Deconvolution of Variable-Rate Reservoir-Performance Data Using B-Splines

Transcription

1 Deconvoltion of Variable-Rate Reservoir-Performance Data Using B-Splines D. Ilk, P.P. Valkó, SPE, and T.A. Blasingame, SPE, Texas A&M U. Smmary We se B-splines for representing the derivative of the nknown nit-rate drawdown pressre and nmerical inversion of the Laplace transform to formlate a new deconvoltion algorithm. When significant errors and inconsistencies are present in the data fnctions, direct and indirect reglarization methods are incorporated. We provide examples of nder- and over-reglarization, and we discss procedres for ensring proper reglarization. We validate or method sing synthetic examples generated withot and with errors (p to 10%). Upon validation, we then demonstrate or deconvoltion method sing a variety of field cases, inclding traditional well tests, permanent downhole gage data, and prodction data. Or work sggests that the new deconvoltion method has broad applicability in variable rate/pressre problems and can be implemented in typical well-test and prodction-data-analysis applications. Introdction The constant-rate drawdown pressre behavior of a well/reservoir system is the primary signatre sed to classify/establish the characteristic reservoir model. Transient-well-test procedres typically are designed to create a pair of controlled flow periods (a pressredrawdown/-bildp seqence) and to convert the last part of the response (the pressre bildp) to an eqivalent constant-rate drawdown by means of special time transforms. However, the presence of wellbore storage, previos flow history, and rate variations may mask or distort characteristic featres in the pressre and rate responses. With the ever-increasing ability to observe downhole rates, it has long been recognized that variable-rate deconvoltion shold be a viable option to traditional well-testing methods becase deconvoltion can provide an eqivalent constant-rate response for the entire time span of observation. This potential advantage of variable-rate deconvoltion has become particlarly obvios with the appearance of permanent downhole instrmentation. First and foremost, variable-rate deconvoltion is mathematically ill-conditioned; while nmeros methods have been developed and applied to deconvolve ideal data, very few deconvoltion methods perform well in practice. The ill-conditioned natre of the deconvoltion problem means that small changes in the inpt data case large variations in the deconvolved constant-rate pressres. Mathematically, we are attempting to solve a first-kind Volterra eqation [see Lamm (2000)] that is ill-posed. However, in or case the kernel of the Volterra-type eqation is the flow-rate fnction (i.e., the generating fnction); this fnction is not known analytically bt, rather, is approximated from the observed flow rates. In practical terms, this isse adds to the complexity of the problem (Stewart et al. 1983). In the literatre related to variable-rate deconvoltion, we find the development of two basic concepts. One concept is to incorporate an a priori knowledge regarding the properties of the deconvolved constant-rate response. The observations of Coats et al. Copyright 2006 Society of Petrolem Engineers This paper (SPE 95571) was first presented at the 2005 SPE Annal Technical Conference and Exhibition, Dallas, 9 12 October, and revised for pblication. Original manscript received for review 17 Jly Revised manscript received 10 Jly Paper peer approved 18 Jly (1964) on the strict monotonicity of the soltion led Kchk et al. (1990) to impose a nonpositive second derivative constraint on pressre response. In some respects, this tradition is maintained in the work given by von Schroeter et al. (2004), Levitan (2003), and Gringarten et al. (2003) when they incorporate non-negativity in the encoding of the soltion. We note that in the examples given, this concept (non-negativity/monotonicity of the soltion) reqires less-straightforward nmerical methods (e.g., nonlinear leastsqares minimization). The second concept is to se a certain level of reglarization (von Schroeter et al. 2004; Levitan 2003; Gringarten et al. 2003), where reglarization is defined as the act or process of making a system reglar or standard (smoothing or eliminating nonstandard or irreglar response featres). Reglarization can be performed indirectly, by representing the desired soltion with a restricted nmber of elements, or directly, by penalizing the nonsmoothness of the soltion. In either case, the additional degree of freedom (the reglarization parameter) has to be established, where this is facilitated by the discrepancy principle (effectively tning the reglarization parameter to a maximm vale while not casing intolerable deviation between the model and the observations). In some fashion, each deconvoltion algorithm developed to date combines these two concepts (non-negativity/monotonicity of the soltion or reglarization). In addition to these main featres, individal deconvoltion algorithms may have other distinctive featres. From a nmerical standpoint, recent deconvoltion methods tend to se advanced techniqes to solve the nderlying system of linear eqations [e.g., singlar vale decomposition, the eqivalent concept of psedoinverse, etc. for example, see Cheney and Kincaid (2003)]. One class of approaches makes extensive se of transformations (i.e., the Laplace transform, the Forier transform, etc.). Several cases consider nmerical Laplace transformation of observed tablated data combined with nmerical inversion (Rombotsos and Stewart 1988; Onr and Reynolds 1998). In addition, Cheng et al. (2003) consider the application of the Forier transformation to solve the deconvoltion problem. Error-minimization methods, typically driven by some type of least-sqares algorithm, inclde von Schroeter s total least sqares method [von Schroeter et al. (2004); Gringarten et al. (2003); Levitan (2003)], as well as a grop of methods that rely on the se of spline fnctions (in varios steps of the deconvoltion algorithm) [e.g., Gillot and Horne (1986) and Mendes et al. (1989)]. Formally, it wold be easy to place or proposed method into existing categories. Or proposed approach ses B-splines; nmerical inversion of the Laplace transform is also sed for certain components of the soltion, and reglarization is provided indirectly (i.e., by the nmber of knots sed in the selected B-spline) and directly (by penalizing the nonsmoothness of the logarithmic derivative of the reconstrcted constant-rate response). However, in detail or approach is radically different from any of the existing methods in contrast to previos methods, or approach does not fit B-splines to observed data. Rather, B-splines are sed to represent the nknown response soltion (i.e., as a linear combination of B- splines). With respect to the Laplace transform, or approach does not se the Laplace transform to transform the observed pressre data series, nor does or approach se nmerical inversion to provide the objective response (i.e., the constant-rate response), as do other spectral methods. In or approach, the application of 582 October 2006 SPE Reservoir Evalation & Engineering

2 nmerical inversion of the Laplace transform is restricted to the constrction of the sensitivity matrix for the least-sqares problem. Or new techniqe represents the derivative of the nknown constant-rate drawdown pressre response as a weighted sm of B-splines, sing logarithmically distribted knots. For the discrete flow-rate fnction, we se piecewise constant, piecewise linear, or any other appropriate representation for which the Laplace transform can be obtained easily. Taking the Laplace transform of the B-splines, we then apply the convoltion theorem in the Laplace domain, and we calclate the sensitivities of the observed pressre response with respect to the B-spline weights by nmerical inversion of the Laplace transform. Finally, the sensitivity matrix is sed in conjnction with a least-sqares criterion to yield the soght B-spline weights and the nit-rate response (the spline formlation), which then yields the well-testing derivative fnctions in the real domain. The combined approach of B-splines, Laplace-domain convoltion, and the leastsqares procedre are innovative and robst and shold find nmeros applications in the petrolem indstry (prodction- and well-test-data analysis, inversion of water inflx behavior, etc.). This new deconvoltion approach is made possible by or ability to solve a certain class of convoltion problems with any desired accracy, in particlar, by the availability of a robst nmerical inversion procedre for the Laplace transform [i.e., the Gaver-Wynn-Rho (GWR) algorithm (Valkó and Abate 2004; Abate and Valkó 2004)]. The following are the objectives of this work: Develop and validate a new deconvoltion method based on B-spline representations of the derivative of nknown constant-rate drawdown pressre response (i.e., the ndistorted pressre response). Create a practical and robst deconvoltion tool that can tolerate relatively large (random) errors in the inpt rate and pressre fnctions. The proposed process also shold be capable of tolerating small systematic errors in the inpt fnctions (by means of calibrated reglarization). Apply this new method to traditional variable-rate/pressre problems, sch as wellbore-storage distortion, long-term prodction data, permanent downhole (pressre) gage data, and well tests having mltiple flow seqences. Statement of the Problem Dhamel s principle states that the observed pressre drop is the convoltion of the inpt-rate fnction and the derivative of the constant-rate pressre response at t 0, the system is assmed to be in eqilibrim [i.e., p(r,t 0) p i ]. For reference, the convoltion integral is defined as: t q t p d = p t....(1) 0 The goal of variable-rate deconvoltion is to estimate the nit (i.e., constant) -rate reservoir response with observed measrements of pressre drop and flow rate. This objective leads s to the inverse problem, in which the time-dependent inpt signal (constant-rate pressre response) mst be extracted from the otpt signal (pressre-drop response distorted by the specified variable-rate profile). To reconstrct the nknown fnction (i.e., the derivative of the nit-rate reservoir pressre response [K(t) p (t)], we reqire the soltion of the Volterra integral eqation of the first kind (i.e., the convoltion integral), where the role of the convoltion kernel is represented by the sandface rate. Lamm (2000) shows that for q(t ) =t 0 that is, when the rate does not jmp instantaneosly at the first moment the left side of the eqation is smoothing at least of order two, and, hence, the problem is severely ill-conditioned. In addition, the sandface flow rate is also observed in discrete points and is always corrpted in practice by some form of error; therefore, the reslts will frther vary depending on the way in which we make the transition from discrete observations into the convoltion kernel: q(t ). As noted by Lamm, the convoltion operation has a natral smoothing effect; therefore, significantly different K(t) fnctions can (and may) reconstrct essentially the same p(t) response for a given rate history. Ideally, K(t) 0 and decreases monotonically [Coats et al. (1964) showed that derivatives of a higher order mst remain monotonic]. Unfortnately, even sing downhole measrements, measred (observed) flow rates contain effects that may case deviation from monotonic behavior (e.g., wellbore storage or other variable-rate conditions). If the response contains a skin effect and no wellbore storage, then the implse response, K(t), will contain a Dirac delta component (van Everdingen and Hrst 1949). Therefore, the conceptal representation of the K(t) fnction is not trivial, a point often established by nmeros failed attempts to represent this fnction. In particlar, many algorithms sbstitte the convoltion integral in Eq. 1 by the trapezoidal rle (or some similar approach), which is difficlt to jstify near the origin. Development of a New B-Spline-Based Deconvoltion Method Splines Over Logarithmically Distribted Knots. Spline fnctions (Cheney and Kincaid 2003) are piecewise polynomial fnctions that are defined on sbintervals connected by points called knots (t i ) and have the additional property of continity (of the fnction and its derivatives). In this work, we represent the nknown K(t) fnction as a second-order spline with logarithmically spaced knots. To reveal characteristic reservoir behavior, the nmber of knots shold be on the order of at least 2 to 6 knots per log cycle [for typical kernels of interest (i.e., inpt flow-rate fnctions observed in reservoir engineering). Therefore, we never se more knots, bt we may be forced to redce the nmber of knots in cases in which the data qality does not jstify the prsit of sophisticated (reservoir pressre) signatres. A spline fnction can be represented effectively with a linear combination of basic spline fnctions called B-splines (Cheney and Kincaid 2003). Once the knots (or continity points) are set, generation of B-splines is easy becase of their intrinsic recrrence relation. Distribting the knots logarithmically, we have t i b i,b>1 i 0, ±1, ±2,...; with a sitable selected b>1 basis, the B-spline of degree 0 is defined by B 0 i t = 1 t i t t i+1 0 otherwise....(2) Higher-degree B-splines are generated recrsively sing B k i t = t t i t i+k t i i B k 1 t + t i+k+1 t t i+k+1 t i+1 i+1 B k 1 t,...(3) where k 1,2,...Theith B-spline has a nonzero vale (spport) between t i and t i+k+1, as is illstrated in Appendix B. Any kthdegree spline fnction (over the given system of knots) can be represented as a linear combination of kth-order B-splines. S t = i=l c i B i k t,...(4) where the nmber of B-splines involved is l+1. For second-order splines having spport in an interval (t i1, t i2 ), we obtain l i 1 2, i 2 1, and the nmber of B-splines involved is n i 2 i By linearity, the Laplace transform of Eq. 4 is S s = i=l c i B i k s....(5) We will represent K(t) as a weighted sm of B-splines of degree 2, defined over logarithmically evenly spaced knots: K t = i=l c i B i 2 t....(6) Sbstitting Eq. 6 into Eq. 1, we have: t p t = 0 i=l c i B i 2 q t d....(7) October 2006 SPE Reservoir Evalation & Engineering 583

3 If we have m drawdown observations collected in a vector p that were observed at times (t 1, t 2,... t m), then the problem can be written as an overdetermined system of linear eqations: Xc = p,...(8) where X is the m n sensitivity matrix (or design matrix.) Before solving Eq. 8 in the least-sqares sense, each row of the overdetermined system is mltiplied by a weight factor. Typically, we se the reciprocal of the observed vale; in other words, we assme the responses are corrpted by a constant level of relative errors. Also, additional rows are added; we will discss this operation in the section on reglarization. Theoretically, the elements of the sensitivity matrix are defined as t j X j,i = B 2 i q t j d....(9) 0 In contrast to other methods, we do not assme that the rate fnction is given in a stepwise manner. Rather, we allow for a general rate fnction with a Laplace transform of q(s). We calclate the elements of the X matrix sing nmerical inverse Laplace transformation: X j,i = L 1 q s B 2 i s t j....(10) Here, B 2 i (s), the Laplace transform of the ith B-spline, is known in closed form (which is given in Appendix A). We exclde B-splines with indices i for which b i >t j becase those splines start later than the time of the given observation. Rate Representation. To obtain the Laplace transform of the rate fnction, we have varios options. The one we prefer is to dissect the rate into n q segments with starting times t q k, k 1,...,n q.in each segment,we describe the incremental contribtion by an exponential term and,hence,the total rate is obtained in the form n q q t t q t = a q k=1 k + b q k k exp q c k t t q k,...(11) where ( ) is the Heaviside (nit-step) fnction. The parameters for each segment are fond by linear least-sqares fit combined by direct search on the nonlinear parameter c q k. A reasonable range for this time constant is between t 1 and t m. The fewer segments we se, the more we smooth the observations. A sht-in period is always represented as an individal segment. The selection of flow segments is completely independent from the location of the pressre observations. With the special form of the rate given by Eq. 11, the elements of the sensitivity matrix are calclated as n q X j,i = k=1 L 1 q k s B 2 i s t j t q k,...(12) where q k s = a q k s + b q q kc k 1 + c q k s....(13) For strongly varying flow rates, we can se a piecewise constant representation of the flow rates with many segments, as given in Eq. 14. n q q t = k=1 q k q k 1 t t q k....(14) In sch a case, however, there is no need for inverse Laplace transformation becase Eq. 9 can be calclated by se of closed formlas in which the analytical integral of the second-order B- spline is available (see Appendix A). Nmerical Laplace Inversion. The sccess or failre of or proposed approach depends primarily on the nmerical Laplacetransform inversion. For the previos three decades, fixed precision compting has defined the stats of nmerical inversion [see Davies and Martin (1979) and Narayanan and Beskos (1982)]. Recently, several new algorithms were introdced sing mltiprecision methods (Abate and Valkó 2004). In this work, we se the pblicly available GWR algorithm. With this algorithm, it is possible to invert a large class of Laplace transforms with essentially any desired accracy. For more details on the particlar algorithm and Laplace transform inversion in general, see Appendix B. Decopling the Unit Response and Its Derivative. As we already mentioned, in general, the earliest part of the nknown K(t) fnction is critical, especially if the observed variable-rate response contains a skin effect [bt minimal (or no) wellbore-storage effects]. In sch cases, the K(t) fnction contains a Dirac delta component. To reconcile this condition, we add additional anchor B-splines to the left of the first observed time point. After considerable experimentation, we established that for additional B- splines are sfficient for virtally any scenario. These anchor B-splines do not affect the reconstrcted p d (t) fnction (that is, the logarithmic derivative at the observation points) directly, bt they do affect the reconstrcted p (t) (constant-rate) response at all observation points [i.e., the integral of the p d (t) fnction]. In fact, these additional B-splines are a convenient vehicle to represent the Dirac delta component of the fnction K(t). Reglarization. We fond that sing the concept of a psedoinverse (a ctoff for small singlar vales) does not provide a sfficient reglarization as the noise level in the data increases. Therefore, we reqire an additional reglarization that makes sense in a physical context and ensres the relevance of the spline representation. Appended to the overdetermined system (Eq. 8) to be solved by least sqares are the following two conditions for each spline interval: t i=l c i B i t t=b k t i=l c i B i = 0 t t=b 2 k+1 k+1 and t i=l c i B i t i=l c i B i = 0...(15) t t=b k+1 2 t t=b In other words, we reqire that the vale of the logarithmic derivative of the constant-rate response differ only slightly between the knot and the middle location between knots. Becase the entire system is overdetermined, these eqations will not be satisfied exactly, and their inflence on the soltion will depend on how large is selected. When 0, there is no reglarization, and with generated data with practically no error, 0 reslts in the reqired smooth soltion for the constant-rate response and its logarithmic derivative. In the presence of random noise and/or other inconsistencies, a positive is selected on the basis of an informal interpretation of the discrepancy principle that is, we increase the vale of the reglarization parameter ntil the calclated (model) pressre difference begins to deviate from the observed pressre difference in a specific manner. The mean and standard deviation of the arithmetic difference of the compted and inpt pressre fnctions are also compted, bt algorithmic rles (e.g., L-crve method) for selecting are not recommended, for reasons discssed by von Schroeter et al. (2004) and Gringarten et al. (2003). Soltion Using Psedoinverse. Once the agmented sensitivity matrix X a has been constrcted, the estimate of the nknown parameter vector (c) is obtained from ĉ = X a + p,...(16) where the sperscript pls sign denotes psedoinverse calclated throgh singlar vale decomposition. (For simplicity, here we show the case in which all the weights are nity.) To create the psedoinverse, small singlar vales are removed from the calclations; in or comptations, the atomatic ctoff selection of Mathematica (2005) was sed. Using the c coefficient estimates 584 October 2006 SPE Reservoir Evalation & Engineering

4 Fig. 1 Inpt data for Validation Case 1 [no wellbore-storage effects, skin factor (s x )=0]. for the validation phase, we convolve the known nit-rate pressre soltion with the flow history, which incldes two variable-rateprodction and two pressre-bildp (PBU) periods. In or validation, we se the following variable-rate profile: q t = e t e t t... obtained from this process, we reconstrct the constant-rate response as follows: p t = i=l 2 ĉ B i,int t....(17) The logarithmic derivative of the constant-rate response is given as p d t = t i=l ĉ B i 2 t,...(18) where the integral of the second-order B-spline is easily obtained in symbolic form (see Appendix A). Hence, the nit response and its logarithmic derivative can be calclated anywhere between the start and the end of the test seqence. Validation In this section, we provide validation of the new method by comparing the reslts from the deconvoltion procedre with the exact responses obtained from a known reservoir model. As a base case (specifically, the case of a reservoir model with strong characteristic behavior), we selected a dal-porosity reservoir with circlar bondary ( , , and r ed 1,600; see Table 1 for details). To generate the synthetic pressre histories reqired 0 t 60 hors t 100 hors 300 e 100 t t 200 hors t 300 hors...(19) Becase we ltimately intend to process permanent gage data, this selected rate profile was sed to mimic typical pressredrawdown/-bildp test seqences. In the validation process, we increase the complexity of the reservoir model step by step to assess what inflence the nderlying reservoir model may have on the performance of or deconvoltion procedre. We first consider a model withot wellbore storage and skin effects (Fig. 1); we then incorporate a skin factor of 5 (Fig. 2), and, finally, we se a dimensionless wellbore-storage coefficient of 100 and a skin factor of 5 (Fig. 3). In the first example, we perform deconvoltion with the inpt data shown in Fig. 1. Fig. 4 presents or deconvoltion reslts for this case, and we note that when the deconvoltion reslts are compared with the exact inpt response, one hardly sees any difference. All the characteristic featres of a dal-porosity reservoir model are present in the reslts, inclding the bondary-dominated flow regime. Fig. 2 Inpt data for Validation Case 2 [no wellbore-storage effects, skin factor (s x )=+5]. Fig. 3 Inpt data for Validation Case 3 [dimensionless wellbore-storage coefficient (C D )=100, and skin factor (s x )=+5]. October 2006 SPE Reservoir Evalation & Engineering 585

5 Fig. 4 B-spline deconvoltion reslts for Validation Case 1 (all data sed); no wellbore-storage or skin effects. Fig. 5 B-spline deconvoltion reslts for Validation Case 2 (all data sed); no wellbore-storage effects, skin factor (s x )=+5. In Fig. 5, we present the deconvoltion reslts for the case in which a skin factor of 5 is inclded in the inpt pressre response; we note an additional increase in the nit-rate drawdown pressre response (from the positive skin factor), and we see that the welltesting derivative fnction remains essentially identical to the trends observed in Fig. 4 (as wold be expected). In Fig. 6, we present the deconvoltion reslts when wellborestorage and skin effects are incorporated into the base model. We note that the deconvoltion shown in Fig. 6 is performed sing srface rates, which do not reflect wellbore-storage conditions (althogh wellbore storage effects are, obviosly, inclded in the observed pressre data). Fig. 6 is of particlar practical interest becase we clearly mst recognize that althogh deconvoltion to remove wellbore-storage effects is a worthwhile goal, in crrent practice, it is very nlikely that we wold be able to obtain the downhole flow-rate data with sfficient accracy sch that we cold remove wellbore-storage effects. However, given a general variable-rate/variable-pressre data seqence, we can se deconvoltion to reconstrct the constant-rate drawdown response, which wold inclde wellborestorage effects (as shown qite effectively in Fig. 6). In the validation cases considered to this point, only ideal data have been sed as inpt (i.e., noiseless/consistent data), and reglarization was not necessary. As noted, the deconvolved reslts are essentially the same as the (inpt) ideal model responses. Reglarization is not reqired for synthetic cases, bt for most (if not all) field cases (which sally have inconsistencies and significant measrement errors), we do reqire some sort of reglarization procedre. We provide a synthetic case in which we have added random and systematic data errors (Fig. 7); we note that we have sed the same prodction history and reservoir model with wellborestorage and skin effects [base performance (no error) is shown in Figs. 3 and 6)]. In this particlar example, we have added 10% random error to both the pressre and rate data, and we also seeded the data with a small systematic error in pressre (see Fig. 7 for a smmary of the inpt data). We can still perform deconvoltion on these data, bt in this case, we mst se reglarization to address the combined effects of random and systematic errors in the data. The vale of the reglarization parameter has a profond effect on the soltion for this case. Figs. 8 throgh 10 illstrate the discrepancy principle by which the reglarization parameter,, is chosen. When reglarization is not sed (Fig. 8), there is a random error between the observed response ( p ) and its reconstrcted Fig. 6 B-spline deconvoltion reslts for Validation Case 3 (all data sed); dimensionless wellbore-storage coefficient (C D )=100, and skin factor (s x )=+5. Fig. 7 Validation Case 4: inpt data with systematic error and noise; dimensionless wellbore-storage coefficient (C D )=100, and skin factor (s x )= October 2006 SPE Reservoir Evalation & Engineering

6 Fig. 8 Validation Case 4: noisy pressre observations and the B-spline p wf model response (no reglarization). Fig. 9 Validation Case 4: noisy pressre observations and the B-spline p wf model response [optimal reglarization ( =0.0007)]. vale (Xĉ). If a higher vale of reglarization parameter is sed, oversmoothing is obvios from the one-sided deviations (Fig. 10). The optimm vale of the reglarization parameter (in this case, ) is obtained as the largest vale still not casing systematic discrepancy between observed and reconstrcted responses (Fig. 9). Fig. 11 presents the reslts of deconvoltion when errors and inconsistencies are present in the data. Clearly, some artifacts cased by inconsistent data are seen (particlarly in the welltesting derivative), bt deconvoltion reconstrcts the characteristic featres of the reservoir model for the dration of the entire test seqence. (From another point of view, deconvoltion withot reglarization can effectively reveal inconsistencies present in the data.) From or validation experiments, we can conclde that very accrate (essentially exact) reslts can be obtained sing any major event in the test seqence (i.e., a prodction test, a sht-in, or the complete history). We also note that when the inpt data have errors or inconsistencies, we need some form of reglarization to provide meaningfl reslts. Later, we show that or deconvoltion approach can address relatively high levels of noise and other data inconsistencies. In the deconvoltion process, the initial pressre is reqired to compte the observed pressre drop response ( p ). Therefore, the initial reservoir pressre is an inpt parameter for the B-spline deconvoltion algorithm, and an accrate estimate of the initial pressre is a critical component of deconvoltion. Using an erroneos estimate of initial reservoir pressre will case a systematic error in the deconvolved, eqivalent constant-rate pressre response. In addition, there is disagreement between the base pressre data and the reconstrcted pressres. For example, sing an initial pressre larger than the actal initial pressre will reslt in an apparent psedosteady-state flow regime at the end of the deconvolved test seqence or prodction data set. Ths, we note that an apparent psedosteady-state flow regime is a possible artifact of sing an incorrect estimate of the initial reservoir pressre. It is possible to recover the initial reservoir pressre sing deconvoltion [see von Schroeter et al. (2004) and Levitan et al. (2004)]. We also are able to recover the initial reservoir pressre sing an iterative process in which we start with an initial estimate to minimize the error between base pressre data and reconstrcted pressres. This is not an atomated process; ser intervention is reqired to realize the discrepancy between reconstrcted pressres and the base pressre data. Nevertheless, we conclde that this procedre is an effective way to estimate initial reservoir pressre. Validation 2: Test of the New Deconvoltion Method Using Only the PBU Data Portion of the Test Seqence In this section, we will test the methodology sing individal parts of the well-test seqence. For or prposes, we have generated two synthetic cases in which we select a simple homogeneos reservoir model with wellbore-storage effects for simplicity. The properties for the base reservoir model are provided in Table 2. Fig. 10 Validation Case 4: noisy pressre observations and the B-spline p wf model response [ over-reglarization case ( =0.005)]. Fig. 11 Validation Case 4: deconvolved pressre response [optimal reglarization ( =0.0007)]. October 2006 SPE Reservoir Evalation & Engineering 587

7 In the first case, a constant-rate prodction of 50 STB/D is held for 5 hors and followed by 500 hors sht-in. We specifically set the dration of the bildp to be 100 times larger than the dration of the drawdown to establish the difference between deconvoltion and conventional PBU analysis when the drations of prodction seqences are not consistent. We recognize that this is an extreme test of the algorithm (i.e., to se only the data from the PBU portion, pls the prodction history, to yield the eqivalent constant-rate drawdown response by deconvoltion). However, in practice, the primary tility of or algorithm is likely to be the resoltion of PBU data. While obvios, it is worth restating that an accrate prodction history (i.e., accrate flow rates) is absoltely critical for deconvoltion. Fig. 12a presents the inpt data for deconvoltion and PBU analysis. In Fig. 12b, we immediately note artifacts in the verylate-time pressre derivative of the PBU case (conventional PBU plot), where these artifacts are cased by the fact that the reservoir pressre has completely bilt p to the limiting vale of the average reservoir pressre. However, or deconvoltion method is not affected by this behavior. Using only the PBU data for this case, we perform deconvoltion, and we then constrct the constant-rate pressredrawdown response for the entire history. We present the reslts of the deconvoltion as well as the conventional PBU data fnctions in Fig. 12b, and we compared these data fnctions to the correct model responses for each case. As shown in Fig. 13a, for or second case, we add another prodction and sht-in period to the well-test seqence. The dration ratio between both drawdown and bildp seqences is preserved (sht-in dration is 100 times larger than the dration of the drawdown). In this case, we se only the data from the final PBU portion for the deconvoltion process. As noted previosly, we can constrct the constant-rate pressre-drawdown response for the dration of the bildp sing deconvoltion. In or work, we set the weights of the inpt pressre data to zero, except for the data from the bildp portion [or any individal portion of a test in which data are consistent; PBU data (zero rate case) are the most consistent data becase they are naffected by flow rate dring the test]. As sch, we se the data from the second (final) PBU portion and set the weights of prior data to zero. The reslts for this case are presented in Fig. 13b. It is worth noting that sing only the PBU data from the second (final) PBU test, we can recover the eqivalent constant-rate pressre-drawdown response for the entire well-test seqence sing deconvoltion. Fig.12 (a) Case 1: inpt data for the first case, where the dration of the bildp is 100 times larger than the dration of drawdown. (b) Case 1: B-spline deconvoltion reslts for the first case with conventional PBU analysis and exact reslts (only PBU data are sed for deconvoltion). Applications of the New Deconvoltion Techniqe Having verified or new deconvoltion techniqe, we can now apply the same procedre to field data. For convenience, we classify or field examples into for grops: Wellbore-storage distorted pressre data. Well-test data, inclding mltiple flow seqences. Permanent downhole gage data (several flow seqences). Long-term prodction data (rates and pressres). These examples illstrate the se of B-spline deconvoltion for analyzing varios events (prodction/sht-in seqences) and shold be considered typical of cases that cold be encontered in field operations. Field Case 1: B-Spline Deconvoltion of Pressre-Transient Data Distorted by Wellbore Storage. The most straightforward application of deconvoltion is to deconvolve PBU data to eliminate the effects of wellbore storage. Fetkovich and Vienot (1984) pblished data on an oil well that was hydralically fractred at initial completion. The early-time pressre response is distorted by wellbore-storage effects, bt sandface rates are available. Using this inpt, we perform deconvoltion, the reslts of which are shown in Fig. 14. We compare the deconvolved response along with reservoirmodel responses (simlation) for consistency. Fig. 14 presents the distorted pressre data, the deconvolved constant-rate pressre 588 October 2006 SPE Reservoir Evalation & Engineering

8 Fig. 14 Field Case 1: wellbore-storage distorted pressre response, deconvolved response, and model match; data from Fetkovich and Vienot (1984). pressres generated by the B-spline model indicate that pressre data and reconstrcted pressres by deconvoltion are consistent. Therefore, we note that deconvoltion sing the entire test seqence is sccessfl in this case (Fig. 15b). We proceed and constrct the nit-rate pressre and well-testing derivative fnctions in Fig. 16, together with model matches of the deconvolved pressre drawdown. Fig. 13 (a) Case 2: inpt data for the second case, where the drations of the bildps are 100 times larger than the drations of drawdowns. (b) Case 2: B-spline deconvoltion reslts for the second case with conventional PBU analysis and exact reslts (only data from the final PBU are sed for deconvoltion). drawdown and its logarithmic derivative, and the model matches for the deconvolved response. We sed two models to match the deconvolved pressre response: the niform flx fractre (Fetkovich and Vienot sed this model in their analysis) and the finitecondctivity vertical fractre model. Or deconvoltion reslts compare extraordinarily well with these model responses and serve as confirmation of previos interpretations. In this case, we have sed deconvoltion to eliminate wellborestorage effects, and we sggest that this case serves to verify or contention that or method is sfficiently accrate for se in eliminating wellbore-storage effects (when sandface rates are available). Field Case 2: B-Spline Deconvoltion of Conventional Well Tests Inclding Mltiple Flow Periods (Complex Gas Reservoir). In Fig. 15a, we present the rate and pressre data obtained dring the for-point test of a gas well, which was followed by an extended sht-in. In this case, the PBU data seem to be of sfficient dration for a conventional PBU analysis; therefore, we simply cold proceed and analyze the individal PBU portion of these data. However, or intention is not to replace PBU analysis by deconvoltion bt, rather, to implement or deconvoltion algorithm as an option for well-test analysis. We choose to analyze all the available data, inclding the prodction periods. First, the plot of the base pressre data and the Fig. 15 (a) Field Case 2: moderate-pressre gas well in a complex reservoir. (b) Field Case 2: pressre base data and the B-spline p wf model response. October 2006 SPE Reservoir Evalation & Engineering 589

9 Fig. 16 Field Case 2: deconvolved response and model match (entire history sed in this deconvoltion). Becase the flowing flid is gas, we mst perform the psedopressre transform to adhere to theoretical considerations (we note that psedotime shold also be considered, bt to be theoretically rigoros, psedotime reqires the average reservoir pressre history, which is never available in practice). The deconvolved pressre response sggests that a complex, channel-type reservoir geometry is possible. Therefore, we proceed and match the deconvolved response with two models to verify this possibility. The models considered inclde a well in a homogeneos reservoir bonded by parallel falts and a well in a homogeneos reservoir with closed rectanglar bondaries. The reslts of or model matches validate the complexity of the reservoir. We do observe the familiar halfslope trend of the pressre and pressre-derivative fnctions on the log-log plot, which indicates a channel-type reservoir geometry. On the other hand, we also note that the signatre of a closed reservoir is evident in the psedopressre drop and psedopressre-drop derivative fnctions at late times. In this application, we have sed deconvoltion to extract information from a time interval larger than jst the PBU test seqence. Analysis of the entire testing seqence sing deconvoltion may reveal reservoir characteristics not seen by a single event in the seqence. Field Case 3: B-Spline Deconvoltion of Conventional Well Tests Inclding Mltiple Flow Periods (Low-Permeability Oil Reservoir). In or third example, we perform deconvoltion sing the data from a selected flow period dring a well-test seqence, as shown for the selected oil well in Fig. 17a. This test seqence incldes 170 hors of data, where we have a 50-hor constant-rate prodction (at 200 STB/D), followed by a sht-in for 16 hors; then, prodction resmes at a constant rate of 160 STB/D for 24 hors and is then sht in for 80 hors. Inconsistencies are observed; in particlar, we sspect that in the first prodction period, the given rate is not correct (we note flctations in the drawdown pressre profile). We have two options in this case: first, we can deconvolve the entire seqence, bt we have to se a high vale of reglarization parameter to overcome the effects of inconsistencies that most likely will introdce bias. We also can se the data from the final bildp portion to perform deconvoltion bt take the preceding flow history into accont. We choose the second option and perform deconvoltion sing only the PBU data. In Fig. 17b, reconstrcted pressres obtained from the B-spline model and base pressre data are shown together. It is obvios that the data are not consistent. We note that for constrction of Fig. 16b, we set the weights of pressre data to zero except for the data from the last bildp; as expected, we have a very good match of the Fig. 17 (a) Field Case 3: inpt data for fractred oil well. (b) Field Case 3: pressre base data and the B-spline p wf model response. reconstrcted pressres for both sets of PBU data that are assmed consistent. These reslts sggest that, most probably, reported rates are wrong, and performing deconvoltion sing the entire set of test data will not prodce meaningfl reslts. Nevertheless, the deconvolved pressre response (obtained sing the final PBU data, which are considered consistent with the given rates) and a model match are shown in Fig. 18. The deconvolved well-testing pressre-derivative fnction sggests that the well has a finite- Fig. 18 Field Case 3: deconvolved response and model match (only final PBU data sed in this deconvoltion). 590 October 2006 SPE Reservoir Evalation & Engineering

10 Fig. 19 Field Case 4: permanent downhole gage data for 10 hors. condctivity vertical fractre and that this portion of the reservoir is bonded by parallel falts. We confirm this spposed reservoir model by matching the deconvolved pressres with jst sch a model. Field Case 4: B-Spline Deconvoltion of Permanent Downhole Gage (PDG) Data. Or next example is taken from an oil well eqipped with a permanent downhole measrement system, which is sed to provide continos (downhole) pressre and rate data. In this case, a tremendos amont of data is available for a very short time 10 hors (as shown in Fig. 19). Nevertheless, PBU analysis or other traditional-type analysis is not a realistic option becase the bildps are very short and, in addition, other phenomena (most likely temperatre change and/or phase resegregation) case nonintitive behavior (e.g., between 2 and 4 hors, the well is sht in, bt the observed pressre is decreasing). Fig. 20 presents the reslts of deconvoltion for this case. The apparent oscillations in the deconvolved pressre-derivative fnction indicate that the data are somewhat inconsistent, bt we can proceed and match the deconvolved pressre with simple models [a well in an infinite-acting homogeneos reservoir with wellbore storage and skin, and a well in a homogeneos reservoir with a single sealing falt (also inclding wellbore-storage and skin effects)]. Using analytic simlation, we reprodce the pressre response relative to the observed rate history for the entire test seqence, as shown in Fig. 21. The pressres generated by the model fail to honor the observed data at some points (especially in the intervals in which we have already established that inconsistencies exist in the data); however, these simlations do honor the observed pressres at later times, in agreement with or expectations. Field Case 5: B-Spline Deconvoltion Applied to Prodction Data (Tight Gas Reservoir). Or deconvoltion method is not limited to analyzing well tests; we can also apply this methodology to analyze and interpret traditional (long-term) prodction data. Prodction data are regarded as low-freqency and lowresoltion data. For these reasons, rigoros variable-rate/variablepressre analysis of prodction data is rarely attempted. Or objective is to convert the entire prodction seqence (inclding poor-qality data) into an eqivalent constant flow-rate pressre response. We corroborate or reslts with material-balance deconvoltion (Johnston and Lee 1991), which incldes plotting p p /q g (rate-normalized psedopressre) vs. material-balance psedotime and comparing or deconvoltion reslts and the material-balance deconvoltion reslts with a reservoir model established from the deconvolved data. This example considers a tight gas reservoir (Pratikno et al. 2003). The data are of good (to excellent) qality, and we note the effect of sht-ins as well as daily prodction flctations (Fig. 22a). Fig. 22b presents the reconstrcted pressres by B-spline deconvoltion compared with base pressre data. As seen, pressres obtained sing the B-spline model honor the measred data over a considerable portion of the prodction history. We can conclde that Fig. 20 Field Case 4: deconvolved response and model match (entire data seqence sed). prodction data are consistent and that or B-spline deconvoltion based on the entire long-term prodction history was sccessfl. In Fig. 23, we present the deconvolved responses (or method and material-balance time) compared with an appropriate reservoir model (a well with a finite-condctivity vertical fractre in a bonded reservoir); we note good agreement for this case. We note some anomalies in the well-testing pressre-derivative fnction at early times, where these anomalies were presmably cased by inconsistencies in the pressre data (or simply a scarcity of data at early times). From the derivative trend, we find that bondary-dominated flow is in transition to fll development, and there does exist a slight difference in or interpretation of the data and model selected for this portion of the analysis. Field Case 6: B-Spline Deconvoltion Applied to Prodction Data (Low-Prodctivity Gas Reservoir). The second field example is also a gas field case (Palacio et al. 1993). The daily prodction data are qite erratic (Fig. 24a) becase of liqid loading; in spite of these featres, the B-spline deconvoltion method performs reasonably well (see Fig. 24b). In this case, the margin of error between data and reconstrction is acceptable, and we expect that errors will reslt in the form of artifacts, especially in the deconvolved well-testing derivative fnction. In Fig. 25, we again compare or deconvoltion reslts Fig. 21 Pressre history match sing the models shown in Fig. 19. October 2006 SPE Reservoir Evalation & Engineering 591

11 Fig. 23 Field Case 5: deconvoltion response fnctions, model match, and material-balance time-normalized data. Conclsions Or proposed deconvoltion techniqe is robst becase of the constrction of the coefficient (sensitivity) matrix of the nderlying least-sqares problem sing B-splines. Becase of the (semilogarithmic) B-spline representation of the nknown implse response and the high reliability of the nmerical Laplace transform inversion techniqe sed, the elements of the coefficient matrix are compted accrately, and the reslting deconvoltion process is stable. On the other hand, modeling of the rates reqires a significant effort for accrate comptation of the sensitivity matrix. Rates are reqired to be in fnctional form, and if the rate history indicates discontinities, it shold be decomposed into smooth segments. An increasing nmber of segments will increase the comptational time. With the appropriate description of rates, the proposed techniqe is robst and reliable. Fig. 22 (a) Field Case 5: prodction data history plot for east Texas gas well. (b) Field Case 5: pressre base data and the B-spline p wf model response. with traditional prodction analysis based on material-balance time deconvoltion. Artifacts cased by inconsistencies are seen at early times in the well-testing pressre-derivative fnction, again (we believe) becase of sparse/erratic data. The late-time well-testing pressrederivative fnction trend confirms that the bondary-dominated flow regime has been achieved. The soltion for a homogeneos reservoir with a closed circlar bondary matches all of the deconvolved pressre-response fnctions reasonably well, where this observation also confirms or analysis. We believe that B-spline deconvoltion can be applied sccessflly to prodction data (reglarly measred flow-rate and pressre data). Processing low-qality prodction data with deconvoltion can provide additional insight into reservoir performancebased analysis. It is worth noting that nmeros other field cases of long-term flow-rate and (srface) pressre data have been analyzed and interpreted sccessflly sing the proposed B-spline deconvoltion techniqe. In fact, we believe that B-spline deconvoltion may be sed more often for the analysis of prodction data, compared to its potential se for well-test data. This hypothesis arises from the fact that while often of poor qality, prodction data are generally available. Fig. 24 (a) Field Case 6: prodction-data history plot for a midcontinent gas well. (b) Field Case 6: pressre base data and the B-spline p wf model response. 592 October 2006 SPE Reservoir Evalation & Engineering

12 p drawdown (with respect to p i ) interporosity flow parameter viscosity, cp dmmy variable porosity, fraction storativity parameter Sperscripts ^ estimated (from least sqares) observed Laplace transform + psedoinverse T transpose of a matrix Special Fnctions K 0 (t) modified Bessel fnction of the second kind, zero order Fig. 25 Field Case 6: deconvoltion response fnctions, model match, and material-balance time-normalized data. For the deconvoltion of data in practice, it is necessary to incorporate a physically sond reglarization scheme into or proposed deconvoltion method. Reglarization might be seen as an add-on to the proposed algorithm, bt in a practical sense (i.e., for field applications), or proposed deconvoltion scheme is likely to perform poorly withot the se of reglarization methods. We have sccessflly demonstrated the sed of B-spline deconvoltion for the analysis of a wide range of field data, from traditional PBU data, to permanent downhole gage data, to traditional prodction data. Nomenclatre a q k rate fnction fitting parameter b base of logarithmically evenly distribted knots b q k rate fnction fitting parameter B formation volme factor, RB/STB B k i (t) kth degree B-spline starting at b i k B i,int (t) integral of the kth degree B-spline c vector of nknown coefficients c q k rate fnction fitting parameter c t total system compressibility, psi 1 C D dimensionless wellbore-storage coefficient F c fractre condctivity, md-ft G p cmlative gas prodction, Mscf h pay thickness, ft k permeability, md K(t) implse response ( p d /t) p(t) pressre, psi p constant (nit) -rate pressre response, psi p d well-testing (log) derivative of p, psi p i initial reservoir pressre, psi q(t) rate, STB/D or Mscf/D r e reservoir oter-bondary radis, ft r w wellbore radis, ft r ed oter reservoir-bondary radis, dimensionless s Laplace transform variable s x skin factor S(t) spline fnction t flowing time, hors t q k segment starting time (in rate fnction) t sh sht-in time, hors X sensitivity matrix reglarization parameter References Abate, J. and Valkó, P.P Mlti-precision Laplace transform inversion. Intl. J. for Nmerical Methods in Engineering 60 (5): DOI: Cheney, E.W. and Kincaid, D.R Nmerical Mathematics and Compting. Pacific Grove, California: Brooks Cole. Cheng, Y., Lee, W.J., and McVay, D.A A Deconvoltion Techniqe Using Fast-Forier Transforms. Paper SPE presented at the SPE Annal Technical Conference and Exhibition, Denver, 5 8 October. DOI: /84471-MS. Coats, K.H., Rapoport, L.A., McCord, J.R., and Drews, W.P Determination of Aqifer Inflence Fnctions From Field Data. JPT 16 (12): ; Trans., AIME, 231. SPE-897-PA. DOI: /897- PA. Davies, B. and Martin, B Nmerical inversion of the Laplace transform: A srvey and comparison of methods. J. of Comptational Physics 33 (1): DOI: Fetkovich, M.J. and Vienot, M.E Rate Normalization of Bildp Pressre by Using Afterflow Data. JPT 36 (12): SPE PA. DOI: /12179-PA. Gringarten, A.C., von Schroeter, T., Rolfsvaag, T., and Brner, J Use of Downhole Permanent Pressre Gage Data To Diagnose Prodction Problems in a North Sea Horizontal Well. Paper SPE presented at the SPE Annal Technical Conference and Exhibition, Denver, 5 8 October. DOI: /84470-MS. Gillot, A.Y. and Horne, R.N Using Simltaneos Downhole Flow- Rate and Pressre Measrements To Improve Analysis of Well Tests. SPEFE 1 (3): SPE PA. DOI: /12958-PA. Johnston, J.L. and Lee, W.J Interpreting Short-Term Bildp Tests From Low-Prodctivity Gas Wells Using Deconvoltion. Paper SPE presented at the SPE Gas Technology Symposim, Hoston, Janary. DOI: /21503-MS. Kchk, F.J. Carter, R.G., and Ayestaran, L Deconvoltion of Wellbore Pressre and Flow Rate. SPEFE 5 (1): SPE PA. DOI: /13960-PA. Lamm, P.K A Srvey of Reglarization Methods for First-Kind Volterra Eqations. In Srveys on Soltion Methods for Inverse Problems, eds. D. Colton, H.W. Engl, A. Lois, J.R. McLaghlin, and W. Rndell, New York City: Springer. Levitan, M.M Practical Application of Pressre-Rate Deconvoltion to Analysis of Real Well Tests. Paper SPE presented at the SPE Annal Technical Conference and Exhibition, Denver, 5 8 October. DOI: /84290-MS. Levitan, M.M., Crawford, G.E., and Hardwick, A Practical Considerations for Pressre-Rate Deconvoltion of Well Test Data. Paper SPE presented at the SPE Annal Technical Conference and Exhibition, Hoston, September. DOI: /90680-MS. Mathematica (software) Version Champaign-Urbana, Illinois: Wolfram Research. Mendes, L.C.C., Tygel, M., and Correa, A.C.F A Deconvoltion Algorithm for Analysis of Variable-Rate Well Test Pressre Data. Paper SPE presented at the SPE Annal Technical Conference and October 2006 SPE Reservoir Evalation & Engineering 593