Evaluation Of The Curve-fitting Method And The Horner-plot Method For .
PROCEEDINGS, Twentieth Workshop on Geothermal Reservoir EngineeringStanford University, Stanford, California, January 24-26 1995SGP-TR-150EVALUATION OF THE CURVE-FITTING METHOD AND THE HORNER-PLOT METHODFOR ESTIMATION OF THE TRUE FORMATION TEMPERATUREUSING TEMPERATURE RECOVERY LOGGING DATAMasami Hyodo and Shinji TakasugiGeothermal Energy Research and Development Co., Ltd., Tokyo 103, JapanABSTRACTtrue formation temperature, temperature recovery logs are most often carried out. Also, another important information of the fluid entry depthin the well will be obtained by these logs. If wecould estimate the formation temperature usingshort period of logging data, it is very preferableto save the rig time and logging cost.This paper describes the method to estimate thetrue formation temperature using temperaturerecovery logging data after the well reach tototal depth (T.D.). The method designated as"Curve-fitting method (CFM)" is based onmathematical model proposed by Middleton(1979, 1982). The accuracy and applicability ofthis method are evaluated with several field dataand compared advantageously with the Hornerplot method.The Horner-plot method has been popularlyused for estimating the formation temperature(Dowdle and Cobb, 1975). This method is easyto utilize even in the field, however on the otherhand, it requires relatively long period of temperature recover data up to 120 hours to estimate correctly. It is also pointed out that thefluid circulation time as the Horner-time is verycritical to the estimated result, therefore theHorner-time must be decided carefully.Then, real-time data acquisition system including interpretation software has also been successfully developed.As a conclusion, the followings are confirmed:1) The developed CFM can be applicable to theestimation of the true formation temperatureeven using 24 hours temperature recoverydata, although the Horner-plot method mightneed up to 120 hours recovery data, usually.Mathematical models for borehole temperaturestabilization were proposed to estimate the trueformation temperature. In this paper, thesemodels were evaluated for the applicability tothe estimation of the true formation temperatureand the recovery temperature.-2) Though depending upon the quality of thedata and/or number of the temperature recovery logging data, i t might be possible toestimate the true formation temperature usingless than 24 hours recovery data. Because,the computer program of this system has thefunction to decide when the measurement oftemperature recovery logging should befinished.-2.1 HornerLPlot MethodJ. The Horner-plot method as a formation temperature estimation which is based on an empiricalanalysis method that the phenomenon of temperature recovery after circulation has ceased inheat conductive geothermal well, is similar tothe pressure build-up behavior of geothermalreservoir. The Horner-plot method proposed byDowdle and Cobb (1975) gives a reliable staticformation temperature in region of low geothermal gradient, and can be expressed thefollowing formula: -.T h e true formation tempqrature is one of theimportant parameter for geothermal reservoirevaluation and it will be used if drilling of thewell will be made complete or not when the wellis reached to planned T.D. In order to obtain the-23-
B H T h Tf C 1 o g{ ( t f d t ) / d t }recovery after circulation has ceased are examined. The "Circular model" proposed by Luikovis assuming the physical well model as a circularcylinder (Fig. 1 (a)). T h e " S q u a r e m o d e l "proposed by Carslaw and Jaeger is assumingthat a well can be approximated by a squarecylinder as a physical well model (Fig. 1 (b)).Middleton proposed that a conductive heattransfer into a vertical, approximately cylindrical region of small diameter in rectangularcoordinates leads to an expression for temperature which is very much simpler than the corresponding expression in cylindrical coordinates(Fig. 1 (c)). Through c a s e s t u d i e s o f thesemodels using non-linear least squares fitting,results of calculated formation temperature fromthe model proposed by Middleton are evaluatedmore reliable and more accurate in comparisonwith field data. The reason why the most unrealMiddleton's well model in three models calculates most reliable and accurate results is expected that the Middleton's m o d e l m a y b eexpressed realistic wellbore such as very roughshape and not gauged (Fig. 2).(1)Tf is the true formation temperature; t is thecirculation time; dt is the elapsed time aftercirculation has ceased.According to Dowdle and Cobb, comparativeand analytical studies of the temperature buildup and pressure build-up using the diffusibilityequation showed that the two methods are notcompletely analogous and Equ. (1) is not correcttheoretically. Therefore, the method requiredlong period of temperature recover time.2.2 Curve-Fitting Method(1) Curve-fitting methodT h e analytical C F M based on mathematicaltemperature stabilization has been proposed tocalculate the true formation temperature withbetter accuracy even from shorter period oftemperature recovery data by Luikov (1968),Carslaw and Jaeger (1959), and Middleton(1979, 1982). CFM calculates a static temperature of the equilibrium formation by mathematically represented the physical well model fortemperature recover. Proposed mathematicalmodel by Middleton was the temperature distribution in the center of the well with a verticalcylinder of infinite length, after circulation ofdrilling mud has ceased; BHTc(t) can be expressed the following formula:B H T c ( t ) Ti R i(3) Inversion for CFMForward and inverse techniques a s fittingmethod are examined. In forward technique, theformation temperature can be obtained by superimposing a set of master curve, based onEqu. (2), on observed temperature data plottedat the same scale (similar to type-curve matching). However, many master curves are necessary to obtain accurate result, and it is verycomplicated work of trial and error, and also it'stakes time. ( T f -Ti n iThen, inverse technique which is non-liner leastsquares fitting method expressed in Equ. (3) isapplied to obtain the formation temperature.t is the elapsed time after circulation has ceased;Tini is the initial temperature in the borehole at t 0; Tf is the true formation temperature; R isthe effective radius of the region affected bydrilling; K is the thermal diffusibility of the wellcontents.Fluid circulation time in drilling, which is required for the Horner-plot method and also is alittle uncertainty about adequate number, is notnecessary in this model.BHTo(ti) is the observed temperature at a time;BHTc(ti) is the calculated temperature at a timefrom Equ. ( 3 ) ; n is number of measured data.Where the effective radius; R and thermal diffusibility; K are expressed simply assumptionfollowing value a;(2) Mathematical well modelsProposed several physical well models to analyze the behavior of wellbore temperature-24-
3.3 Comparison of Estimated Temperature(4)Figure 5 s h o w s the c r o s s plot o f estimatedformation temperature by the Horner-plotmethod (Tfbuild) and by CFM (Tffit) using 31temperature recovery logs at 13 experimentalwells in eight geothermal areas. Figure 5 (a)illustrates the cross-plot with the temperaturerecovery data up to 120 hours of elapsed timeafter the circulation ceased. The average errorbetween both methods (Tffit - Tfbuild) is about-8.6"C (standard variation; (5 7.0). Figure 5(b) illustrates the cross-plot with the temperature recovery until eight hours. The averageerror of Tffit - Tfbuild is approximately-13.7"C ((5 9.4). As shown in Fig. 5, the errorbecomes bigger in region where geothermalgradient is higher.h is the thermal conductivity; p is the density; cis t h e s p e c i f i c heat capacity. Therefore, ininverse technique, estimated true formationtemperature can be obtained by iterate to minimize the sum of squares between BHTo andBHTc at a time. Features of a non-liner leastsquares method, in comparison with type-curvematching, are in its rapidity and objectivity.Therefore, we call this inverse method as CFM.Figure 3 illustrates an example of the non-linerleast squares fitting result by inversion.4. OUAl , m O N OF T H E T H O L E3. EVA1,UATION O F M E T H O D S FORNG THE TRUE FORMATIONTEMPERATUREThrough the case studies, qualification of themethods can be concluded as follows.3.1 Estimated Formation Temperature by theHorner-Plot Method4.1 Horner-Plot MethodEstimation of the true formation temperature bythe Horner-plot method using experimentaltemperature recovery logging data which wasperformed at the LC-1 well in Kyushu, Japan isshown in Fig. 4. T h i s graph shows that theHorner-plot method needs the temperaturerecovery logging data at least up to 48 hours,usually 120 hours of elapsed time to obtain theliner region in the Horner-plot. It is also confirmed that the value of circulation time is veryessential to estimate correctly, otherwise it isvery easily to miss-estimate 20-50 C.One of the reason of difference between estimated temperatures by both methods shown inFig. 5 can be understood that a main cause forthe error in estimated temperatures by t h eHorner-plot method is insufficient temperaturerecovery log data and inexact circulation time.Therefore, i t is possible to derive that biggererrors in estimated temperature by the Hornerplot method are the defect of this method forestimation of the true formation temperature ingeothermal area where geothermal gradients arerelatively high.3.2 Est i m at e d Form at i o n T e m p era t u re byCurve-Fitting Method (CFM)4.2 Curve-Fitting Method (CFM)The mathematical model for CFM is based onthe heat conductive formation model, thereforeCFM cannot be applied to estimate formationtemperature in non-conductive temperaturedistribution zone. From the same reason, CFMalso cannot estimate the true formation temperature at lost circulation zone. If it is important toestimate the temperature in this zone, w e needmore study.Evaluation of CFM was performed using experimental temperature recovery data (Takai et al.,1994). Reliability and accuracy were exhaustively evaluated using elapsed time of temperature recovery log versus number of data andtime interval as explained in Table 1. Throughthese case studies, it is verified that CFM can besuitable to estimate the formation temperature.That is, the accuracy of the estimated formationtemperature using until 24.5 hours data is atmost 5"C, and even using until 12.5 hours data;is at most 10 C. The computation time for CFMis less than 10 seconds using recent lap-top PC.However, estimated temperature is not sensitiveto the circulation time, therefore it is easier touse i n the field and m o r e accurate than theHorner-plot method.-25-
estimation method of the true formationtemperature has been successfully developed.CFM is based on mathematical model fortemperature recovery and includes the inversion scheme with the non-liner least squaresmethod. Because of these schemes, estimated temperature can be computed immediatelyafter the temperature recovery log has beendone.APPLICATIONIn order to apply CFM as a method for the trueformation temperature estimation and/or recovertemperature estimation at any elapsed timeduring temperature recovery logging in the field,real-time data acquisition system has beendeveloped. Acquired data on field computerare processed almost real-timely, accordinglythe estimated temperature can be calculated atevery depth just after the temperature recoverylog has been run.2) In comparison with the Homer-plot method,C F M is more reliable and useful f o r t h etemperature recovery and the true formationtemperature estimation.This computer program has the function forevaluation of the reliability of the estimatedtemperature. Then, the engineer can use thisinformation f o r his decision making whentemperature recovery logs shall be finished.This function will help to keep the accuracy ofthe measurement, and to save the rig time andlogging cost. As a function to decide termination of temperature recovery log, the followingparameters are calculated and plotted;3) The accuracy for the true formation temperature estimation is less than 5 C even using 24hours temperature recovery data, accordingto our case studies.4) Developed data acquisition system can givevarious parameters for decision making whenthe measurement of temperature recoveryshould be finished.5) Through these case studies, it is expected to1) standard deviation,2) transition of difference of the estimatedreduce the logging and rig costs using CFMas an estimation method of the true formationtemperature and recovery temperature.temperature (Tffit(n) - Tffit(n-1)),3) standardized sum of squares by number ofdata, and4) transition of the estimated true formationtemperature.ACKNOWLEDGMENTSThe authors express their appreciation of theNew Energy and Industrial Technology Development Organization (NEDO) for giving uspermission to present the results obtained in the"Development of Geothermal Hot Water PowerGeneration Plant" in MITI's the Sunshine Project, Japan.Figure 6 illustrates examples of these features.Also, using continuous temperature logging datain depth, this program can estimate the formation temperature and/or recovery temperature atany elapsed time versus depth continuously.This means the true formation temperatureprofile and its reliability are plotted just aftertemperature recovery log has been run.REFERENCESCarslaw, H.S. and Jaeger, J.C. (1959), "Conduction of heat in solids," Oxford University Press.Chiba, M., Takasugi, S., Hachino, Y., andMuramatsu, S. (1988), "Estimating of equilibrium formation temperature by c u r v e fittingmethod," Proceedings of the International Symposium on Geothermal Energy, 383-386.The true formation temperature is essentiallyimportant value for geothermal reservoir evaluation. Estimation of the true formation temperature with good accuracy using short period oflogging time is preferable economically andhelp to m a k e a quick decision, that i s veryimportant for the well drilling. Through thesestudies, we can conclude as follows:Dowdle, W.L. and Cobb, W.M. (1975), "Staticformation temperature from well logs - anempirical method," J. Petrol. Tech., 27, 13261330.1) T h e Curve-fitting method (CFM) as an-26-
. . . . . . . . -.- .- . .-.--.-- .ture stabilization with continued circulation ofdrilling mud," Geophysics, 47, 1716-1723.Hyodo, M., Takai, K., and Takasugi, S. (1994),"Evaluation of curve-fitting method for estimating the formation temperature from loggingdata," Proceedings of the 90th SEGJ Conference, 285-289.New Energy Industrial Technology Development Organization (1992), "Summary of development of techniques to control lost circulationin geothermal wells," 18-27.Luikov, A.V. (1968), "Analytical heat diffusiontheory," Academic Press Inc.Takai, K., Hyodo, M., and Takasugi, S. (1994),"Estimating of equilibrium formation temperature by curve fitting method and its problems,"Stanford Nineteenth Annual Workshop on Geothermal Reservoir Engineering.Middleton, M.F. (1979), "A model for bottomhole temperature stabilization," Geophysics, 44,1458-1462.Middleton, M.F. (1982), "Bottom-hole tempera-BHTc ( X , Y , 0)'T i n i(A) The CircularW e l l Model(B) T h e S q u a r eW e l l Model I .(C) M i d d l e t o n ' sRectangularW e l l Modell lIIlFigure 1. Physical well models as a mathematical model.IFigure 2. Schematic ideal and realistic wellbore configuration.-27- II-I-
E l a p s e d Time(Hours)15. 5No.00126.6126.6@126.6126.618. 5132.8132.8132.8132. 824. 5142.4142.4142.4142.4o f Dataa9E. F. T ( T jfor 72,5 hrs170.5-0.5: ; . 2 : 5 T 27001'''760-2 0 - 30 - 1126.6142.40 - 1.5159.7160.9-0.6 0.2-0.3 0.2-1.9 3.9 1.4 1.0 0.5-11.3-10.11""'ITffitVcx.0.0010170.423 070126.6@II0.010.IHorner-Tirne((t dt)/dt)i-TfbuildP-8.6 degree C/'100200E.F.T by Horner-Plot Method (Degree C )I1Figure 4. Example of estimation of the trueformation temperature by the Horner-plot method.These graphs show effects betweenthe various circulation time at 700 mdepth of the LC-1 well (Tffit 215 C). C: Assumed circulation timeis 40 d a y s (Tfbuild 243 C). S:Assumed circulation time is fivedays (Tfbuild 205 C). T: Assumed circulation time is two days(Tfbuild 196 C).E.F.T by Horner-Plot Method (Degree C)Figure 5. Cross-plot of estimated true temperature by the Horner-plot methodversus CFM (a) using elapsed time of120 hours and (b) using elapsed timeof eight hours.-28-
IFitting curveFitting curveMeasured dataaBl! 8,2 8.41I2I4 618 20TIHE (hours)14011Measured data1180 288480FI :SIR-N 12 :INU-N F3 :r-N F4 z EDlI F5 :AXIS F6 :1URN FIE :RUNF1 :CHANCE:PRK(MITNTlHE (honrs)F3 :PRN-(RESULT) Prt-Scr:H-COPYFIE :UIDFigure 6. Example of parameters for evaluation of reliability of estimated temperatures. (a) Exampleof standard deviation of estimated temperatures. (b) Example of transition of estimatedtemperatures.-29-
mize the sum of squares between BHTo and BHTc at a time. Features of a non-liner least squares method, in comparison with type-curve matching, are in its rapidity and objectivity. Therefore, we call this inverse method as CFM. Figure 3 illustrates an example of the non-liner least squares fitting result by inversion. 3. EVA1,UATION OF METHODS FOR
May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .
On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
behringer ultra-curve pro dsp 24 a/d- d/a dsp ultra-curve pro ultra- curve pro 1.1 behringer ultra-curve pro 24 ad/da 24 dsp ultra-curve pro dsp8024 smd (surface mounted device) iso9000 ultra-curve pro 1.2 ultra-curve pro ultra-curve pro 19 2u 10 ultra-curve pro ultra-curve pro iec . 7 ultra-curve pro dsp8024 .
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Part 5 - CURVE FITTING Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates. There are two general approaches for curve fitting: Least Squares regression: Data exhibit a significant degree of scatter. The strategy is to derive a single curve that represents the general trend of the data .
