IDENTIFYING ABBERANT SEGMENTS IN PERMANENT DOWNHOLE GAUGE DATA

Transcription

1 IDENTIFYING ABBERANT SEGMENTS IN PERMANENT DOWNHOLE GAUGE DATA A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE By Emuejevoke Origbo June 2010

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3 I certify tht I hve red this report nd tht in my opinion it is fully dequte, in scope nd in qulity, s prtil fulfillment of the degree of Mster of Science in Petroleum Engineering. Prof. Rolnd Horne (Principl Advisor) iii

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5 Abstrct Dt from permnent downhole guges re needed for interprettion of subsurfce conditions in well. The dt from permnent downhole guges re voluminous nd usully contins berrnt segments. Using this berrnt dt to chrcterize the reservoir leds to genertion of inccurte reservoir prmeters (permebility, skin nd storge). The pproch used in this work to solve the problem of interprettion of permnent downhole guge dt ws by genertion of multisegment synthetic pressure dt using the pressure eqution with ll the reservoir prmeters known. An berrtion ws introduced in the form of pressure segment tht went ginst the reservoir physics; it decresed with production shut in, when it should increse. An lgorithm bsed on direct Klmn filtering technique ws developed which ws independent of the reservoir model nd extrcted signl with the minimum error (men squre devition) from the noisy/berrnt signls. In this wy, berrnt segments were successfully identified, removed nd the originl signl with ctul reservoir prmeters recovered. v

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7 Acknowledgments I m grteful to the member compnies of SUPRI-D for finncil support for this work. To my dvisor, Professor Rolnd Horne for his dvice, ptience nd mentorship throughout the course of my work, I sy thnk you. To Professor Hmdi Tchelepi, thnk you for your encourgement. To my friends t the deprtment of Energy Resources Engineering, thnk you for mking my work enjoyble. To my prents, I sy thnk you for your support throughout the yers. Drlington, Ufuom, Efes nd Jite, you re the best siblings in the world! Dniel Elstein, the single best thing to hppen to me; thnk you for mking me lugh nd for cusing time to pss effortlessly. I m becuse we re vii

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9 Contents Abstrct... v Acknowledgments... vii Contents... ix List of Figures.xi 1 Introduction Bckground Problem Sttement Literture Review 16 3 Methodology Dt Genertion Degree of Aberrtion The Klmn Filter Results 28 5 Conclusion nd Recommendtion Conclusion Recommendtion for Future Work Nomenclture 39 References 41 ix

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11 List of Figures Figure 1-1: Pressure Dt showing originl nd fitted dt. Reproduced from Athichngorn (2000) Figure 3-2: Synthetic pressure dt Figure 3-2: Synthetic flow rte dt with corresponding pressure dt Figure 3-3: Synthetic flow rte dt with corresponding pressure dt super-imposed with n berrnt segment Figure 3-4: Rtio of pressure nd flow rte derivtive versus time super-imposed on one nother Figure 3-5: Close-up of rtio of pressure nd flow rte derivtive versus time superimposed on one nother Figure 3-6: Rtio of pressure nd flow rte derivtive versus time (including the berrnt segment time step), super-imposed on one nother Figure 3-7: Position of cr estimted using the Klmn filter. Reproduced from Simon (2009) Figure 4-1: True, mesured (noisy) nd filtered (denoised) pressure dt Figure 4-2: Pressure kernel for true (synthetic) nd mesured (noisy) dt Figure 4-3: Estimted pressure kernel using the Klmn filter Figure 4-4: True, mesured (noisy) nd filtered (denoised) pressure dt with berrtion strting t the 6th time step (60-70hours) Figure 4-5: Pressure kernel for true (synthetic) nd mesured (noisy & berrtion) dt. 32 Figure 4-6: Estimted pressure kernel using the Klmn filter Figure 4-7: Pressure kernel for true (synthetic) nd mesured (clipped) dt Figure 4-8: Estimted pressure kernel using the Klmn filter Figure 4-9: Estimted pressure kernel using the Klmn filter Figure 4-10: True, mesured (noisy) nd estimted pressure dt Figure 4-11: Pressure kernel for true (synthetic) nd mesured (noisy) dt for cse Figure 4-12: Estimted pressure kernel for cse 3 using the Klmn filter xi

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13 Chpter 1 Introduction 1.1. Bckground In 1972, Schlumberger instlled the first permnent downhole guge on logging cble in West Afric. Tody, there re over 7000 permnent downhole guge instlled in wells ll over the world supplying continuous rel time dt bout subsurfce reservoir conditions. Permnent downhole guges re used in reservoir monitoring nd mngement by interpreting the pressure, flow rte nd temperture dt red from the guges. Dt from the guges re voluminous nd since they re collected over long periods of time, prone to errors. Mking decisions bsed on the dt without first removing these errors my led to wrong conclusions being reched on the reservoir conditions. There is need to vlidte the guge dt by compring with expected pressure trnsients generted using the flow rte dt. After the comprison, the dt cn be improved by removing segments of the dt tht re berrnt Problem Sttement Severl ttempts hve been mde to interpret dt from permnent downhole guges. These ttempts hve utilized pressure dt from permnent downhole guges directly without performing the criticl first step of berrnt segments removl. Flow rte nd pressure dt were nlyzed together in these studies. 13

14 Tble 1-1: Reservoir prmeters mtched using berrnt nd filtered dt. Reproduced from Athichngorn (1999) Prmeter Actul vlues First mtch Finl mtch Permebility, k Skin, S Storge, C Reservoir rdius, re Reservoir rdius, re As shown in Tble 1-1, n illustrtion from Athichngorn (1999), interprettion of pressure dt with berrnt segments led to inccurte estimtes of reservoir prmeters. The originl prmeters from which Figure 1-1 ws generted re shown in the second column of Tble 1-1. The first mtch chieved, without filtering the signl ws clculted s shown in the third column of Tble 1-1. There ws mrked distinction between the first mtch nd the ctul reservoir vlues. As the signl ws successively filtered close mtch ws mde between the filtered dt nd the originl reservoir prmeters. The finl mtch of the reservoir prmeters were resonbly pproximtions to the ctul vlues. The originl dt with berrnt segments nd the fitted dt with close estimtes of ctul reservoir prmeters were plotted s shown in Figure 1-1. The fitted dt signl ws generted using the vlues derived from the finl mtch. The method used in filtering the dt will not be discussed here. Although this ws multisegment signl with only two berrnt segments, the effect on reservoir prmeter estimtion ws significnt. In the cse of field dt from permnent downhole guges with mny hours of production nd severl pressure trnsients, the error in the prmeters estimted would be mgnified. In 14

15 this study, the pressure trnsient dt ws filtered to identify berrnt segments. Initilly, the method chosen to identify berrnt segments ws the visul chrcteristics method combined with the rtio of derivtives method. However, due to limittions of these methods, severl other methods were tested. As will be shown in subsequent chpters, the Klmn filter technique ws found to identify berrnt segments stisfctorily in multisegment pressure dt Pressure, psi Time, hrs Figure 1-1: Pressure Dt showing originl nd fitted dt. Reproduced from Athichngorn (2000) 15

16 Chpter 2 Literture Review Over the lst decde, severl uthors hve developed methods of interpreting dt from permnent downhole guges. These methods hve improved the use of guge dt in reservoir mngement significntly. Gilly nd Horne (1998) studied the integrtion of flow rte history nd pressure history dt in well test nlysis. The study used the convolution eqution, Lplce trnsforms nd deconvolution to increse the qulity nd quntity of informtion extrcted from pressure dt. In ddition, the study provided mens for interprettion of longer pressure response. Athichngorn (1999) developed seven step pproch. Athichngorn (1999) utilized the convolution eqution, wvelets, Fourier trnsform, regression nlysis nd dt selected with sliding window in interpreting dt from permnent downhole guges. Athichngorn (1999) detected outliers in the dt, denoised the dt nd identified both bbernt trnsients nd brek points. Thoms (2002) conducted work in berrnt trnsient removl from permnent downhole guge dt. Thoms (2002) utilized the convolution eqution, regression nlysis nd the pressure eqution. A pttern-recognition technique ws developed which ided removl of berrnt trnsients. Furthermore, Thoms (2002) proposed tht the pressure derivtive dt be nlyzed on logrithmic scle to id in removl of berrnt trnsients in guge 16

17 dt. In this current study, the method proposed by Thoms (2002) ws explored s the rtio of derivtives method. Nomur nd Horne (2009) utilized wvelets, deconvolution nd visul chrcteristics in trnsient identifiction nd flow rte estimtion. A method of identifying brek points in trnsient dt ws developed. Welch nd Bishop (2006) provided n introduction to technique used in interpreting pressure signls, the Klmn filter: The Klmn filter is set of mthemticl equtions tht provides n efficient computtionl (recursive) mens to estimte the stte of process, in wy tht minimizes the men of the squred error. The Klmn filter hs been pplied in signl processing in the fields of medicine nd engineering. In the medicl field the Klmn filter hs been used to interpret blood flow rte from hert monitoring devices nd pressure in rteries of the hert. In the field of engineering, the Klmn filter hs been used in erospce engineering for trcking the trjectory of stellites. The Klmn filter hs lso been utilized in the erth sciences in interprettion of seismic nd pressure dt nd in predicting pressure dt profiles from reservoirs. Yu et l. (2009), studied lekge detection in crude oil pipelines. Yu et l. (2009) interpreted pressure nd flow rte signls using the combined Klmn filter-discrete wvelet trnsform method. The result of the study ws method for denoising pressure dt nd for extrcting lekge loctions in crude oil pipelines bsed on the extrcted filtered signl. 17

18 Chpter 3 Methodology 3.1. Dt Genertion A reltionship exists between the flow rte history nd the pressure history tht is bsed on the reservoir physics. Tht reltionship ws used in this work in generting synthetic pressure dt with known reservoir prmeters using the pressure eqution given s Eqution (3.1). p wf qbµ k = p + + log( t) log( ) s i 2 kh φµ c r t w (3.1) Three ssumptions were mde regrding the reservoir conditions in this study: Flow rte dt is noiseless nd constnt in ech time step; Flow rte represents the reservoir physics ccurtely; Brek points re known for ech pressure trnsient. Figure 3-2: Synthetic pressure dt 18

19 Typiclly, drw-down response is recorded from reservoir during production. As production is decresed or stopped, pressure build-up response is recorded s shown in Figure 3-2. Figure 3-2: Synthetic flow rte dt with corresponding pressure dt. To investigte the presence of berrnt segments in the pressure dt, n berrtion ws introduced nd superimposed in the fifth time step (40 hour-50 hour) s shown in Figure 3-3. A method ws developed to identify this berrtion. The method utilized in identifying berrtions in the pressure dt ws the ppliction of pttern recognition technique bsed on visul chrcteristics s stted in previous literture. 19

20 Figure 3-3: Synthetic flow rte dt with corresponding pressure dt superimposed with n berrnt segment. Visully, it ws possible to identify the berrnt segment. A further investigtion explored ws the possibility of berrtions in segments of the pressure dt tht ppered to obey the reservoir physics. The curves for drw down nd build up of the pressure dt were ll tested to scertin if the steepness of the curves mtched expected reservoir responses t corresponding time steps. The test ws to determine the degree of complince of these segments with the reservoir physics Degree of Aberrtion To clculte the degree of berrtion in the pressure trnsients, the pressure derivtives nd the flow rte derivtives were computed for ech time step. The rtio of derivtives 20

21 method ws then introduced. As the nme suggests, the method involves computing the rtio of the pressure derivtive nd tht of the flow rte derivtive with the pressure derivte tken s the denomintor. From Figure 3-3, s flow rte incresed pressure decresed. Thus, negtive pressure derivtive ws clculted for the cse of incresing flow rte. The flow rte derivtive, s flow rte incresed ws positive. On the other hnd, flow rte decrese or stoppge resulted in n increse in pressure. A positive pressure derivtive ws clculted for the cse of decresing flow rte. The flow rte derivtive, s flow rte decresed ws negtive. Figure 3-4: Rtio of pressure nd flow rte derivtive versus time super-imposed on one nother. Applying the rtio of derivtives method, negtive number ws clculted ech time s the pressure nd flow rte derivtives for ech time step hd lternte signs. A plot of the rtio of derivtes versus time step is given in Figure 3-4. The time steps were of equl 21

22 lengths of ten hours nd the plots for ech time step ws super-imposed on the plot for the previous time step to llow for visul comprison. As time incresed pst one hour, no significnt visul chrcteristic differences were observed in the combined plot of ech time step. A closer look ws tken of the section on the rtio of derivtives curve where the curves did not fully overlp. The differences in the plots of the vrious time steps were quite subtle. Figure 3-5: Close-up of rtio of pressure nd flow rte derivtive versus time super-imposed on one nother. As expected, in the segment of the pressure dt where the berrtion ws introduced s in Figure 3-3, the rtio of the derivtives ws positive. In this segment, incresed flow 22

23 rte response corresponded to n incresed pressure response. Thus the rtio of the derivtives for this time step ws positive s shown in Figure 3-6. Figure 3-6: Rtio of pressure nd flow rte derivtive versus time (including the berrnt segment time step), superimposed on one nother. After utilizing the visul chrcteristic method, to better estimte the degree of berrtion in ech pressure trnsient, the men squred devition between trnsients ws clculted. The results were stored s elements of mtrix. The mtrix generted ws symmetric mtrix with zero on the digonl s ech element is exctly similr to itself. However, it ws difficult to set threshold men squred devition vlue from which to estblish the degree of berrtion, s the vlues were smll nd setting the threshold would be highly subjective process. 23

24 To overcome this chllenge, new method ws sort to identify berrtions in pressure trnsients. Studies suggested tht the Klmn filter ws n efficient computtionl (recursive) mens to estimte the stte of process, in wy tht minimizes the men of the squred error, Welch nd Bishop (2006). As it ws necessry to compute the men of the squred error in this work without recourse to subject method of determining berrtion in pressure dt, the Klmn filter ws used to identify berrnt segments The Klmn Filter The Klmn filter estimtes the stte x of discrete-time controlled process tht is governed by liner stochstic difference eqution. The Klmn filter consists of two min components: A discrete process model, described by liner stochstic difference eqution which reltes chnge in stte with time; x k = Axk 1 + w k (3.2) A mesurement model described by liner function which estblishes the reltionship between the stte of process nd mesurement. (3.3) A is the mtrix (n n), tht describes how the stte evolves from time k to k-1 without noise. z k = Hxk + v k H is the mtrix (m n) tht describes how to mp the stte x k to n observtion z k wk nd vk re rndom vribles representing the process nd mesurement noise tht re ssumed to be independent nd normlly distributed with covrince R k nd Q k respectively. 24

25 xˆ k R n is the estimted stte t time step k nd xˆ k R n is the stte fter prediction before observtion. e ˆ (3.4) k = xk xk e k = x xˆ (3.5) k k The clculted errors re given by Equtions (3.4) nd (3.5). The error covrince mtrices re clculted using Equtions (3.6) nd (3.7). T P = E[ e e ] (3.6) k k k T P k = E[ ekek ] (3.7) The Klmn filter estimtes xˆ k nd P k. In this study, the pressure trnsient dt from the permnent downhole guges were ssumed sttionry. The form of the time updte eqution lso known s the predictor equtions, used is given by Eqution (3.8). This form of the eqution does not updte the stte with time nd the mtrix A is the identity mtrix. The updte error covrince mtrix P is given by Eqution (3.9). x ˆ = Axˆ (3.8) k k 1 T P k = APk 1 A + Q (3.9) The mesurement eqution, lso clled the corrector eqution used to clculte the expected vlue of x is given by Eqution (3.10). The updte error covrince mtrix is clculted using Eqution (3.11). x ˆ xˆ + K ( z Hxˆ ) (3.10) k = k k k k 25

26 P (3.11) k = (I K kh) Pk The optiml Klmn gin K k ws clculted using Eqution (3.12). T T 1 K k = Pk H ( HPk H + R) (3.12) The Klmn filter opertes s series of predictions, using the time updte, nd corrections, using the mesurement updte, to estimte the expected vlue of the stte of system. Figure 3-7: Position of cr estimted using the Klmn filter. Reproduced from Simon (2009). In the exmple illustrtion in Figure 3-7, series of noisy mesurements of the position of cr were filtered using the Klmn filter. The filtered signl, the ctul signl nd the mesured signl re referred to s predicted, true nd mesured respectively. This exmple is nlogous to the problem being solved in this work. The pressure dt from the permnent downhole guge is similr to the noisy mesurements of the position of cr, the mesured dt. The true dt could be tken s the synthetic pressure dt 26

27 generted nd the predicted position of the cr, the filtered pressure signl. The problem of identifying berrnt segments in permnent downhole guge dt ws solved with method nlogous to tht used to generte the illustrtion in Figure 3.7. In this study, x k nd zk re the ctul pressure trnsient dt nd mesured pressure trnsient dt respectively. H is the flow rte dt. Q the process noise covrince nd R, the mesurement noise covrince control the effectiveness of the filter. The rtio Q/R should be reltively smll to ensure optiml reproduction of the ctul dt by the Klmn filter. The predicted dt is generted by substituting the pressure kernel xk nd the flow rte dt, H in Eqution (3.3). In implementing the Klmn filter lgorithm, the flow rte ws ssumed constnt for ech time step. Ech column in the mtrix of flow rte dt, H, ws generted s vector of constnts for ech time step. The pressure kernel is extrcted from the pressure trnsients by utilizing the Mtlb Klmn toolbox s developed by Murphy (1998). To generte the pressure kernel, Eqution (3.1) tkes the form given below: y + t = Iyt 1 wt for t = 1 n (3.13) The pressure kernel, xk nd the flow rte re substituted in Eqution (3.3) to solve for the predicted dt from the noisy mesurements nd Eqution (3.3) tkes the form of Eqution (3.14). The mtrix H is s given in Eqution (3.15). p + t = Hyt vt for t = 1 n (3.14) H q1 = M q 1 L M L qn M qn (3.15) 27

28 Chpter 4 Results The Klmn filter ws used to denoise synthetic dt generted using known reservoir prmeters nd to identify sections of the dt with berrnt segments. Three cses were exmined in this study. Cse1: Filtering pressure dt with 5% Gussin noise using the Klmn filter; Cse 2: Identifying n berrnt segment introduced in the pressure dt in Cse 1; Cse 3: Sensitivity nlysis of Cse 1 with 5% Gussin noise introduced in the flow rte dt. For Cse 1, synthetic pressure dt representing the pressure dt from permnent downhole guge ws generted nd lbeled true dt. 5% Gussin noise ws dded to the generted synthetic pressure dt nd lbeled mesured dt. The Klmn filter ws then used to filter the mesured dt nd the result lbeled filtered dt. These results re shown in Figure 4-1. In ddition, plots of the pressure kernel corresponding to the true, mesured nd filtered dt were generted. These plots of the pressure kernels re the key mkers for determining if n berrnt segment ws present in the dt or not. The berrnt segments were identified using pttern recognition method. 28

29 Figure 4-1: True, mesured (noisy) nd filtered (denoised) pressure dt. The pressure kernel plots for the true nd mesured dt were plotted together to id visul comprison s shown in Figure 4-2. Any visul discrepncies in the plot of the pressure kernel generted from pressure dt with tht of the expected kernel plot s shown in Figure 4-3, ws tken to signify the possibility of n berrtion in the pressure dt. 29

30 Figure 4-2: Pressure kernel for true (synthetic) nd mesured (noisy) dt. Figure 4-3: Estimted pressure kernel using the Klmn filter. 30

31 In Cse 2, n berrtion ws dded in the hour segment of the mesured pressure dt used in Cse 1 in the form of pressure segment tht went ginst the reservoir physics s shown in Figure 4-4. The Klmn filter ws pplied nd the plots of the pressure dt, true, mesured nd filtered, given s shown in Figure 4-4. Figure 4-4: True, mesured (noisy) nd filtered (denoised) pressure dt with berrtion strting t the 6th time step (60-70hours). The Klmn filter ccurtely filtered the mesured dt nd reproduced pressure signl profile tht mtched the mesured dt. The pressure kernel plots for the true nd mesured dt shown in Figure 4-5 displyed instbility in the pressure kernel plots beginning t 60 hours tht continued till the end of the signl run t 100 hours. The pproch used to serch for the berrtion in the pressure dt ws to clip out the time step t which the berrtion ws first noticed nd to remove time steps sequentilly fter 31

32 tht until the berrtion ws identified. The evidence of successful identifiction of the berrnt segment ws return to stbility of the pressure kernel plot. Figure 4-5: Pressure kernel for true (synthetic) nd mesured (noisy nd berrnt) dt. Figure 4-6: Estimted pressure kernel using the Klmn filter. 32

33 The removl of the pressure dt segment in the time step in which the berrtion ws first noticed cused the pressure kernel curve to immeditely stbilize s shown in Figures 4-7 nd 4-8 respectively. A check ws mde by moving the berrtion to different time steps nd observing the effect on the pressure kernel plot. In ech of the cses with the berrtion in different time steps, similr response to tht observed in Cse 2 ws recorded. Instbility in the pressure kernel plot ws noticed which signified the presence of n berrnt segment in the pressure dt. Clipping out the time step t which the berrtion ws first noticed led to removl of the berrnt segment. Figure 4-7: Pressure kernel for true (synthetic) nd mesured (clipped) dt. 33

34 Figure 4-8: Estimted pressure kernel using the Klmn filter. Figure 4-9: Estimted pressure kernel using the Klmn filter. 34

35 The full plot of the pressure trnsient dt fter removing the berrnt segment from Figure 4-4 is given in Figure 4-9. Cse 3, sensitivity nlysis on the synthetic pressure dt, ws crried out to investigte the effect of the presence of noise in the flow rte dt used to generte the pressure dt in cse 1. 5% Gussin noise ws introduced in the flow rte dt. The presence of noise in the flow rte dt did not chnge the response of the pressure dt to the filter lgorithm s shown in Figures 4-10, 4-11 nd 4-12 respectively. The only observble difference between Figures 4-1, 4-2 nd 4-3 nd Figures 4-10, 4-11 nd 4-12 respectively, is the presence of the rises nd flls in the pressure dt plot. No difference ws observed in the plot of the pressure kernel curves for Cse 3 from tht of Cse 1. 35

36 Figure 4-10: True, mesured (noisy) nd estimted pressure dt. Figure 4-11: Pressure kernel for true (synthetic) nd mesured (noisy) dt for Cse 3. 36

37 Figure 4-12: Estimted pressure kernel for Cse 3 using the Klmn filter. 37

38 Chpter 5 Conclusion nd Recommendtion 5.1 Conclusion Aberrnt segments in permnent downhole guge dt were identified nd removed. The results from this study serve s necessry first step in ny interprettion of pressure nd flow rte dt from permnent downhole guges. The Klmn filter nd the method of deconvolution were utilized in identifying the berrnt segments. 5.2 Recommendtion for Future Work In this study, the flow rte dt were ssumed ccurte in ll cses. The flow rte dt were then used to predict wht the pressure dt should be. When the pressure dt did not mtch the flow rte dt prediction pressure profile, it ws ssumed to be berrnt. However, in the field, flow rte dt would often be inccurte. The reverse cse should be simulted; the cse of pressure being ssumed ccurte nd sections of the flow rte dt tht go ginst the reservoir physics, termed berrnt. In ddition, in this study, the reservoir prmeters were ssumed constnt with time; sttionry. The scenrio of reservoir prmeters chnging with time should be simulted. 38

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40 Nomenclture A = stte trnsition mtrix B = formtion volume fctor (res vol/std vol) c t = totl system compressibility (/psi) e k = error vector h = thickness (ft) H = mesurement mtrix k = time K = Klmn gin mtrix p i = initil reservoir pressure (psi) P = updte error covrince mtrix p wf = well flowing pressure (psi) q = flowrte rte (STB/d) Q = process noise covrince mtrix r w = wellbore rdius (ft) R = mesurement noise covrince mtrix s = skin t = time v k = mesurement noise w k = process noise x k = stte vector y = pressure kernel z k = mesurement vector µ = viscosity (cp) ø = porosity (pore volume/bulk volume) 40

41 References Athichngorn, S., 1999, Development of n interprettion methodology for longterm pressure dt from permnent downhole guges. Stnford University PhD thesis. Athichngorn, S., Horne, R.N., nd Kikni, J., 2002, Processing nd interprettion of long-term dt cquired from permnent pressure guges, SPE Reservoir Evlution & Engineering (October 2002), Gilly, P. nd Horne, R.N. 1998, Anlysis of pressure/flowrte dt using the pressure history recovery method, pper SPE presented t the 1998 SPE Annul technicl conference & exhibition, New Orlens, Louisin, September Murphy, K., 1998, Klmn filter toolbox for Mtlb. Nomur, M., 2006, Processing nd interprettion of pressure trnsient dt from permnent downhole guges. Stnford University PhD thesis. Nomur, M. nd Horne, R. N., 2009, Dt processing nd interprettion of well test dt s non-prmetric problem, SPE pper presented t the SPE western regionl meeting, Sn Jose, CA, Mrch, Simon, D., 2009, Klmn filtering Thoms, O., 2002, The dt s the model: Interpreting permnent downhole guge dt without knowing the reservoir model. Stnford University MS thesis. Welch, G. nd Bishop G., 2006, An introduction to the Klmn filter. Yu, Z., Ji, L., Zhoumo Z., nd Jin, S., 2009, A combined Klmn filter-discrete wvelet trnsform method for lekge detection of crude oil pipelines, pper presented t the ninth interntionl conference on electronic mesurement nd instruments. 41