but s only has C contnuty and usually data that beng nterpolated by PCHIP s overshoot at certan nterval [, 8, 9]. Ths unwanted overshoot s not vsual pleasng and may remove the mportant characterstcs of the data. Furthermore PCHIP s sutable for nterpolaton data that s monotone []. In general the data obtaned n Ol and Gas ndustry s not monotonc and there exsts thousands data sets that need to be takng care by the engneers. Ths s where cubc splne nterpolaton provdes a very good alternatve to the exstng methods such as lnear regresson and non-lnear regresson. Ths paper s contnuaton from the work of Karm and Yahya [1]. The man dfferent s that n ths paper both cubc splnes nterpolaton and PCHIP wll be used to nterpolate the gven data sets, meanwhle n [1] the cubc smoothng splne and PCHIP are used for SBL data fttng (smoothng) and data nterpolaton. Furthermore an error analyss for data nterpolaton by usng cubc splne nterpolaton and PCHIP also wll be dscussed n detals. Ths paper s organzed by four sectons whch are descrbed as follows. Frst Secton s about the ntroducton of data nterpolaton method used to nterpolate ol and gas data. Second secton descrbes the detal of the theory of cubc splnes nterpolaton as well PCHIP. Data collecton and expermental setup are dscussed n detals through Secton 3.Fourth Secton descrbes the numercal results obtaned by usng cubc splne and PCHIP for Petroleum Engneerng data nterpolaton. Secton 5 dscusses the error analyss by usng three dfferent true functons. Conclusons and some future researchs recommendaton wll be dscussed n the fnal secton.. CUBIC SPLINES INTERPOLATION Ths secton s devoted for cubc splne nterpolaton and PCHIP..1 Cubc splne nterpolaton The equaton of the cubc splne n the th nterval[ x 1, x], s show n Equaton (1). f x a b x c x d x n (1) 3 ( ) ; 1,,..., where a, b, c,and d s the 4n coeffcents for 1,,... n, The cubc splne nterpolaton n (1) satsfes the followng condtons: f ( x ) f ()

3 Cubc splne nterpolaton for petroleum engneerng data 5085 Equaton () untl (5) gves a total of f ( x ) f ( x ) (3) f ( x ) f ( x ) (4) ' ' f ( x ) f ( x ) (5) " " n condtons. For cubc splne nterpolaton, we need 4 n parameters to be determned. Therefore, we need two more condtons that can be ether clamped, natural boundary condtons etc. In ths paper wll be used the natural boundary condtons. It s gven as follows: " " f0 ( x0 ) fn 1( xn) 0 (6) After some substtuton to the Equaton () untl (6), we can extend t to nclude an extra parameter a that we actually not nterested n, to gve the followngs tr-dagonal lnear equatons: n c 0 h0 ( h0 h1 ) h1 c1 0 h1 ( h1 h ) h c 0 0 hn ( hn 1 hn ) h n c n ( a a1 ) ( a1 a0) h1 h ( an an 1) ( an 1 an) hn1 hn 0 If the grd ponts are equally spaced wth h h for some number h, then (7) 1 0 0c0 0 h 4h h 0 c1 a0 a1 a 0 h 4h h 0 c 3 (8) h 0 0 h 4h h an an 1a n c n 0 The tr-dagonal lnear equatons n (7) and (8) can be solved effcently by usng Gauss-Sedel teraton or LU decomposton or Thomas algorthm. Below s example for cubc splne nterpolaton by usng car acceleraton data.

4 5086 Samsul Arffn Abdul Karm et al. Example 1: Car acceleratng data Table 1. Car Acceleratng Tme (second) Velocty (m.p.h.) Here we apply cubc splne nterpolaton on [,3], [3,4] and [4,5]. Thus we have three (3) pecewse cubc splne nterpolaton that satsfes condton () untl (6) above. The cubc splne nterpolaton s gven as follows: x 11.58x 3.3x 8.58,,3 3 f ( x) 4.67x 47.83x 146.0x , 3, 4 (10) 3.73x 40.96x 09.11x ,5 Fgure 1 shows the cubc splne nterpolaton for data lsted n Table 1 above. Fgure : Cubc Splne Interpolaton Car Acceleratng

5 Cubc splne nterpolaton for petroleum engneerng data PCHIP PCHIP s ntated by Frstch and Carlson [8] and later these methods have been extended by Frstch and Butland [9] for local control of the nterpolatng curves. PCHIP functons can be defned as follows: For xx, x1, 1,,..., n 1, px s a cubc polynomal whch can be represented as follows: p x f H f H h D H h D H (11) Where D px, j, 1 and H, k 0,1,,3. are the usual cubc Hermte bass functons gven j k by H 0 x 1, H3 x, H 1 and H3 and 1 wth 3 3 3, and x x, 0,1. h x x, h PCHIP have been used for monotoncty-, postvty- and convexty-preservng data nterpolaton by Frstch and Carlson [8] and Frstch and Butland [9]. They derved the suffcent and necessary condton for the PCHIP to be monotonc on entre gven nterval of data that to be nterpolated. Orgnal cubc Hermte splne cannot guarantee to preserves the shape of the gven data. Thus 1 PCHIP wll guarantee to preserves the shape of the gven data wth C contnuty..3 Examples of Cubc Splne Interpolaton and PCHIP To show the dfferent between nterpolatng the gven data sets by usng (a) cubc splne and (b) PCHIP, we use the famous data sets taken from Frstch and Carlson [8] gven n Table. Ths data s comng from LLL radochemcal calculaton. Example. PCHIP and cubc splne nterpolaton x f x f Table. Interpolaton data from Frstch and Carlson [8]

6 5088 Samsul Arffn Abdul Karm et al. Fgure 3. Interpolaton usng cubc splne (dashed) and PCHIP (green). Clearly PCHIP preserves the geometrc shape of the data e.g. monotoncty. But from Fgure 3, the nterpolaton curves are overshoot and not very vsual pleasng. Even though cubc splne nterpolates the gven data wthout preservng the monotoncty of the gven data, the fnal nterpolatng s very smooth and n the cases where the geometrc shape of the data s not a crteron n nterpretng the ol and gas data, so cubc splne s more sutable compare to PCHIP. More detals on comparson between cubc splne nterpolaton and PCHIP can be found n the forthcomng monograph by Karm []. 3. DATA COLLECTION AND EXPERIMENTAL SETUP Nowadays, drllng become more challenges day by day. One of the technologes that have been ntroduced to mprove drllng operaton s casng whle drllng technque. Ths technque s usng casng to drll the well nstead of the drll ppe that usually use n conventonal drllng. Thus, t wll save the tme and cost from the trppng tme. Recently, casng drllng can only be used n shallow depth and soft formatons. Because hgh stress caused by the hard formaton wll lead the damage of the casng. So, desgn of the casng need to be analyzed carefully to gve the best drllng performance. Data that has been collected for ths report are measured depth, rate of penetraton (ROP), and drllng flud flow and pump pressure. All of the data are mportance because ts need to be accurate durng the plannng to am for a successful drllng operaton. The sze of the casng s 9-5/8 n and the depth of the well that has been drlled by ths technque s about 3401 ft. Now we

7 Cubc splne nterpolaton for petroleum engneerng data 5089 arrved at the man objectve of the paper to use cubc splne nterpolaton and PCHIP for Petroleum Engneerng data. The comparson between both splne also wll be dscussed n detals. 4. NUMERICAL RESULTS AND DISCUSSION In ths secton, we wll dscuss the applcaton of cubc splne nterpolaton and PCHIP to nterpolate the Petroleum Engneerng data. We dscussed three examples (1) Measured Depth vs Torque () Measured Depth vs Pump Pressure and (3) Measured Depth vs Flud Flow. Example 3: Measured Depth vs Torque Torque measurement s mportant n drllng because too much torque wll leads to the nablty to reach the target. Fgure 4. PCHIP Interpolaton (Measured Depth vs Torque)

8 5090 Samsul Arffn Abdul Karm et al. Fgure 5. Cubc Splne Interpolaton (Measured Depth vs Torque) Fgure 6. Combned Interpolaton (Measured Depth vs Torque): cubc splne (blue), PCHIP (red) and lnear splne (green).

9 Cubc splne nterpolaton for petroleum engneerng data 5091 Example 4: Measured Depth vs Pump Pressure Pump s used to pump the drllng flud nto the well whle drllng. Pump pressure can determned the pressure nsde the well. It needs to be measured accurately to avod any problem such as gas kck or blowout happened. Fgure 7: PCHIP Interpolaton (Measured Depth vs Pump Pressure) Fgure 8: Cubc Splne Interpolaton (Measured Depth vs Pump Pressure)

10 509 Samsul Arffn Abdul Karm et al. Fgure 9: Combned Interpolaton (Measured Depth vs Pump Pressure): cubc splne (blue), PCHIP (red) and lnear splne (green) Example 5: Measured Depth vs Flud Flow Flud flow n the well s beng measured to prevent any formaton flud flows nto the well because f the f the flud has been nvaded, t wll consdered a kck and wll lead to blow out of the well s not shut-n.

11 Cubc splne nterpolaton for petroleum engneerng data 5093 Fgure 10: PCHIP Interpolaton (Measured Depth vs Flud Flow) Fgure 11: Cubc Splne Interpolaton (Measured Depth vs Flud Flow)

12 5094 Samsul Arffn Abdul Karm et al. Fgure 14: Combned Interpolaton (Measured Depth vs Flud Flow): cubc splne (blue), PCHIP (red) and lnear splne (green). The followng algorthm can be used for data nterpolaton by usng PCHIP and cubc splne nterpolaton. Data nterpolaton algorthm 1. Input the nterpolaton data x, f, 1,,..., n 1.. (a) For cubc splne nterpolaton do the followng: () Calculate the cubc splne coeffcents for each nterval () Form cubc splne nterpolaton functon for each nterval x, x1, 1,,..., n 1. ()Generate the cubc splne nterpolaton that can be used to nterpret the gven data. (b) For PCHIP do the followng: () Calculate the new dervatve (from suffcent and necessary condton that gven n Frstch and Carlson [8] and Frstch and Butland [9]) for each nterval. () For 1,,..., n 1, Construct the PCHIP nterpolaton functon for each nterval x, x1, 1,,..., n 1.

13 Cubc splne nterpolaton for petroleum engneerng data ERROR ANALYSIS In ths secton we wll dscuss the error analyss for both splnes; cubc splne nterpolaton and PCHIP. We use three data sets taken from true functon x f x 8, f ( x) e x x and 10 f ( x) cos ( x) respectvely. Table 3, Table 4 and Table 5 summarzed the error analyss for data nterpolaton by usng PCHIP ( P x) and cubc splne ( S x) for all three functons. For the error measurements we use (1) Absolute error and () Root Mean Square Error (RMSE) gven by the followng formula: RMSE = n 1 y yˆ N. (1) where y s a true data and x f( x ) S( x) P( x ) S( x) f ( x) ŷ s a observed data and N s a total number of data. ( S( x) f ( x)) P( x) f ( x) ( P( x) f ( x)) SUM Table 3. Error for f x x 8

14 5096 Samsul Arffn Abdul Karm et al. x f( x ) S( x) P( x ) S( x) f ( x) ( S( x) f ( x)) P( x) f ( x) ( P( x) f ( x)) SUM Table 4. Error analyss for f ( x) e x x 6. CONCLUSIONS Ths paper dscussed the use of cubc splne nterpolaton and PCHIP for Petroleum Engneerng (PE) data nterpolaton. The man dfferent between both cubc splne s ther contnuty. From the numercal results, we can be concluded that there s trade-off between the order of contnuty and the shape of the fnal nterpolatng curves. Cubc splne nterpolaton gves much more smooth nterpolatng curves compare to the nterpolatng curves by usng PCHIP. The nterpolatng curves by usng PCHIP tend to overshoot on certan gven nterval. Ths fact can be explaned through ther contnuty as well as PCHIP tend to modfy the frst dervatve n order to preserves the shape of the gven data. Overall both cubc splne works well for all tested data sets. Error analyses by usng PCHIP and cubc splne for data nterpolaton also have been dscussed n detals. The author s n the fnal stages to proposed new ways to compute the frst dervatve values. Any such a fndng wll be publshed n our forthcomng paper. We wll report the related fndngs n our forthcomng papers.

15 Cubc splne nterpolaton for petroleum engneerng data 5097 x f( x ) S( x) P( x ) S( x) f ( x) ( S( x) f ( x)) P( x) f ( x) ( P( x) f ( x)) SUM Table 5. Error analyss for 10 f ( x) cos ( x) Bl 1 3 Root Mean Square Absolute Error Error (RMSE) Functon Cubc Cubc PCHIP PCHIP Splne Splne f ( x) x f ( x) e x x 10 x f ( x) cos ( ) Table 6. Comparson of error between cubc splne and PCHIP From the Table 6, the absolute error and root mean square error (RMSE) of cubc splne nterpolatons s less than the error by usng PCHIP nterpolaton for functon f ( x) x 8 and

16 5098 Samsul Arffn Abdul Karm et al. f ( x) e x 10 x whle for the functon f ( x) cos ( x), the absolute and RMSE of the cubc splne s hgher that the PCHIP nterpolaton. Ths s due to the fact that at the knots Generally the cubc splne nterpolaton wll gve less error compare to the PCHIP nterpolaton. ACKNOWLEDGMENT The authors would lke to acknowledge Unverst Teknolog PETRONAS (UTP) for the fnancal support receved n the form of a research grant: Short Term Internal Research Fundng (STIRF) No. 35/01. REFERENCES 1. Karm, S.A.A. and Yahya, N. (013). Seabed Logng Data Curve Fttng usng cubc Splnes. Appled Mathematcal Scences,. Karm, S.A.A. Data Interpolaton, Smoothng and Approxmaton usng Cubc Splne and Polynomal. (014). Book manuscrpt. 3. Akma, H. A new method and smooth curve fttng based on local procedures. J. Assoc. Comput. Mech. 17: (1970) 4. Bartels, R. H., Beatty, J. C. and Barsky, B. A. (1987). An Introducton to Splnes for use n Computer Graphcs and Geometrc Modelng, Morgan Kaufmann Publshers. 5. Bézer, P. Numercal Control: Mathematcs and Applcatons, Wley, New York. (197). 6. Derckx, P. Curve and Surface Fttng wth Splnes. (1996). Oxford Unversty Press, New York. 7. Farn, G. Curves and Surfaces for Computer Aded Geometrc Desgn. (00). A Practcal Gude. 5 th Edton, Morgan Kaufmann. 8. Frtsch, F.N. and Carlson, R.E. Monotone pecewse cubc nterpolaton. (1980). SIAM J. Numer. Anal. 17: Frtsch, F.N. and Butland, J. (1984). A method for constructng local monotone pecewse cubc nterpolants. SIAM J. Sc. and Statst. Comput. 5: Wang, Y. Smoothng Splnes: Methods and Applcatons. (01). (Chapman & Hall/CRC Monographs on Statstcs & Appled Probablty), Chapman and Hall/CRC, Hansen, P.C., Pereyra, V. and Scherer, G. Least Squares Data Fttng wth Applcatons. (01). The Johns Hopkns Unversty Press (December 5, 01). Receved: Aprl 5, 014